src/HOL/Hoare_Parallel/RG_Examples.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 44890 22f665a2e91c child 51121 34dbeb8f16a9 permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
```     1 header {* \section{Examples} *}
```
```     2
```
```     3 theory RG_Examples
```
```     4 imports RG_Syntax
```
```     5 begin
```
```     6
```
```     7 lemmas definitions [simp]= stable_def Pre_def Rely_def Guar_def Post_def Com_def
```
```     8
```
```     9 subsection {* Set Elements of an Array to Zero *}
```
```    10
```
```    11 lemma le_less_trans2: "\<lbrakk>(j::nat)<k; i\<le> j\<rbrakk> \<Longrightarrow> i<k"
```
```    12 by simp
```
```    13
```
```    14 lemma add_le_less_mono: "\<lbrakk> (a::nat) < c; b\<le>d \<rbrakk> \<Longrightarrow> a + b < c + d"
```
```    15 by simp
```
```    16
```
```    17 record Example1 =
```
```    18   A :: "nat list"
```
```    19
```
```    20 lemma Example1:
```
```    21  "\<turnstile> COBEGIN
```
```    22       SCHEME [0 \<le> i < n]
```
```    23      (\<acute>A := \<acute>A [i := 0],
```
```    24      \<lbrace> n < length \<acute>A \<rbrace>,
```
```    25      \<lbrace> length \<ordmasculine>A = length \<ordfeminine>A \<and> \<ordmasculine>A ! i = \<ordfeminine>A ! i \<rbrace>,
```
```    26      \<lbrace> length \<ordmasculine>A = length \<ordfeminine>A \<and> (\<forall>j<n. i \<noteq> j \<longrightarrow> \<ordmasculine>A ! j = \<ordfeminine>A ! j) \<rbrace>,
```
```    27      \<lbrace> \<acute>A ! i = 0 \<rbrace>)
```
```    28     COEND
```
```    29  SAT [\<lbrace> n < length \<acute>A \<rbrace>, \<lbrace> \<ordmasculine>A = \<ordfeminine>A \<rbrace>, \<lbrace> True \<rbrace>, \<lbrace> \<forall>i < n. \<acute>A ! i = 0 \<rbrace>]"
```
```    30 apply(rule Parallel)
```
```    31 apply (auto intro!: Basic)
```
```    32 done
```
```    33
```
```    34 lemma Example1_parameterized:
```
```    35 "k < t \<Longrightarrow>
```
```    36   \<turnstile> COBEGIN
```
```    37     SCHEME [k*n\<le>i<(Suc k)*n] (\<acute>A:=\<acute>A[i:=0],
```
```    38    \<lbrace>t*n < length \<acute>A\<rbrace>,
```
```    39    \<lbrace>t*n < length \<ordmasculine>A \<and> length \<ordmasculine>A=length \<ordfeminine>A \<and> \<ordmasculine>A!i = \<ordfeminine>A!i\<rbrace>,
```
```    40    \<lbrace>t*n < length \<ordmasculine>A \<and> length \<ordmasculine>A=length \<ordfeminine>A \<and> (\<forall>j<length \<ordmasculine>A . i\<noteq>j \<longrightarrow> \<ordmasculine>A!j = \<ordfeminine>A!j)\<rbrace>,
```
```    41    \<lbrace>\<acute>A!i=0\<rbrace>)
```
```    42    COEND
```
```    43  SAT [\<lbrace>t*n < length \<acute>A\<rbrace>,
```
```    44       \<lbrace>t*n < length \<ordmasculine>A \<and> length \<ordmasculine>A=length \<ordfeminine>A \<and> (\<forall>i<n. \<ordmasculine>A!(k*n+i)=\<ordfeminine>A!(k*n+i))\<rbrace>,
```
```    45       \<lbrace>t*n < length \<ordmasculine>A \<and> length \<ordmasculine>A=length \<ordfeminine>A \<and>
```
```    46       (\<forall>i<length \<ordmasculine>A . (i<k*n \<longrightarrow> \<ordmasculine>A!i = \<ordfeminine>A!i) \<and> ((Suc k)*n \<le> i\<longrightarrow> \<ordmasculine>A!i = \<ordfeminine>A!i))\<rbrace>,
```
```    47       \<lbrace>\<forall>i<n. \<acute>A!(k*n+i) = 0\<rbrace>]"
```
```    48 apply(rule Parallel)
```
```    49     apply auto
```
```    50   apply(erule_tac x="k*n +i" in allE)
```
```    51   apply(subgoal_tac "k*n+i <length (A b)")
```
```    52    apply force
```
```    53   apply(erule le_less_trans2)
```
```    54   apply(case_tac t,simp+)
```
```    55   apply (simp add:add_commute)
```
```    56   apply(simp add: add_le_mono)
```
```    57 apply(rule Basic)
```
```    58    apply simp
```
```    59    apply clarify
```
```    60    apply (subgoal_tac "k*n+i< length (A x)")
```
```    61     apply simp
```
```    62    apply(erule le_less_trans2)
```
```    63    apply(case_tac t,simp+)
```
```    64    apply (simp add:add_commute)
```
```    65    apply(rule add_le_mono, auto)
```
```    66 done
```
```    67
```
```    68
```
```    69 subsection {* Increment a Variable in Parallel *}
```
```    70
```
```    71 subsubsection {* Two components *}
```
```    72
```
```    73 record Example2 =
```
```    74   x  :: nat
```
```    75   c_0 :: nat
```
```    76   c_1 :: nat
```
```    77
```
```    78 lemma Example2:
```
```    79  "\<turnstile>  COBEGIN
```
```    80     (\<langle> \<acute>x:=\<acute>x+1;; \<acute>c_0:=\<acute>c_0 + 1 \<rangle>,
```
```    81      \<lbrace>\<acute>x=\<acute>c_0 + \<acute>c_1  \<and> \<acute>c_0=0\<rbrace>,
```
```    82      \<lbrace>\<ordmasculine>c_0 = \<ordfeminine>c_0 \<and>
```
```    83         (\<ordmasculine>x=\<ordmasculine>c_0 + \<ordmasculine>c_1
```
```    84         \<longrightarrow> \<ordfeminine>x = \<ordfeminine>c_0 + \<ordfeminine>c_1)\<rbrace>,
```
```    85      \<lbrace>\<ordmasculine>c_1 = \<ordfeminine>c_1 \<and>
```
```    86          (\<ordmasculine>x=\<ordmasculine>c_0 + \<ordmasculine>c_1
```
```    87          \<longrightarrow> \<ordfeminine>x =\<ordfeminine>c_0 + \<ordfeminine>c_1)\<rbrace>,
```
```    88      \<lbrace>\<acute>x=\<acute>c_0 + \<acute>c_1 \<and> \<acute>c_0=1 \<rbrace>)
```
```    89   \<parallel>
```
```    90       (\<langle> \<acute>x:=\<acute>x+1;; \<acute>c_1:=\<acute>c_1+1 \<rangle>,
```
```    91      \<lbrace>\<acute>x=\<acute>c_0 + \<acute>c_1 \<and> \<acute>c_1=0 \<rbrace>,
```
```    92      \<lbrace>\<ordmasculine>c_1 = \<ordfeminine>c_1 \<and>
```
```    93         (\<ordmasculine>x=\<ordmasculine>c_0 + \<ordmasculine>c_1
```
```    94         \<longrightarrow> \<ordfeminine>x = \<ordfeminine>c_0 + \<ordfeminine>c_1)\<rbrace>,
```
```    95      \<lbrace>\<ordmasculine>c_0 = \<ordfeminine>c_0 \<and>
```
```    96          (\<ordmasculine>x=\<ordmasculine>c_0 + \<ordmasculine>c_1
```
```    97         \<longrightarrow> \<ordfeminine>x =\<ordfeminine>c_0 + \<ordfeminine>c_1)\<rbrace>,
```
```    98      \<lbrace>\<acute>x=\<acute>c_0 + \<acute>c_1 \<and> \<acute>c_1=1\<rbrace>)
```
```    99  COEND
```
```   100  SAT [\<lbrace>\<acute>x=0 \<and> \<acute>c_0=0 \<and> \<acute>c_1=0\<rbrace>,
```
```   101       \<lbrace>\<ordmasculine>x=\<ordfeminine>x \<and>  \<ordmasculine>c_0= \<ordfeminine>c_0 \<and> \<ordmasculine>c_1=\<ordfeminine>c_1\<rbrace>,
```
```   102       \<lbrace>True\<rbrace>,
```
```   103       \<lbrace>\<acute>x=2\<rbrace>]"
```
```   104 apply(rule Parallel)
```
```   105    apply simp_all
```
```   106    apply clarify
```
```   107    apply(case_tac i)
```
```   108     apply simp
```
```   109     apply(rule conjI)
```
```   110      apply clarify
```
```   111      apply simp
```
```   112     apply clarify
```
```   113     apply simp
```
```   114    apply simp
```
```   115    apply(rule conjI)
```
```   116     apply clarify
```
```   117     apply simp
```
```   118    apply clarify
```
```   119    apply simp
```
```   120    apply(subgoal_tac "j=0")
```
```   121     apply (simp)
```
```   122    apply arith
```
```   123   apply clarify
```
```   124   apply(case_tac i,simp,simp)
```
```   125  apply clarify
```
```   126  apply simp
```
```   127  apply(erule_tac x=0 in all_dupE)
```
```   128  apply(erule_tac x=1 in allE,simp)
```
```   129 apply clarify
```
```   130 apply(case_tac i,simp)
```
```   131  apply(rule Await)
```
```   132   apply simp_all
```
```   133  apply(clarify)
```
```   134  apply(rule Seq)
```
```   135   prefer 2
```
```   136   apply(rule Basic)
```
```   137    apply simp_all
```
```   138   apply(rule subset_refl)
```
```   139  apply(rule Basic)
```
```   140  apply simp_all
```
```   141  apply clarify
```
```   142  apply simp
```
```   143 apply(rule Await)
```
```   144  apply simp_all
```
```   145 apply(clarify)
```
```   146 apply(rule Seq)
```
```   147  prefer 2
```
```   148  apply(rule Basic)
```
```   149   apply simp_all
```
```   150  apply(rule subset_refl)
```
```   151 apply(auto intro!: Basic)
```
```   152 done
```
```   153
```
```   154 subsubsection {* Parameterized *}
```
```   155
```
```   156 lemma Example2_lemma2_aux: "j<n \<Longrightarrow>
```
```   157  (\<Sum>i=0..<n. (b i::nat)) =
```
```   158  (\<Sum>i=0..<j. b i) + b j + (\<Sum>i=0..<n-(Suc j) . b (Suc j + i))"
```
```   159 apply(induct n)
```
```   160  apply simp_all
```
```   161 apply(simp add:less_Suc_eq)
```
```   162  apply(auto)
```
```   163 apply(subgoal_tac "n - j = Suc(n- Suc j)")
```
```   164   apply simp
```
```   165 apply arith
```
```   166 done
```
```   167
```
```   168 lemma Example2_lemma2_aux2:
```
```   169   "j\<le> s \<Longrightarrow> (\<Sum>i::nat=0..<j. (b (s:=t)) i) = (\<Sum>i=0..<j. b i)"
```
```   170 apply(induct j)
```
```   171  apply (simp_all cong:setsum_cong)
```
```   172 done
```
```   173
```
```   174 lemma Example2_lemma2:
```
```   175  "\<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)"
```
```   176 apply(frule_tac b="(b (j:=(Suc 0)))" in Example2_lemma2_aux)
```
```   177 apply(erule_tac  t="setsum (b(j := (Suc 0))) {0..<n}" in ssubst)
```
```   178 apply(frule_tac b=b in Example2_lemma2_aux)
```
```   179 apply(erule_tac  t="setsum b {0..<n}" in ssubst)
```
```   180 apply(subgoal_tac "Suc (setsum b {0..<j} + b j + (\<Sum>i=0..<n - Suc j. b (Suc j + i)))=(setsum b {0..<j} + Suc (b j) + (\<Sum>i=0..<n - Suc j. b (Suc j + i)))")
```
```   181 apply(rotate_tac -1)
```
```   182 apply(erule ssubst)
```
```   183 apply(subgoal_tac "j\<le>j")
```
```   184  apply(drule_tac b="b" and t="(Suc 0)" in Example2_lemma2_aux2)
```
```   185 apply(rotate_tac -1)
```
```   186 apply(erule ssubst)
```
```   187 apply simp_all
```
```   188 done
```
```   189
```
```   190 lemma Example2_lemma2_Suc0: "\<lbrakk>j<n; b j=0\<rbrakk> \<Longrightarrow>
```
```   191  Suc (\<Sum>i::nat=0..< n. b i)=(\<Sum>i=0..< n. (b (j:=Suc 0)) i)"
```
```   192 by(simp add:Example2_lemma2)
```
```   193
```
```   194 record Example2_parameterized =
```
```   195   C :: "nat \<Rightarrow> nat"
```
```   196   y  :: nat
```
```   197
```
```   198 lemma Example2_parameterized: "0<n \<Longrightarrow>
```
```   199   \<turnstile> COBEGIN SCHEME  [0\<le>i<n]
```
```   200      (\<langle> \<acute>y:=\<acute>y+1;; \<acute>C:=\<acute>C (i:=1) \<rangle>,
```
```   201      \<lbrace>\<acute>y=(\<Sum>i=0..<n. \<acute>C i) \<and> \<acute>C i=0\<rbrace>,
```
```   202      \<lbrace>\<ordmasculine>C i = \<ordfeminine>C i \<and>
```
```   203       (\<ordmasculine>y=(\<Sum>i=0..<n. \<ordmasculine>C i) \<longrightarrow> \<ordfeminine>y =(\<Sum>i=0..<n. \<ordfeminine>C i))\<rbrace>,
```
```   204      \<lbrace>(\<forall>j<n. i\<noteq>j \<longrightarrow> \<ordmasculine>C j = \<ordfeminine>C j) \<and>
```
```   205        (\<ordmasculine>y=(\<Sum>i=0..<n. \<ordmasculine>C i) \<longrightarrow> \<ordfeminine>y =(\<Sum>i=0..<n. \<ordfeminine>C i))\<rbrace>,
```
```   206      \<lbrace>\<acute>y=(\<Sum>i=0..<n. \<acute>C i) \<and> \<acute>C i=1\<rbrace>)
```
```   207     COEND
```
```   208  SAT [\<lbrace>\<acute>y=0 \<and> (\<Sum>i=0..<n. \<acute>C i)=0 \<rbrace>, \<lbrace>\<ordmasculine>C=\<ordfeminine>C \<and> \<ordmasculine>y=\<ordfeminine>y\<rbrace>, \<lbrace>True\<rbrace>, \<lbrace>\<acute>y=n\<rbrace>]"
```
```   209 apply(rule Parallel)
```
```   210 apply force
```
```   211 apply force
```
```   212 apply(force)
```
```   213 apply clarify
```
```   214 apply simp
```
```   215 apply(simp cong:setsum_ivl_cong)
```
```   216 apply clarify
```
```   217 apply simp
```
```   218 apply(rule Await)
```
```   219 apply simp_all
```
```   220 apply clarify
```
```   221 apply(rule Seq)
```
```   222 prefer 2
```
```   223 apply(rule Basic)
```
```   224 apply(rule subset_refl)
```
```   225 apply simp+
```
```   226 apply(rule Basic)
```
```   227 apply simp
```
```   228 apply clarify
```
```   229 apply simp
```
```   230 apply(simp add:Example2_lemma2_Suc0 cong:if_cong)
```
```   231 apply simp+
```
```   232 done
```
```   233
```
```   234 subsection {* Find Least Element *}
```
```   235
```
```   236 text {* A previous lemma: *}
```
```   237
```
```   238 lemma mod_aux :"\<lbrakk>i < (n::nat); a mod n = i;  j < a + n; j mod n = i; a < j\<rbrakk> \<Longrightarrow> False"
```
```   239 apply(subgoal_tac "a=a div n*n + a mod n" )
```
```   240  prefer 2 apply (simp (no_asm_use))
```
```   241 apply(subgoal_tac "j=j div n*n + j mod n")
```
```   242  prefer 2 apply (simp (no_asm_use))
```
```   243 apply simp
```
```   244 apply(subgoal_tac "a div n*n < j div n*n")
```
```   245 prefer 2 apply arith
```
```   246 apply(subgoal_tac "j div n*n < (a div n + 1)*n")
```
```   247 prefer 2 apply simp
```
```   248 apply (simp only:mult_less_cancel2)
```
```   249 apply arith
```
```   250 done
```
```   251
```
```   252 record Example3 =
```
```   253   X :: "nat \<Rightarrow> nat"
```
```   254   Y :: "nat \<Rightarrow> nat"
```
```   255
```
```   256 lemma Example3: "m mod n=0 \<Longrightarrow>
```
```   257  \<turnstile> COBEGIN
```
```   258  SCHEME [0\<le>i<n]
```
```   259  (WHILE (\<forall>j<n. \<acute>X i < \<acute>Y j)  DO
```
```   260    IF P(B!(\<acute>X i)) THEN \<acute>Y:=\<acute>Y (i:=\<acute>X i)
```
```   261    ELSE \<acute>X:= \<acute>X (i:=(\<acute>X i)+ n) FI
```
```   262   OD,
```
```   263  \<lbrace>(\<acute>X i) mod n=i \<and> (\<forall>j<\<acute>X i. j mod n=i \<longrightarrow> \<not>P(B!j)) \<and> (\<acute>Y i<m \<longrightarrow> P(B!(\<acute>Y i)) \<and> \<acute>Y i\<le> m+i)\<rbrace>,
```
```   264  \<lbrace>(\<forall>j<n. i\<noteq>j \<longrightarrow> \<ordfeminine>Y j \<le> \<ordmasculine>Y j) \<and> \<ordmasculine>X i = \<ordfeminine>X i \<and>
```
```   265    \<ordmasculine>Y i = \<ordfeminine>Y i\<rbrace>,
```
```   266  \<lbrace>(\<forall>j<n. i\<noteq>j \<longrightarrow> \<ordmasculine>X j = \<ordfeminine>X j \<and> \<ordmasculine>Y j = \<ordfeminine>Y j) \<and>
```
```   267    \<ordfeminine>Y i \<le> \<ordmasculine>Y i\<rbrace>,
```
```   268  \<lbrace>(\<acute>X i) mod n=i \<and> (\<forall>j<\<acute>X i. j mod n=i \<longrightarrow> \<not>P(B!j)) \<and> (\<acute>Y i<m \<longrightarrow> P(B!(\<acute>Y i)) \<and> \<acute>Y i\<le> m+i) \<and> (\<exists>j<n. \<acute>Y j \<le> \<acute>X i) \<rbrace>)
```
```   269  COEND
```
```   270  SAT [\<lbrace> \<forall>i<n. \<acute>X i=i \<and> \<acute>Y i=m+i \<rbrace>,\<lbrace>\<ordmasculine>X=\<ordfeminine>X \<and> \<ordmasculine>Y=\<ordfeminine>Y\<rbrace>,\<lbrace>True\<rbrace>,
```
```   271   \<lbrace>\<forall>i<n. (\<acute>X i) mod n=i \<and> (\<forall>j<\<acute>X i. j mod n=i \<longrightarrow> \<not>P(B!j)) \<and>
```
```   272     (\<acute>Y i<m \<longrightarrow> P(B!(\<acute>Y i)) \<and> \<acute>Y i\<le> m+i) \<and> (\<exists>j<n. \<acute>Y j \<le> \<acute>X i)\<rbrace>]"
```
```   273 apply(rule Parallel)
```
```   274 --{*5 subgoals left *}
```
```   275 apply force+
```
```   276 apply clarify
```
```   277 apply simp
```
```   278 apply(rule While)
```
```   279     apply force
```
```   280    apply force
```
```   281   apply force
```
```   282  apply(rule_tac pre'="\<lbrace> \<acute>X i mod n = i \<and> (\<forall>j. j<\<acute>X i \<longrightarrow> j mod n = i \<longrightarrow> \<not>P(B!j)) \<and> (\<acute>Y i < n * q \<longrightarrow> P (B!(\<acute>Y i))) \<and> \<acute>X i<\<acute>Y i\<rbrace>" in Conseq)
```
```   283      apply force
```
```   284     apply(rule subset_refl)+
```
```   285  apply(rule Cond)
```
```   286     apply force
```
```   287    apply(rule Basic)
```
```   288       apply force
```
```   289      apply fastforce
```
```   290     apply force
```
```   291    apply force
```
```   292   apply(rule Basic)
```
```   293      apply simp
```
```   294      apply clarify
```
```   295      apply simp
```
```   296      apply (case_tac "X x (j mod n) \<le> j")
```
```   297      apply (drule le_imp_less_or_eq)
```
```   298      apply (erule disjE)
```
```   299      apply (drule_tac j=j and n=n and i="j mod n" and a="X x (j mod n)" in mod_aux)
```
```   300      apply auto
```
```   301 done
```
```   302
```
```   303 text {* Same but with a list as auxiliary variable: *}
```
```   304
```
```   305 record Example3_list =
```
```   306   X :: "nat list"
```
```   307   Y :: "nat list"
```
```   308
```
```   309 lemma Example3_list: "m mod n=0 \<Longrightarrow> \<turnstile> (COBEGIN SCHEME [0\<le>i<n]
```
```   310  (WHILE (\<forall>j<n. \<acute>X!i < \<acute>Y!j)  DO
```
```   311      IF P(B!(\<acute>X!i)) THEN \<acute>Y:=\<acute>Y[i:=\<acute>X!i] ELSE \<acute>X:= \<acute>X[i:=(\<acute>X!i)+ n] FI
```
```   312   OD,
```
```   313  \<lbrace>n<length \<acute>X \<and> n<length \<acute>Y \<and> (\<acute>X!i) mod n=i \<and> (\<forall>j<\<acute>X!i. j mod n=i \<longrightarrow> \<not>P(B!j)) \<and> (\<acute>Y!i<m \<longrightarrow> P(B!(\<acute>Y!i)) \<and> \<acute>Y!i\<le> m+i)\<rbrace>,
```
```   314  \<lbrace>(\<forall>j<n. i\<noteq>j \<longrightarrow> \<ordfeminine>Y!j \<le> \<ordmasculine>Y!j) \<and> \<ordmasculine>X!i = \<ordfeminine>X!i \<and>
```
```   315    \<ordmasculine>Y!i = \<ordfeminine>Y!i \<and> length \<ordmasculine>X = length \<ordfeminine>X \<and> length \<ordmasculine>Y = length \<ordfeminine>Y\<rbrace>,
```
```   316  \<lbrace>(\<forall>j<n. i\<noteq>j \<longrightarrow> \<ordmasculine>X!j = \<ordfeminine>X!j \<and> \<ordmasculine>Y!j = \<ordfeminine>Y!j) \<and>
```
```   317    \<ordfeminine>Y!i \<le> \<ordmasculine>Y!i \<and> length \<ordmasculine>X = length \<ordfeminine>X \<and> length \<ordmasculine>Y = length \<ordfeminine>Y\<rbrace>,
```
```   318  \<lbrace>(\<acute>X!i) mod n=i \<and> (\<forall>j<\<acute>X!i. j mod n=i \<longrightarrow> \<not>P(B!j)) \<and> (\<acute>Y!i<m \<longrightarrow> P(B!(\<acute>Y!i)) \<and> \<acute>Y!i\<le> m+i) \<and> (\<exists>j<n. \<acute>Y!j \<le> \<acute>X!i) \<rbrace>) COEND)
```
```   319  SAT [\<lbrace>n<length \<acute>X \<and> n<length \<acute>Y \<and> (\<forall>i<n. \<acute>X!i=i \<and> \<acute>Y!i=m+i) \<rbrace>,
```
```   320       \<lbrace>\<ordmasculine>X=\<ordfeminine>X \<and> \<ordmasculine>Y=\<ordfeminine>Y\<rbrace>,
```
```   321       \<lbrace>True\<rbrace>,
```
```   322       \<lbrace>\<forall>i<n. (\<acute>X!i) mod n=i \<and> (\<forall>j<\<acute>X!i. j mod n=i \<longrightarrow> \<not>P(B!j)) \<and>
```
```   323         (\<acute>Y!i<m \<longrightarrow> P(B!(\<acute>Y!i)) \<and> \<acute>Y!i\<le> m+i) \<and> (\<exists>j<n. \<acute>Y!j \<le> \<acute>X!i)\<rbrace>]"
```
```   324 apply(rule Parallel)
```
```   325 --{* 5 subgoals left *}
```
```   326 apply force+
```
```   327 apply clarify
```
```   328 apply simp
```
```   329 apply(rule While)
```
```   330     apply force
```
```   331    apply force
```
```   332   apply force
```
```   333  apply(rule_tac pre'="\<lbrace>n<length \<acute>X \<and> n<length \<acute>Y \<and> \<acute>X ! i mod n = i \<and> (\<forall>j. j < \<acute>X ! i \<longrightarrow> j mod n = i \<longrightarrow> \<not> P (B ! j)) \<and> (\<acute>Y ! i < n * q \<longrightarrow> P (B ! (\<acute>Y ! i))) \<and> \<acute>X!i<\<acute>Y!i\<rbrace>" in Conseq)
```
```   334      apply force
```
```   335     apply(rule subset_refl)+
```
```   336  apply(rule Cond)
```
```   337     apply force
```
```   338    apply(rule Basic)
```
```   339       apply force
```
```   340      apply force
```
```   341     apply force
```
```   342    apply force
```
```   343   apply(rule Basic)
```
```   344      apply simp
```
```   345      apply clarify
```
```   346      apply simp
```
```   347      apply(rule allI)
```
```   348      apply(rule impI)+
```
```   349      apply(case_tac "X x ! i\<le> j")
```
```   350       apply(drule le_imp_less_or_eq)
```
```   351       apply(erule disjE)
```
```   352        apply(drule_tac j=j and n=n and i=i and a="X x ! i" in mod_aux)
```
```   353      apply auto
```
```   354 done
```
```   355
```
```   356 end
```