src/HOL/Algebra/Lattice.thy
 author wenzelm Thu Apr 22 11:01:34 2004 +0200 (2004-04-22) changeset 14651 02b8f3bcf7fe parent 14577 dbb95b825244 child 14666 65f8680c3f16 permissions -rw-r--r--
improved notation;
```     1 (*
```
```     2   Title:     Orders and Lattices
```
```     3   Id:        \$Id\$
```
```     4   Author:    Clemens Ballarin, started 7 November 2003
```
```     5   Copyright: Clemens Ballarin
```
```     6 *)
```
```     7
```
```     8 header {* Order and Lattices *}
```
```     9
```
```    10 theory Lattice = Group:
```
```    11
```
```    12 subsection {* Partial Orders *}
```
```    13
```
```    14 record 'a order = "'a partial_object" +
```
```    15   le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50)
```
```    16
```
```    17 locale order_syntax = struct L
```
```    18
```
```    19 locale partial_order = order_syntax +
```
```    20   assumes refl [intro, simp]:
```
```    21                   "x \<in> carrier L ==> x \<sqsubseteq> x"
```
```    22     and anti_sym [intro]:
```
```    23                   "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y"
```
```    24     and trans [trans]:
```
```    25                   "[| x \<sqsubseteq> y; y \<sqsubseteq> z;
```
```    26                    x \<in> carrier L; y \<in> carrier L; z \<in> carrier L |] ==> x \<sqsubseteq> z"
```
```    27
```
```    28 constdefs (structure L)
```
```    29   less :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50)
```
```    30   "x \<sqsubset> y == x \<sqsubseteq> y & x ~= y"
```
```    31
```
```    32   -- {* Upper and lower bounds of a set. *}
```
```    33   Upper :: "[_, 'a set] => 'a set"
```
```    34   "Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> le L x u)} \<inter>
```
```    35                 carrier L"
```
```    36
```
```    37   Lower :: "[_, 'a set] => 'a set"
```
```    38   "Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> le L l x)} \<inter>
```
```    39                 carrier L"
```
```    40
```
```    41   -- {* Least and greatest, as predicate. *}
```
```    42   least :: "[_, 'a, 'a set] => bool"
```
```    43   "least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. le L l x)"
```
```    44
```
```    45   greatest :: "[_, 'a, 'a set] => bool"
```
```    46   "greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. le L x g)"
```
```    47
```
```    48   -- {* Supremum and infimum *}
```
```    49   sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_"  90)
```
```    50   "\<Squnion>A == THE x. least L x (Upper L A)"
```
```    51
```
```    52   inf :: "[_, 'a set] => 'a" ("\<Sqinter>\<index>_"  90)
```
```    53   "\<Sqinter>A == THE x. greatest L x (Lower L A)"
```
```    54
```
```    55   join :: "[_, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65)
```
```    56   "x \<squnion> y == sup L {x, y}"
```
```    57
```
```    58   meet :: "[_, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 65)
```
```    59   "x \<sqinter> y == inf L {x, y}"
```
```    60
```
```    61
```
```    62 subsubsection {* Upper *}
```
```    63
```
```    64 lemma Upper_closed [intro, simp]:
```
```    65   "Upper L A \<subseteq> carrier L"
```
```    66   by (unfold Upper_def) clarify
```
```    67
```
```    68 lemma UpperD [dest]:
```
```    69   includes order_syntax
```
```    70   shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u"
```
```    71   by (unfold Upper_def) blast
```
```    72
```
```    73 lemma Upper_memI:
```
```    74   includes order_syntax
```
```    75   shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A"
```
```    76   by (unfold Upper_def) blast
```
```    77
```
```    78 lemma Upper_antimono:
```
```    79   "A \<subseteq> B ==> Upper L B \<subseteq> Upper L A"
```
```    80   by (unfold Upper_def) blast
```
```    81
```
```    82
```
```    83 subsubsection {* Lower *}
```
```    84
```
```    85 lemma Lower_closed [intro, simp]:
```
```    86   "Lower L A \<subseteq> carrier L"
```
```    87   by (unfold Lower_def) clarify
```
```    88
```
```    89 lemma LowerD [dest]:
```
```    90   includes order_syntax
```
```    91   shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x"
```
```    92   by (unfold Lower_def) blast
```
```    93
```
```    94 lemma Lower_memI:
```
```    95   includes order_syntax
```
```    96   shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A"
```
```    97   by (unfold Lower_def) blast
```
```    98
```
```    99 lemma Lower_antimono:
```
```   100   "A \<subseteq> B ==> Lower L B \<subseteq> Lower L A"
```
```   101   by (unfold Lower_def) blast
```
```   102
```
```   103
```
```   104 subsubsection {* least *}
```
```   105
```
```   106 lemma least_carrier [intro, simp]:
```
```   107   shows "least L l A ==> l \<in> carrier L"
```
```   108   by (unfold least_def) fast
```
```   109
```
```   110 lemma least_mem:
```
```   111   "least L l A ==> l \<in> A"
```
```   112   by (unfold least_def) fast
```
```   113
```
```   114 lemma (in partial_order) least_unique:
```
```   115   "[| least L x A; least L y A |] ==> x = y"
```
```   116   by (unfold least_def) blast
```
```   117
```
```   118 lemma least_le:
```
```   119   includes order_syntax
```
```   120   shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a"
```
```   121   by (unfold least_def) fast
```
```   122
```
```   123 lemma least_UpperI:
```
```   124   includes order_syntax
```
```   125   assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s"
```
```   126     and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y"
```
```   127     and L: "A \<subseteq> carrier L" "s \<in> carrier L"
```
```   128   shows "least L s (Upper L A)"
```
```   129 proof (unfold least_def, intro conjI)
```
```   130   show "Upper L A \<subseteq> carrier L" by simp
```
```   131 next
```
```   132   from above L show "s \<in> Upper L A" by (simp add: Upper_def)
```
```   133 next
```
```   134   from below show "ALL x : Upper L A. s \<sqsubseteq> x" by fast
```
```   135 qed
```
```   136
```
```   137
```
```   138 subsubsection {* greatest *}
```
```   139
```
```   140 lemma greatest_carrier [intro, simp]:
```
```   141   shows "greatest L l A ==> l \<in> carrier L"
```
```   142   by (unfold greatest_def) fast
```
```   143
```
```   144 lemma greatest_mem:
```
```   145   "greatest L l A ==> l \<in> A"
```
```   146   by (unfold greatest_def) fast
```
```   147
```
```   148 lemma (in partial_order) greatest_unique:
```
```   149   "[| greatest L x A; greatest L y A |] ==> x = y"
```
```   150   by (unfold greatest_def) blast
```
```   151
```
```   152 lemma greatest_le:
```
```   153   includes order_syntax
```
```   154   shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x"
```
```   155   by (unfold greatest_def) fast
```
```   156
```
```   157 lemma greatest_LowerI:
```
```   158   includes order_syntax
```
```   159   assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x"
```
```   160     and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i"
```
```   161     and L: "A \<subseteq> carrier L" "i \<in> carrier L"
```
```   162   shows "greatest L i (Lower L A)"
```
```   163 proof (unfold greatest_def, intro conjI)
```
```   164   show "Lower L A \<subseteq> carrier L" by simp
```
```   165 next
```
```   166   from below L show "i \<in> Lower L A" by (simp add: Lower_def)
```
```   167 next
```
```   168   from above show "ALL x : Lower L A. x \<sqsubseteq> i" by fast
```
```   169 qed
```
```   170
```
```   171 subsection {* Lattices *}
```
```   172
```
```   173 locale lattice = partial_order +
```
```   174   assumes sup_of_two_exists:
```
```   175     "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. least L s (Upper L {x, y})"
```
```   176     and inf_of_two_exists:
```
```   177     "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"
```
```   178
```
```   179 lemma least_Upper_above:
```
```   180   includes order_syntax
```
```   181   shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
```
```   182   by (unfold least_def) blast
```
```   183
```
```   184 lemma greatest_Lower_above:
```
```   185   includes order_syntax
```
```   186   shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x"
```
```   187   by (unfold greatest_def) blast
```
```   188
```
```   189 subsubsection {* Supremum *}
```
```   190
```
```   191 lemma (in lattice) joinI:
```
```   192   "[| !!l. least L l (Upper L {x, y}) ==> P l; x \<in> carrier L; y \<in> carrier L |]
```
```   193   ==> P (x \<squnion> y)"
```
```   194 proof (unfold join_def sup_def)
```
```   195   assume L: "x \<in> carrier L" "y \<in> carrier L"
```
```   196     and P: "!!l. least L l (Upper L {x, y}) ==> P l"
```
```   197   with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
```
```   198   with L show "P (THE l. least L l (Upper L {x, y}))"
```
```   199   by (fast intro: theI2 least_unique P)
```
```   200 qed
```
```   201
```
```   202 lemma (in lattice) join_closed [simp]:
```
```   203   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L"
```
```   204   by (rule joinI) (rule least_carrier)
```
```   205
```
```   206 lemma (in partial_order) sup_of_singletonI:      (* only reflexivity needed ? *)
```
```   207   "x \<in> carrier L ==> least L x (Upper L {x})"
```
```   208   by (rule least_UpperI) fast+
```
```   209
```
```   210 lemma (in partial_order) sup_of_singleton [simp]:
```
```   211   includes order_syntax
```
```   212   shows "x \<in> carrier L ==> \<Squnion> {x} = x"
```
```   213   by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI)
```
```   214
```
```   215 text {* Condition on A: supremum exists. *}
```
```   216
```
```   217 lemma (in lattice) sup_insertI:
```
```   218   "[| !!s. least L s (Upper L (insert x A)) ==> P s;
```
```   219   least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |]
```
```   220   ==> P (\<Squnion> (insert x A))"
```
```   221 proof (unfold sup_def)
```
```   222   assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
```
```   223     and P: "!!l. least L l (Upper L (insert x A)) ==> P l"
```
```   224     and least_a: "least L a (Upper L A)"
```
```   225   from L least_a have La: "a \<in> carrier L" by simp
```
```   226   from L sup_of_two_exists least_a
```
```   227   obtain s where least_s: "least L s (Upper L {a, x})" by blast
```
```   228   show "P (THE l. least L l (Upper L (insert x A)))"
```
```   229   proof (rule theI2 [where a = s])
```
```   230     show "least L s (Upper L (insert x A))"
```
```   231     proof (rule least_UpperI)
```
```   232       fix z
```
```   233       assume xA: "z \<in> insert x A"
```
```   234       show "z \<sqsubseteq> s"
```
```   235       proof -
```
```   236 	{
```
```   237 	  assume "z = x" then have ?thesis
```
```   238 	    by (simp add: least_Upper_above [OF least_s] L La)
```
```   239         }
```
```   240 	moreover
```
```   241         {
```
```   242 	  assume "z \<in> A"
```
```   243           with L least_s least_a have ?thesis
```
```   244 	    by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
```
```   245         }
```
```   246       moreover note xA
```
```   247       ultimately show ?thesis by blast
```
```   248     qed
```
```   249   next
```
```   250     fix y
```
```   251     assume y: "y \<in> Upper L (insert x A)"
```
```   252     show "s \<sqsubseteq> y"
```
```   253     proof (rule least_le [OF least_s], rule Upper_memI)
```
```   254       fix z
```
```   255       assume z: "z \<in> {a, x}"
```
```   256       show "z \<sqsubseteq> y"
```
```   257       proof -
```
```   258 	{
```
```   259           have y': "y \<in> Upper L A"
```
```   260 	    apply (rule subsetD [where A = "Upper L (insert x A)"])
```
```   261 	    apply (rule Upper_antimono) apply clarify apply assumption
```
```   262 	    done
```
```   263 	  assume "z = a"
```
```   264 	  with y' least_a have ?thesis by (fast dest: least_le)
```
```   265         }
```
```   266 	moreover
```
```   267 	{
```
```   268            assume "z = x"
```
```   269            with y L have ?thesis by blast
```
```   270         }
```
```   271         moreover note z
```
```   272         ultimately show ?thesis by blast
```
```   273       qed
```
```   274     qed (rule Upper_closed [THEN subsetD])
```
```   275   next
```
```   276     from L show "insert x A \<subseteq> carrier L" by simp
```
```   277   next
```
```   278     from least_s show "s \<in> carrier L" by simp
```
```   279   qed
```
```   280 next
```
```   281     fix l
```
```   282     assume least_l: "least L l (Upper L (insert x A))"
```
```   283     show "l = s"
```
```   284     proof (rule least_unique)
```
```   285       show "least L s (Upper L (insert x A))"
```
```   286       proof (rule least_UpperI)
```
```   287 	fix z
```
```   288 	assume xA: "z \<in> insert x A"
```
```   289 	show "z \<sqsubseteq> s"
```
```   290       proof -
```
```   291 	{
```
```   292 	  assume "z = x" then have ?thesis
```
```   293 	    by (simp add: least_Upper_above [OF least_s] L La)
```
```   294         }
```
```   295 	moreover
```
```   296         {
```
```   297 	  assume "z \<in> A"
```
```   298           with L least_s least_a have ?thesis
```
```   299 	    by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
```
```   300         }
```
```   301 	  moreover note xA
```
```   302 	  ultimately show ?thesis by blast
```
```   303 	qed
```
```   304       next
```
```   305 	fix y
```
```   306 	assume y: "y \<in> Upper L (insert x A)"
```
```   307 	show "s \<sqsubseteq> y"
```
```   308 	proof (rule least_le [OF least_s], rule Upper_memI)
```
```   309 	  fix z
```
```   310 	  assume z: "z \<in> {a, x}"
```
```   311 	  show "z \<sqsubseteq> y"
```
```   312 	  proof -
```
```   313 	    {
```
```   314           have y': "y \<in> Upper L A"
```
```   315 	    apply (rule subsetD [where A = "Upper L (insert x A)"])
```
```   316 	    apply (rule Upper_antimono) apply clarify apply assumption
```
```   317 	    done
```
```   318 	  assume "z = a"
```
```   319 	  with y' least_a have ?thesis by (fast dest: least_le)
```
```   320         }
```
```   321 	moreover
```
```   322 	{
```
```   323            assume "z = x"
```
```   324            with y L have ?thesis by blast
```
```   325             }
```
```   326             moreover note z
```
```   327             ultimately show ?thesis by blast
```
```   328 	  qed
```
```   329 	qed (rule Upper_closed [THEN subsetD])
```
```   330       next
```
```   331 	from L show "insert x A \<subseteq> carrier L" by simp
```
```   332       next
```
```   333 	from least_s show "s \<in> carrier L" by simp
```
```   334       qed
```
```   335     qed
```
```   336   qed
```
```   337 qed
```
```   338
```
```   339 lemma (in lattice) finite_sup_least:
```
```   340   "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion> A) (Upper L A)"
```
```   341 proof (induct set: Finites)
```
```   342   case empty then show ?case by simp
```
```   343 next
```
```   344   case (insert A x)
```
```   345   show ?case
```
```   346   proof (cases "A = {}")
```
```   347     case True
```
```   348     with insert show ?thesis by (simp add: sup_of_singletonI)
```
```   349   next
```
```   350     case False
```
```   351     from insert show ?thesis
```
```   352     proof (rule_tac sup_insertI)
```
```   353       from False insert show "least L (\<Squnion> A) (Upper L A)" by simp
```
```   354     qed simp_all
```
```   355   qed
```
```   356 qed
```
```   357
```
```   358 lemma (in lattice) finite_sup_insertI:
```
```   359   assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
```
```   360     and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L"
```
```   361   shows "P (\<Squnion> (insert x A))"
```
```   362 proof (cases "A = {}")
```
```   363   case True with P and xA show ?thesis
```
```   364     by (simp add: sup_of_singletonI)
```
```   365 next
```
```   366   case False with P and xA show ?thesis
```
```   367     by (simp add: sup_insertI finite_sup_least)
```
```   368 qed
```
```   369
```
```   370 lemma (in lattice) finite_sup_closed:
```
```   371   "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion> A \<in> carrier L"
```
```   372 proof (induct set: Finites)
```
```   373   case empty then show ?case by simp
```
```   374 next
```
```   375   case (insert A x) then show ?case
```
```   376     by (rule_tac finite_sup_insertI) (simp_all)
```
```   377 qed
```
```   378
```
```   379 lemma (in lattice) join_left:
```
```   380   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
```
```   381   by (rule joinI [folded join_def]) (blast dest: least_mem )
```
```   382
```
```   383 lemma (in lattice) join_right:
```
```   384   "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
```
```   385   by (rule joinI [folded join_def]) (blast dest: least_mem )
```
```   386
```
```   387 lemma (in lattice) sup_of_two_least:
```
```   388   "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion> {x, y}) (Upper L {x, y})"
```
```   389 proof (unfold sup_def)
```
```   390   assume L: "x \<in> carrier L" "y \<in> carrier L"
```
```   391   with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
```
```   392   with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})"
```
```   393   by (fast intro: theI2 least_unique)  (* blast fails *)
```
```   394 qed
```
```   395
```
```   396 lemma (in lattice) join_le:
```
```   397   assumes sub: "x \<sqsubseteq> z" "y \<sqsubseteq> z"
```
```   398     and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
```
```   399   shows "x \<squnion> y \<sqsubseteq> z"
```
```   400 proof (rule joinI)
```
```   401   fix s
```
```   402   assume "least L s (Upper L {x, y})"
```
```   403   with sub L show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI)
```
```   404 qed
```
```   405
```
```   406 lemma (in lattice) join_assoc_lemma:
```
```   407   assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
```
```   408   shows "x \<squnion> (y \<squnion> z) = \<Squnion> {x, y, z}"
```
```   409 proof (rule finite_sup_insertI)
```
```   410   -- {* The textbook argument in Jacobson I, p 457 *}
```
```   411   fix s
```
```   412   assume sup: "least L s (Upper L {x, y, z})"
```
```   413   show "x \<squnion> (y \<squnion> z) = s"
```
```   414   proof (rule anti_sym)
```
```   415     from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s"
```
```   416       by (fastsimp intro!: join_le elim: least_Upper_above)
```
```   417   next
```
```   418     from sup L show "s \<sqsubseteq> x \<squnion> (y \<squnion> z)"
```
```   419     by (erule_tac least_le)
```
```   420       (blast intro!: Upper_memI intro: trans join_left join_right join_closed)
```
```   421   qed (simp_all add: L least_carrier [OF sup])
```
```   422 qed (simp_all add: L)
```
```   423
```
```   424 lemma join_comm:
```
```   425   includes order_syntax
```
```   426   shows "x \<squnion> y = y \<squnion> x"
```
```   427   by (unfold join_def) (simp add: insert_commute)
```
```   428
```
```   429 lemma (in lattice) join_assoc:
```
```   430   assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
```
```   431   shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
```
```   432 proof -
```
```   433   have "(x \<squnion> y) \<squnion> z = z \<squnion> (x \<squnion> y)" by (simp only: join_comm)
```
```   434   also from L have "... = \<Squnion> {z, x, y}" by (simp add: join_assoc_lemma)
```
```   435   also from L have "... = \<Squnion> {x, y, z}" by (simp add: insert_commute)
```
```   436   also from L have "... = x \<squnion> (y \<squnion> z)" by (simp add: join_assoc_lemma)
```
```   437   finally show ?thesis .
```
```   438 qed
```
```   439
```
```   440 subsubsection {* Infimum *}
```
```   441
```
```   442 lemma (in lattice) meetI:
```
```   443   "[| !!i. greatest L i (Lower L {x, y}) ==> P i;
```
```   444   x \<in> carrier L; y \<in> carrier L |]
```
```   445   ==> P (x \<sqinter> y)"
```
```   446 proof (unfold meet_def inf_def)
```
```   447   assume L: "x \<in> carrier L" "y \<in> carrier L"
```
```   448     and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
```
```   449   with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast
```
```   450   with L show "P (THE g. greatest L g (Lower L {x, y}))"
```
```   451   by (fast intro: theI2 greatest_unique P)
```
```   452 qed
```
```   453
```
```   454 lemma (in lattice) meet_closed [simp]:
```
```   455   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L"
```
```   456   by (rule meetI) (rule greatest_carrier)
```
```   457
```
```   458 lemma (in partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
```
```   459   "x \<in> carrier L ==> greatest L x (Lower L {x})"
```
```   460   by (rule greatest_LowerI) fast+
```
```   461
```
```   462 lemma (in partial_order) inf_of_singleton [simp]:
```
```   463   includes order_syntax
```
```   464   shows "x \<in> carrier L ==> \<Sqinter> {x} = x"
```
```   465   by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI)
```
```   466
```
```   467 text {* Condition on A: infimum exists. *}
```
```   468
```
```   469 lemma (in lattice) inf_insertI:
```
```   470   "[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
```
```   471   greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |]
```
```   472   ==> P (\<Sqinter> (insert x A))"
```
```   473 proof (unfold inf_def)
```
```   474   assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
```
```   475     and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g"
```
```   476     and greatest_a: "greatest L a (Lower L A)"
```
```   477   from L greatest_a have La: "a \<in> carrier L" by simp
```
```   478   from L inf_of_two_exists greatest_a
```
```   479   obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
```
```   480   show "P (THE g. greatest L g (Lower L (insert x A)))"
```
```   481   proof (rule theI2 [where a = i])
```
```   482     show "greatest L i (Lower L (insert x A))"
```
```   483     proof (rule greatest_LowerI)
```
```   484       fix z
```
```   485       assume xA: "z \<in> insert x A"
```
```   486       show "i \<sqsubseteq> z"
```
```   487       proof -
```
```   488 	{
```
```   489 	  assume "z = x" then have ?thesis
```
```   490 	    by (simp add: greatest_Lower_above [OF greatest_i] L La)
```
```   491         }
```
```   492 	moreover
```
```   493         {
```
```   494 	  assume "z \<in> A"
```
```   495           with L greatest_i greatest_a have ?thesis
```
```   496 	    by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
```
```   497         }
```
```   498       moreover note xA
```
```   499       ultimately show ?thesis by blast
```
```   500     qed
```
```   501   next
```
```   502     fix y
```
```   503     assume y: "y \<in> Lower L (insert x A)"
```
```   504     show "y \<sqsubseteq> i"
```
```   505     proof (rule greatest_le [OF greatest_i], rule Lower_memI)
```
```   506       fix z
```
```   507       assume z: "z \<in> {a, x}"
```
```   508       show "y \<sqsubseteq> z"
```
```   509       proof -
```
```   510 	{
```
```   511           have y': "y \<in> Lower L A"
```
```   512 	    apply (rule subsetD [where A = "Lower L (insert x A)"])
```
```   513 	    apply (rule Lower_antimono) apply clarify apply assumption
```
```   514 	    done
```
```   515 	  assume "z = a"
```
```   516 	  with y' greatest_a have ?thesis by (fast dest: greatest_le)
```
```   517         }
```
```   518 	moreover
```
```   519 	{
```
```   520            assume "z = x"
```
```   521            with y L have ?thesis by blast
```
```   522         }
```
```   523         moreover note z
```
```   524         ultimately show ?thesis by blast
```
```   525       qed
```
```   526     qed (rule Lower_closed [THEN subsetD])
```
```   527   next
```
```   528     from L show "insert x A \<subseteq> carrier L" by simp
```
```   529   next
```
```   530     from greatest_i show "i \<in> carrier L" by simp
```
```   531   qed
```
```   532 next
```
```   533     fix g
```
```   534     assume greatest_g: "greatest L g (Lower L (insert x A))"
```
```   535     show "g = i"
```
```   536     proof (rule greatest_unique)
```
```   537       show "greatest L i (Lower L (insert x A))"
```
```   538       proof (rule greatest_LowerI)
```
```   539 	fix z
```
```   540 	assume xA: "z \<in> insert x A"
```
```   541 	show "i \<sqsubseteq> z"
```
```   542       proof -
```
```   543 	{
```
```   544 	  assume "z = x" then have ?thesis
```
```   545 	    by (simp add: greatest_Lower_above [OF greatest_i] L La)
```
```   546         }
```
```   547 	moreover
```
```   548         {
```
```   549 	  assume "z \<in> A"
```
```   550           with L greatest_i greatest_a have ?thesis
```
```   551 	    by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
```
```   552         }
```
```   553 	  moreover note xA
```
```   554 	  ultimately show ?thesis by blast
```
```   555 	qed
```
```   556       next
```
```   557 	fix y
```
```   558 	assume y: "y \<in> Lower L (insert x A)"
```
```   559 	show "y \<sqsubseteq> i"
```
```   560 	proof (rule greatest_le [OF greatest_i], rule Lower_memI)
```
```   561 	  fix z
```
```   562 	  assume z: "z \<in> {a, x}"
```
```   563 	  show "y \<sqsubseteq> z"
```
```   564 	  proof -
```
```   565 	    {
```
```   566           have y': "y \<in> Lower L A"
```
```   567 	    apply (rule subsetD [where A = "Lower L (insert x A)"])
```
```   568 	    apply (rule Lower_antimono) apply clarify apply assumption
```
```   569 	    done
```
```   570 	  assume "z = a"
```
```   571 	  with y' greatest_a have ?thesis by (fast dest: greatest_le)
```
```   572         }
```
```   573 	moreover
```
```   574 	{
```
```   575            assume "z = x"
```
```   576            with y L have ?thesis by blast
```
```   577             }
```
```   578             moreover note z
```
```   579             ultimately show ?thesis by blast
```
```   580 	  qed
```
```   581 	qed (rule Lower_closed [THEN subsetD])
```
```   582       next
```
```   583 	from L show "insert x A \<subseteq> carrier L" by simp
```
```   584       next
```
```   585 	from greatest_i show "i \<in> carrier L" by simp
```
```   586       qed
```
```   587     qed
```
```   588   qed
```
```   589 qed
```
```   590
```
```   591 lemma (in lattice) finite_inf_greatest:
```
```   592   "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter> A) (Lower L A)"
```
```   593 proof (induct set: Finites)
```
```   594   case empty then show ?case by simp
```
```   595 next
```
```   596   case (insert A x)
```
```   597   show ?case
```
```   598   proof (cases "A = {}")
```
```   599     case True
```
```   600     with insert show ?thesis by (simp add: inf_of_singletonI)
```
```   601   next
```
```   602     case False
```
```   603     from insert show ?thesis
```
```   604     proof (rule_tac inf_insertI)
```
```   605       from False insert show "greatest L (\<Sqinter> A) (Lower L A)" by simp
```
```   606     qed simp_all
```
```   607   qed
```
```   608 qed
```
```   609
```
```   610 lemma (in lattice) finite_inf_insertI:
```
```   611   assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
```
```   612     and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L"
```
```   613   shows "P (\<Sqinter> (insert x A))"
```
```   614 proof (cases "A = {}")
```
```   615   case True with P and xA show ?thesis
```
```   616     by (simp add: inf_of_singletonI)
```
```   617 next
```
```   618   case False with P and xA show ?thesis
```
```   619     by (simp add: inf_insertI finite_inf_greatest)
```
```   620 qed
```
```   621
```
```   622 lemma (in lattice) finite_inf_closed:
```
```   623   "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter> A \<in> carrier L"
```
```   624 proof (induct set: Finites)
```
```   625   case empty then show ?case by simp
```
```   626 next
```
```   627   case (insert A x) then show ?case
```
```   628     by (rule_tac finite_inf_insertI) (simp_all)
```
```   629 qed
```
```   630
```
```   631 lemma (in lattice) meet_left:
```
```   632   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
```
```   633   by (rule meetI [folded meet_def]) (blast dest: greatest_mem )
```
```   634
```
```   635 lemma (in lattice) meet_right:
```
```   636   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
```
```   637   by (rule meetI [folded meet_def]) (blast dest: greatest_mem )
```
```   638
```
```   639 lemma (in lattice) inf_of_two_greatest:
```
```   640   "[| x \<in> carrier L; y \<in> carrier L |] ==>
```
```   641   greatest L (\<Sqinter> {x, y}) (Lower L {x, y})"
```
```   642 proof (unfold inf_def)
```
```   643   assume L: "x \<in> carrier L" "y \<in> carrier L"
```
```   644   with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
```
```   645   with L
```
```   646   show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})"
```
```   647   by (fast intro: theI2 greatest_unique)  (* blast fails *)
```
```   648 qed
```
```   649
```
```   650 lemma (in lattice) meet_le:
```
```   651   assumes sub: "z \<sqsubseteq> x" "z \<sqsubseteq> y"
```
```   652     and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
```
```   653   shows "z \<sqsubseteq> x \<sqinter> y"
```
```   654 proof (rule meetI)
```
```   655   fix i
```
```   656   assume "greatest L i (Lower L {x, y})"
```
```   657   with sub L show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
```
```   658 qed
```
```   659
```
```   660 lemma (in lattice) meet_assoc_lemma:
```
```   661   assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
```
```   662   shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter> {x, y, z}"
```
```   663 proof (rule finite_inf_insertI)
```
```   664   txt {* The textbook argument in Jacobson I, p 457 *}
```
```   665   fix i
```
```   666   assume inf: "greatest L i (Lower L {x, y, z})"
```
```   667   show "x \<sqinter> (y \<sqinter> z) = i"
```
```   668   proof (rule anti_sym)
```
```   669     from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
```
```   670       by (fastsimp intro!: meet_le elim: greatest_Lower_above)
```
```   671   next
```
```   672     from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i"
```
```   673     by (erule_tac greatest_le)
```
```   674       (blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed)
```
```   675   qed (simp_all add: L greatest_carrier [OF inf])
```
```   676 qed (simp_all add: L)
```
```   677
```
```   678 lemma meet_comm:
```
```   679   includes order_syntax
```
```   680   shows "x \<sqinter> y = y \<sqinter> x"
```
```   681   by (unfold meet_def) (simp add: insert_commute)
```
```   682
```
```   683 lemma (in lattice) meet_assoc:
```
```   684   assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
```
```   685   shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
```
```   686 proof -
```
```   687   have "(x \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm)
```
```   688   also from L have "... = \<Sqinter> {z, x, y}" by (simp add: meet_assoc_lemma)
```
```   689   also from L have "... = \<Sqinter> {x, y, z}" by (simp add: insert_commute)
```
```   690   also from L have "... = x \<sqinter> (y \<sqinter> z)" by (simp add: meet_assoc_lemma)
```
```   691   finally show ?thesis .
```
```   692 qed
```
```   693
```
```   694 subsection {* Total Orders *}
```
```   695
```
```   696 locale total_order = lattice +
```
```   697   assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
```
```   698
```
```   699 text {* Introduction rule: the usual definition of total order *}
```
```   700
```
```   701 lemma (in partial_order) total_orderI:
```
```   702   assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
```
```   703   shows "total_order L"
```
```   704 proof (rule total_order.intro)
```
```   705   show "lattice_axioms L"
```
```   706   proof (rule lattice_axioms.intro)
```
```   707     fix x y
```
```   708     assume L: "x \<in> carrier L" "y \<in> carrier L"
```
```   709     show "EX s. least L s (Upper L {x, y})"
```
```   710     proof -
```
```   711       note total L
```
```   712       moreover
```
```   713       {
```
```   714 	assume "x \<sqsubseteq> y"
```
```   715         with L have "least L y (Upper L {x, y})"
```
```   716 	  by (rule_tac least_UpperI) auto
```
```   717       }
```
```   718       moreover
```
```   719       {
```
```   720 	assume "y \<sqsubseteq> x"
```
```   721         with L have "least L x (Upper L {x, y})"
```
```   722 	  by (rule_tac least_UpperI) auto
```
```   723       }
```
```   724       ultimately show ?thesis by blast
```
```   725     qed
```
```   726   next
```
```   727     fix x y
```
```   728     assume L: "x \<in> carrier L" "y \<in> carrier L"
```
```   729     show "EX i. greatest L i (Lower L {x, y})"
```
```   730     proof -
```
```   731       note total L
```
```   732       moreover
```
```   733       {
```
```   734 	assume "y \<sqsubseteq> x"
```
```   735         with L have "greatest L y (Lower L {x, y})"
```
```   736 	  by (rule_tac greatest_LowerI) auto
```
```   737       }
```
```   738       moreover
```
```   739       {
```
```   740 	assume "x \<sqsubseteq> y"
```
```   741         with L have "greatest L x (Lower L {x, y})"
```
```   742 	  by (rule_tac greatest_LowerI) auto
```
```   743       }
```
```   744       ultimately show ?thesis by blast
```
```   745     qed
```
```   746   qed
```
```   747 qed (assumption | rule total_order_axioms.intro)+
```
```   748
```
```   749 subsection {* Complete lattices *}
```
```   750
```
```   751 locale complete_lattice = lattice +
```
```   752   assumes sup_exists:
```
```   753     "[| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
```
```   754     and inf_exists:
```
```   755     "[| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
```
```   756
```
```   757 text {* Introduction rule: the usual definition of complete lattice *}
```
```   758
```
```   759 lemma (in partial_order) complete_latticeI:
```
```   760   assumes sup_exists:
```
```   761     "!!A. [| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
```
```   762     and inf_exists:
```
```   763     "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
```
```   764   shows "complete_lattice L"
```
```   765 proof (rule complete_lattice.intro)
```
```   766   show "lattice_axioms L"
```
```   767   by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
```
```   768 qed (assumption | rule complete_lattice_axioms.intro)+
```
```   769
```
```   770 constdefs (structure L)
```
```   771   top :: "_ => 'a" ("\<top>\<index>")
```
```   772   "\<top> == sup L (carrier L)"
```
```   773
```
```   774   bottom :: "_ => 'a" ("\<bottom>\<index>")
```
```   775   "\<bottom> == inf L (carrier L)"
```
```   776
```
```   777
```
```   778 lemma (in complete_lattice) supI:
```
```   779   "[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |]
```
```   780   ==> P (\<Squnion>A)"
```
```   781 proof (unfold sup_def)
```
```   782   assume L: "A \<subseteq> carrier L"
```
```   783     and P: "!!l. least L l (Upper L A) ==> P l"
```
```   784   with sup_exists obtain s where "least L s (Upper L A)" by blast
```
```   785   with L show "P (THE l. least L l (Upper L A))"
```
```   786   by (fast intro: theI2 least_unique P)
```
```   787 qed
```
```   788
```
```   789 lemma (in complete_lattice) sup_closed [simp]:
```
```   790   "A \<subseteq> carrier L ==> \<Squnion> A \<in> carrier L"
```
```   791   by (rule supI) simp_all
```
```   792
```
```   793 lemma (in complete_lattice) top_closed [simp, intro]:
```
```   794   "\<top> \<in> carrier L"
```
```   795   by (unfold top_def) simp
```
```   796
```
```   797 lemma (in complete_lattice) infI:
```
```   798   "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
```
```   799   ==> P (\<Sqinter> A)"
```
```   800 proof (unfold inf_def)
```
```   801   assume L: "A \<subseteq> carrier L"
```
```   802     and P: "!!l. greatest L l (Lower L A) ==> P l"
```
```   803   with inf_exists obtain s where "greatest L s (Lower L A)" by blast
```
```   804   with L show "P (THE l. greatest L l (Lower L A))"
```
```   805   by (fast intro: theI2 greatest_unique P)
```
```   806 qed
```
```   807
```
```   808 lemma (in complete_lattice) inf_closed [simp]:
```
```   809   "A \<subseteq> carrier L ==> \<Sqinter> A \<in> carrier L"
```
```   810   by (rule infI) simp_all
```
```   811
```
```   812 lemma (in complete_lattice) bottom_closed [simp, intro]:
```
```   813   "\<bottom> \<in> carrier L"
```
```   814   by (unfold bottom_def) simp
```
```   815
```
```   816 text {* Jacobson: Theorem 8.1 *}
```
```   817
```
```   818 lemma Lower_empty [simp]:
```
```   819   "Lower L {} = carrier L"
```
```   820   by (unfold Lower_def) simp
```
```   821
```
```   822 lemma Upper_empty [simp]:
```
```   823   "Upper L {} = carrier L"
```
```   824   by (unfold Upper_def) simp
```
```   825
```
```   826 theorem (in partial_order) complete_lattice_criterion1:
```
```   827   assumes top_exists: "EX g. greatest L g (carrier L)"
```
```   828     and inf_exists:
```
```   829       "!!A. [| A \<subseteq> carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)"
```
```   830   shows "complete_lattice L"
```
```   831 proof (rule complete_latticeI)
```
```   832   from top_exists obtain top where top: "greatest L top (carrier L)" ..
```
```   833   fix A
```
```   834   assume L: "A \<subseteq> carrier L"
```
```   835   let ?B = "Upper L A"
```
```   836   from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le)
```
```   837   then have B_non_empty: "?B ~= {}" by fast
```
```   838   have B_L: "?B \<subseteq> carrier L" by simp
```
```   839   from inf_exists [OF B_L B_non_empty]
```
```   840   obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
```
```   841   have "least L b (Upper L A)"
```
```   842 apply (rule least_UpperI)
```
```   843    apply (rule greatest_le [where A = "Lower L ?B"])
```
```   844     apply (rule b_inf_B)
```
```   845    apply (rule Lower_memI)
```
```   846     apply (erule UpperD)
```
```   847      apply assumption
```
```   848     apply (rule L)
```
```   849    apply (fast intro: L [THEN subsetD])
```
```   850   apply (erule greatest_Lower_above [OF b_inf_B])
```
```   851   apply simp
```
```   852  apply (rule L)
```
```   853 apply (rule greatest_carrier [OF b_inf_B]) (* rename rule: _closed *)
```
```   854 done
```
```   855   then show "EX s. least L s (Upper L A)" ..
```
```   856 next
```
```   857   fix A
```
```   858   assume L: "A \<subseteq> carrier L"
```
```   859   show "EX i. greatest L i (Lower L A)"
```
```   860   proof (cases "A = {}")
```
```   861     case True then show ?thesis
```
```   862       by (simp add: top_exists)
```
```   863   next
```
```   864     case False with L show ?thesis
```
```   865       by (rule inf_exists)
```
```   866   qed
```
```   867 qed
```
```   868
```
```   869 (* TODO: prove dual version *)
```
```   870
```
```   871 subsection {* Examples *}
```
```   872
```
```   873 subsubsection {* Powerset of a set is a complete lattice *}
```
```   874
```
```   875 theorem powerset_is_complete_lattice:
```
```   876   "complete_lattice (| carrier = Pow A, le = op \<subseteq> |)"
```
```   877   (is "complete_lattice ?L")
```
```   878 proof (rule partial_order.complete_latticeI)
```
```   879   show "partial_order ?L"
```
```   880     by (rule partial_order.intro) auto
```
```   881 next
```
```   882   fix B
```
```   883   assume "B \<subseteq> carrier ?L"
```
```   884   then have "least ?L (\<Union> B) (Upper ?L B)"
```
```   885     by (fastsimp intro!: least_UpperI simp: Upper_def)
```
```   886   then show "EX s. least ?L s (Upper ?L B)" ..
```
```   887 next
```
```   888   fix B
```
```   889   assume "B \<subseteq> carrier ?L"
```
```   890   then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)"
```
```   891     txt {* @{term "\<Inter> B"} is not the infimum of @{term B}:
```
```   892       @{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! *}
```
```   893     by (fastsimp intro!: greatest_LowerI simp: Lower_def)
```
```   894   then show "EX i. greatest ?L i (Lower ?L B)" ..
```
```   895 qed
```
```   896
```
```   897 subsubsection {* Lattice of subgroups of a group *}
```
```   898
```
```   899 theorem (in group) subgroups_partial_order:
```
```   900   "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
```
```   901   by (rule partial_order.intro) simp_all
```
```   902
```
```   903 lemma (in group) subgroup_self:
```
```   904   "subgroup (carrier G) G"
```
```   905   by (rule subgroupI) auto
```
```   906
```
```   907 lemma (in group) subgroup_imp_group:
```
```   908   "subgroup H G ==> group (G(| carrier := H |))"
```
```   909   using subgroup.groupI [OF _ group.intro] .
```
```   910
```
```   911 lemma (in group) is_monoid [intro, simp]:
```
```   912   "monoid G"
```
```   913   by (rule monoid.intro)
```
```   914
```
```   915 lemma (in group) subgroup_inv_equality:
```
```   916   "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
```
```   917 apply (rule_tac inv_equality [THEN sym])
```
```   918   apply (rule group.l_inv [OF subgroup_imp_group, simplified])
```
```   919    apply assumption+
```
```   920  apply (rule subsetD [OF subgroup.subset])
```
```   921   apply assumption+
```
```   922 apply (rule subsetD [OF subgroup.subset])
```
```   923  apply assumption
```
```   924 apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified])
```
```   925   apply assumption+
```
```   926 done
```
```   927
```
```   928 theorem (in group) subgroups_Inter:
```
```   929   assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"
```
```   930     and not_empty: "A ~= {}"
```
```   931   shows "subgroup (\<Inter>A) G"
```
```   932 proof (rule subgroupI)
```
```   933   from subgr [THEN subgroup.subset] and not_empty
```
```   934   show "\<Inter>A \<subseteq> carrier G" by blast
```
```   935 next
```
```   936   from subgr [THEN subgroup.one_closed]
```
```   937   show "\<Inter>A ~= {}" by blast
```
```   938 next
```
```   939   fix x assume "x \<in> \<Inter>A"
```
```   940   with subgr [THEN subgroup.m_inv_closed]
```
```   941   show "inv x \<in> \<Inter>A" by blast
```
```   942 next
```
```   943   fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
```
```   944   with subgr [THEN subgroup.m_closed]
```
```   945   show "x \<otimes> y \<in> \<Inter>A" by blast
```
```   946 qed
```
```   947
```
```   948 theorem (in group) subgroups_complete_lattice:
```
```   949   "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
```
```   950     (is "complete_lattice ?L")
```
```   951 proof (rule partial_order.complete_lattice_criterion1)
```
```   952   show "partial_order ?L" by (rule subgroups_partial_order)
```
```   953 next
```
```   954   have "greatest ?L (carrier G) (carrier ?L)"
```
```   955     by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
```
```   956   then show "EX G. greatest ?L G (carrier ?L)" ..
```
```   957 next
```
```   958   fix A
```
```   959   assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
```
```   960   then have Int_subgroup: "subgroup (\<Inter>A) G"
```
```   961     by (fastsimp intro: subgroups_Inter)
```
```   962   have "greatest ?L (\<Inter>A) (Lower ?L A)"
```
```   963     (is "greatest ?L ?Int _")
```
```   964   proof (rule greatest_LowerI)
```
```   965     fix H
```
```   966     assume H: "H \<in> A"
```
```   967     with L have subgroupH: "subgroup H G" by auto
```
```   968     from subgroupH have submagmaH: "submagma H G" by (rule subgroup.axioms)
```
```   969     from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
```
```   970       by (rule subgroup_imp_group)
```
```   971     from groupH have monoidH: "monoid ?H"
```
```   972       by (rule group.is_monoid)
```
```   973     from H have Int_subset: "?Int \<subseteq> H" by fastsimp
```
```   974     then show "le ?L ?Int H" by simp
```
```   975   next
```
```   976     fix H
```
```   977     assume H: "H \<in> Lower ?L A"
```
```   978     with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)
```
```   979   next
```
```   980     show "A \<subseteq> carrier ?L" by (rule L)
```
```   981   next
```
```   982     show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
```
```   983   qed
```
```   984   then show "EX I. greatest ?L I (Lower ?L A)" ..
```
```   985 qed
```
```   986
```
`   987 end`