src/HOL/Algebra/Lattice.thy
 author wenzelm Thu May 06 14:14:18 2004 +0200 (2004-05-06) changeset 14706 71590b7733b7 parent 14693 4deda204e1d8 child 14751 0d7850e27fed permissions -rw-r--r--
tuned document;
```     1 (*
```
```     2   Title:     HOL/Algebra/Lattice.thy
```
```     3   Id:        \$Id\$
```
```     4   Author:    Clemens Ballarin, started 7 November 2003
```
```     5   Copyright: Clemens Ballarin
```
```     6 *)
```
```     7
```
```     8 header {* Orders 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 partial_order = struct L +
```
```    18   assumes refl [intro, simp]:
```
```    19                   "x \<in> carrier L ==> x \<sqsubseteq> x"
```
```    20     and anti_sym [intro]:
```
```    21                   "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y"
```
```    22     and trans [trans]:
```
```    23                   "[| x \<sqsubseteq> y; y \<sqsubseteq> z;
```
```    24                    x \<in> carrier L; y \<in> carrier L; z \<in> carrier L |] ==> x \<sqsubseteq> z"
```
```    25
```
```    26 constdefs (structure L)
```
```    27   less :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50)
```
```    28   "x \<sqsubset> y == x \<sqsubseteq> y & x ~= y"
```
```    29
```
```    30   -- {* Upper and lower bounds of a set. *}
```
```    31   Upper :: "[_, 'a set] => 'a set"
```
```    32   "Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> x \<sqsubseteq> u)} \<inter>
```
```    33                 carrier L"
```
```    34
```
```    35   Lower :: "[_, 'a set] => 'a set"
```
```    36   "Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> l \<sqsubseteq> x)} \<inter>
```
```    37                 carrier L"
```
```    38
```
```    39   -- {* Least and greatest, as predicate. *}
```
```    40   least :: "[_, 'a, 'a set] => bool"
```
```    41   "least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. l \<sqsubseteq> x)"
```
```    42
```
```    43   greatest :: "[_, 'a, 'a set] => bool"
```
```    44   "greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. x \<sqsubseteq> g)"
```
```    45
```
```    46   -- {* Supremum and infimum *}
```
```    47   sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_"  90)
```
```    48   "\<Squnion>A == THE x. least L x (Upper L A)"
```
```    49
```
```    50   inf :: "[_, 'a set] => 'a" ("\<Sqinter>\<index>_"  90)
```
```    51   "\<Sqinter>A == THE x. greatest L x (Lower L A)"
```
```    52
```
```    53   join :: "[_, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65)
```
```    54   "x \<squnion> y == sup L {x, y}"
```
```    55
```
```    56   meet :: "[_, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 65)
```
```    57   "x \<sqinter> y == inf L {x, y}"
```
```    58
```
```    59
```
```    60 subsubsection {* Upper *}
```
```    61
```
```    62 lemma Upper_closed [intro, simp]:
```
```    63   "Upper L A \<subseteq> carrier L"
```
```    64   by (unfold Upper_def) clarify
```
```    65
```
```    66 lemma UpperD [dest]:
```
```    67   includes struct L
```
```    68   shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u"
```
```    69   by (unfold Upper_def) blast
```
```    70
```
```    71 lemma Upper_memI:
```
```    72   includes struct L
```
```    73   shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A"
```
```    74   by (unfold Upper_def) blast
```
```    75
```
```    76 lemma Upper_antimono:
```
```    77   "A \<subseteq> B ==> Upper L B \<subseteq> Upper L A"
```
```    78   by (unfold Upper_def) blast
```
```    79
```
```    80
```
```    81 subsubsection {* Lower *}
```
```    82
```
```    83 lemma Lower_closed [intro, simp]:
```
```    84   "Lower L A \<subseteq> carrier L"
```
```    85   by (unfold Lower_def) clarify
```
```    86
```
```    87 lemma LowerD [dest]:
```
```    88   includes struct L
```
```    89   shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x"
```
```    90   by (unfold Lower_def) blast
```
```    91
```
```    92 lemma Lower_memI:
```
```    93   includes struct L
```
```    94   shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A"
```
```    95   by (unfold Lower_def) blast
```
```    96
```
```    97 lemma Lower_antimono:
```
```    98   "A \<subseteq> B ==> Lower L B \<subseteq> Lower L A"
```
```    99   by (unfold Lower_def) blast
```
```   100
```
```   101
```
```   102 subsubsection {* least *}
```
```   103
```
```   104 lemma least_carrier [intro, simp]:
```
```   105   shows "least L l A ==> l \<in> carrier L"
```
```   106   by (unfold least_def) fast
```
```   107
```
```   108 lemma least_mem:
```
```   109   "least L l A ==> l \<in> A"
```
```   110   by (unfold least_def) fast
```
```   111
```
```   112 lemma (in partial_order) least_unique:
```
```   113   "[| least L x A; least L y A |] ==> x = y"
```
```   114   by (unfold least_def) blast
```
```   115
```
```   116 lemma least_le:
```
```   117   includes struct L
```
```   118   shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a"
```
```   119   by (unfold least_def) fast
```
```   120
```
```   121 lemma least_UpperI:
```
```   122   includes struct L
```
```   123   assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s"
```
```   124     and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y"
```
```   125     and L: "A \<subseteq> carrier L"  "s \<in> carrier L"
```
```   126   shows "least L s (Upper L A)"
```
```   127 proof -
```
```   128   have "Upper L A \<subseteq> carrier L" by simp
```
```   129   moreover from above L have "s \<in> Upper L A" by (simp add: Upper_def)
```
```   130   moreover from below have "ALL x : Upper L A. s \<sqsubseteq> x" by fast
```
```   131   ultimately show ?thesis by (simp add: least_def)
```
```   132 qed
```
```   133
```
```   134
```
```   135 subsubsection {* greatest *}
```
```   136
```
```   137 lemma greatest_carrier [intro, simp]:
```
```   138   shows "greatest L l A ==> l \<in> carrier L"
```
```   139   by (unfold greatest_def) fast
```
```   140
```
```   141 lemma greatest_mem:
```
```   142   "greatest L l A ==> l \<in> A"
```
```   143   by (unfold greatest_def) fast
```
```   144
```
```   145 lemma (in partial_order) greatest_unique:
```
```   146   "[| greatest L x A; greatest L y A |] ==> x = y"
```
```   147   by (unfold greatest_def) blast
```
```   148
```
```   149 lemma greatest_le:
```
```   150   includes struct L
```
```   151   shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x"
```
```   152   by (unfold greatest_def) fast
```
```   153
```
```   154 lemma greatest_LowerI:
```
```   155   includes struct L
```
```   156   assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x"
```
```   157     and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i"
```
```   158     and L: "A \<subseteq> carrier L"  "i \<in> carrier L"
```
```   159   shows "greatest L i (Lower L A)"
```
```   160 proof -
```
```   161   have "Lower L A \<subseteq> carrier L" by simp
```
```   162   moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def)
```
```   163   moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast
```
```   164   ultimately show ?thesis by (simp add: greatest_def)
```
```   165 qed
```
```   166
```
```   167
```
```   168 subsection {* Lattices *}
```
```   169
```
```   170 locale lattice = partial_order +
```
```   171   assumes sup_of_two_exists:
```
```   172     "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. least L s (Upper L {x, y})"
```
```   173     and inf_of_two_exists:
```
```   174     "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"
```
```   175
```
```   176 lemma least_Upper_above:
```
```   177   includes struct L
```
```   178   shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
```
```   179   by (unfold least_def) blast
```
```   180
```
```   181 lemma greatest_Lower_above:
```
```   182   includes struct L
```
```   183   shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x"
```
```   184   by (unfold greatest_def) blast
```
```   185
```
```   186
```
```   187 subsubsection {* Supremum *}
```
```   188
```
```   189 lemma (in lattice) joinI:
```
```   190   "[| !!l. least L l (Upper L {x, y}) ==> P l; x \<in> carrier L; y \<in> carrier L |]
```
```   191   ==> P (x \<squnion> y)"
```
```   192 proof (unfold join_def sup_def)
```
```   193   assume L: "x \<in> carrier L"  "y \<in> carrier L"
```
```   194     and P: "!!l. least L l (Upper L {x, y}) ==> P l"
```
```   195   with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
```
```   196   with L show "P (THE l. least L l (Upper L {x, y}))"
```
```   197     by (fast intro: theI2 least_unique P)
```
```   198 qed
```
```   199
```
```   200 lemma (in lattice) join_closed [simp]:
```
```   201   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L"
```
```   202   by (rule joinI) (rule least_carrier)
```
```   203
```
```   204 lemma (in partial_order) sup_of_singletonI:      (* only reflexivity needed ? *)
```
```   205   "x \<in> carrier L ==> least L x (Upper L {x})"
```
```   206   by (rule least_UpperI) fast+
```
```   207
```
```   208 lemma (in partial_order) sup_of_singleton [simp]:
```
```   209   includes struct L
```
```   210   shows "x \<in> carrier L ==> \<Squnion>{x} = x"
```
```   211   by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI)
```
```   212
```
```   213
```
```   214 text {* Condition on @{text A}: supremum exists. *}
```
```   215
```
```   216 lemma (in lattice) sup_insertI:
```
```   217   "[| !!s. least L s (Upper L (insert x A)) ==> P s;
```
```   218   least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |]
```
```   219   ==> P (\<Squnion>(insert x A))"
```
```   220 proof (unfold sup_def)
```
```   221   assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
```
```   222     and P: "!!l. least L l (Upper L (insert x A)) ==> P l"
```
```   223     and least_a: "least L a (Upper L A)"
```
```   224   from L least_a have La: "a \<in> carrier L" by simp
```
```   225   from L sup_of_two_exists least_a
```
```   226   obtain s where least_s: "least L s (Upper L {a, x})" by blast
```
```   227   show "P (THE l. least L l (Upper L (insert x A)))"
```
```   228   proof (rule theI2)
```
```   229     show "least L s (Upper L (insert x A))"
```
```   230     proof (rule least_UpperI)
```
```   231       fix z
```
```   232       assume "z \<in> insert x A"
```
```   233       then show "z \<sqsubseteq> s"
```
```   234       proof
```
```   235         assume "z = x" then show ?thesis
```
```   236           by (simp add: least_Upper_above [OF least_s] L La)
```
```   237       next
```
```   238         assume "z \<in> A"
```
```   239         with L least_s least_a show ?thesis
```
```   240           by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
```
```   241       qed
```
```   242     next
```
```   243       fix y
```
```   244       assume y: "y \<in> Upper L (insert x A)"
```
```   245       show "s \<sqsubseteq> y"
```
```   246       proof (rule least_le [OF least_s], rule Upper_memI)
```
```   247 	fix z
```
```   248 	assume z: "z \<in> {a, x}"
```
```   249 	then show "z \<sqsubseteq> y"
```
```   250 	proof
```
```   251           have y': "y \<in> Upper L A"
```
```   252             apply (rule subsetD [where A = "Upper L (insert x A)"])
```
```   253             apply (rule Upper_antimono) apply clarify apply assumption
```
```   254             done
```
```   255           assume "z = a"
```
```   256           with y' least_a show ?thesis by (fast dest: least_le)
```
```   257 	next
```
```   258 	  assume "z \<in> {x}"  (* FIXME "z = x"; declare specific elim rule for "insert x {}" (!?) *)
```
```   259           with y L show ?thesis by blast
```
```   260 	qed
```
```   261       qed (rule Upper_closed [THEN subsetD])
```
```   262     next
```
```   263       from L show "insert x A \<subseteq> carrier L" by simp
```
```   264       from least_s show "s \<in> carrier L" by simp
```
```   265     qed
```
```   266   next
```
```   267     fix l
```
```   268     assume least_l: "least L l (Upper L (insert x A))"
```
```   269     show "l = s"
```
```   270     proof (rule least_unique)
```
```   271       show "least L s (Upper L (insert x A))"
```
```   272       proof (rule least_UpperI)
```
```   273         fix z
```
```   274         assume "z \<in> insert x A"
```
```   275         then show "z \<sqsubseteq> s"
```
```   276 	proof
```
```   277           assume "z = x" then show ?thesis
```
```   278             by (simp add: least_Upper_above [OF least_s] L La)
```
```   279 	next
```
```   280           assume "z \<in> A"
```
```   281           with L least_s least_a show ?thesis
```
```   282             by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
```
```   283 	qed
```
```   284       next
```
```   285         fix y
```
```   286         assume y: "y \<in> Upper L (insert x A)"
```
```   287         show "s \<sqsubseteq> y"
```
```   288         proof (rule least_le [OF least_s], rule Upper_memI)
```
```   289           fix z
```
```   290           assume z: "z \<in> {a, x}"
```
```   291           then show "z \<sqsubseteq> y"
```
```   292           proof
```
```   293             have y': "y \<in> Upper L A"
```
```   294 	      apply (rule subsetD [where A = "Upper L (insert x A)"])
```
```   295 	      apply (rule Upper_antimono) apply clarify apply assumption
```
```   296 	      done
```
```   297             assume "z = a"
```
```   298             with y' least_a show ?thesis by (fast dest: least_le)
```
```   299 	  next
```
```   300             assume "z \<in> {x}"
```
```   301             with y L show ?thesis by blast
```
```   302           qed
```
```   303         qed (rule Upper_closed [THEN subsetD])
```
```   304       next
```
```   305         from L show "insert x A \<subseteq> carrier L" by simp
```
```   306         from least_s show "s \<in> carrier L" by simp
```
```   307       qed
```
```   308     qed
```
```   309   qed
```
```   310 qed
```
```   311
```
```   312 lemma (in lattice) finite_sup_least:
```
```   313   "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion>A) (Upper L A)"
```
```   314 proof (induct set: Finites)
```
```   315   case empty
```
```   316   then show ?case by simp
```
```   317 next
```
```   318   case (insert A x)
```
```   319   show ?case
```
```   320   proof (cases "A = {}")
```
```   321     case True
```
```   322     with insert show ?thesis by (simp add: sup_of_singletonI)
```
```   323   next
```
```   324     case False
```
```   325     with insert have "least L (\<Squnion>A) (Upper L A)" by simp
```
```   326     with _ show ?thesis
```
```   327       by (rule sup_insertI) (simp_all add: insert [simplified])
```
```   328   qed
```
```   329 qed
```
```   330
```
```   331 lemma (in lattice) finite_sup_insertI:
```
```   332   assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
```
```   333     and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
```
```   334   shows "P (\<Squnion> (insert x A))"
```
```   335 proof (cases "A = {}")
```
```   336   case True with P and xA show ?thesis
```
```   337     by (simp add: sup_of_singletonI)
```
```   338 next
```
```   339   case False with P and xA show ?thesis
```
```   340     by (simp add: sup_insertI finite_sup_least)
```
```   341 qed
```
```   342
```
```   343 lemma (in lattice) finite_sup_closed:
```
```   344   "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion>A \<in> carrier L"
```
```   345 proof (induct set: Finites)
```
```   346   case empty then show ?case by simp
```
```   347 next
```
```   348   case (insert A x) then show ?case
```
```   349     by - (rule finite_sup_insertI, simp_all)
```
```   350 qed
```
```   351
```
```   352 lemma (in lattice) join_left:
```
```   353   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
```
```   354   by (rule joinI [folded join_def]) (blast dest: least_mem)
```
```   355
```
```   356 lemma (in lattice) join_right:
```
```   357   "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
```
```   358   by (rule joinI [folded join_def]) (blast dest: least_mem)
```
```   359
```
```   360 lemma (in lattice) sup_of_two_least:
```
```   361   "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion>{x, y}) (Upper L {x, y})"
```
```   362 proof (unfold sup_def)
```
```   363   assume L: "x \<in> carrier L"  "y \<in> carrier L"
```
```   364   with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
```
```   365   with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})"
```
```   366   by (fast intro: theI2 least_unique)  (* blast fails *)
```
```   367 qed
```
```   368
```
```   369 lemma (in lattice) join_le:
```
```   370   assumes sub: "x \<sqsubseteq> z"  "y \<sqsubseteq> z"
```
```   371     and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
```
```   372   shows "x \<squnion> y \<sqsubseteq> z"
```
```   373 proof (rule joinI)
```
```   374   fix s
```
```   375   assume "least L s (Upper L {x, y})"
```
```   376   with sub L show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI)
```
```   377 qed
```
```   378
```
```   379 lemma (in lattice) join_assoc_lemma:
```
```   380   assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
```
```   381   shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}"
```
```   382 proof (rule finite_sup_insertI)
```
```   383   -- {* The textbook argument in Jacobson I, p 457 *}
```
```   384   fix s
```
```   385   assume sup: "least L s (Upper L {x, y, z})"
```
```   386   show "x \<squnion> (y \<squnion> z) = s"
```
```   387   proof (rule anti_sym)
```
```   388     from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s"
```
```   389       by (fastsimp intro!: join_le elim: least_Upper_above)
```
```   390   next
```
```   391     from sup L show "s \<sqsubseteq> x \<squnion> (y \<squnion> z)"
```
```   392     by (erule_tac least_le)
```
```   393       (blast intro!: Upper_memI intro: trans join_left join_right join_closed)
```
```   394   qed (simp_all add: L least_carrier [OF sup])
```
```   395 qed (simp_all add: L)
```
```   396
```
```   397 lemma join_comm:
```
```   398   includes struct L
```
```   399   shows "x \<squnion> y = y \<squnion> x"
```
```   400   by (unfold join_def) (simp add: insert_commute)
```
```   401
```
```   402 lemma (in lattice) join_assoc:
```
```   403   assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
```
```   404   shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
```
```   405 proof -
```
```   406   have "(x \<squnion> y) \<squnion> z = z \<squnion> (x \<squnion> y)" by (simp only: join_comm)
```
```   407   also from L have "... = \<Squnion>{z, x, y}" by (simp add: join_assoc_lemma)
```
```   408   also from L have "... = \<Squnion>{x, y, z}" by (simp add: insert_commute)
```
```   409   also from L have "... = x \<squnion> (y \<squnion> z)" by (simp add: join_assoc_lemma)
```
```   410   finally show ?thesis .
```
```   411 qed
```
```   412
```
```   413
```
```   414 subsubsection {* Infimum *}
```
```   415
```
```   416 lemma (in lattice) meetI:
```
```   417   "[| !!i. greatest L i (Lower L {x, y}) ==> P i;
```
```   418   x \<in> carrier L; y \<in> carrier L |]
```
```   419   ==> P (x \<sqinter> y)"
```
```   420 proof (unfold meet_def inf_def)
```
```   421   assume L: "x \<in> carrier L"  "y \<in> carrier L"
```
```   422     and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
```
```   423   with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast
```
```   424   with L show "P (THE g. greatest L g (Lower L {x, y}))"
```
```   425   by (fast intro: theI2 greatest_unique P)
```
```   426 qed
```
```   427
```
```   428 lemma (in lattice) meet_closed [simp]:
```
```   429   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L"
```
```   430   by (rule meetI) (rule greatest_carrier)
```
```   431
```
```   432 lemma (in partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
```
```   433   "x \<in> carrier L ==> greatest L x (Lower L {x})"
```
```   434   by (rule greatest_LowerI) fast+
```
```   435
```
```   436 lemma (in partial_order) inf_of_singleton [simp]:
```
```   437   includes struct L
```
```   438   shows "x \<in> carrier L ==> \<Sqinter> {x} = x"
```
```   439   by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI)
```
```   440
```
```   441 text {* Condition on A: infimum exists. *}
```
```   442
```
```   443 lemma (in lattice) inf_insertI:
```
```   444   "[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
```
```   445   greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |]
```
```   446   ==> P (\<Sqinter>(insert x A))"
```
```   447 proof (unfold inf_def)
```
```   448   assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
```
```   449     and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g"
```
```   450     and greatest_a: "greatest L a (Lower L A)"
```
```   451   from L greatest_a have La: "a \<in> carrier L" by simp
```
```   452   from L inf_of_two_exists greatest_a
```
```   453   obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
```
```   454   show "P (THE g. greatest L g (Lower L (insert x A)))"
```
```   455   proof (rule theI2)
```
```   456     show "greatest L i (Lower L (insert x A))"
```
```   457     proof (rule greatest_LowerI)
```
```   458       fix z
```
```   459       assume "z \<in> insert x A"
```
```   460       then show "i \<sqsubseteq> z"
```
```   461       proof
```
```   462         assume "z = x" then show ?thesis
```
```   463           by (simp add: greatest_Lower_above [OF greatest_i] L La)
```
```   464       next
```
```   465         assume "z \<in> A"
```
```   466         with L greatest_i greatest_a show ?thesis
```
```   467           by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
```
```   468       qed
```
```   469     next
```
```   470       fix y
```
```   471       assume y: "y \<in> Lower L (insert x A)"
```
```   472       show "y \<sqsubseteq> i"
```
```   473       proof (rule greatest_le [OF greatest_i], rule Lower_memI)
```
```   474 	fix z
```
```   475 	assume z: "z \<in> {a, x}"
```
```   476 	then show "y \<sqsubseteq> z"
```
```   477 	proof
```
```   478           have y': "y \<in> Lower L A"
```
```   479             apply (rule subsetD [where A = "Lower L (insert x A)"])
```
```   480             apply (rule Lower_antimono) apply clarify apply assumption
```
```   481             done
```
```   482           assume "z = a"
```
```   483           with y' greatest_a show ?thesis by (fast dest: greatest_le)
```
```   484 	next
```
```   485           assume "z \<in> {x}"
```
```   486           with y L show ?thesis by blast
```
```   487 	qed
```
```   488       qed (rule Lower_closed [THEN subsetD])
```
```   489     next
```
```   490       from L show "insert x A \<subseteq> carrier L" by simp
```
```   491       from greatest_i show "i \<in> carrier L" by simp
```
```   492     qed
```
```   493   next
```
```   494     fix g
```
```   495     assume greatest_g: "greatest L g (Lower L (insert x A))"
```
```   496     show "g = i"
```
```   497     proof (rule greatest_unique)
```
```   498       show "greatest L i (Lower L (insert x A))"
```
```   499       proof (rule greatest_LowerI)
```
```   500         fix z
```
```   501         assume "z \<in> insert x A"
```
```   502         then show "i \<sqsubseteq> z"
```
```   503 	proof
```
```   504           assume "z = x" then show ?thesis
```
```   505             by (simp add: greatest_Lower_above [OF greatest_i] L La)
```
```   506 	next
```
```   507           assume "z \<in> A"
```
```   508           with L greatest_i greatest_a show ?thesis
```
```   509             by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
```
```   510         qed
```
```   511       next
```
```   512         fix y
```
```   513         assume y: "y \<in> Lower L (insert x A)"
```
```   514         show "y \<sqsubseteq> i"
```
```   515         proof (rule greatest_le [OF greatest_i], rule Lower_memI)
```
```   516           fix z
```
```   517           assume z: "z \<in> {a, x}"
```
```   518           then show "y \<sqsubseteq> z"
```
```   519           proof
```
```   520             have y': "y \<in> Lower L A"
```
```   521 	      apply (rule subsetD [where A = "Lower L (insert x A)"])
```
```   522 	      apply (rule Lower_antimono) apply clarify apply assumption
```
```   523 	      done
```
```   524             assume "z = a"
```
```   525             with y' greatest_a show ?thesis by (fast dest: greatest_le)
```
```   526 	  next
```
```   527             assume "z \<in> {x}"
```
```   528             with y L show ?thesis by blast
```
```   529 	  qed
```
```   530         qed (rule Lower_closed [THEN subsetD])
```
```   531       next
```
```   532         from L show "insert x A \<subseteq> carrier L" by simp
```
```   533         from greatest_i show "i \<in> carrier L" by simp
```
```   534       qed
```
```   535     qed
```
```   536   qed
```
```   537 qed
```
```   538
```
```   539 lemma (in lattice) finite_inf_greatest:
```
```   540   "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter>A) (Lower L A)"
```
```   541 proof (induct set: Finites)
```
```   542   case empty then show ?case by simp
```
```   543 next
```
```   544   case (insert A x)
```
```   545   show ?case
```
```   546   proof (cases "A = {}")
```
```   547     case True
```
```   548     with insert show ?thesis by (simp add: inf_of_singletonI)
```
```   549   next
```
```   550     case False
```
```   551     from insert show ?thesis
```
```   552     proof (rule_tac inf_insertI)
```
```   553       from False insert show "greatest L (\<Sqinter>A) (Lower L A)" by simp
```
```   554     qed simp_all
```
```   555   qed
```
```   556 qed
```
```   557
```
```   558 lemma (in lattice) finite_inf_insertI:
```
```   559   assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
```
```   560     and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
```
```   561   shows "P (\<Sqinter> (insert x A))"
```
```   562 proof (cases "A = {}")
```
```   563   case True with P and xA show ?thesis
```
```   564     by (simp add: inf_of_singletonI)
```
```   565 next
```
```   566   case False with P and xA show ?thesis
```
```   567     by (simp add: inf_insertI finite_inf_greatest)
```
```   568 qed
```
```   569
```
```   570 lemma (in lattice) finite_inf_closed:
```
```   571   "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter>A \<in> carrier L"
```
```   572 proof (induct set: Finites)
```
```   573   case empty then show ?case by simp
```
```   574 next
```
```   575   case (insert A x) then show ?case
```
```   576     by (rule_tac finite_inf_insertI) (simp_all)
```
```   577 qed
```
```   578
```
```   579 lemma (in lattice) meet_left:
```
```   580   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
```
```   581   by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
```
```   582
```
```   583 lemma (in lattice) meet_right:
```
```   584   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
```
```   585   by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
```
```   586
```
```   587 lemma (in lattice) inf_of_two_greatest:
```
```   588   "[| x \<in> carrier L; y \<in> carrier L |] ==>
```
```   589   greatest L (\<Sqinter> {x, y}) (Lower L {x, y})"
```
```   590 proof (unfold inf_def)
```
```   591   assume L: "x \<in> carrier L"  "y \<in> carrier L"
```
```   592   with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
```
```   593   with L
```
```   594   show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})"
```
```   595   by (fast intro: theI2 greatest_unique)  (* blast fails *)
```
```   596 qed
```
```   597
```
```   598 lemma (in lattice) meet_le:
```
```   599   assumes sub: "z \<sqsubseteq> x"  "z \<sqsubseteq> y"
```
```   600     and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
```
```   601   shows "z \<sqsubseteq> x \<sqinter> y"
```
```   602 proof (rule meetI)
```
```   603   fix i
```
```   604   assume "greatest L i (Lower L {x, y})"
```
```   605   with sub L show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
```
```   606 qed
```
```   607
```
```   608 lemma (in lattice) meet_assoc_lemma:
```
```   609   assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
```
```   610   shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}"
```
```   611 proof (rule finite_inf_insertI)
```
```   612   txt {* The textbook argument in Jacobson I, p 457 *}
```
```   613   fix i
```
```   614   assume inf: "greatest L i (Lower L {x, y, z})"
```
```   615   show "x \<sqinter> (y \<sqinter> z) = i"
```
```   616   proof (rule anti_sym)
```
```   617     from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
```
```   618       by (fastsimp intro!: meet_le elim: greatest_Lower_above)
```
```   619   next
```
```   620     from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i"
```
```   621     by (erule_tac greatest_le)
```
```   622       (blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed)
```
```   623   qed (simp_all add: L greatest_carrier [OF inf])
```
```   624 qed (simp_all add: L)
```
```   625
```
```   626 lemma meet_comm:
```
```   627   includes struct L
```
```   628   shows "x \<sqinter> y = y \<sqinter> x"
```
```   629   by (unfold meet_def) (simp add: insert_commute)
```
```   630
```
```   631 lemma (in lattice) meet_assoc:
```
```   632   assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
```
```   633   shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
```
```   634 proof -
```
```   635   have "(x \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm)
```
```   636   also from L have "... = \<Sqinter> {z, x, y}" by (simp add: meet_assoc_lemma)
```
```   637   also from L have "... = \<Sqinter> {x, y, z}" by (simp add: insert_commute)
```
```   638   also from L have "... = x \<sqinter> (y \<sqinter> z)" by (simp add: meet_assoc_lemma)
```
```   639   finally show ?thesis .
```
```   640 qed
```
```   641
```
```   642
```
```   643 subsection {* Total Orders *}
```
```   644
```
```   645 locale total_order = lattice +
```
```   646   assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
```
```   647
```
```   648 text {* Introduction rule: the usual definition of total order *}
```
```   649
```
```   650 lemma (in partial_order) total_orderI:
```
```   651   assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
```
```   652   shows "total_order L"
```
```   653 proof (rule total_order.intro)
```
```   654   show "lattice_axioms L"
```
```   655   proof (rule lattice_axioms.intro)
```
```   656     fix x y
```
```   657     assume L: "x \<in> carrier L"  "y \<in> carrier L"
```
```   658     show "EX s. least L s (Upper L {x, y})"
```
```   659     proof -
```
```   660       note total L
```
```   661       moreover
```
```   662       {
```
```   663         assume "x \<sqsubseteq> y"
```
```   664         with L have "least L y (Upper L {x, y})"
```
```   665           by (rule_tac least_UpperI) auto
```
```   666       }
```
```   667       moreover
```
```   668       {
```
```   669         assume "y \<sqsubseteq> x"
```
```   670         with L have "least L x (Upper L {x, y})"
```
```   671           by (rule_tac least_UpperI) auto
```
```   672       }
```
```   673       ultimately show ?thesis by blast
```
```   674     qed
```
```   675   next
```
```   676     fix x y
```
```   677     assume L: "x \<in> carrier L"  "y \<in> carrier L"
```
```   678     show "EX i. greatest L i (Lower L {x, y})"
```
```   679     proof -
```
```   680       note total L
```
```   681       moreover
```
```   682       {
```
```   683         assume "y \<sqsubseteq> x"
```
```   684         with L have "greatest L y (Lower L {x, y})"
```
```   685           by (rule_tac greatest_LowerI) auto
```
```   686       }
```
```   687       moreover
```
```   688       {
```
```   689         assume "x \<sqsubseteq> y"
```
```   690         with L have "greatest L x (Lower L {x, y})"
```
```   691           by (rule_tac greatest_LowerI) auto
```
```   692       }
```
```   693       ultimately show ?thesis by blast
```
```   694     qed
```
```   695   qed
```
```   696 qed (assumption | rule total_order_axioms.intro)+
```
```   697
```
```   698
```
```   699 subsection {* Complete lattices *}
```
```   700
```
```   701 locale complete_lattice = lattice +
```
```   702   assumes sup_exists:
```
```   703     "[| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
```
```   704     and inf_exists:
```
```   705     "[| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
```
```   706
```
```   707 text {* Introduction rule: the usual definition of complete lattice *}
```
```   708
```
```   709 lemma (in partial_order) complete_latticeI:
```
```   710   assumes sup_exists:
```
```   711     "!!A. [| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
```
```   712     and inf_exists:
```
```   713     "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
```
```   714   shows "complete_lattice L"
```
```   715 proof (rule complete_lattice.intro)
```
```   716   show "lattice_axioms L"
```
```   717     by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
```
```   718 qed (assumption | rule complete_lattice_axioms.intro)+
```
```   719
```
```   720 constdefs (structure L)
```
```   721   top :: "_ => 'a" ("\<top>\<index>")
```
```   722   "\<top> == sup L (carrier L)"
```
```   723
```
```   724   bottom :: "_ => 'a" ("\<bottom>\<index>")
```
```   725   "\<bottom> == inf L (carrier L)"
```
```   726
```
```   727
```
```   728 lemma (in complete_lattice) supI:
```
```   729   "[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |]
```
```   730   ==> P (\<Squnion>A)"
```
```   731 proof (unfold sup_def)
```
```   732   assume L: "A \<subseteq> carrier L"
```
```   733     and P: "!!l. least L l (Upper L A) ==> P l"
```
```   734   with sup_exists obtain s where "least L s (Upper L A)" by blast
```
```   735   with L show "P (THE l. least L l (Upper L A))"
```
```   736   by (fast intro: theI2 least_unique P)
```
```   737 qed
```
```   738
```
```   739 lemma (in complete_lattice) sup_closed [simp]:
```
```   740   "A \<subseteq> carrier L ==> \<Squnion>A \<in> carrier L"
```
```   741   by (rule supI) simp_all
```
```   742
```
```   743 lemma (in complete_lattice) top_closed [simp, intro]:
```
```   744   "\<top> \<in> carrier L"
```
```   745   by (unfold top_def) simp
```
```   746
```
```   747 lemma (in complete_lattice) infI:
```
```   748   "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
```
```   749   ==> P (\<Sqinter>A)"
```
```   750 proof (unfold inf_def)
```
```   751   assume L: "A \<subseteq> carrier L"
```
```   752     and P: "!!l. greatest L l (Lower L A) ==> P l"
```
```   753   with inf_exists obtain s where "greatest L s (Lower L A)" by blast
```
```   754   with L show "P (THE l. greatest L l (Lower L A))"
```
```   755   by (fast intro: theI2 greatest_unique P)
```
```   756 qed
```
```   757
```
```   758 lemma (in complete_lattice) inf_closed [simp]:
```
```   759   "A \<subseteq> carrier L ==> \<Sqinter>A \<in> carrier L"
```
```   760   by (rule infI) simp_all
```
```   761
```
```   762 lemma (in complete_lattice) bottom_closed [simp, intro]:
```
```   763   "\<bottom> \<in> carrier L"
```
```   764   by (unfold bottom_def) simp
```
```   765
```
```   766 text {* Jacobson: Theorem 8.1 *}
```
```   767
```
```   768 lemma Lower_empty [simp]:
```
```   769   "Lower L {} = carrier L"
```
```   770   by (unfold Lower_def) simp
```
```   771
```
```   772 lemma Upper_empty [simp]:
```
```   773   "Upper L {} = carrier L"
```
```   774   by (unfold Upper_def) simp
```
```   775
```
```   776 theorem (in partial_order) complete_lattice_criterion1:
```
```   777   assumes top_exists: "EX g. greatest L g (carrier L)"
```
```   778     and inf_exists:
```
```   779       "!!A. [| A \<subseteq> carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)"
```
```   780   shows "complete_lattice L"
```
```   781 proof (rule complete_latticeI)
```
```   782   from top_exists obtain top where top: "greatest L top (carrier L)" ..
```
```   783   fix A
```
```   784   assume L: "A \<subseteq> carrier L"
```
```   785   let ?B = "Upper L A"
```
```   786   from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le)
```
```   787   then have B_non_empty: "?B ~= {}" by fast
```
```   788   have B_L: "?B \<subseteq> carrier L" by simp
```
```   789   from inf_exists [OF B_L B_non_empty]
```
```   790   obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
```
```   791   have "least L b (Upper L A)"
```
```   792 apply (rule least_UpperI)
```
```   793    apply (rule greatest_le [where A = "Lower L ?B"])
```
```   794     apply (rule b_inf_B)
```
```   795    apply (rule Lower_memI)
```
```   796     apply (erule UpperD)
```
```   797      apply assumption
```
```   798     apply (rule L)
```
```   799    apply (fast intro: L [THEN subsetD])
```
```   800   apply (erule greatest_Lower_above [OF b_inf_B])
```
```   801   apply simp
```
```   802  apply (rule L)
```
```   803 apply (rule greatest_carrier [OF b_inf_B]) (* rename rule: _closed *)
```
```   804 done
```
```   805   then show "EX s. least L s (Upper L A)" ..
```
```   806 next
```
```   807   fix A
```
```   808   assume L: "A \<subseteq> carrier L"
```
```   809   show "EX i. greatest L i (Lower L A)"
```
```   810   proof (cases "A = {}")
```
```   811     case True then show ?thesis
```
```   812       by (simp add: top_exists)
```
```   813   next
```
```   814     case False with L show ?thesis
```
```   815       by (rule inf_exists)
```
```   816   qed
```
```   817 qed
```
```   818
```
```   819 (* TODO: prove dual version *)
```
```   820
```
```   821 subsection {* Examples *}
```
```   822
```
```   823 subsubsection {* Powerset of a set is a complete lattice *}
```
```   824
```
```   825 theorem powerset_is_complete_lattice:
```
```   826   "complete_lattice (| carrier = Pow A, le = op \<subseteq> |)"
```
```   827   (is "complete_lattice ?L")
```
```   828 proof (rule partial_order.complete_latticeI)
```
```   829   show "partial_order ?L"
```
```   830     by (rule partial_order.intro) auto
```
```   831 next
```
```   832   fix B
```
```   833   assume "B \<subseteq> carrier ?L"
```
```   834   then have "least ?L (\<Union> B) (Upper ?L B)"
```
```   835     by (fastsimp intro!: least_UpperI simp: Upper_def)
```
```   836   then show "EX s. least ?L s (Upper ?L B)" ..
```
```   837 next
```
```   838   fix B
```
```   839   assume "B \<subseteq> carrier ?L"
```
```   840   then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)"
```
```   841     txt {* @{term "\<Inter> B"} is not the infimum of @{term B}:
```
```   842       @{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! *}
```
```   843     by (fastsimp intro!: greatest_LowerI simp: Lower_def)
```
```   844   then show "EX i. greatest ?L i (Lower ?L B)" ..
```
```   845 qed
```
```   846
```
```   847 subsubsection {* Lattice of subgroups of a group *}
```
```   848
```
```   849 theorem (in group) subgroups_partial_order:
```
```   850   "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
```
```   851   by (rule partial_order.intro) simp_all
```
```   852
```
```   853 lemma (in group) subgroup_self:
```
```   854   "subgroup (carrier G) G"
```
```   855   by (rule subgroupI) auto
```
```   856
```
```   857 lemma (in group) subgroup_imp_group:
```
```   858   "subgroup H G ==> group (G(| carrier := H |))"
```
```   859   using subgroup.groupI [OF _ group.intro] .
```
```   860
```
```   861 lemma (in group) is_monoid [intro, simp]:
```
```   862   "monoid G"
```
```   863   by (rule monoid.intro)
```
```   864
```
```   865 lemma (in group) subgroup_inv_equality:
```
```   866   "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
```
```   867 apply (rule_tac inv_equality [THEN sym])
```
```   868   apply (rule group.l_inv [OF subgroup_imp_group, simplified])
```
```   869    apply assumption+
```
```   870  apply (rule subsetD [OF subgroup.subset])
```
```   871   apply assumption+
```
```   872 apply (rule subsetD [OF subgroup.subset])
```
```   873  apply assumption
```
```   874 apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified])
```
```   875   apply assumption+
```
```   876 done
```
```   877
```
```   878 theorem (in group) subgroups_Inter:
```
```   879   assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"
```
```   880     and not_empty: "A ~= {}"
```
```   881   shows "subgroup (\<Inter>A) G"
```
```   882 proof (rule subgroupI)
```
```   883   from subgr [THEN subgroup.subset] and not_empty
```
```   884   show "\<Inter>A \<subseteq> carrier G" by blast
```
```   885 next
```
```   886   from subgr [THEN subgroup.one_closed]
```
```   887   show "\<Inter>A ~= {}" by blast
```
```   888 next
```
```   889   fix x assume "x \<in> \<Inter>A"
```
```   890   with subgr [THEN subgroup.m_inv_closed]
```
```   891   show "inv x \<in> \<Inter>A" by blast
```
```   892 next
```
```   893   fix x y assume "x \<in> \<Inter>A"  "y \<in> \<Inter>A"
```
```   894   with subgr [THEN subgroup.m_closed]
```
```   895   show "x \<otimes> y \<in> \<Inter>A" by blast
```
```   896 qed
```
```   897
```
```   898 theorem (in group) subgroups_complete_lattice:
```
```   899   "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
```
```   900     (is "complete_lattice ?L")
```
```   901 proof (rule partial_order.complete_lattice_criterion1)
```
```   902   show "partial_order ?L" by (rule subgroups_partial_order)
```
```   903 next
```
```   904   have "greatest ?L (carrier G) (carrier ?L)"
```
```   905     by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
```
```   906   then show "EX G. greatest ?L G (carrier ?L)" ..
```
```   907 next
```
```   908   fix A
```
```   909   assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
```
```   910   then have Int_subgroup: "subgroup (\<Inter>A) G"
```
```   911     by (fastsimp intro: subgroups_Inter)
```
```   912   have "greatest ?L (\<Inter>A) (Lower ?L A)"
```
```   913     (is "greatest ?L ?Int _")
```
```   914   proof (rule greatest_LowerI)
```
```   915     fix H
```
```   916     assume H: "H \<in> A"
```
```   917     with L have subgroupH: "subgroup H G" by auto
```
```   918     from subgroupH have submagmaH: "submagma H G" by (rule subgroup.axioms)
```
```   919     from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
```
```   920       by (rule subgroup_imp_group)
```
```   921     from groupH have monoidH: "monoid ?H"
```
```   922       by (rule group.is_monoid)
```
```   923     from H have Int_subset: "?Int \<subseteq> H" by fastsimp
```
```   924     then show "le ?L ?Int H" by simp
```
```   925   next
```
```   926     fix H
```
```   927     assume H: "H \<in> Lower ?L A"
```
```   928     with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)
```
```   929   next
```
```   930     show "A \<subseteq> carrier ?L" by (rule L)
```
```   931   next
```
```   932     show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
```
```   933   qed
```
```   934   then show "EX I. greatest ?L I (Lower ?L A)" ..
```
```   935 qed
```
```   936
```
```   937 end
```