src/HOL/Finite_Set.thy
 author nipkow Sun Jun 17 18:47:03 2007 +0200 (2007-06-17) changeset 23413 5caa2710dd5b parent 23398 0b5a400c7595 child 23477 f4b83f03cac9 permissions -rw-r--r--
tuned laws for cancellation in divisions for fields.
```     1 (*  Title:      HOL/Finite_Set.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
```
```     4                 with contributions by Jeremy Avigad
```
```     5 *)
```
```     6
```
```     7 header {* Finite sets *}
```
```     8
```
```     9 theory Finite_Set
```
```    10 imports Divides
```
```    11 begin
```
```    12
```
```    13 subsection {* Definition and basic properties *}
```
```    14
```
```    15 inductive2 finite :: "'a set => bool"
```
```    16   where
```
```    17     emptyI [simp, intro!]: "finite {}"
```
```    18   | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
```
```    19
```
```    20 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
```
```    21   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    22   shows "\<exists>a::'a. a \<notin> A"
```
```    23 proof -
```
```    24   from prems have "A \<noteq> UNIV" by blast
```
```    25   thus ?thesis by blast
```
```    26 qed
```
```    27
```
```    28 lemma finite_induct [case_names empty insert, induct set: finite]:
```
```    29   "finite F ==>
```
```    30     P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
```
```    31   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
```
```    32 proof -
```
```    33   assume "P {}" and
```
```    34     insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
```
```    35   assume "finite F"
```
```    36   thus "P F"
```
```    37   proof induct
```
```    38     show "P {}" by fact
```
```    39     fix x F assume F: "finite F" and P: "P F"
```
```    40     show "P (insert x F)"
```
```    41     proof cases
```
```    42       assume "x \<in> F"
```
```    43       hence "insert x F = F" by (rule insert_absorb)
```
```    44       with P show ?thesis by (simp only:)
```
```    45     next
```
```    46       assume "x \<notin> F"
```
```    47       from F this P show ?thesis by (rule insert)
```
```    48     qed
```
```    49   qed
```
```    50 qed
```
```    51
```
```    52 lemma finite_ne_induct[case_names singleton insert, consumes 2]:
```
```    53 assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
```
```    54  \<lbrakk> \<And>x. P{x};
```
```    55    \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
```
```    56  \<Longrightarrow> P F"
```
```    57 using fin
```
```    58 proof induct
```
```    59   case empty thus ?case by simp
```
```    60 next
```
```    61   case (insert x F)
```
```    62   show ?case
```
```    63   proof cases
```
```    64     assume "F = {}"
```
```    65     thus ?thesis using `P {x}` by simp
```
```    66   next
```
```    67     assume "F \<noteq> {}"
```
```    68     thus ?thesis using insert by blast
```
```    69   qed
```
```    70 qed
```
```    71
```
```    72 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```    73   assumes "finite F" and "F \<subseteq> A"
```
```    74     and empty: "P {}"
```
```    75     and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
```
```    76   shows "P F"
```
```    77 proof -
```
```    78   from `finite F` and `F \<subseteq> A`
```
```    79   show ?thesis
```
```    80   proof induct
```
```    81     show "P {}" by fact
```
```    82   next
```
```    83     fix x F
```
```    84     assume "finite F" and "x \<notin> F" and
```
```    85       P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
```
```    86     show "P (insert x F)"
```
```    87     proof (rule insert)
```
```    88       from i show "x \<in> A" by blast
```
```    89       from i have "F \<subseteq> A" by blast
```
```    90       with P show "P F" .
```
```    91       show "finite F" by fact
```
```    92       show "x \<notin> F" by fact
```
```    93     qed
```
```    94   qed
```
```    95 qed
```
```    96
```
```    97 text{* Finite sets are the images of initial segments of natural numbers: *}
```
```    98
```
```    99 lemma finite_imp_nat_seg_image_inj_on:
```
```   100   assumes fin: "finite A"
```
```   101   shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
```
```   102 using fin
```
```   103 proof induct
```
```   104   case empty
```
```   105   show ?case
```
```   106   proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp
```
```   107   qed
```
```   108 next
```
```   109   case (insert a A)
```
```   110   have notinA: "a \<notin> A" by fact
```
```   111   from insert.hyps obtain n f
```
```   112     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
```
```   113   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
```
```   114         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
```
```   115     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```   116   thus ?case by blast
```
```   117 qed
```
```   118
```
```   119 lemma nat_seg_image_imp_finite:
```
```   120   "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
```
```   121 proof (induct n)
```
```   122   case 0 thus ?case by simp
```
```   123 next
```
```   124   case (Suc n)
```
```   125   let ?B = "f ` {i. i < n}"
```
```   126   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```   127   show ?case
```
```   128   proof cases
```
```   129     assume "\<exists>k<n. f n = f k"
```
```   130     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   131     thus ?thesis using finB by simp
```
```   132   next
```
```   133     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   134     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   135     thus ?thesis using finB by simp
```
```   136   qed
```
```   137 qed
```
```   138
```
```   139 lemma finite_conv_nat_seg_image:
```
```   140   "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
```
```   141 by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   142
```
```   143 subsubsection{* Finiteness and set theoretic constructions *}
```
```   144
```
```   145 lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
```
```   146   -- {* The union of two finite sets is finite. *}
```
```   147   by (induct set: finite) simp_all
```
```   148
```
```   149 lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
```
```   150   -- {* Every subset of a finite set is finite. *}
```
```   151 proof -
```
```   152   assume "finite B"
```
```   153   thus "!!A. A \<subseteq> B ==> finite A"
```
```   154   proof induct
```
```   155     case empty
```
```   156     thus ?case by simp
```
```   157   next
```
```   158     case (insert x F A)
```
```   159     have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" by fact+
```
```   160     show "finite A"
```
```   161     proof cases
```
```   162       assume x: "x \<in> A"
```
```   163       with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   164       with r have "finite (A - {x})" .
```
```   165       hence "finite (insert x (A - {x}))" ..
```
```   166       also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
```
```   167       finally show ?thesis .
```
```   168     next
```
```   169       show "A \<subseteq> F ==> ?thesis" by fact
```
```   170       assume "x \<notin> A"
```
```   171       with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   172     qed
```
```   173   qed
```
```   174 qed
```
```   175
```
```   176 lemma finite_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}"
```
```   177 using finite_subset[of "{x \<in> A. P x}" "A"] by blast
```
```   178
```
```   179 lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
```
```   180   by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
```
```   181
```
```   182 lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
```
```   183   -- {* The converse obviously fails. *}
```
```   184   by (blast intro: finite_subset)
```
```   185
```
```   186 lemma finite_insert [simp]: "finite (insert a A) = finite A"
```
```   187   apply (subst insert_is_Un)
```
```   188   apply (simp only: finite_Un, blast)
```
```   189   done
```
```   190
```
```   191 lemma finite_Union[simp, intro]:
```
```   192  "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
```
```   193 by (induct rule:finite_induct) simp_all
```
```   194
```
```   195 lemma finite_empty_induct:
```
```   196   assumes "finite A"
```
```   197     and "P A"
```
```   198     and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
```
```   199   shows "P {}"
```
```   200 proof -
```
```   201   have "P (A - A)"
```
```   202   proof -
```
```   203     {
```
```   204       fix c b :: "'a set"
```
```   205       assume c: "finite c" and b: "finite b"
```
```   206 	and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
```
```   207       have "c \<subseteq> b ==> P (b - c)"
```
```   208 	using c
```
```   209       proof induct
```
```   210 	case empty
```
```   211 	from P1 show ?case by simp
```
```   212       next
```
```   213 	case (insert x F)
```
```   214 	have "P (b - F - {x})"
```
```   215 	proof (rule P2)
```
```   216           from _ b show "finite (b - F)" by (rule finite_subset) blast
```
```   217           from insert show "x \<in> b - F" by simp
```
```   218           from insert show "P (b - F)" by simp
```
```   219 	qed
```
```   220 	also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
```
```   221 	finally show ?case .
```
```   222       qed
```
```   223     }
```
```   224     then show ?thesis by this (simp_all add: assms)
```
```   225   qed
```
```   226   then show ?thesis by simp
```
```   227 qed
```
```   228
```
```   229 lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
```
```   230   by (rule Diff_subset [THEN finite_subset])
```
```   231
```
```   232 lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
```
```   233   apply (subst Diff_insert)
```
```   234   apply (case_tac "a : A - B")
```
```   235    apply (rule finite_insert [symmetric, THEN trans])
```
```   236    apply (subst insert_Diff, simp_all)
```
```   237   done
```
```   238
```
```   239 lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A"
```
```   240   by simp
```
```   241
```
```   242
```
```   243 text {* Image and Inverse Image over Finite Sets *}
```
```   244
```
```   245 lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
```
```   246   -- {* The image of a finite set is finite. *}
```
```   247   by (induct set: finite) simp_all
```
```   248
```
```   249 lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
```
```   250   apply (frule finite_imageI)
```
```   251   apply (erule finite_subset, assumption)
```
```   252   done
```
```   253
```
```   254 lemma finite_range_imageI:
```
```   255     "finite (range g) ==> finite (range (%x. f (g x)))"
```
```   256   apply (drule finite_imageI, simp)
```
```   257   done
```
```   258
```
```   259 lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
```
```   260 proof -
```
```   261   have aux: "!!A. finite (A - {}) = finite A" by simp
```
```   262   fix B :: "'a set"
```
```   263   assume "finite B"
```
```   264   thus "!!A. f`A = B ==> inj_on f A ==> finite A"
```
```   265     apply induct
```
```   266      apply simp
```
```   267     apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
```
```   268      apply clarify
```
```   269      apply (simp (no_asm_use) add: inj_on_def)
```
```   270      apply (blast dest!: aux [THEN iffD1], atomize)
```
```   271     apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
```
```   272     apply (frule subsetD [OF equalityD2 insertI1], clarify)
```
```   273     apply (rule_tac x = xa in bexI)
```
```   274      apply (simp_all add: inj_on_image_set_diff)
```
```   275     done
```
```   276 qed (rule refl)
```
```   277
```
```   278
```
```   279 lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
```
```   280   -- {* The inverse image of a singleton under an injective function
```
```   281          is included in a singleton. *}
```
```   282   apply (auto simp add: inj_on_def)
```
```   283   apply (blast intro: the_equality [symmetric])
```
```   284   done
```
```   285
```
```   286 lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
```
```   287   -- {* The inverse image of a finite set under an injective function
```
```   288          is finite. *}
```
```   289   apply (induct set: finite)
```
```   290    apply simp_all
```
```   291   apply (subst vimage_insert)
```
```   292   apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
```
```   293   done
```
```   294
```
```   295
```
```   296 text {* The finite UNION of finite sets *}
```
```   297
```
```   298 lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
```
```   299   by (induct set: finite) simp_all
```
```   300
```
```   301 text {*
```
```   302   Strengthen RHS to
```
```   303   @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
```
```   304
```
```   305   We'd need to prove
```
```   306   @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
```
```   307   by induction. *}
```
```   308
```
```   309 lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
```
```   310   by (blast intro: finite_UN_I finite_subset)
```
```   311
```
```   312
```
```   313 lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
```
```   314 by (simp add: Plus_def)
```
```   315
```
```   316 text {* Sigma of finite sets *}
```
```   317
```
```   318 lemma finite_SigmaI [simp]:
```
```   319     "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   320   by (unfold Sigma_def) (blast intro!: finite_UN_I)
```
```   321
```
```   322 lemma finite_cartesian_product: "[| finite A; finite B |] ==>
```
```   323     finite (A <*> B)"
```
```   324   by (rule finite_SigmaI)
```
```   325
```
```   326 lemma finite_Prod_UNIV:
```
```   327     "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
```
```   328   apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
```
```   329    apply (erule ssubst)
```
```   330    apply (erule finite_SigmaI, auto)
```
```   331   done
```
```   332
```
```   333 lemma finite_cartesian_productD1:
```
```   334      "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
```
```   335 apply (auto simp add: finite_conv_nat_seg_image)
```
```   336 apply (drule_tac x=n in spec)
```
```   337 apply (drule_tac x="fst o f" in spec)
```
```   338 apply (auto simp add: o_def)
```
```   339  prefer 2 apply (force dest!: equalityD2)
```
```   340 apply (drule equalityD1)
```
```   341 apply (rename_tac y x)
```
```   342 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   343  prefer 2 apply force
```
```   344 apply clarify
```
```   345 apply (rule_tac x=k in image_eqI, auto)
```
```   346 done
```
```   347
```
```   348 lemma finite_cartesian_productD2:
```
```   349      "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
```
```   350 apply (auto simp add: finite_conv_nat_seg_image)
```
```   351 apply (drule_tac x=n in spec)
```
```   352 apply (drule_tac x="snd o f" in spec)
```
```   353 apply (auto simp add: o_def)
```
```   354  prefer 2 apply (force dest!: equalityD2)
```
```   355 apply (drule equalityD1)
```
```   356 apply (rename_tac x y)
```
```   357 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   358  prefer 2 apply force
```
```   359 apply clarify
```
```   360 apply (rule_tac x=k in image_eqI, auto)
```
```   361 done
```
```   362
```
```   363
```
```   364 text {* The powerset of a finite set *}
```
```   365
```
```   366 lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
```
```   367 proof
```
```   368   assume "finite (Pow A)"
```
```   369   with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
```
```   370   thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   371 next
```
```   372   assume "finite A"
```
```   373   thus "finite (Pow A)"
```
```   374     by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
```
```   375 qed
```
```   376
```
```   377
```
```   378 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   379 by(blast intro: finite_subset[OF subset_Pow_Union])
```
```   380
```
```   381
```
```   382 lemma finite_converse [iff]: "finite (r^-1) = finite r"
```
```   383   apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
```
```   384    apply simp
```
```   385    apply (rule iffI)
```
```   386     apply (erule finite_imageD [unfolded inj_on_def])
```
```   387     apply (simp split add: split_split)
```
```   388    apply (erule finite_imageI)
```
```   389   apply (simp add: converse_def image_def, auto)
```
```   390   apply (rule bexI)
```
```   391    prefer 2 apply assumption
```
```   392   apply simp
```
```   393   done
```
```   394
```
```   395
```
```   396 text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
```
```   397 Ehmety) *}
```
```   398
```
```   399 lemma finite_Field: "finite r ==> finite (Field r)"
```
```   400   -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
```
```   401   apply (induct set: finite)
```
```   402    apply (auto simp add: Field_def Domain_insert Range_insert)
```
```   403   done
```
```   404
```
```   405 lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
```
```   406   apply clarify
```
```   407   apply (erule trancl_induct)
```
```   408    apply (auto simp add: Field_def)
```
```   409   done
```
```   410
```
```   411 lemma finite_trancl: "finite (r^+) = finite r"
```
```   412   apply auto
```
```   413    prefer 2
```
```   414    apply (rule trancl_subset_Field2 [THEN finite_subset])
```
```   415    apply (rule finite_SigmaI)
```
```   416     prefer 3
```
```   417     apply (blast intro: r_into_trancl' finite_subset)
```
```   418    apply (auto simp add: finite_Field)
```
```   419   done
```
```   420
```
```   421
```
```   422 subsection {* A fold functional for finite sets *}
```
```   423
```
```   424 text {* The intended behaviour is
```
```   425 @{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
```
```   426 if @{text f} is associative-commutative. For an application of @{text fold}
```
```   427 se the definitions of sums and products over finite sets.
```
```   428 *}
```
```   429
```
```   430 inductive2
```
```   431   foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a => bool"
```
```   432   for f ::  "'a => 'a => 'a"
```
```   433   and g :: "'b => 'a"
```
```   434   and z :: 'a
```
```   435 where
```
```   436   emptyI [intro]: "foldSet f g z {} z"
```
```   437 | insertI [intro]:
```
```   438      "\<lbrakk> x \<notin> A; foldSet f g z A y \<rbrakk>
```
```   439       \<Longrightarrow> foldSet f g z (insert x A) (f (g x) y)"
```
```   440
```
```   441 inductive_cases2 empty_foldSetE [elim!]: "foldSet f g z {} x"
```
```   442
```
```   443 constdefs
```
```   444   fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
```
```   445   "fold f g z A == THE x. foldSet f g z A x"
```
```   446
```
```   447 text{*A tempting alternative for the definiens is
```
```   448 @{term "if finite A then THE x. foldSet f g e A x else e"}.
```
```   449 It allows the removal of finiteness assumptions from the theorems
```
```   450 @{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
```
```   451 The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
```
```   452
```
```   453
```
```   454 lemma Diff1_foldSet:
```
```   455   "foldSet f g z (A - {x}) y ==> x: A ==> foldSet f g z A (f (g x) y)"
```
```   456 by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
```
```   457
```
```   458 lemma foldSet_imp_finite: "foldSet f g z A x==> finite A"
```
```   459   by (induct set: foldSet) auto
```
```   460
```
```   461 lemma finite_imp_foldSet: "finite A ==> EX x. foldSet f g z A x"
```
```   462   by (induct set: finite) auto
```
```   463
```
```   464
```
```   465 subsubsection {* Commutative monoids *}
```
```   466
```
```   467 (*FIXME integrate with Orderings.thy/OrderedGroup.thy*)
```
```   468 locale ACf =
```
```   469   fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
```
```   470   assumes commute: "x \<cdot> y = y \<cdot> x"
```
```   471     and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
```
```   472 begin
```
```   473
```
```   474 lemma left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```   475 proof -
```
```   476   have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
```
```   477   also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
```
```   478   also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
```
```   479   finally show ?thesis .
```
```   480 qed
```
```   481
```
```   482 lemmas AC = assoc commute left_commute
```
```   483
```
```   484 end
```
```   485
```
```   486 locale ACe = ACf +
```
```   487   fixes e :: 'a
```
```   488   assumes ident [simp]: "x \<cdot> e = x"
```
```   489 begin
```
```   490
```
```   491 lemma left_ident [simp]: "e \<cdot> x = x"
```
```   492 proof -
```
```   493   have "x \<cdot> e = x" by (rule ident)
```
```   494   thus ?thesis by (subst commute)
```
```   495 qed
```
```   496
```
```   497 end
```
```   498
```
```   499 locale ACIf = ACf +
```
```   500   assumes idem: "x \<cdot> x = x"
```
```   501 begin
```
```   502
```
```   503 lemma idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
```
```   504 proof -
```
```   505   have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
```
```   506   also have "\<dots> = x \<cdot> y" by(simp add:idem)
```
```   507   finally show ?thesis .
```
```   508 qed
```
```   509
```
```   510 lemmas ACI = AC idem idem2
```
```   511
```
```   512 end
```
```   513
```
```   514
```
```   515 subsubsection{*From @{term foldSet} to @{term fold}*}
```
```   516
```
```   517 lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
```
```   518   by (auto simp add: less_Suc_eq)
```
```   519
```
```   520 lemma insert_image_inj_on_eq:
```
```   521      "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A;
```
```   522         inj_on h {i. i < Suc m}|]
```
```   523       ==> A = h ` {i. i < m}"
```
```   524 apply (auto simp add: image_less_Suc inj_on_def)
```
```   525 apply (blast intro: less_trans)
```
```   526 done
```
```   527
```
```   528 lemma insert_inj_onE:
```
```   529   assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A"
```
```   530       and inj_on: "inj_on h {i::nat. i<n}"
```
```   531   shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
```
```   532 proof (cases n)
```
```   533   case 0 thus ?thesis using aA by auto
```
```   534 next
```
```   535   case (Suc m)
```
```   536   have nSuc: "n = Suc m" by fact
```
```   537   have mlessn: "m<n" by (simp add: nSuc)
```
```   538   from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
```
```   539   let ?hm = "swap k m h"
```
```   540   have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn
```
```   541     by (simp add: inj_on_swap_iff inj_on)
```
```   542   show ?thesis
```
```   543   proof (intro exI conjI)
```
```   544     show "inj_on ?hm {i. i < m}" using inj_hm
```
```   545       by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
```
```   546     show "m<n" by (rule mlessn)
```
```   547     show "A = ?hm ` {i. i < m}"
```
```   548     proof (rule insert_image_inj_on_eq)
```
```   549       show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
```
```   550       show "?hm m \<notin> A" by (simp add: swap_def hkeq anot)
```
```   551       show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
```
```   552 	using aA hkeq nSuc klessn
```
```   553 	by (auto simp add: swap_def image_less_Suc fun_upd_image
```
```   554 			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
```
```   555     qed
```
```   556   qed
```
```   557 qed
```
```   558
```
```   559 lemma (in ACf) foldSet_determ_aux:
```
```   560   "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n};
```
```   561                 foldSet f g z A x; foldSet f g z A x' \<rbrakk>
```
```   562    \<Longrightarrow> x' = x"
```
```   563 proof (induct n rule: less_induct)
```
```   564   case (less n)
```
```   565     have IH: "!!m h A x x'.
```
```   566                \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m};
```
```   567                 foldSet f g z A x; foldSet f g z A x'\<rbrakk> \<Longrightarrow> x' = x" by fact
```
```   568     have Afoldx: "foldSet f g z A x" and Afoldx': "foldSet f g z A x'"
```
```   569      and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+
```
```   570     show ?case
```
```   571     proof (rule foldSet.cases [OF Afoldx])
```
```   572       assume "A = {}" and "x = z"
```
```   573       with Afoldx' show "x' = x" by blast
```
```   574     next
```
```   575       fix B b u
```
```   576       assume AbB: "A = insert b B" and x: "x = g b \<cdot> u"
```
```   577          and notinB: "b \<notin> B" and Bu: "foldSet f g z B u"
```
```   578       show "x'=x"
```
```   579       proof (rule foldSet.cases [OF Afoldx'])
```
```   580         assume "A = {}" and "x' = z"
```
```   581         with AbB show "x' = x" by blast
```
```   582       next
```
```   583 	fix C c v
```
```   584 	assume AcC: "A = insert c C" and x': "x' = g c \<cdot> v"
```
```   585            and notinC: "c \<notin> C" and Cv: "foldSet f g z C v"
```
```   586 	from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
```
```   587         from insert_inj_onE [OF Beq notinB injh]
```
```   588         obtain hB mB where inj_onB: "inj_on hB {i. i < mB}"
```
```   589                      and Beq: "B = hB ` {i. i < mB}"
```
```   590                      and lessB: "mB < n" by auto
```
```   591 	from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
```
```   592         from insert_inj_onE [OF Ceq notinC injh]
```
```   593         obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
```
```   594                        and Ceq: "C = hC ` {i. i < mC}"
```
```   595                        and lessC: "mC < n" by auto
```
```   596 	show "x'=x"
```
```   597 	proof cases
```
```   598           assume "b=c"
```
```   599 	  then moreover have "B = C" using AbB AcC notinB notinC by auto
```
```   600 	  ultimately show ?thesis  using Bu Cv x x' IH[OF lessC Ceq inj_onC]
```
```   601             by auto
```
```   602 	next
```
```   603 	  assume diff: "b \<noteq> c"
```
```   604 	  let ?D = "B - {c}"
```
```   605 	  have B: "B = insert c ?D" and C: "C = insert b ?D"
```
```   606 	    using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
```
```   607 	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
```
```   608 	  with AbB have "finite ?D" by simp
```
```   609 	  then obtain d where Dfoldd: "foldSet f g z ?D d"
```
```   610 	    using finite_imp_foldSet by iprover
```
```   611 	  moreover have cinB: "c \<in> B" using B by auto
```
```   612 	  ultimately have "foldSet f g z B (g c \<cdot> d)"
```
```   613 	    by(rule Diff1_foldSet)
```
```   614 	  hence "g c \<cdot> d = u" by (rule IH [OF lessB Beq inj_onB Bu])
```
```   615           moreover have "g b \<cdot> d = v"
```
```   616 	  proof (rule IH[OF lessC Ceq inj_onC Cv])
```
```   617 	    show "foldSet f g z C (g b \<cdot> d)" using C notinB Dfoldd
```
```   618 	      by fastsimp
```
```   619 	  qed
```
```   620 	  ultimately show ?thesis using x x' by (auto simp: AC)
```
```   621 	qed
```
```   622       qed
```
```   623     qed
```
```   624   qed
```
```   625
```
```   626
```
```   627 lemma (in ACf) foldSet_determ:
```
```   628   "foldSet f g z A x ==> foldSet f g z A y ==> y = x"
```
```   629 apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on])
```
```   630 apply (blast intro: foldSet_determ_aux [rule_format])
```
```   631 done
```
```   632
```
```   633 lemma (in ACf) fold_equality: "foldSet f g z A y ==> fold f g z A = y"
```
```   634   by (unfold fold_def) (blast intro: foldSet_determ)
```
```   635
```
```   636 text{* The base case for @{text fold}: *}
```
```   637
```
```   638 lemma fold_empty [simp]: "fold f g z {} = z"
```
```   639   by (unfold fold_def) blast
```
```   640
```
```   641 lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
```
```   642     (foldSet f g z (insert x A) v) =
```
```   643     (EX y. foldSet f g z A y & v = f (g x) y)"
```
```   644   apply auto
```
```   645   apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
```
```   646    apply (fastsimp dest: foldSet_imp_finite)
```
```   647   apply (blast intro: foldSet_determ)
```
```   648   done
```
```   649
```
```   650 text{* The recursion equation for @{text fold}: *}
```
```   651
```
```   652 lemma (in ACf) fold_insert[simp]:
```
```   653     "finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
```
```   654   apply (unfold fold_def)
```
```   655   apply (simp add: fold_insert_aux)
```
```   656   apply (rule the_equality)
```
```   657   apply (auto intro: finite_imp_foldSet
```
```   658     cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
```
```   659   done
```
```   660
```
```   661 lemma (in ACf) fold_rec:
```
```   662 assumes fin: "finite A" and a: "a:A"
```
```   663 shows "fold f g z A = f (g a) (fold f g z (A - {a}))"
```
```   664 proof-
```
```   665   have A: "A = insert a (A - {a})" using a by blast
```
```   666   hence "fold f g z A = fold f g z (insert a (A - {a}))" by simp
```
```   667   also have "\<dots> = f (g a) (fold f g z (A - {a}))"
```
```   668     by(rule fold_insert) (simp add:fin)+
```
```   669   finally show ?thesis .
```
```   670 qed
```
```   671
```
```   672
```
```   673 text{* A simplified version for idempotent functions: *}
```
```   674
```
```   675 lemma (in ACIf) fold_insert_idem:
```
```   676 assumes finA: "finite A"
```
```   677 shows "fold f g z (insert a A) = g a \<cdot> fold f g z A"
```
```   678 proof cases
```
```   679   assume "a \<in> A"
```
```   680   then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
```
```   681     by(blast dest: mk_disjoint_insert)
```
```   682   show ?thesis
```
```   683   proof -
```
```   684     from finA A have finB: "finite B" by(blast intro: finite_subset)
```
```   685     have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
```
```   686     also have "\<dots> = (g a) \<cdot> (fold f g z B)"
```
```   687       using finB disj by simp
```
```   688     also have "\<dots> = g a \<cdot> fold f g z A"
```
```   689       using A finB disj by(simp add:idem assoc[symmetric])
```
```   690     finally show ?thesis .
```
```   691   qed
```
```   692 next
```
```   693   assume "a \<notin> A"
```
```   694   with finA show ?thesis by simp
```
```   695 qed
```
```   696
```
```   697 lemma (in ACIf) foldI_conv_id:
```
```   698   "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
```
```   699 by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)
```
```   700
```
```   701 subsubsection{*Lemmas about @{text fold}*}
```
```   702
```
```   703 lemma (in ACf) fold_commute:
```
```   704   "finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
```
```   705   apply (induct set: finite)
```
```   706    apply simp
```
```   707   apply (simp add: left_commute [of x])
```
```   708   done
```
```   709
```
```   710 lemma (in ACf) fold_nest_Un_Int:
```
```   711   "finite A ==> finite B
```
```   712     ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
```
```   713   apply (induct set: finite)
```
```   714    apply simp
```
```   715   apply (simp add: fold_commute Int_insert_left insert_absorb)
```
```   716   done
```
```   717
```
```   718 lemma (in ACf) fold_nest_Un_disjoint:
```
```   719   "finite A ==> finite B ==> A Int B = {}
```
```   720     ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
```
```   721   by (simp add: fold_nest_Un_Int)
```
```   722
```
```   723 lemma (in ACf) fold_reindex:
```
```   724 assumes fin: "finite A"
```
```   725 shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
```
```   726 using fin apply induct
```
```   727  apply simp
```
```   728 apply simp
```
```   729 done
```
```   730
```
```   731 lemma (in ACe) fold_Un_Int:
```
```   732   "finite A ==> finite B ==>
```
```   733     fold f g e A \<cdot> fold f g e B =
```
```   734     fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
```
```   735   apply (induct set: finite, simp)
```
```   736   apply (simp add: AC insert_absorb Int_insert_left)
```
```   737   done
```
```   738
```
```   739 corollary (in ACe) fold_Un_disjoint:
```
```   740   "finite A ==> finite B ==> A Int B = {} ==>
```
```   741     fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
```
```   742   by (simp add: fold_Un_Int)
```
```   743
```
```   744 lemma (in ACe) fold_UN_disjoint:
```
```   745   "\<lbrakk> finite I; ALL i:I. finite (A i);
```
```   746      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
```
```   747    \<Longrightarrow> fold f g e (UNION I A) =
```
```   748        fold f (%i. fold f g e (A i)) e I"
```
```   749   apply (induct set: finite, simp, atomize)
```
```   750   apply (subgoal_tac "ALL i:F. x \<noteq> i")
```
```   751    prefer 2 apply blast
```
```   752   apply (subgoal_tac "A x Int UNION F A = {}")
```
```   753    prefer 2 apply blast
```
```   754   apply (simp add: fold_Un_disjoint)
```
```   755   done
```
```   756
```
```   757 text{*Fusion theorem, as described in
```
```   758 Graham Hutton's paper,
```
```   759 A Tutorial on the Universality and Expressiveness of Fold,
```
```   760 JFP 9:4 (355-372), 1999.*}
```
```   761 lemma (in ACf) fold_fusion:
```
```   762       includes ACf g
```
```   763       shows
```
```   764 	"finite A ==>
```
```   765 	 (!!x y. h (g x y) = f x (h y)) ==>
```
```   766          h (fold g j w A) = fold f j (h w) A"
```
```   767   by (induct set: finite) simp_all
```
```   768
```
```   769 lemma (in ACf) fold_cong:
```
```   770   "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
```
```   771   apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
```
```   772    apply simp
```
```   773   apply (erule finite_induct, simp)
```
```   774   apply (simp add: subset_insert_iff, clarify)
```
```   775   apply (subgoal_tac "finite C")
```
```   776    prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
```
```   777   apply (subgoal_tac "C = insert x (C - {x})")
```
```   778    prefer 2 apply blast
```
```   779   apply (erule ssubst)
```
```   780   apply (drule spec)
```
```   781   apply (erule (1) notE impE)
```
```   782   apply (simp add: Ball_def del: insert_Diff_single)
```
```   783   done
```
```   784
```
```   785 lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   786   fold f (%x. fold f (g x) e (B x)) e A =
```
```   787   fold f (split g) e (SIGMA x:A. B x)"
```
```   788 apply (subst Sigma_def)
```
```   789 apply (subst fold_UN_disjoint, assumption, simp)
```
```   790  apply blast
```
```   791 apply (erule fold_cong)
```
```   792 apply (subst fold_UN_disjoint, simp, simp)
```
```   793  apply blast
```
```   794 apply simp
```
```   795 done
```
```   796
```
```   797 lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
```
```   798    fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
```
```   799 apply (erule finite_induct, simp)
```
```   800 apply (simp add:AC)
```
```   801 done
```
```   802
```
```   803
```
```   804 text{* Interpretation of locales -- see OrderedGroup.thy *}
```
```   805
```
```   806 interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"]
```
```   807   by unfold_locales (auto intro: add_assoc add_commute)
```
```   808
```
```   809 interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"]
```
```   810   by unfold_locales (auto intro: mult_assoc mult_commute)
```
```   811
```
```   812
```
```   813 subsection {* Generalized summation over a set *}
```
```   814
```
```   815 constdefs
```
```   816   setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
```
```   817   "setsum f A == if finite A then fold (op +) f 0 A else 0"
```
```   818
```
```   819 abbreviation
```
```   820   Setsum  ("\<Sum>_" [1000] 999) where
```
```   821   "\<Sum>A == setsum (%x. x) A"
```
```   822
```
```   823 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```   824 written @{text"\<Sum>x\<in>A. e"}. *}
```
```   825
```
```   826 syntax
```
```   827   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```   828 syntax (xsymbols)
```
```   829   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   830 syntax (HTML output)
```
```   831   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   832
```
```   833 translations -- {* Beware of argument permutation! *}
```
```   834   "SUM i:A. b" == "setsum (%i. b) A"
```
```   835   "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
```
```   836
```
```   837 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```   838  @{text"\<Sum>x|P. e"}. *}
```
```   839
```
```   840 syntax
```
```   841   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   842 syntax (xsymbols)
```
```   843   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   844 syntax (HTML output)
```
```   845   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   846
```
```   847 translations
```
```   848   "SUM x|P. t" => "setsum (%x. t) {x. P}"
```
```   849   "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
```
```   850
```
```   851 print_translation {*
```
```   852 let
```
```   853   fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) \$ Abs(y,Ty,P)] =
```
```   854     if x<>y then raise Match
```
```   855     else let val x' = Syntax.mark_bound x
```
```   856              val t' = subst_bound(x',t)
```
```   857              val P' = subst_bound(x',P)
```
```   858          in Syntax.const "_qsetsum" \$ Syntax.mark_bound x \$ P' \$ t' end
```
```   859 in [("setsum", setsum_tr')] end
```
```   860 *}
```
```   861
```
```   862
```
```   863 lemma setsum_empty [simp]: "setsum f {} = 0"
```
```   864   by (simp add: setsum_def)
```
```   865
```
```   866 lemma setsum_insert [simp]:
```
```   867     "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   868   by (simp add: setsum_def)
```
```   869
```
```   870 lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
```
```   871   by (simp add: setsum_def)
```
```   872
```
```   873 lemma setsum_reindex:
```
```   874      "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   875 by(auto simp add: setsum_def AC_add.fold_reindex dest!:finite_imageD)
```
```   876
```
```   877 lemma setsum_reindex_id:
```
```   878      "inj_on f B ==> setsum f B = setsum id (f ` B)"
```
```   879 by (auto simp add: setsum_reindex)
```
```   880
```
```   881 lemma setsum_cong:
```
```   882   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   883 by(fastsimp simp: setsum_def intro: AC_add.fold_cong)
```
```   884
```
```   885 lemma strong_setsum_cong[cong]:
```
```   886   "A = B ==> (!!x. x:B =simp=> f x = g x)
```
```   887    ==> setsum (%x. f x) A = setsum (%x. g x) B"
```
```   888 by(fastsimp simp: simp_implies_def setsum_def intro: AC_add.fold_cong)
```
```   889
```
```   890 lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
```
```   891   by (rule setsum_cong[OF refl], auto);
```
```   892
```
```   893 lemma setsum_reindex_cong:
```
```   894      "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
```
```   895       ==> setsum h B = setsum g A"
```
```   896   by (simp add: setsum_reindex cong: setsum_cong)
```
```   897
```
```   898 lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
```
```   899 apply (clarsimp simp: setsum_def)
```
```   900 apply (erule finite_induct, auto)
```
```   901 done
```
```   902
```
```   903 lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
```
```   904 by(simp add:setsum_cong)
```
```   905
```
```   906 lemma setsum_Un_Int: "finite A ==> finite B ==>
```
```   907   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   908   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   909 by(simp add: setsum_def AC_add.fold_Un_Int [symmetric])
```
```   910
```
```   911 lemma setsum_Un_disjoint: "finite A ==> finite B
```
```   912   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   913 by (subst setsum_Un_Int [symmetric], auto)
```
```   914
```
```   915 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   916   the lhs need not be, since UNION I A could still be finite.*)
```
```   917 lemma setsum_UN_disjoint:
```
```   918     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```   919         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```   920       setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   921 by(simp add: setsum_def AC_add.fold_UN_disjoint cong: setsum_cong)
```
```   922
```
```   923 text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
```
```   924 directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
```
```   925 lemma setsum_Union_disjoint:
```
```   926   "[| (ALL A:C. finite A);
```
```   927       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```   928    ==> setsum f (Union C) = setsum (setsum f) C"
```
```   929 apply (cases "finite C")
```
```   930  prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
```
```   931   apply (frule setsum_UN_disjoint [of C id f])
```
```   932  apply (unfold Union_def id_def, assumption+)
```
```   933 done
```
```   934
```
```   935 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```   936   the rhs need not be, since SIGMA A B could still be finite.*)
```
```   937 lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   938     (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```   939 by(simp add:setsum_def AC_add.fold_Sigma split_def cong:setsum_cong)
```
```   940
```
```   941 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```   942 lemma setsum_cartesian_product:
```
```   943    "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
```
```   944 apply (cases "finite A")
```
```   945  apply (cases "finite B")
```
```   946   apply (simp add: setsum_Sigma)
```
```   947  apply (cases "A={}", simp)
```
```   948  apply (simp)
```
```   949 apply (auto simp add: setsum_def
```
```   950             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   951 done
```
```   952
```
```   953 lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```   954 by(simp add:setsum_def AC_add.fold_distrib)
```
```   955
```
```   956
```
```   957 subsubsection {* Properties in more restricted classes of structures *}
```
```   958
```
```   959 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   960   apply (case_tac "finite A")
```
```   961    prefer 2 apply (simp add: setsum_def)
```
```   962   apply (erule rev_mp)
```
```   963   apply (erule finite_induct, auto)
```
```   964   done
```
```   965
```
```   966 lemma setsum_eq_0_iff [simp]:
```
```   967     "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   968   by (induct set: finite) auto
```
```   969
```
```   970 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   971     (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   972   -- {* For the natural numbers, we have subtraction. *}
```
```   973   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
```
```   974
```
```   975 lemma setsum_Un: "finite A ==> finite B ==>
```
```   976     (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   977       setsum f A + setsum f B - setsum f (A Int B)"
```
```   978   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
```
```   979
```
```   980 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   981     (if a:A then setsum f A - f a else setsum f A)"
```
```   982   apply (case_tac "finite A")
```
```   983    prefer 2 apply (simp add: setsum_def)
```
```   984   apply (erule finite_induct)
```
```   985    apply (auto simp add: insert_Diff_if)
```
```   986   apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   987   done
```
```   988
```
```   989 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   990   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   991   (if a:A then setsum f A - f a else setsum f A)"
```
```   992   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   993
```
```   994 lemma setsum_diff1'[rule_format]: "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
```
```   995   apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
```
```   996   apply (auto simp add: insert_Diff_if add_ac)
```
```   997   done
```
```   998
```
```   999 (* By Jeremy Siek: *)
```
```  1000
```
```  1001 lemma setsum_diff_nat:
```
```  1002   assumes "finite B"
```
```  1003     and "B \<subseteq> A"
```
```  1004   shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```  1005   using prems
```
```  1006 proof induct
```
```  1007   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```  1008 next
```
```  1009   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```  1010     and xFinA: "insert x F \<subseteq> A"
```
```  1011     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```  1012   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```  1013   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```  1014     by (simp add: setsum_diff1_nat)
```
```  1015   from xFinA have "F \<subseteq> A" by simp
```
```  1016   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```  1017   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```  1018     by simp
```
```  1019   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```  1020   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```  1021     by simp
```
```  1022   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```  1023   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```  1024     by simp
```
```  1025   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```  1026 qed
```
```  1027
```
```  1028 lemma setsum_diff:
```
```  1029   assumes le: "finite A" "B \<subseteq> A"
```
```  1030   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```  1031 proof -
```
```  1032   from le have finiteB: "finite B" using finite_subset by auto
```
```  1033   show ?thesis using finiteB le
```
```  1034   proof induct
```
```  1035     case empty
```
```  1036     thus ?case by auto
```
```  1037   next
```
```  1038     case (insert x F)
```
```  1039     thus ?case using le finiteB
```
```  1040       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```  1041   qed
```
```  1042 qed
```
```  1043
```
```  1044 lemma setsum_mono:
```
```  1045   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
```
```  1046   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```  1047 proof (cases "finite K")
```
```  1048   case True
```
```  1049   thus ?thesis using le
```
```  1050   proof induct
```
```  1051     case empty
```
```  1052     thus ?case by simp
```
```  1053   next
```
```  1054     case insert
```
```  1055     thus ?case using add_mono by fastsimp
```
```  1056   qed
```
```  1057 next
```
```  1058   case False
```
```  1059   thus ?thesis
```
```  1060     by (simp add: setsum_def)
```
```  1061 qed
```
```  1062
```
```  1063 lemma setsum_strict_mono:
```
```  1064   fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```  1065   assumes "finite A"  "A \<noteq> {}"
```
```  1066     and "!!x. x:A \<Longrightarrow> f x < g x"
```
```  1067   shows "setsum f A < setsum g A"
```
```  1068   using prems
```
```  1069 proof (induct rule: finite_ne_induct)
```
```  1070   case singleton thus ?case by simp
```
```  1071 next
```
```  1072   case insert thus ?case by (auto simp: add_strict_mono)
```
```  1073 qed
```
```  1074
```
```  1075 lemma setsum_negf:
```
```  1076   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```  1077 proof (cases "finite A")
```
```  1078   case True thus ?thesis by (induct set: finite) auto
```
```  1079 next
```
```  1080   case False thus ?thesis by (simp add: setsum_def)
```
```  1081 qed
```
```  1082
```
```  1083 lemma setsum_subtractf:
```
```  1084   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```  1085     setsum f A - setsum g A"
```
```  1086 proof (cases "finite A")
```
```  1087   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
```
```  1088 next
```
```  1089   case False thus ?thesis by (simp add: setsum_def)
```
```  1090 qed
```
```  1091
```
```  1092 lemma setsum_nonneg:
```
```  1093   assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```  1094   shows "0 \<le> setsum f A"
```
```  1095 proof (cases "finite A")
```
```  1096   case True thus ?thesis using nn
```
```  1097   proof induct
```
```  1098     case empty then show ?case by simp
```
```  1099   next
```
```  1100     case (insert x F)
```
```  1101     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```  1102     with insert show ?case by simp
```
```  1103   qed
```
```  1104 next
```
```  1105   case False thus ?thesis by (simp add: setsum_def)
```
```  1106 qed
```
```  1107
```
```  1108 lemma setsum_nonpos:
```
```  1109   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
```
```  1110   shows "setsum f A \<le> 0"
```
```  1111 proof (cases "finite A")
```
```  1112   case True thus ?thesis using np
```
```  1113   proof induct
```
```  1114     case empty then show ?case by simp
```
```  1115   next
```
```  1116     case (insert x F)
```
```  1117     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```  1118     with insert show ?case by simp
```
```  1119   qed
```
```  1120 next
```
```  1121   case False thus ?thesis by (simp add: setsum_def)
```
```  1122 qed
```
```  1123
```
```  1124 lemma setsum_mono2:
```
```  1125 fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
```
```  1126 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```  1127 shows "setsum f A \<le> setsum f B"
```
```  1128 proof -
```
```  1129   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```  1130     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```  1131   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```  1132     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```  1133   also have "A \<union> (B-A) = B" using sub by blast
```
```  1134   finally show ?thesis .
```
```  1135 qed
```
```  1136
```
```  1137 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```  1138     ALL x: B - A.
```
```  1139       0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
```
```  1140         setsum f A <= setsum f B"
```
```  1141   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```  1142   apply (erule ssubst)
```
```  1143   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```  1144   apply simp
```
```  1145   apply (rule add_left_mono)
```
```  1146   apply (erule setsum_nonneg)
```
```  1147   apply (subst setsum_Un_disjoint [THEN sym])
```
```  1148   apply (erule finite_subset, assumption)
```
```  1149   apply (rule finite_subset)
```
```  1150   prefer 2
```
```  1151   apply assumption
```
```  1152   apply auto
```
```  1153   apply (rule setsum_cong)
```
```  1154   apply auto
```
```  1155 done
```
```  1156
```
```  1157 lemma setsum_right_distrib:
```
```  1158   fixes f :: "'a => ('b::semiring_0)"
```
```  1159   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```  1160 proof (cases "finite A")
```
```  1161   case True
```
```  1162   thus ?thesis
```
```  1163   proof induct
```
```  1164     case empty thus ?case by simp
```
```  1165   next
```
```  1166     case (insert x A) thus ?case by (simp add: right_distrib)
```
```  1167   qed
```
```  1168 next
```
```  1169   case False thus ?thesis by (simp add: setsum_def)
```
```  1170 qed
```
```  1171
```
```  1172 lemma setsum_left_distrib:
```
```  1173   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```  1174 proof (cases "finite A")
```
```  1175   case True
```
```  1176   then show ?thesis
```
```  1177   proof induct
```
```  1178     case empty thus ?case by simp
```
```  1179   next
```
```  1180     case (insert x A) thus ?case by (simp add: left_distrib)
```
```  1181   qed
```
```  1182 next
```
```  1183   case False thus ?thesis by (simp add: setsum_def)
```
```  1184 qed
```
```  1185
```
```  1186 lemma setsum_divide_distrib:
```
```  1187   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```  1188 proof (cases "finite A")
```
```  1189   case True
```
```  1190   then show ?thesis
```
```  1191   proof induct
```
```  1192     case empty thus ?case by simp
```
```  1193   next
```
```  1194     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```  1195   qed
```
```  1196 next
```
```  1197   case False thus ?thesis by (simp add: setsum_def)
```
```  1198 qed
```
```  1199
```
```  1200 lemma setsum_abs[iff]:
```
```  1201   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1202   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```  1203 proof (cases "finite A")
```
```  1204   case True
```
```  1205   thus ?thesis
```
```  1206   proof induct
```
```  1207     case empty thus ?case by simp
```
```  1208   next
```
```  1209     case (insert x A)
```
```  1210     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```  1211   qed
```
```  1212 next
```
```  1213   case False thus ?thesis by (simp add: setsum_def)
```
```  1214 qed
```
```  1215
```
```  1216 lemma setsum_abs_ge_zero[iff]:
```
```  1217   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1218   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```  1219 proof (cases "finite A")
```
```  1220   case True
```
```  1221   thus ?thesis
```
```  1222   proof induct
```
```  1223     case empty thus ?case by simp
```
```  1224   next
```
```  1225     case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg)
```
```  1226   qed
```
```  1227 next
```
```  1228   case False thus ?thesis by (simp add: setsum_def)
```
```  1229 qed
```
```  1230
```
```  1231 lemma abs_setsum_abs[simp]:
```
```  1232   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1233   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```  1234 proof (cases "finite A")
```
```  1235   case True
```
```  1236   thus ?thesis
```
```  1237   proof induct
```
```  1238     case empty thus ?case by simp
```
```  1239   next
```
```  1240     case (insert a A)
```
```  1241     hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
```
```  1242     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```  1243     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```  1244       by (simp del: abs_of_nonneg)
```
```  1245     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```  1246     finally show ?case .
```
```  1247   qed
```
```  1248 next
```
```  1249   case False thus ?thesis by (simp add: setsum_def)
```
```  1250 qed
```
```  1251
```
```  1252
```
```  1253 text {* Commuting outer and inner summation *}
```
```  1254
```
```  1255 lemma swap_inj_on:
```
```  1256   "inj_on (%(i, j). (j, i)) (A \<times> B)"
```
```  1257   by (unfold inj_on_def) fast
```
```  1258
```
```  1259 lemma swap_product:
```
```  1260   "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
```
```  1261   by (simp add: split_def image_def) blast
```
```  1262
```
```  1263 lemma setsum_commute:
```
```  1264   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
```
```  1265 proof (simp add: setsum_cartesian_product)
```
```  1266   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
```
```  1267     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
```
```  1268     (is "?s = _")
```
```  1269     apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
```
```  1270     apply (simp add: split_def)
```
```  1271     done
```
```  1272   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
```
```  1273     (is "_ = ?t")
```
```  1274     apply (simp add: swap_product)
```
```  1275     done
```
```  1276   finally show "?s = ?t" .
```
```  1277 qed
```
```  1278
```
```  1279 lemma setsum_product:
```
```  1280   fixes f :: "'a => ('b::semiring_0)"
```
```  1281   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```  1282   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
```
```  1283
```
```  1284
```
```  1285 subsection {* Generalized product over a set *}
```
```  1286
```
```  1287 constdefs
```
```  1288   setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
```
```  1289   "setprod f A == if finite A then fold (op *) f 1 A else 1"
```
```  1290
```
```  1291 abbreviation
```
```  1292   Setprod  ("\<Prod>_" [1000] 999) where
```
```  1293   "\<Prod>A == setprod (%x. x) A"
```
```  1294
```
```  1295 syntax
```
```  1296   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```  1297 syntax (xsymbols)
```
```  1298   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1299 syntax (HTML output)
```
```  1300   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1301
```
```  1302 translations -- {* Beware of argument permutation! *}
```
```  1303   "PROD i:A. b" == "setprod (%i. b) A"
```
```  1304   "\<Prod>i\<in>A. b" == "setprod (%i. b) A"
```
```  1305
```
```  1306 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```  1307  @{text"\<Prod>x|P. e"}. *}
```
```  1308
```
```  1309 syntax
```
```  1310   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```  1311 syntax (xsymbols)
```
```  1312   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1313 syntax (HTML output)
```
```  1314   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1315
```
```  1316 translations
```
```  1317   "PROD x|P. t" => "setprod (%x. t) {x. P}"
```
```  1318   "\<Prod>x|P. t" => "setprod (%x. t) {x. P}"
```
```  1319
```
```  1320
```
```  1321 lemma setprod_empty [simp]: "setprod f {} = 1"
```
```  1322   by (auto simp add: setprod_def)
```
```  1323
```
```  1324 lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
```
```  1325     setprod f (insert a A) = f a * setprod f A"
```
```  1326   by (simp add: setprod_def)
```
```  1327
```
```  1328 lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
```
```  1329   by (simp add: setprod_def)
```
```  1330
```
```  1331 lemma setprod_reindex:
```
```  1332      "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```  1333 by(auto simp: setprod_def AC_mult.fold_reindex dest!:finite_imageD)
```
```  1334
```
```  1335 lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```  1336 by (auto simp add: setprod_reindex)
```
```  1337
```
```  1338 lemma setprod_cong:
```
```  1339   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```  1340 by(fastsimp simp: setprod_def intro: AC_mult.fold_cong)
```
```  1341
```
```  1342 lemma strong_setprod_cong:
```
```  1343   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
```
```  1344 by(fastsimp simp: simp_implies_def setprod_def intro: AC_mult.fold_cong)
```
```  1345
```
```  1346 lemma setprod_reindex_cong: "inj_on f A ==>
```
```  1347     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```  1348   by (frule setprod_reindex, simp)
```
```  1349
```
```  1350
```
```  1351 lemma setprod_1: "setprod (%i. 1) A = 1"
```
```  1352   apply (case_tac "finite A")
```
```  1353   apply (erule finite_induct, auto simp add: mult_ac)
```
```  1354   done
```
```  1355
```
```  1356 lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
```
```  1357   apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
```
```  1358   apply (erule ssubst, rule setprod_1)
```
```  1359   apply (rule setprod_cong, auto)
```
```  1360   done
```
```  1361
```
```  1362 lemma setprod_Un_Int: "finite A ==> finite B
```
```  1363     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```  1364 by(simp add: setprod_def AC_mult.fold_Un_Int[symmetric])
```
```  1365
```
```  1366 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```  1367   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```  1368 by (subst setprod_Un_Int [symmetric], auto)
```
```  1369
```
```  1370 lemma setprod_UN_disjoint:
```
```  1371     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1372         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1373       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```  1374 by(simp add: setprod_def AC_mult.fold_UN_disjoint cong: setprod_cong)
```
```  1375
```
```  1376 lemma setprod_Union_disjoint:
```
```  1377   "[| (ALL A:C. finite A);
```
```  1378       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```  1379    ==> setprod f (Union C) = setprod (setprod f) C"
```
```  1380 apply (cases "finite C")
```
```  1381  prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
```
```  1382   apply (frule setprod_UN_disjoint [of C id f])
```
```  1383  apply (unfold Union_def id_def, assumption+)
```
```  1384 done
```
```  1385
```
```  1386 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1387     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
```
```  1388     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```  1389 by(simp add:setprod_def AC_mult.fold_Sigma split_def cong:setprod_cong)
```
```  1390
```
```  1391 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1392 lemma setprod_cartesian_product:
```
```  1393      "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
```
```  1394 apply (cases "finite A")
```
```  1395  apply (cases "finite B")
```
```  1396   apply (simp add: setprod_Sigma)
```
```  1397  apply (cases "A={}", simp)
```
```  1398  apply (simp add: setprod_1)
```
```  1399 apply (auto simp add: setprod_def
```
```  1400             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1401 done
```
```  1402
```
```  1403 lemma setprod_timesf:
```
```  1404      "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
```
```  1405 by(simp add:setprod_def AC_mult.fold_distrib)
```
```  1406
```
```  1407
```
```  1408 subsubsection {* Properties in more restricted classes of structures *}
```
```  1409
```
```  1410 lemma setprod_eq_1_iff [simp]:
```
```  1411     "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
```
```  1412   by (induct set: finite) auto
```
```  1413
```
```  1414 lemma setprod_zero:
```
```  1415      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
```
```  1416   apply (induct set: finite, force, clarsimp)
```
```  1417   apply (erule disjE, auto)
```
```  1418   done
```
```  1419
```
```  1420 lemma setprod_nonneg [rule_format]:
```
```  1421      "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1422   apply (case_tac "finite A")
```
```  1423   apply (induct set: finite, force, clarsimp)
```
```  1424   apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
```
```  1425   apply (rule mult_mono, assumption+)
```
```  1426   apply (auto simp add: setprod_def)
```
```  1427   done
```
```  1428
```
```  1429 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
```
```  1430      --> 0 < setprod f A"
```
```  1431   apply (case_tac "finite A")
```
```  1432   apply (induct set: finite, force, clarsimp)
```
```  1433   apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
```
```  1434   apply (rule mult_strict_mono, assumption+)
```
```  1435   apply (auto simp add: setprod_def)
```
```  1436   done
```
```  1437
```
```  1438 lemma setprod_nonzero [rule_format]:
```
```  1439     "(ALL x y. (x::'a::comm_semiring_1) * y = 0 --> x = 0 | y = 0) ==>
```
```  1440       finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
```
```  1441   apply (erule finite_induct, auto)
```
```  1442   done
```
```  1443
```
```  1444 lemma setprod_zero_eq:
```
```  1445     "(ALL x y. (x::'a::comm_semiring_1) * y = 0 --> x = 0 | y = 0) ==>
```
```  1446      finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
```
```  1447   apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
```
```  1448   done
```
```  1449
```
```  1450 lemma setprod_nonzero_field:
```
```  1451     "finite A ==> (ALL x: A. f x \<noteq> (0::'a::idom)) ==> setprod f A \<noteq> 0"
```
```  1452   apply (rule setprod_nonzero, auto)
```
```  1453   done
```
```  1454
```
```  1455 lemma setprod_zero_eq_field:
```
```  1456     "finite A ==> (setprod f A = (0::'a::idom)) = (EX x: A. f x = 0)"
```
```  1457   apply (rule setprod_zero_eq, auto)
```
```  1458   done
```
```  1459
```
```  1460 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1461     (setprod f (A Un B) :: 'a ::{field})
```
```  1462       = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1463   apply (subst setprod_Un_Int [symmetric], auto)
```
```  1464   apply (subgoal_tac "finite (A Int B)")
```
```  1465   apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
```
```  1466   apply (subst times_divide_eq_right [THEN sym], auto)
```
```  1467   done
```
```  1468
```
```  1469 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1470     (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1471       (if a:A then setprod f A / f a else setprod f A)"
```
```  1472 by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```  1473
```
```  1474 lemma setprod_inversef: "finite A ==>
```
```  1475     ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
```
```  1476       setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1477   apply (erule finite_induct)
```
```  1478   apply (simp, simp)
```
```  1479   done
```
```  1480
```
```  1481 lemma setprod_dividef:
```
```  1482      "[|finite A;
```
```  1483         \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
```
```  1484       ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1485   apply (subgoal_tac
```
```  1486          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1487   apply (erule ssubst)
```
```  1488   apply (subst divide_inverse)
```
```  1489   apply (subst setprod_timesf)
```
```  1490   apply (subst setprod_inversef, assumption+, rule refl)
```
```  1491   apply (rule setprod_cong, rule refl)
```
```  1492   apply (subst divide_inverse, auto)
```
```  1493   done
```
```  1494
```
```  1495 subsection {* Finite cardinality *}
```
```  1496
```
```  1497 text {* This definition, although traditional, is ugly to work with:
```
```  1498 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
```
```  1499 But now that we have @{text setsum} things are easy:
```
```  1500 *}
```
```  1501
```
```  1502 constdefs
```
```  1503   card :: "'a set => nat"
```
```  1504   "card A == setsum (%x. 1::nat) A"
```
```  1505
```
```  1506 lemma card_empty [simp]: "card {} = 0"
```
```  1507   by (simp add: card_def)
```
```  1508
```
```  1509 lemma card_infinite [simp]: "~ finite A ==> card A = 0"
```
```  1510   by (simp add: card_def)
```
```  1511
```
```  1512 lemma card_eq_setsum: "card A = setsum (%x. 1) A"
```
```  1513 by (simp add: card_def)
```
```  1514
```
```  1515 lemma card_insert_disjoint [simp]:
```
```  1516   "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
```
```  1517 by(simp add: card_def)
```
```  1518
```
```  1519 lemma card_insert_if:
```
```  1520     "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
```
```  1521   by (simp add: insert_absorb)
```
```  1522
```
```  1523 lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
```
```  1524   apply auto
```
```  1525   apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
```
```  1526   done
```
```  1527
```
```  1528 lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
```
```  1529 by auto
```
```  1530
```
```  1531 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
```
```  1532 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1533 apply(simp del:insert_Diff_single)
```
```  1534 done
```
```  1535
```
```  1536 lemma card_Diff_singleton:
```
```  1537     "finite A ==> x: A ==> card (A - {x}) = card A - 1"
```
```  1538   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1539
```
```  1540 lemma card_Diff_singleton_if:
```
```  1541     "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
```
```  1542   by (simp add: card_Diff_singleton)
```
```  1543
```
```  1544 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1545   by (simp add: card_insert_if card_Suc_Diff1)
```
```  1546
```
```  1547 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1548   by (simp add: card_insert_if)
```
```  1549
```
```  1550 lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
```
```  1551 by (simp add: card_def setsum_mono2)
```
```  1552
```
```  1553 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1554   apply (induct set: finite, simp, clarify)
```
```  1555   apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1556    prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1557   apply (drule_tac x = "A - {x}" in spec)
```
```  1558   apply (simp add: card_Diff_singleton_if split add: split_if_asm)
```
```  1559   apply (case_tac "card A", auto)
```
```  1560   done
```
```  1561
```
```  1562 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1563   apply (simp add: psubset_def linorder_not_le [symmetric])
```
```  1564   apply (blast dest: card_seteq)
```
```  1565   done
```
```  1566
```
```  1567 lemma card_Un_Int: "finite A ==> finite B
```
```  1568     ==> card A + card B = card (A Un B) + card (A Int B)"
```
```  1569 by(simp add:card_def setsum_Un_Int)
```
```  1570
```
```  1571 lemma card_Un_disjoint: "finite A ==> finite B
```
```  1572     ==> A Int B = {} ==> card (A Un B) = card A + card B"
```
```  1573   by (simp add: card_Un_Int)
```
```  1574
```
```  1575 lemma card_Diff_subset:
```
```  1576   "finite B ==> B <= A ==> card (A - B) = card A - card B"
```
```  1577 by(simp add:card_def setsum_diff_nat)
```
```  1578
```
```  1579 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1580   apply (rule Suc_less_SucD)
```
```  1581   apply (simp add: card_Suc_Diff1)
```
```  1582   done
```
```  1583
```
```  1584 lemma card_Diff2_less:
```
```  1585     "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1586   apply (case_tac "x = y")
```
```  1587    apply (simp add: card_Diff1_less)
```
```  1588   apply (rule less_trans)
```
```  1589    prefer 2 apply (auto intro!: card_Diff1_less)
```
```  1590   done
```
```  1591
```
```  1592 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1593   apply (case_tac "x : A")
```
```  1594    apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1595   done
```
```  1596
```
```  1597 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1598 by (erule psubsetI, blast)
```
```  1599
```
```  1600 lemma insert_partition:
```
```  1601   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1602   \<Longrightarrow> x \<inter> \<Union> F = {}"
```
```  1603 by auto
```
```  1604
```
```  1605 text{* main cardinality theorem *}
```
```  1606 lemma card_partition [rule_format]:
```
```  1607      "finite C ==>
```
```  1608         finite (\<Union> C) -->
```
```  1609         (\<forall>c\<in>C. card c = k) -->
```
```  1610         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  1611         k * card(C) = card (\<Union> C)"
```
```  1612 apply (erule finite_induct, simp)
```
```  1613 apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition
```
```  1614        finite_subset [of _ "\<Union> (insert x F)"])
```
```  1615 done
```
```  1616
```
```  1617
```
```  1618 text{*The form of a finite set of given cardinality*}
```
```  1619
```
```  1620 lemma card_eq_SucD:
```
```  1621   assumes cardeq: "card A = Suc k" and fin: "finite A"
```
```  1622   shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k"
```
```  1623 proof -
```
```  1624   have "card A \<noteq> 0" using cardeq by auto
```
```  1625   then obtain b where b: "b \<in> A" using fin by auto
```
```  1626   show ?thesis
```
```  1627   proof (intro exI conjI)
```
```  1628     show "A = insert b (A-{b})" using b by blast
```
```  1629     show "b \<notin> A - {b}" by blast
```
```  1630     show "card (A - {b}) = k" by (simp add: fin cardeq b card_Diff_singleton)
```
```  1631   qed
```
```  1632 qed
```
```  1633
```
```  1634
```
```  1635 lemma card_Suc_eq:
```
```  1636   "finite A ==>
```
```  1637    (card A = Suc k) = (\<exists>b B. A = insert b B & b \<notin> B & card B = k)"
```
```  1638 by (auto dest!: card_eq_SucD)
```
```  1639
```
```  1640 lemma card_1_eq:
```
```  1641   "finite A ==> (card A = Suc 0) = (\<exists>x. A = {x})"
```
```  1642 by (auto dest!: card_eq_SucD)
```
```  1643
```
```  1644 lemma card_2_eq:
```
```  1645   "finite A ==> (card A = Suc(Suc 0)) = (\<exists>x y. x\<noteq>y & A = {x,y})"
```
```  1646 by (auto dest!: card_eq_SucD, blast)
```
```  1647
```
```  1648
```
```  1649 lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
```
```  1650 apply (cases "finite A")
```
```  1651 apply (erule finite_induct)
```
```  1652 apply (auto simp add: ring_distrib add_ac)
```
```  1653 done
```
```  1654
```
```  1655 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{recpower, comm_monoid_mult})) = y^(card A)"
```
```  1656   apply (erule finite_induct)
```
```  1657   apply (auto simp add: power_Suc)
```
```  1658   done
```
```  1659
```
```  1660 lemma setsum_bounded:
```
```  1661   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, pordered_ab_semigroup_add})"
```
```  1662   shows "setsum f A \<le> of_nat(card A) * K"
```
```  1663 proof (cases "finite A")
```
```  1664   case True
```
```  1665   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```  1666 next
```
```  1667   case False thus ?thesis by (simp add: setsum_def)
```
```  1668 qed
```
```  1669
```
```  1670
```
```  1671 subsubsection {* Cardinality of unions *}
```
```  1672
```
```  1673 lemma card_UN_disjoint:
```
```  1674     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1675         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1676       card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```  1677   apply (simp add: card_def del: setsum_constant)
```
```  1678   apply (subgoal_tac
```
```  1679            "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
```
```  1680   apply (simp add: setsum_UN_disjoint del: setsum_constant)
```
```  1681   apply (simp cong: setsum_cong)
```
```  1682   done
```
```  1683
```
```  1684 lemma card_Union_disjoint:
```
```  1685   "finite C ==> (ALL A:C. finite A) ==>
```
```  1686         (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
```
```  1687       card (Union C) = setsum card C"
```
```  1688   apply (frule card_UN_disjoint [of C id])
```
```  1689   apply (unfold Union_def id_def, assumption+)
```
```  1690   done
```
```  1691
```
```  1692 subsubsection {* Cardinality of image *}
```
```  1693
```
```  1694 text{*The image of a finite set can be expressed using @{term fold}.*}
```
```  1695 lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
```
```  1696   apply (erule finite_induct, simp)
```
```  1697   apply (subst ACf.fold_insert)
```
```  1698   apply (auto simp add: ACf_def)
```
```  1699   done
```
```  1700
```
```  1701 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
```
```  1702   apply (induct set: finite)
```
```  1703    apply simp
```
```  1704   apply (simp add: le_SucI finite_imageI card_insert_if)
```
```  1705   done
```
```  1706
```
```  1707 lemma card_image: "inj_on f A ==> card (f ` A) = card A"
```
```  1708 by(simp add:card_def setsum_reindex o_def del:setsum_constant)
```
```  1709
```
```  1710 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  1711   by (simp add: card_seteq card_image)
```
```  1712
```
```  1713 lemma eq_card_imp_inj_on:
```
```  1714   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
```
```  1715 apply (induct rule:finite_induct)
```
```  1716 apply simp
```
```  1717 apply(frule card_image_le[where f = f])
```
```  1718 apply(simp add:card_insert_if split:if_splits)
```
```  1719 done
```
```  1720
```
```  1721 lemma inj_on_iff_eq_card:
```
```  1722   "finite A ==> inj_on f A = (card(f ` A) = card A)"
```
```  1723 by(blast intro: card_image eq_card_imp_inj_on)
```
```  1724
```
```  1725
```
```  1726 lemma card_inj_on_le:
```
```  1727     "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
```
```  1728 apply (subgoal_tac "finite A")
```
```  1729  apply (force intro: card_mono simp add: card_image [symmetric])
```
```  1730 apply (blast intro: finite_imageD dest: finite_subset)
```
```  1731 done
```
```  1732
```
```  1733 lemma card_bij_eq:
```
```  1734     "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  1735        finite A; finite B |] ==> card A = card B"
```
```  1736   by (auto intro: le_anti_sym card_inj_on_le)
```
```  1737
```
```  1738
```
```  1739 subsubsection {* Cardinality of products *}
```
```  1740
```
```  1741 (*
```
```  1742 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1743   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1744   by auto
```
```  1745 *)
```
```  1746
```
```  1747 lemma card_SigmaI [simp]:
```
```  1748   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```  1749   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1750 by(simp add:card_def setsum_Sigma del:setsum_constant)
```
```  1751
```
```  1752 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```  1753 apply (cases "finite A")
```
```  1754 apply (cases "finite B")
```
```  1755 apply (auto simp add: card_eq_0_iff
```
```  1756             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1757 done
```
```  1758
```
```  1759 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```  1760 by (simp add: card_cartesian_product)
```
```  1761
```
```  1762
```
```  1763
```
```  1764 subsubsection {* Cardinality of the Powerset *}
```
```  1765
```
```  1766 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
```
```  1767   apply (induct set: finite)
```
```  1768    apply (simp_all add: Pow_insert)
```
```  1769   apply (subst card_Un_disjoint, blast)
```
```  1770     apply (blast intro: finite_imageI, blast)
```
```  1771   apply (subgoal_tac "inj_on (insert x) (Pow F)")
```
```  1772    apply (simp add: card_image Pow_insert)
```
```  1773   apply (unfold inj_on_def)
```
```  1774   apply (blast elim!: equalityE)
```
```  1775   done
```
```  1776
```
```  1777 text {* Relates to equivalence classes.  Based on a theorem of
```
```  1778 F. Kammüller's.  *}
```
```  1779
```
```  1780 lemma dvd_partition:
```
```  1781   "finite (Union C) ==>
```
```  1782     ALL c : C. k dvd card c ==>
```
```  1783     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
```
```  1784   k dvd card (Union C)"
```
```  1785 apply(frule finite_UnionD)
```
```  1786 apply(rotate_tac -1)
```
```  1787   apply (induct set: finite, simp_all, clarify)
```
```  1788   apply (subst card_Un_disjoint)
```
```  1789   apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
```
```  1790   done
```
```  1791
```
```  1792
```
```  1793 subsection{* A fold functional for non-empty sets *}
```
```  1794
```
```  1795 text{* Does not require start value. *}
```
```  1796
```
```  1797 inductive2
```
```  1798   fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
```
```  1799   for f :: "'a => 'a => 'a"
```
```  1800 where
```
```  1801   fold1Set_insertI [intro]:
```
```  1802    "\<lbrakk> foldSet f id a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
```
```  1803
```
```  1804 constdefs
```
```  1805   fold1 :: "('a => 'a => 'a) => 'a set => 'a"
```
```  1806   "fold1 f A == THE x. fold1Set f A x"
```
```  1807
```
```  1808 lemma fold1Set_nonempty:
```
```  1809   "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
```
```  1810   by(erule fold1Set.cases, simp_all)
```
```  1811
```
```  1812 inductive_cases2 empty_fold1SetE [elim!]: "fold1Set f {} x"
```
```  1813
```
```  1814 inductive_cases2 insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
```
```  1815
```
```  1816
```
```  1817 lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
```
```  1818   by (blast intro: foldSet.intros elim: foldSet.cases)
```
```  1819
```
```  1820 lemma fold1_singleton [simp]: "fold1 f {a} = a"
```
```  1821   by (unfold fold1_def) blast
```
```  1822
```
```  1823 lemma finite_nonempty_imp_fold1Set:
```
```  1824   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
```
```  1825 apply (induct A rule: finite_induct)
```
```  1826 apply (auto dest: finite_imp_foldSet [of _ f id])
```
```  1827 done
```
```  1828
```
```  1829 text{*First, some lemmas about @{term foldSet}.*}
```
```  1830
```
```  1831 lemma (in ACf) foldSet_insert_swap:
```
```  1832 assumes fold: "foldSet f id b A y"
```
```  1833 shows "b \<notin> A \<Longrightarrow> foldSet f id z (insert b A) (z \<cdot> y)"
```
```  1834 using fold
```
```  1835 proof (induct rule: foldSet.induct)
```
```  1836   case emptyI thus ?case by (force simp add: fold_insert_aux commute)
```
```  1837 next
```
```  1838   case (insertI x A y)
```
```  1839     have "foldSet f (\<lambda>u. u) z (insert x (insert b A)) (x \<cdot> (z \<cdot> y))"
```
```  1840       using insertI by force  --{*how does @{term id} get unfolded?*}
```
```  1841     thus ?case by (simp add: insert_commute AC)
```
```  1842 qed
```
```  1843
```
```  1844 lemma (in ACf) foldSet_permute_diff:
```
```  1845 assumes fold: "foldSet f id b A x"
```
```  1846 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> foldSet f id a (insert b (A-{a})) x"
```
```  1847 using fold
```
```  1848 proof (induct rule: foldSet.induct)
```
```  1849   case emptyI thus ?case by simp
```
```  1850 next
```
```  1851   case (insertI x A y)
```
```  1852   have "a = x \<or> a \<in> A" using insertI by simp
```
```  1853   thus ?case
```
```  1854   proof
```
```  1855     assume "a = x"
```
```  1856     with insertI show ?thesis
```
```  1857       by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap)
```
```  1858   next
```
```  1859     assume ainA: "a \<in> A"
```
```  1860     hence "foldSet f id a (insert x (insert b (A - {a}))) (x \<cdot> y)"
```
```  1861       using insertI by (force simp: id_def)
```
```  1862     moreover
```
```  1863     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
```
```  1864       using ainA insertI by blast
```
```  1865     ultimately show ?thesis by (simp add: id_def)
```
```  1866   qed
```
```  1867 qed
```
```  1868
```
```  1869 lemma (in ACf) fold1_eq_fold:
```
```  1870      "[|finite A; a \<notin> A|] ==> fold1 f (insert a A) = fold f id a A"
```
```  1871 apply (simp add: fold1_def fold_def)
```
```  1872 apply (rule the_equality)
```
```  1873 apply (best intro: foldSet_determ theI dest: finite_imp_foldSet [of _ f id])
```
```  1874 apply (rule sym, clarify)
```
```  1875 apply (case_tac "Aa=A")
```
```  1876  apply (best intro: the_equality foldSet_determ)
```
```  1877 apply (subgoal_tac "foldSet f id a A x")
```
```  1878  apply (best intro: the_equality foldSet_determ)
```
```  1879 apply (subgoal_tac "insert aa (Aa - {a}) = A")
```
```  1880  prefer 2 apply (blast elim: equalityE)
```
```  1881 apply (auto dest: foldSet_permute_diff [where a=a])
```
```  1882 done
```
```  1883
```
```  1884 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
```
```  1885 apply safe
```
```  1886 apply simp
```
```  1887 apply (drule_tac x=x in spec)
```
```  1888 apply (drule_tac x="A-{x}" in spec, auto)
```
```  1889 done
```
```  1890
```
```  1891 lemma (in ACf) fold1_insert:
```
```  1892   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
```
```  1893   shows "fold1 f (insert x A) = f x (fold1 f A)"
```
```  1894 proof -
```
```  1895   from nonempty obtain a A' where "A = insert a A' & a ~: A'"
```
```  1896     by (auto simp add: nonempty_iff)
```
```  1897   with A show ?thesis
```
```  1898     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
```
```  1899 qed
```
```  1900
```
```  1901 lemma (in ACIf) fold1_insert_idem [simp]:
```
```  1902   assumes nonempty: "A \<noteq> {}" and A: "finite A"
```
```  1903   shows "fold1 f (insert x A) = f x (fold1 f A)"
```
```  1904 proof -
```
```  1905   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
```
```  1906     by (auto simp add: nonempty_iff)
```
```  1907   show ?thesis
```
```  1908   proof cases
```
```  1909     assume "a = x"
```
```  1910     thus ?thesis
```
```  1911     proof cases
```
```  1912       assume "A' = {}"
```
```  1913       with prems show ?thesis by (simp add: idem)
```
```  1914     next
```
```  1915       assume "A' \<noteq> {}"
```
```  1916       with prems show ?thesis
```
```  1917 	by (simp add: fold1_insert assoc [symmetric] idem)
```
```  1918     qed
```
```  1919   next
```
```  1920     assume "a \<noteq> x"
```
```  1921     with prems show ?thesis
```
```  1922       by (simp add: insert_commute fold1_eq_fold fold_insert_idem)
```
```  1923   qed
```
```  1924 qed
```
```  1925
```
```  1926 lemma (in ACIf) hom_fold1_commute:
```
```  1927 assumes hom: "!!x y. h(f x y) = f (h x) (h y)"
```
```  1928 and N: "finite N" "N \<noteq> {}" shows "h(fold1 f N) = fold1 f (h ` N)"
```
```  1929 using N proof (induct rule: finite_ne_induct)
```
```  1930   case singleton thus ?case by simp
```
```  1931 next
```
```  1932   case (insert n N)
```
```  1933   then have "h(fold1 f (insert n N)) = h(f n (fold1 f N))" by simp
```
```  1934   also have "\<dots> = f (h n) (h(fold1 f N))" by(rule hom)
```
```  1935   also have "h(fold1 f N) = fold1 f (h ` N)" by(rule insert)
```
```  1936   also have "f (h n) \<dots> = fold1 f (insert (h n) (h ` N))"
```
```  1937     using insert by(simp)
```
```  1938   also have "insert (h n) (h ` N) = h ` insert n N" by simp
```
```  1939   finally show ?case .
```
```  1940 qed
```
```  1941
```
```  1942
```
```  1943 text{* Now the recursion rules for definitions: *}
```
```  1944
```
```  1945 lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
```
```  1946 by(simp add:fold1_singleton)
```
```  1947
```
```  1948 lemma (in ACf) fold1_insert_def:
```
```  1949   "\<lbrakk> g = fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x \<cdot> (g A)"
```
```  1950 by(simp add:fold1_insert)
```
```  1951
```
```  1952 lemma (in ACIf) fold1_insert_idem_def:
```
```  1953   "\<lbrakk> g = fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x \<cdot> (g A)"
```
```  1954 by(simp add:fold1_insert_idem)
```
```  1955
```
```  1956 subsubsection{* Determinacy for @{term fold1Set} *}
```
```  1957
```
```  1958 text{*Not actually used!!*}
```
```  1959
```
```  1960 lemma (in ACf) foldSet_permute:
```
```  1961   "[|foldSet f id b (insert a A) x; a \<notin> A; b \<notin> A|]
```
```  1962    ==> foldSet f id a (insert b A) x"
```
```  1963 apply (case_tac "a=b")
```
```  1964 apply (auto dest: foldSet_permute_diff)
```
```  1965 done
```
```  1966
```
```  1967 lemma (in ACf) fold1Set_determ:
```
```  1968   "fold1Set f A x ==> fold1Set f A y ==> y = x"
```
```  1969 proof (clarify elim!: fold1Set.cases)
```
```  1970   fix A x B y a b
```
```  1971   assume Ax: "foldSet f id a A x"
```
```  1972   assume By: "foldSet f id b B y"
```
```  1973   assume anotA:  "a \<notin> A"
```
```  1974   assume bnotB:  "b \<notin> B"
```
```  1975   assume eq: "insert a A = insert b B"
```
```  1976   show "y=x"
```
```  1977   proof cases
```
```  1978     assume same: "a=b"
```
```  1979     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
```
```  1980     thus ?thesis using Ax By same by (blast intro: foldSet_determ)
```
```  1981   next
```
```  1982     assume diff: "a\<noteq>b"
```
```  1983     let ?D = "B - {a}"
```
```  1984     have B: "B = insert a ?D" and A: "A = insert b ?D"
```
```  1985      and aB: "a \<in> B" and bA: "b \<in> A"
```
```  1986       using eq anotA bnotB diff by (blast elim!:equalityE)+
```
```  1987     with aB bnotB By
```
```  1988     have "foldSet f id a (insert b ?D) y"
```
```  1989       by (auto intro: foldSet_permute simp add: insert_absorb)
```
```  1990     moreover
```
```  1991     have "foldSet f id a (insert b ?D) x"
```
```  1992       by (simp add: A [symmetric] Ax)
```
```  1993     ultimately show ?thesis by (blast intro: foldSet_determ)
```
```  1994   qed
```
```  1995 qed
```
```  1996
```
```  1997 lemma (in ACf) fold1Set_equality: "fold1Set f A y ==> fold1 f A = y"
```
```  1998   by (unfold fold1_def) (blast intro: fold1Set_determ)
```
```  1999
```
```  2000 declare
```
```  2001   empty_foldSetE [rule del]   foldSet.intros [rule del]
```
```  2002   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
```
```  2003   -- {* No more proofs involve these relations. *}
```
```  2004
```
```  2005
```
```  2006 subsubsection{* Semi-Lattices *}
```
```  2007
```
```  2008 locale ACIfSL = ord + ACIf +
```
```  2009   assumes below_def: "x \<sqsubseteq> y \<longleftrightarrow> x \<cdot> y = x"
```
```  2010   and strict_below_def: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
```
```  2011 begin
```
```  2012
```
```  2013 lemma below_refl [simp]: "x \<^loc>\<le> x"
```
```  2014   by (simp add: below_def idem)
```
```  2015
```
```  2016 lemma below_antisym:
```
```  2017   assumes xy: "x \<^loc>\<le> y" and yx: "y \<^loc>\<le> x"
```
```  2018   shows "x = y"
```
```  2019   using xy [unfolded below_def, symmetric]
```
```  2020     yx [unfolded below_def commute]
```
```  2021   by (rule trans)
```
```  2022
```
```  2023 lemma below_trans:
```
```  2024   assumes xy: "x \<^loc>\<le> y" and yz: "y \<^loc>\<le> z"
```
```  2025   shows "x \<^loc>\<le> z"
```
```  2026 proof -
```
```  2027   from xy have x_xy: "x \<cdot> y = x" by (simp add: below_def)
```
```  2028   from yz have y_yz: "y \<cdot> z = y" by (simp add: below_def)
```
```  2029   from y_yz have "x \<cdot> y \<cdot> z = x \<cdot> y" by (simp add: assoc)
```
```  2030   with x_xy have "x \<cdot> y \<cdot> z = x"  by simp
```
```  2031   moreover from x_xy have "x \<cdot> z = x \<cdot> y \<cdot> z" by simp
```
```  2032   ultimately have "x \<cdot> z = x" by simp
```
```  2033   then show ?thesis by (simp add: below_def)
```
```  2034 qed
```
```  2035
```
```  2036 lemma below_f_conv [simp]: "x \<sqsubseteq> y \<cdot> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
```
```  2037 proof
```
```  2038   assume "x \<sqsubseteq> y \<cdot> z"
```
```  2039   hence xyzx: "x \<cdot> (y \<cdot> z) = x"  by(simp add: below_def)
```
```  2040   have "x \<cdot> y = x"
```
```  2041   proof -
```
```  2042     have "x \<cdot> y = (x \<cdot> (y \<cdot> z)) \<cdot> y" by(rule subst[OF xyzx], rule refl)
```
```  2043     also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  2044     also have "\<dots> = x" by(rule xyzx)
```
```  2045     finally show ?thesis .
```
```  2046   qed
```
```  2047   moreover have "x \<cdot> z = x"
```
```  2048   proof -
```
```  2049     have "x \<cdot> z = (x \<cdot> (y \<cdot> z)) \<cdot> z" by(rule subst[OF xyzx], rule refl)
```
```  2050     also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  2051     also have "\<dots> = x" by(rule xyzx)
```
```  2052     finally show ?thesis .
```
```  2053   qed
```
```  2054   ultimately show "x \<sqsubseteq> y \<and> x \<sqsubseteq> z" by(simp add: below_def)
```
```  2055 next
```
```  2056   assume a: "x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
```
```  2057   hence y: "x \<cdot> y = x" and z: "x \<cdot> z = x" by(simp_all add: below_def)
```
```  2058   have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by(simp add:assoc)
```
```  2059   also have "x \<cdot> y = x" using a by(simp_all add: below_def)
```
```  2060   also have "x \<cdot> z = x" using a by(simp_all add: below_def)
```
```  2061   finally show "x \<sqsubseteq> y \<cdot> z" by(simp_all add: below_def)
```
```  2062 qed
```
```  2063
```
```  2064 end
```
```  2065
```
```  2066 interpretation ACIfSL < order
```
```  2067 by unfold_locales
```
```  2068   (simp add: strict_below_def, auto intro: below_refl below_trans below_antisym)
```
```  2069
```
```  2070 locale ACIfSLlin = ACIfSL +
```
```  2071   assumes lin: "x\<cdot>y \<in> {x,y}"
```
```  2072 begin
```
```  2073
```
```  2074 lemma above_f_conv:
```
```  2075  "x \<cdot> y \<sqsubseteq> z = (x \<sqsubseteq> z \<or> y \<sqsubseteq> z)"
```
```  2076 proof
```
```  2077   assume a: "x \<cdot> y \<sqsubseteq> z"
```
```  2078   have "x \<cdot> y = x \<or> x \<cdot> y = y" using lin[of x y] by simp
```
```  2079   thus "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
```
```  2080   proof
```
```  2081     assume "x \<cdot> y = x" hence "x \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
```
```  2082   next
```
```  2083     assume "x \<cdot> y = y" hence "y \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
```
```  2084   qed
```
```  2085 next
```
```  2086   assume "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
```
```  2087   thus "x \<cdot> y \<sqsubseteq> z"
```
```  2088   proof
```
```  2089     assume a: "x \<sqsubseteq> z"
```
```  2090     have "(x \<cdot> y) \<cdot> z = (x \<cdot> z) \<cdot> y" by(simp add:ACI)
```
```  2091     also have "x \<cdot> z = x" using a by(simp add:below_def)
```
```  2092     finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
```
```  2093   next
```
```  2094     assume a: "y \<sqsubseteq> z"
```
```  2095     have "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  2096     also have "y \<cdot> z = y" using a by(simp add:below_def)
```
```  2097     finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
```
```  2098   qed
```
```  2099 qed
```
```  2100
```
```  2101 lemma strict_below_f_conv[simp]: "x \<sqsubset> y \<cdot> z = (x \<sqsubset> y \<and> x \<sqsubset> z)"
```
```  2102 apply(simp add: strict_below_def)
```
```  2103 using lin[of y z] by (auto simp:below_def ACI)
```
```  2104
```
```  2105 lemma strict_above_f_conv:
```
```  2106   "x \<cdot> y \<sqsubset> z = (x \<sqsubset> z \<or> y \<sqsubset> z)"
```
```  2107 apply(simp add: strict_below_def above_f_conv)
```
```  2108 using lin[of y z] lin[of x z] by (auto simp:below_def ACI)
```
```  2109
```
```  2110 end
```
```  2111
```
```  2112 interpretation ACIfSLlin < linorder
```
```  2113   by unfold_locales
```
```  2114     (insert lin [simplified insert_iff], simp add: below_def commute)
```
```  2115
```
```  2116
```
```  2117 subsubsection{* Lemmas about @{text fold1} *}
```
```  2118
```
```  2119 lemma (in ACf) fold1_Un:
```
```  2120 assumes A: "finite A" "A \<noteq> {}"
```
```  2121 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
```
```  2122        fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
```
```  2123 using A
```
```  2124 proof(induct rule:finite_ne_induct)
```
```  2125   case singleton thus ?case by(simp add:fold1_insert)
```
```  2126 next
```
```  2127   case insert thus ?case by (simp add:fold1_insert assoc)
```
```  2128 qed
```
```  2129
```
```  2130 lemma (in ACIf) fold1_Un2:
```
```  2131 assumes A: "finite A" "A \<noteq> {}"
```
```  2132 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
```
```  2133        fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
```
```  2134 using A
```
```  2135 proof(induct rule:finite_ne_induct)
```
```  2136   case singleton thus ?case by(simp add:fold1_insert_idem)
```
```  2137 next
```
```  2138   case insert thus ?case by (simp add:fold1_insert_idem assoc)
```
```  2139 qed
```
```  2140
```
```  2141 lemma (in ACf) fold1_in:
```
```  2142   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}"
```
```  2143   shows "fold1 f A \<in> A"
```
```  2144 using A
```
```  2145 proof (induct rule:finite_ne_induct)
```
```  2146   case singleton thus ?case by simp
```
```  2147 next
```
```  2148   case insert thus ?case using elem by (force simp add:fold1_insert)
```
```  2149 qed
```
```  2150
```
```  2151 lemma (in ACIfSL) below_fold1_iff:
```
```  2152 assumes A: "finite A" "A \<noteq> {}"
```
```  2153 shows "x \<sqsubseteq> fold1 f A = (\<forall>a\<in>A. x \<sqsubseteq> a)"
```
```  2154 using A
```
```  2155 by(induct rule:finite_ne_induct) simp_all
```
```  2156
```
```  2157 lemma (in ACIfSLlin) strict_below_fold1_iff:
```
```  2158   "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> x \<sqsubset> fold1 f A = (\<forall>a\<in>A. x \<sqsubset> a)"
```
```  2159 by(induct rule:finite_ne_induct) simp_all
```
```  2160
```
```  2161
```
```  2162 lemma (in ACIfSL) fold1_belowI:
```
```  2163 assumes A: "finite A" "A \<noteq> {}"
```
```  2164 shows "a \<in> A \<Longrightarrow> fold1 f A \<sqsubseteq> a"
```
```  2165 using A
```
```  2166 proof (induct rule:finite_ne_induct)
```
```  2167   case singleton thus ?case by simp
```
```  2168 next
```
```  2169   case (insert x F)
```
```  2170   from insert(5) have "a = x \<or> a \<in> F" by simp
```
```  2171   thus ?case
```
```  2172   proof
```
```  2173     assume "a = x" thus ?thesis using insert by(simp add:below_def ACI)
```
```  2174   next
```
```  2175     assume "a \<in> F"
```
```  2176     hence bel: "fold1 f F \<sqsubseteq> a" by(rule insert)
```
```  2177     have "fold1 f (insert x F) \<cdot> a = x \<cdot> (fold1 f F \<cdot> a)"
```
```  2178       using insert by(simp add:below_def ACI)
```
```  2179     also have "fold1 f F \<cdot> a = fold1 f F"
```
```  2180       using bel  by(simp add:below_def ACI)
```
```  2181     also have "x \<cdot> \<dots> = fold1 f (insert x F)"
```
```  2182       using insert by(simp add:below_def ACI)
```
```  2183     finally show ?thesis  by(simp add:below_def)
```
```  2184   qed
```
```  2185 qed
```
```  2186
```
```  2187 lemma (in ACIfSLlin) fold1_below_iff:
```
```  2188 assumes A: "finite A" "A \<noteq> {}"
```
```  2189 shows "fold1 f A \<sqsubseteq> x = (\<exists>a\<in>A. a \<sqsubseteq> x)"
```
```  2190 using A
```
```  2191 by(induct rule:finite_ne_induct)(simp_all add:above_f_conv)
```
```  2192
```
```  2193 lemma (in ACIfSLlin) fold1_strict_below_iff:
```
```  2194 assumes A: "finite A" "A \<noteq> {}"
```
```  2195 shows "fold1 f A \<sqsubset> x = (\<exists>a\<in>A. a \<sqsubset> x)"
```
```  2196 using A
```
```  2197 by(induct rule:finite_ne_induct)(simp_all add:strict_above_f_conv)
```
```  2198
```
```  2199 lemma (in ACIfSLlin) fold1_antimono:
```
```  2200 assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
```
```  2201 shows "fold1 f B \<sqsubseteq> fold1 f A"
```
```  2202 proof(cases)
```
```  2203   assume "A = B" thus ?thesis by simp
```
```  2204 next
```
```  2205   assume "A \<noteq> B"
```
```  2206   have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
```
```  2207   have "fold1 f B = fold1 f (A \<union> (B-A))" by(subst B)(rule refl)
```
```  2208   also have "\<dots> = f (fold1 f A) (fold1 f (B-A))"
```
```  2209   proof -
```
```  2210     have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
```
```  2211     moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *)
```
```  2212     moreover have "(B-A) \<noteq> {}" using prems by blast
```
```  2213     moreover have "A Int (B-A) = {}" using prems by blast
```
```  2214     ultimately show ?thesis using `A \<noteq> {}` by(rule_tac fold1_Un)
```
```  2215   qed
```
```  2216   also have "\<dots> \<sqsubseteq> fold1 f A" by(simp add: above_f_conv)
```
```  2217   finally show ?thesis .
```
```  2218 qed
```
```  2219
```
```  2220
```
```  2221 subsubsection {* Fold1 in lattices with @{const inf} and @{const sup} *}
```
```  2222
```
```  2223 text{*
```
```  2224   As an application of @{text fold1} we define infimum
```
```  2225   and supremum in (not necessarily complete!) lattices
```
```  2226   over (non-empty) sets by means of @{text fold1}.
```
```  2227 *}
```
```  2228
```
```  2229 lemma (in lower_semilattice) ACf_inf: "ACf (op \<sqinter>)"
```
```  2230   by (blast intro: ACf.intro inf_commute inf_assoc)
```
```  2231
```
```  2232 lemma (in upper_semilattice) ACf_sup: "ACf (op \<squnion>)"
```
```  2233   by (blast intro: ACf.intro sup_commute sup_assoc)
```
```  2234
```
```  2235 lemma (in lower_semilattice) ACIf_inf: "ACIf (op \<sqinter>)"
```
```  2236 apply(rule ACIf.intro)
```
```  2237 apply(rule ACf_inf)
```
```  2238 apply(rule ACIf_axioms.intro)
```
```  2239 apply(rule inf_idem)
```
```  2240 done
```
```  2241
```
```  2242 lemma (in upper_semilattice) ACIf_sup: "ACIf (op \<squnion>)"
```
```  2243 apply(rule ACIf.intro)
```
```  2244 apply(rule ACf_sup)
```
```  2245 apply(rule ACIf_axioms.intro)
```
```  2246 apply(rule sup_idem)
```
```  2247 done
```
```  2248
```
```  2249 lemma (in lower_semilattice) ACIfSL_inf: "ACIfSL (op \<^loc>\<le>) (op \<^loc><) (op \<sqinter>)"
```
```  2250 apply(rule ACIfSL.intro)
```
```  2251 apply(rule ACIf.intro)
```
```  2252 apply(rule ACf_inf)
```
```  2253 apply(rule ACIf.axioms[OF ACIf_inf])
```
```  2254 apply(rule ACIfSL_axioms.intro)
```
```  2255 apply(rule iffI)
```
```  2256  apply(blast intro: antisym inf_le1 inf_le2 inf_greatest refl)
```
```  2257 apply(erule subst)
```
```  2258 apply(rule inf_le2)
```
```  2259 apply(rule less_le)
```
```  2260 done
```
```  2261
```
```  2262 lemma (in upper_semilattice) ACIfSL_sup: "ACIfSL (%x y. y \<^loc>\<le> x) (%x y. y \<^loc>< x) (op \<squnion>)"
```
```  2263 apply(rule ACIfSL.intro)
```
```  2264 apply(rule ACIf.intro)
```
```  2265 apply(rule ACf_sup)
```
```  2266 apply(rule ACIf.axioms[OF ACIf_sup])
```
```  2267 apply(rule ACIfSL_axioms.intro)
```
```  2268 apply(rule iffI)
```
```  2269  apply(blast intro: antisym sup_ge1 sup_ge2 sup_least refl)
```
```  2270 apply(erule subst)
```
```  2271 apply(rule sup_ge2)
```
```  2272 apply(simp add: neq_commute less_le)
```
```  2273 done
```
```  2274
```
```  2275 locale Lattice = lattice -- {* we do not pollute the @{text lattice} clas *}
```
```  2276 begin
```
```  2277
```
```  2278 definition
```
```  2279   Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
```
```  2280 where
```
```  2281   "Inf = fold1 (op \<sqinter>)"
```
```  2282
```
```  2283 definition
```
```  2284   Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```  2285 where
```
```  2286   "Sup = fold1 (op \<squnion>)"
```
```  2287
```
```  2288 end
```
```  2289
```
```  2290 locale Distrib_Lattice = distrib_lattice + Lattice
```
```  2291
```
```  2292 lemma (in Lattice) Inf_le_Sup[simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Squnion>A"
```
```  2293 apply(unfold Sup_def Inf_def)
```
```  2294 apply(subgoal_tac "EX a. a:A")
```
```  2295 prefer 2 apply blast
```
```  2296 apply(erule exE)
```
```  2297 apply(rule order_trans)
```
```  2298 apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_inf])
```
```  2299 apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup])
```
```  2300 done
```
```  2301
```
```  2302 lemma (in Lattice) sup_Inf_absorb[simp]:
```
```  2303   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<squnion> \<Sqinter>A) = a"
```
```  2304 apply(subst sup_commute)
```
```  2305 apply(simp add:Inf_def sup_absorb2 ACIfSL.fold1_belowI[OF ACIfSL_inf])
```
```  2306 done
```
```  2307
```
```  2308 lemma (in Lattice) inf_Sup_absorb[simp]:
```
```  2309   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<sqinter> \<Squnion>A) = a"
```
```  2310 by(simp add:Sup_def inf_absorb1 ACIfSL.fold1_belowI[OF ACIfSL_sup])
```
```  2311
```
```  2312 lemma (in Distrib_Lattice) sup_Inf1_distrib:
```
```  2313  "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (x \<squnion> \<Sqinter>A) = \<Sqinter>{x \<squnion> a|a. a \<in> A}"
```
```  2314 apply(simp add:Inf_def image_def
```
```  2315   ACIf.hom_fold1_commute[OF ACIf_inf, where h="sup x", OF sup_inf_distrib1])
```
```  2316 apply(rule arg_cong, blast)
```
```  2317 done
```
```  2318
```
```  2319
```
```  2320 lemma (in Distrib_Lattice) sup_Inf2_distrib:
```
```  2321 assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  2322 shows "(\<Sqinter>A \<squnion> \<Sqinter>B) = \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
```
```  2323 using A
```
```  2324 proof (induct rule: finite_ne_induct)
```
```  2325   case singleton thus ?case
```
```  2326     by (simp add: sup_Inf1_distrib[OF B] fold1_singleton_def[OF Inf_def])
```
```  2327 next
```
```  2328   case (insert x A)
```
```  2329   have finB: "finite {x \<squnion> b |b. b \<in> B}"
```
```  2330     by(rule finite_surj[where f = "%b. x \<squnion> b", OF B(1)], auto)
```
```  2331   have finAB: "finite {a \<squnion> b |a b. a \<in> A \<and> b \<in> B}"
```
```  2332   proof -
```
```  2333     have "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<squnion> b})"
```
```  2334       by blast
```
```  2335     thus ?thesis by(simp add: insert(1) B(1))
```
```  2336   qed
```
```  2337   have ne: "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  2338   have "\<Sqinter>(insert x A) \<squnion> \<Sqinter>B = (x \<sqinter> \<Sqinter>A) \<squnion> \<Sqinter>B"
```
```  2339     using insert
```
```  2340     thm ACIf.fold1_insert_idem_def
```
```  2341  by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def])
```
```  2342   also have "\<dots> = (x \<squnion> \<Sqinter>B) \<sqinter> (\<Sqinter>A \<squnion> \<Sqinter>B)" by(rule sup_inf_distrib2)
```
```  2343   also have "\<dots> = \<Sqinter>{x \<squnion> b|b. b \<in> B} \<sqinter> \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
```
```  2344     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  2345   also have "\<dots> = \<Sqinter>({x\<squnion>b |b. b \<in> B} \<union> {a\<squnion>b |a b. a \<in> A \<and> b \<in> B})"
```
```  2346     (is "_ = \<Sqinter>?M")
```
```  2347     using B insert
```
```  2348     by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne])
```
```  2349   also have "?M = {a \<squnion> b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  2350     by blast
```
```  2351   finally show ?case .
```
```  2352 qed
```
```  2353
```
```  2354
```
```  2355 lemma (in Distrib_Lattice) inf_Sup1_distrib:
```
```  2356  "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (x \<sqinter> \<Squnion>A) = \<Squnion>{x \<sqinter> a|a. a \<in> A}"
```
```  2357 apply(simp add:Sup_def image_def
```
```  2358   ACIf.hom_fold1_commute[OF ACIf_sup, where h="inf x", OF inf_sup_distrib1])
```
```  2359 apply(rule arg_cong, blast)
```
```  2360 done
```
```  2361
```
```  2362
```
```  2363 lemma (in Distrib_Lattice) inf_Sup2_distrib:
```
```  2364 assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  2365 shows "(\<Squnion>A \<sqinter> \<Squnion>B) = \<Squnion>{a \<sqinter> b|a b. a \<in> A \<and> b \<in> B}"
```
```  2366 using A
```
```  2367 proof (induct rule: finite_ne_induct)
```
```  2368   case singleton thus ?case
```
```  2369     by(simp add: inf_Sup1_distrib[OF B] fold1_singleton_def[OF Sup_def])
```
```  2370 next
```
```  2371   case (insert x A)
```
```  2372   have finB: "finite {x \<sqinter> b |b. b \<in> B}"
```
```  2373     by(rule finite_surj[where f = "%b. x \<sqinter> b", OF B(1)], auto)
```
```  2374   have finAB: "finite {a \<sqinter> b |a b. a \<in> A \<and> b \<in> B}"
```
```  2375   proof -
```
```  2376     have "{a \<sqinter> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<sqinter> b})"
```
```  2377       by blast
```
```  2378     thus ?thesis by(simp add: insert(1) B(1))
```
```  2379   qed
```
```  2380   have ne: "{a \<sqinter> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  2381   have "\<Squnion>(insert x A) \<sqinter> \<Squnion>B = (x \<squnion> \<Squnion>A) \<sqinter> \<Squnion>B"
```
```  2382     using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_sup Sup_def])
```
```  2383   also have "\<dots> = (x \<sqinter> \<Squnion>B) \<squnion> (\<Squnion>A \<sqinter> \<Squnion>B)" by(rule inf_sup_distrib2)
```
```  2384   also have "\<dots> = \<Squnion>{x \<sqinter> b|b. b \<in> B} \<squnion> \<Squnion>{a \<sqinter> b|a b. a \<in> A \<and> b \<in> B}"
```
```  2385     using insert by(simp add:inf_Sup1_distrib[OF B])
```
```  2386   also have "\<dots> = \<Squnion>({x\<sqinter>b |b. b \<in> B} \<union> {a\<sqinter>b |a b. a \<in> A \<and> b \<in> B})"
```
```  2387     (is "_ = \<Squnion>?M")
```
```  2388     using B insert
```
```  2389     by(simp add:Sup_def ACIf.fold1_Un2[OF ACIf_sup finB _ finAB ne])
```
```  2390   also have "?M = {a \<sqinter> b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  2391     by blast
```
```  2392   finally show ?case .
```
```  2393 qed
```
```  2394
```
```  2395 text {*
```
```  2396   Infimum and supremum in complete lattices may also
```
```  2397   be characterized by @{const fold1}:
```
```  2398 *}
```
```  2399
```
```  2400 lemma (in complete_lattice) Inf_fold1:
```
```  2401   "finite A \<Longrightarrow>  A \<noteq> {} \<Longrightarrow> \<Sqinter>A = fold1 (op \<sqinter>) A"
```
```  2402 by (induct A set: finite)
```
```  2403    (simp_all add: Inf_insert_simp ACIf.fold1_insert_idem [OF ACIf_inf])
```
```  2404
```
```  2405 lemma (in complete_lattice) Sup_fold1:
```
```  2406   "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Squnion>A = fold1 (op \<squnion>) A"
```
```  2407 by (induct A set: finite)
```
```  2408    (simp_all add: Sup_insert_simp ACIf.fold1_insert_idem [OF ACIf_sup])
```
```  2409
```
```  2410
```
```  2411 subsubsection {* Fold1 in linear orders with @{const min} and @{const max} *}
```
```  2412
```
```  2413 text{*
```
```  2414   As an application of @{text fold1} we define minimum
```
```  2415   and maximum in (not necessarily complete!) linear orders
```
```  2416   over (non-empty) sets by means of @{text fold1}.
```
```  2417 *}
```
```  2418
```
```  2419 locale Linorder = linorder -- {* we do not pollute the @{text linorder} class *}
```
```  2420 begin
```
```  2421
```
```  2422 definition
```
```  2423   Min :: "'a set \<Rightarrow> 'a"
```
```  2424 where
```
```  2425   "Min = fold1 min"
```
```  2426
```
```  2427 definition
```
```  2428   Max :: "'a set \<Rightarrow> 'a"
```
```  2429 where
```
```  2430   "Max = fold1 max"
```
```  2431
```
```  2432 text {* recall: @{term min} and @{term max} behave like @{const inf} and @{const sup} *}
```
```  2433
```
```  2434 lemma ACIf_min: "ACIf min"
```
```  2435   by (rule lower_semilattice.ACIf_inf,
```
```  2436     rule lattice.axioms,
```
```  2437     rule distrib_lattice.axioms,
```
```  2438     rule distrib_lattice_min_max)
```
```  2439
```
```  2440 lemma ACf_min: "ACf min"
```
```  2441   by (rule lower_semilattice.ACf_inf,
```
```  2442     rule lattice.axioms,
```
```  2443     rule distrib_lattice.axioms,
```
```  2444     rule distrib_lattice_min_max)
```
```  2445
```
```  2446 lemma ACIfSL_min: "ACIfSL (op \<^loc>\<le>) (op \<^loc><) min"
```
```  2447   by (rule lower_semilattice.ACIfSL_inf,
```
```  2448     rule lattice.axioms,
```
```  2449     rule distrib_lattice.axioms,
```
```  2450     rule distrib_lattice_min_max)
```
```  2451
```
```  2452 lemma ACIfSLlin_min: "ACIfSLlin (op \<^loc>\<le>) (op \<^loc><) min"
```
```  2453   by (rule ACIfSLlin.intro,
```
```  2454     rule lower_semilattice.ACIfSL_inf,
```
```  2455     rule lattice.axioms,
```
```  2456     rule distrib_lattice.axioms,
```
```  2457     rule distrib_lattice_min_max)
```
```  2458     (unfold_locales, simp add: min_def)
```
```  2459
```
```  2460 lemma ACIf_max: "ACIf max"
```
```  2461   by (rule upper_semilattice.ACIf_sup,
```
```  2462     rule lattice.axioms,
```
```  2463     rule distrib_lattice.axioms,
```
```  2464     rule distrib_lattice_min_max)
```
```  2465
```
```  2466 lemma ACf_max: "ACf max"
```
```  2467   by (rule upper_semilattice.ACf_sup,
```
```  2468     rule lattice.axioms,
```
```  2469     rule distrib_lattice.axioms,
```
```  2470     rule distrib_lattice_min_max)
```
```  2471
```
```  2472 lemma ACIfSL_max: "ACIfSL (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x) max"
```
```  2473   by (rule upper_semilattice.ACIfSL_sup,
```
```  2474     rule lattice.axioms,
```
```  2475     rule distrib_lattice.axioms,
```
```  2476     rule distrib_lattice_min_max)
```
```  2477
```
```  2478 lemma ACIfSLlin_max: "ACIfSLlin (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x) max"
```
```  2479   by (rule ACIfSLlin.intro,
```
```  2480     rule upper_semilattice.ACIfSL_sup,
```
```  2481     rule lattice.axioms,
```
```  2482     rule distrib_lattice.axioms,
```
```  2483     rule distrib_lattice_min_max)
```
```  2484     (unfold_locales, simp add: max_def)
```
```  2485
```
```  2486 lemmas Min_singleton [simp] = fold1_singleton_def [OF Min_def]
```
```  2487 lemmas Max_singleton [simp] = fold1_singleton_def [OF Max_def]
```
```  2488 lemmas Min_insert [simp] = ACIf.fold1_insert_idem_def [OF ACIf_min Min_def]
```
```  2489 lemmas Max_insert [simp] = ACIf.fold1_insert_idem_def [OF ACIf_max Max_def]
```
```  2490
```
```  2491 lemma Min_in [simp]:
```
```  2492   shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A"
```
```  2493   using ACf.fold1_in [OF ACf_min]
```
```  2494   by (fastsimp simp: Min_def min_def)
```
```  2495
```
```  2496 lemma Max_in [simp]:
```
```  2497   shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A"
```
```  2498   using ACf.fold1_in [OF ACf_max]
```
```  2499   by (fastsimp simp: Max_def max_def)
```
```  2500
```
```  2501 lemma Min_antimono: "\<lbrakk> M \<subseteq> N; M \<noteq> {}; finite N \<rbrakk> \<Longrightarrow> Min N \<^loc>\<le> Min M"
```
```  2502   by (simp add: Min_def ACIfSLlin.fold1_antimono [OF ACIfSLlin_min])
```
```  2503
```
```  2504 lemma Max_mono: "\<lbrakk> M \<subseteq> N; M \<noteq> {}; finite N \<rbrakk> \<Longrightarrow> Max M \<^loc>\<le> Max N"
```
```  2505   by (simp add: Max_def ACIfSLlin.fold1_antimono [OF ACIfSLlin_max])
```
```  2506
```
```  2507 lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<^loc>\<le> x"
```
```  2508   by (simp add: Min_def ACIfSL.fold1_belowI [OF ACIfSL_min])
```
```  2509
```
```  2510 lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<^loc>\<le> Max A"
```
```  2511   by (simp add: Max_def ACIfSL.fold1_belowI [OF ACIfSL_max])
```
```  2512
```
```  2513 lemma Min_ge_iff [simp]:
```
```  2514   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> x \<^loc>\<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<^loc>\<le> a)"
```
```  2515   by (simp add: Min_def ACIfSL.below_fold1_iff [OF ACIfSL_min])
```
```  2516
```
```  2517 lemma Max_le_iff [simp]:
```
```  2518   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Max A \<^loc>\<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<^loc>\<le> x)"
```
```  2519   by (simp add: Max_def ACIfSL.below_fold1_iff [OF ACIfSL_max])
```
```  2520
```
```  2521 lemma Min_gr_iff [simp]:
```
```  2522   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> x \<^loc>< Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<^loc>< a)"
```
```  2523   by (simp add: Min_def ACIfSLlin.strict_below_fold1_iff [OF ACIfSLlin_min])
```
```  2524
```
```  2525 lemma Max_less_iff [simp]:
```
```  2526   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Max A \<^loc>< x \<longleftrightarrow> (\<forall>a\<in>A. a \<^loc>< x)"
```
```  2527   by (simp add: Max_def ACIfSLlin.strict_below_fold1_iff [OF ACIfSLlin_max])
```
```  2528
```
```  2529 lemma Min_le_iff:
```
```  2530   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Min A \<^loc>\<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<^loc>\<le> x)"
```
```  2531   by (simp add: Min_def ACIfSLlin.fold1_below_iff [OF ACIfSLlin_min])
```
```  2532
```
```  2533 lemma Max_ge_iff:
```
```  2534   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> x \<^loc>\<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<^loc>\<le> a)"
```
```  2535   by (simp add: Max_def ACIfSLlin.fold1_below_iff [OF ACIfSLlin_max])
```
```  2536
```
```  2537 lemma Min_less_iff:
```
```  2538   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Min A \<^loc>< x \<longleftrightarrow> (\<exists>a\<in>A. a \<^loc>< x)"
```
```  2539   by (simp add: Min_def ACIfSLlin.fold1_strict_below_iff [OF ACIfSLlin_min])
```
```  2540
```
```  2541 lemma Max_gr_iff:
```
```  2542   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> x \<^loc>< Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<^loc>< a)"
```
```  2543   by (simp add: Max_def ACIfSLlin.fold1_strict_below_iff [OF ACIfSLlin_max])
```
```  2544
```
```  2545 lemma Min_Un: "\<lbrakk>finite A; A \<noteq> {}; finite B; B \<noteq> {}\<rbrakk>
```
```  2546   \<Longrightarrow> Min (A \<union> B) = min (Min A) (Min B)"
```
```  2547   by (simp add: Min_def ACIf.fold1_Un2 [OF ACIf_min])
```
```  2548
```
```  2549 lemma Max_Un: "\<lbrakk>finite A; A \<noteq> {}; finite B; B \<noteq> {}\<rbrakk>
```
```  2550   \<Longrightarrow> Max (A \<union> B) = max (Max A) (Max B)"
```
```  2551   by (simp add: Max_def ACIf.fold1_Un2 [OF ACIf_max])
```
```  2552
```
```  2553 lemma hom_Min_commute:
```
```  2554  "(\<And>x y. h (min x y) = min (h x) (h y))
```
```  2555   \<Longrightarrow> finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> h (Min N) = Min (h ` N)"
```
```  2556   by (simp add: Min_def ACIf.hom_fold1_commute [OF ACIf_min])
```
```  2557
```
```  2558 lemma hom_Max_commute:
```
```  2559  "(\<And>x y. h (max x y) = max (h x) (h y))
```
```  2560   \<Longrightarrow> finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> h (Max N) = Max (h ` N)"
```
```  2561   by (simp add: Max_def ACIf.hom_fold1_commute [OF ACIf_max])
```
```  2562
```
```  2563 end
```
```  2564
```
```  2565 locale Linorder_ab_semigroup_add = Linorder + pordered_ab_semigroup_add
```
```  2566 begin
```
```  2567
```
```  2568 lemma add_Min_commute:
```
```  2569   fixes k
```
```  2570   shows "finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> k \<^loc>+ Min N = Min {k \<^loc>+ m | m. m \<in> N}"
```
```  2571   apply (subgoal_tac "\<And>x y. k \<^loc>+ min x y = min (k \<^loc>+ x) (k \<^loc>+ y)")
```
```  2572   using hom_Min_commute [of "(op \<^loc>+) k" N]
```
```  2573   apply simp apply (rule arg_cong [where f = Min]) apply blast
```
```  2574   apply (simp add: min_def not_le)
```
```  2575   apply (blast intro: antisym less_imp_le add_left_mono)
```
```  2576   done
```
```  2577
```
```  2578 lemma add_Max_commute:
```
```  2579   fixes k
```
```  2580   shows "finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> k \<^loc>+ Max N = Max {k \<^loc>+ m | m. m \<in> N}"
```
```  2581   apply (subgoal_tac "\<And>x y. k \<^loc>+ max x y = max (k \<^loc>+ x) (k \<^loc>+ y)")
```
```  2582   using hom_Max_commute [of "(op \<^loc>+) k" N]
```
```  2583   apply simp apply (rule arg_cong [where f = Max]) apply blast
```
```  2584   apply (simp add: max_def not_le)
```
```  2585   apply (blast intro: antisym less_imp_le add_left_mono)
```
```  2586   done
```
```  2587
```
```  2588 end
```
```  2589
```
```  2590 definition
```
```  2591   Min :: "'a set \<Rightarrow> 'a\<Colon>linorder"
```
```  2592 where
```
```  2593   "Min = fold1 min"
```
```  2594
```
```  2595 definition
```
```  2596   Max :: "'a set \<Rightarrow> 'a\<Colon>linorder"
```
```  2597 where
```
```  2598   "Max = fold1 max"
```
```  2599
```
```  2600 lemma Linorder_Min:
```
```  2601   "Linorder.Min (op \<le>) = Min"
```
```  2602 proof
```
```  2603   fix A :: "'a set"
```
```  2604   show "Linorder.Min (op \<le>) A = Min A"
```
```  2605   by (simp add: Min_def Linorder.Min_def [OF Linorder.intro, OF linorder_axioms]
```
```  2606     ord_class.min)
```
```  2607 qed
```
```  2608
```
```  2609 lemma Linorder_Max:
```
```  2610   "Linorder.Max (op \<le>) = Max"
```
```  2611 proof
```
```  2612   fix A :: "'a set"
```
```  2613   show "Linorder.Max (op \<le>) A = Max A"
```
```  2614   by (simp add: Max_def Linorder.Max_def [OF Linorder.intro, OF linorder_axioms]
```
```  2615     ord_class.max)
```
```  2616 qed
```
```  2617
```
```  2618 (*FIXME: temporary solution - doesn't work properly*)
```
```  2619 interpretation [folded ord_class.min ord_class.max, unfolded Linorder_Min Linorder_Max]:
```
```  2620   Linorder_ab_semigroup_add ["op \<le> \<Colon> 'a\<Colon>{linorder, pordered_ab_semigroup_add} \<Rightarrow> 'a \<Rightarrow> bool" "op <" "op +"]
```
```  2621   by (rule Linorder_ab_semigroup_add.intro,
```
```  2622     rule Linorder.intro, rule linorder_axioms, rule pordered_ab_semigroup_add_axioms)
```
```  2623 hide const Min Max
```
```  2624
```
```  2625 interpretation [folded ord_class.min ord_class.max, unfolded Linorder_Min Linorder_Max]:
```
```  2626   Linorder ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <"]
```
```  2627   by (rule Linorder.intro, rule linorder_axioms)
```
```  2628 declare Min_singleton [simp]
```
```  2629 declare Max_singleton [simp]
```
```  2630 declare Min_insert [simp]
```
```  2631 declare Max_insert [simp]
```
```  2632 declare Min_in [simp]
```
```  2633 declare Max_in [simp]
```
```  2634 declare Min_le [simp]
```
```  2635 declare Max_ge [simp]
```
```  2636 declare Min_ge_iff [simp]
```
```  2637 declare Max_le_iff [simp]
```
```  2638 declare Min_gr_iff [simp]
```
```  2639 declare Max_less_iff [simp]
```
```  2640 declare Max_less_iff [simp]
```
```  2641
```
```  2642
```
```  2643 subsection {* Class @{text finite} *}
```
```  2644
```
```  2645 setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}
```
```  2646 class finite (attach UNIV) = type +
```
```  2647   assumes finite: "finite UNIV"
```
```  2648 setup {* Sign.parent_path *}
```
```  2649 hide const finite
```
```  2650
```
```  2651 lemma finite_set: "finite (A::'a::finite set)"
```
```  2652   by (rule finite_subset [OF subset_UNIV finite])
```
```  2653
```
```  2654 lemma univ_unit:
```
```  2655   "UNIV = {()}" by auto
```
```  2656
```
```  2657 instance unit :: finite
```
```  2658 proof
```
```  2659   have "finite {()}" by simp
```
```  2660   also note univ_unit [symmetric]
```
```  2661   finally show "finite (UNIV :: unit set)" .
```
```  2662 qed
```
```  2663
```
```  2664 lemmas [code func] = univ_unit
```
```  2665
```
```  2666 lemma univ_bool:
```
```  2667   "UNIV = {False, True}" by auto
```
```  2668
```
```  2669 instance bool :: finite
```
```  2670 proof
```
```  2671   have "finite {False, True}" by simp
```
```  2672   also note univ_bool [symmetric]
```
```  2673   finally show "finite (UNIV :: bool set)" .
```
```  2674 qed
```
```  2675
```
```  2676 lemmas [code func] = univ_bool
```
```  2677
```
```  2678 instance * :: (finite, finite) finite
```
```  2679 proof
```
```  2680   show "finite (UNIV :: ('a \<times> 'b) set)"
```
```  2681   proof (rule finite_Prod_UNIV)
```
```  2682     show "finite (UNIV :: 'a set)" by (rule finite)
```
```  2683     show "finite (UNIV :: 'b set)" by (rule finite)
```
```  2684   qed
```
```  2685 qed
```
```  2686
```
```  2687 lemma univ_prod [code func]:
```
```  2688   "UNIV = (UNIV \<Colon> 'a\<Colon>finite set) \<times> (UNIV \<Colon> 'b\<Colon>finite set)"
```
```  2689   unfolding UNIV_Times_UNIV ..
```
```  2690
```
```  2691 instance "+" :: (finite, finite) finite
```
```  2692 proof
```
```  2693   have a: "finite (UNIV :: 'a set)" by (rule finite)
```
```  2694   have b: "finite (UNIV :: 'b set)" by (rule finite)
```
```  2695   from a b have "finite ((UNIV :: 'a set) <+> (UNIV :: 'b set))"
```
```  2696     by (rule finite_Plus)
```
```  2697   thus "finite (UNIV :: ('a + 'b) set)" by simp
```
```  2698 qed
```
```  2699
```
```  2700 lemma univ_sum [code func]:
```
```  2701   "UNIV = (UNIV \<Colon> 'a\<Colon>finite set) <+> (UNIV \<Colon> 'b\<Colon>finite set)"
```
```  2702   unfolding UNIV_Plus_UNIV ..
```
```  2703
```
```  2704 lemma insert_None_conv_UNIV: "insert None (range Some) = UNIV"
```
```  2705   by (rule set_ext, case_tac x, auto)
```
```  2706
```
```  2707 instance option :: (finite) finite
```
```  2708 proof
```
```  2709   have "finite (UNIV :: 'a set)" by (rule finite)
```
```  2710   hence "finite (insert None (Some ` (UNIV :: 'a set)))" by simp
```
```  2711   also have "insert None (Some ` (UNIV :: 'a set)) = UNIV"
```
```  2712     by (rule insert_None_conv_UNIV)
```
```  2713   finally show "finite (UNIV :: 'a option set)" .
```
```  2714 qed
```
```  2715
```
```  2716 lemma univ_option [code func]:
```
```  2717   "UNIV = insert (None \<Colon> 'a\<Colon>finite option) (image Some UNIV)"
```
```  2718   unfolding insert_None_conv_UNIV ..
```
```  2719
```
```  2720 instance set :: (finite) finite
```
```  2721 proof
```
```  2722   have "finite (UNIV :: 'a set)" by (rule finite)
```
```  2723   hence "finite (Pow (UNIV :: 'a set))"
```
```  2724     by (rule finite_Pow_iff [THEN iffD2])
```
```  2725   thus "finite (UNIV :: 'a set set)" by simp
```
```  2726 qed
```
```  2727
```
```  2728 lemma univ_set [code func]:
```
```  2729   "UNIV = Pow (UNIV \<Colon> 'a\<Colon>finite set)" unfolding Pow_UNIV ..
```
```  2730
```
```  2731 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```  2732   by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
```
```  2733
```
```  2734 instance "fun" :: (finite, finite) finite
```
```  2735 proof
```
```  2736   show "finite (UNIV :: ('a => 'b) set)"
```
```  2737   proof (rule finite_imageD)
```
```  2738     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```  2739     show "finite (range ?graph)" by (rule finite_set)
```
```  2740     show "inj ?graph" by (rule inj_graph)
```
```  2741   qed
```
```  2742 qed
```
```  2743
```
```  2744
```
```  2745 subsection {* Equality and order on functions *}
```
```  2746
```
```  2747 instance "fun" :: (finite, eq) eq ..
```
```  2748
```
```  2749 lemma eq_fun [code func]:
```
```  2750   "f = g \<longleftrightarrow> (\<forall>x\<Colon>'a\<Colon>finite \<in> UNIV. (f x \<Colon> 'b\<Colon>eq) = g x)"
```
```  2751   unfolding expand_fun_eq by auto
```
```  2752
```
```  2753 lemma order_fun [code func]:
```
```  2754   "f \<le> g \<longleftrightarrow> (\<forall>x\<Colon>'a\<Colon>finite \<in> UNIV. (f x \<Colon> 'b\<Colon>order) \<le> g x)"
```
```  2755   "f < g \<longleftrightarrow> f \<le> g \<and> (\<exists>x\<Colon>'a\<Colon>finite \<in> UNIV. (f x \<Colon> 'b\<Colon>order) < g x)"
```
```  2756   unfolding le_fun_def less_fun_def less_le
```
```  2757   by (auto simp add: expand_fun_eq)
```
```  2758
```
```  2759 end
```