src/HOL/Finite_Set.thy
 author nipkow Tue Oct 04 23:30:46 2005 +0200 (2005-10-04) changeset 17761 2c42d0a94f58 parent 17589 58eeffd73be1 child 17782 b3846df9d643 permissions -rw-r--r--
new lemmas
```     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 Power Inductive Lattice_Locales
```
```    11 begin
```
```    12
```
```    13 subsection {* Definition and basic properties *}
```
```    14
```
```    15 consts Finites :: "'a set set"
```
```    16 syntax
```
```    17   finite :: "'a set => bool"
```
```    18 translations
```
```    19   "finite A" == "A : Finites"
```
```    20
```
```    21 inductive Finites
```
```    22   intros
```
```    23     emptyI [simp, intro!]: "{} : Finites"
```
```    24     insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
```
```    25
```
```    26 axclass finite \<subseteq> type
```
```    27   finite: "finite UNIV"
```
```    28
```
```    29 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
```
```    30   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    31   shows "\<exists>a::'a. a \<notin> A"
```
```    32 proof -
```
```    33   from prems have "A \<noteq> UNIV" by blast
```
```    34   thus ?thesis by blast
```
```    35 qed
```
```    36
```
```    37 lemma finite_induct [case_names empty insert, induct set: Finites]:
```
```    38   "finite F ==>
```
```    39     P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
```
```    40   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
```
```    41 proof -
```
```    42   assume "P {}" and
```
```    43     insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
```
```    44   assume "finite F"
```
```    45   thus "P F"
```
```    46   proof induct
```
```    47     show "P {}" .
```
```    48     fix x F assume F: "finite F" and P: "P F"
```
```    49     show "P (insert x F)"
```
```    50     proof cases
```
```    51       assume "x \<in> F"
```
```    52       hence "insert x F = F" by (rule insert_absorb)
```
```    53       with P show ?thesis by (simp only:)
```
```    54     next
```
```    55       assume "x \<notin> F"
```
```    56       from F this P show ?thesis by (rule insert)
```
```    57     qed
```
```    58   qed
```
```    59 qed
```
```    60
```
```    61 lemma finite_ne_induct[case_names singleton insert, consumes 2]:
```
```    62 assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
```
```    63  \<lbrakk> \<And>x. P{x};
```
```    64    \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
```
```    65  \<Longrightarrow> P F"
```
```    66 using fin
```
```    67 proof induct
```
```    68   case empty thus ?case by simp
```
```    69 next
```
```    70   case (insert x F)
```
```    71   show ?case
```
```    72   proof cases
```
```    73     assume "F = {}" thus ?thesis using insert(4) by simp
```
```    74   next
```
```    75     assume "F \<noteq> {}" thus ?thesis using insert by blast
```
```    76   qed
```
```    77 qed
```
```    78
```
```    79 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```    80   "finite F ==> F \<subseteq> A ==>
```
```    81     P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
```
```    82     P F"
```
```    83 proof -
```
```    84   assume "P {}" and insert:
```
```    85     "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
```
```    86   assume "finite F"
```
```    87   thus "F \<subseteq> A ==> P F"
```
```    88   proof induct
```
```    89     show "P {}" .
```
```    90     fix x F assume "finite F" and "x \<notin> F"
```
```    91       and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
```
```    92     show "P (insert x F)"
```
```    93     proof (rule insert)
```
```    94       from i show "x \<in> A" by blast
```
```    95       from i have "F \<subseteq> A" by blast
```
```    96       with P show "P F" .
```
```    97     qed
```
```    98   qed
```
```    99 qed
```
```   100
```
```   101 text{* Finite sets are the images of initial segments of natural numbers: *}
```
```   102
```
```   103 lemma finite_imp_nat_seg_image_inj_on:
```
```   104   assumes fin: "finite A"
```
```   105   shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
```
```   106 using fin
```
```   107 proof induct
```
```   108   case empty
```
```   109   show ?case
```
```   110   proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp
```
```   111   qed
```
```   112 next
```
```   113   case (insert a A)
```
```   114   have notinA: "a \<notin> A" .
```
```   115   from insert.hyps obtain n f
```
```   116     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
```
```   117   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
```
```   118         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
```
```   119     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```   120   thus ?case by blast
```
```   121 qed
```
```   122
```
```   123 lemma nat_seg_image_imp_finite:
```
```   124   "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
```
```   125 proof (induct n)
```
```   126   case 0 thus ?case by simp
```
```   127 next
```
```   128   case (Suc n)
```
```   129   let ?B = "f ` {i. i < n}"
```
```   130   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```   131   show ?case
```
```   132   proof cases
```
```   133     assume "\<exists>k<n. f n = f k"
```
```   134     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   135     thus ?thesis using finB by simp
```
```   136   next
```
```   137     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   138     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   139     thus ?thesis using finB by simp
```
```   140   qed
```
```   141 qed
```
```   142
```
```   143 lemma finite_conv_nat_seg_image:
```
```   144   "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
```
```   145 by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   146
```
```   147 subsubsection{* Finiteness and set theoretic constructions *}
```
```   148
```
```   149 lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
```
```   150   -- {* The union of two finite sets is finite. *}
```
```   151   by (induct set: Finites) simp_all
```
```   152
```
```   153 lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
```
```   154   -- {* Every subset of a finite set is finite. *}
```
```   155 proof -
```
```   156   assume "finite B"
```
```   157   thus "!!A. A \<subseteq> B ==> finite A"
```
```   158   proof induct
```
```   159     case empty
```
```   160     thus ?case by simp
```
```   161   next
```
```   162     case (insert x F A)
```
```   163     have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
```
```   164     show "finite A"
```
```   165     proof cases
```
```   166       assume x: "x \<in> A"
```
```   167       with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   168       with r have "finite (A - {x})" .
```
```   169       hence "finite (insert x (A - {x}))" ..
```
```   170       also have "insert x (A - {x}) = A" by (rule insert_Diff)
```
```   171       finally show ?thesis .
```
```   172     next
```
```   173       show "A \<subseteq> F ==> ?thesis" .
```
```   174       assume "x \<notin> A"
```
```   175       with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   176     qed
```
```   177   qed
```
```   178 qed
```
```   179
```
```   180 lemma finite_Collect_subset: "finite A \<Longrightarrow> finite{x \<in> A. P x}"
```
```   181 using finite_subset[of "{x \<in> A. P x}" "A"] by blast
```
```   182
```
```   183 lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
```
```   184   by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
```
```   185
```
```   186 lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
```
```   187   -- {* The converse obviously fails. *}
```
```   188   by (blast intro: finite_subset)
```
```   189
```
```   190 lemma finite_insert [simp]: "finite (insert a A) = finite A"
```
```   191   apply (subst insert_is_Un)
```
```   192   apply (simp only: finite_Un, blast)
```
```   193   done
```
```   194
```
```   195 lemma finite_Union[simp, intro]:
```
```   196  "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
```
```   197 by (induct rule:finite_induct) simp_all
```
```   198
```
```   199 lemma finite_empty_induct:
```
```   200   "finite A ==>
```
```   201   P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
```
```   202 proof -
```
```   203   assume "finite A"
```
```   204     and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
```
```   205   have "P (A - A)"
```
```   206   proof -
```
```   207     fix c b :: "'a set"
```
```   208     presume c: "finite c" and b: "finite b"
```
```   209       and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
```
```   210     from c show "c \<subseteq> b ==> P (b - c)"
```
```   211     proof induct
```
```   212       case empty
```
```   213       from P1 show ?case by simp
```
```   214     next
```
```   215       case (insert x F)
```
```   216       have "P (b - F - {x})"
```
```   217       proof (rule P2)
```
```   218         from _ b show "finite (b - F)" by (rule finite_subset) blast
```
```   219         from insert show "x \<in> b - F" by simp
```
```   220         from insert show "P (b - F)" by simp
```
```   221       qed
```
```   222       also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
```
```   223       finally show ?case .
```
```   224     qed
```
```   225   next
```
```   226     show "A \<subseteq> A" ..
```
```   227   qed
```
```   228   thus "P {}" by simp
```
```   229 qed
```
```   230
```
```   231 lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
```
```   232   by (rule Diff_subset [THEN finite_subset])
```
```   233
```
```   234 lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
```
```   235   apply (subst Diff_insert)
```
```   236   apply (case_tac "a : A - B")
```
```   237    apply (rule finite_insert [symmetric, THEN trans])
```
```   238    apply (subst insert_Diff, simp_all)
```
```   239   done
```
```   240
```
```   241
```
```   242 text {* Image and Inverse Image over Finite Sets *}
```
```   243
```
```   244 lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
```
```   245   -- {* The image of a finite set is finite. *}
```
```   246   by (induct set: Finites) simp_all
```
```   247
```
```   248 lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
```
```   249   apply (frule finite_imageI)
```
```   250   apply (erule finite_subset, assumption)
```
```   251   done
```
```   252
```
```   253 lemma finite_range_imageI:
```
```   254     "finite (range g) ==> finite (range (%x. f (g x)))"
```
```   255   apply (drule finite_imageI, simp)
```
```   256   done
```
```   257
```
```   258 lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
```
```   259 proof -
```
```   260   have aux: "!!A. finite (A - {}) = finite A" by simp
```
```   261   fix B :: "'a set"
```
```   262   assume "finite B"
```
```   263   thus "!!A. f`A = B ==> inj_on f A ==> finite A"
```
```   264     apply induct
```
```   265      apply simp
```
```   266     apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
```
```   267      apply clarify
```
```   268      apply (simp (no_asm_use) add: inj_on_def)
```
```   269      apply (blast dest!: aux [THEN iffD1], atomize)
```
```   270     apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
```
```   271     apply (frule subsetD [OF equalityD2 insertI1], clarify)
```
```   272     apply (rule_tac x = xa in bexI)
```
```   273      apply (simp_all add: inj_on_image_set_diff)
```
```   274     done
```
```   275 qed (rule refl)
```
```   276
```
```   277
```
```   278 lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
```
```   279   -- {* The inverse image of a singleton under an injective function
```
```   280          is included in a singleton. *}
```
```   281   apply (auto simp add: inj_on_def)
```
```   282   apply (blast intro: the_equality [symmetric])
```
```   283   done
```
```   284
```
```   285 lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
```
```   286   -- {* The inverse image of a finite set under an injective function
```
```   287          is finite. *}
```
```   288   apply (induct set: Finites, simp_all)
```
```   289   apply (subst vimage_insert)
```
```   290   apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
```
```   291   done
```
```   292
```
```   293
```
```   294 text {* The finite UNION of finite sets *}
```
```   295
```
```   296 lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
```
```   297   by (induct set: Finites) simp_all
```
```   298
```
```   299 text {*
```
```   300   Strengthen RHS to
```
```   301   @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
```
```   302
```
```   303   We'd need to prove
```
```   304   @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
```
```   305   by induction. *}
```
```   306
```
```   307 lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
```
```   308   by (blast intro: finite_UN_I finite_subset)
```
```   309
```
```   310
```
```   311 lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
```
```   312 by (simp add: Plus_def)
```
```   313
```
```   314 text {* Sigma of finite sets *}
```
```   315
```
```   316 lemma finite_SigmaI [simp]:
```
```   317     "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   318   by (unfold Sigma_def) (blast intro!: finite_UN_I)
```
```   319
```
```   320 lemma finite_cartesian_product: "[| finite A; finite B |] ==>
```
```   321     finite (A <*> B)"
```
```   322   by (rule finite_SigmaI)
```
```   323
```
```   324 lemma finite_Prod_UNIV:
```
```   325     "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
```
```   326   apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
```
```   327    apply (erule ssubst)
```
```   328    apply (erule finite_SigmaI, auto)
```
```   329   done
```
```   330
```
```   331 lemma finite_cartesian_productD1:
```
```   332      "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
```
```   333 apply (auto simp add: finite_conv_nat_seg_image)
```
```   334 apply (drule_tac x=n in spec)
```
```   335 apply (drule_tac x="fst o f" in spec)
```
```   336 apply (auto simp add: o_def)
```
```   337  prefer 2 apply (force dest!: equalityD2)
```
```   338 apply (drule equalityD1)
```
```   339 apply (rename_tac y x)
```
```   340 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   341  prefer 2 apply force
```
```   342 apply clarify
```
```   343 apply (rule_tac x=k in image_eqI, auto)
```
```   344 done
```
```   345
```
```   346 lemma finite_cartesian_productD2:
```
```   347      "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
```
```   348 apply (auto simp add: finite_conv_nat_seg_image)
```
```   349 apply (drule_tac x=n in spec)
```
```   350 apply (drule_tac x="snd o f" in spec)
```
```   351 apply (auto simp add: o_def)
```
```   352  prefer 2 apply (force dest!: equalityD2)
```
```   353 apply (drule equalityD1)
```
```   354 apply (rename_tac x y)
```
```   355 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   356  prefer 2 apply force
```
```   357 apply clarify
```
```   358 apply (rule_tac x=k in image_eqI, auto)
```
```   359 done
```
```   360
```
```   361
```
```   362 text {* The powerset of a finite set *}
```
```   363
```
```   364 lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
```
```   365 proof
```
```   366   assume "finite (Pow A)"
```
```   367   with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
```
```   368   thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   369 next
```
```   370   assume "finite A"
```
```   371   thus "finite (Pow A)"
```
```   372     by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
```
```   373 qed
```
```   374
```
```   375
```
```   376 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   377 by(blast intro: finite_subset[OF subset_Pow_Union])
```
```   378
```
```   379
```
```   380 lemma finite_converse [iff]: "finite (r^-1) = finite r"
```
```   381   apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
```
```   382    apply simp
```
```   383    apply (rule iffI)
```
```   384     apply (erule finite_imageD [unfolded inj_on_def])
```
```   385     apply (simp split add: split_split)
```
```   386    apply (erule finite_imageI)
```
```   387   apply (simp add: converse_def image_def, auto)
```
```   388   apply (rule bexI)
```
```   389    prefer 2 apply assumption
```
```   390   apply simp
```
```   391   done
```
```   392
```
```   393
```
```   394 text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
```
```   395 Ehmety) *}
```
```   396
```
```   397 lemma finite_Field: "finite r ==> finite (Field r)"
```
```   398   -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
```
```   399   apply (induct set: Finites)
```
```   400    apply (auto simp add: Field_def Domain_insert Range_insert)
```
```   401   done
```
```   402
```
```   403 lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
```
```   404   apply clarify
```
```   405   apply (erule trancl_induct)
```
```   406    apply (auto simp add: Field_def)
```
```   407   done
```
```   408
```
```   409 lemma finite_trancl: "finite (r^+) = finite r"
```
```   410   apply auto
```
```   411    prefer 2
```
```   412    apply (rule trancl_subset_Field2 [THEN finite_subset])
```
```   413    apply (rule finite_SigmaI)
```
```   414     prefer 3
```
```   415     apply (blast intro: r_into_trancl' finite_subset)
```
```   416    apply (auto simp add: finite_Field)
```
```   417   done
```
```   418
```
```   419
```
```   420 subsection {* A fold functional for finite sets *}
```
```   421
```
```   422 text {* The intended behaviour is
```
```   423 @{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
```
```   424 if @{text f} is associative-commutative. For an application of @{text fold}
```
```   425 se the definitions of sums and products over finite sets.
```
```   426 *}
```
```   427
```
```   428 consts
```
```   429   foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
```
```   430
```
```   431 inductive "foldSet f g z"
```
```   432 intros
```
```   433 emptyI [intro]: "({}, z) : foldSet f g z"
```
```   434 insertI [intro]:
```
```   435      "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
```
```   436       \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
```
```   437
```
```   438 inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
```
```   439
```
```   440 constdefs
```
```   441   fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
```
```   442   "fold f g z A == THE x. (A, x) : foldSet f g z"
```
```   443
```
```   444 text{*A tempting alternative for the definiens is
```
```   445 @{term "if finite A then THE x. (A, x) : foldSet f g e else e"}.
```
```   446 It allows the removal of finiteness assumptions from the theorems
```
```   447 @{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
```
```   448 The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
```
```   449
```
```   450
```
```   451 lemma Diff1_foldSet:
```
```   452   "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
```
```   453 by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
```
```   454
```
```   455 lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
```
```   456   by (induct set: foldSet) auto
```
```   457
```
```   458 lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
```
```   459   by (induct set: Finites) auto
```
```   460
```
```   461
```
```   462 subsubsection {* Commutative monoids *}
```
```   463
```
```   464 locale ACf =
```
```   465   fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
```
```   466   assumes commute: "x \<cdot> y = y \<cdot> x"
```
```   467     and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
```
```   468
```
```   469 locale ACe = ACf +
```
```   470   fixes e :: 'a
```
```   471   assumes ident [simp]: "x \<cdot> e = x"
```
```   472
```
```   473 locale ACIf = ACf +
```
```   474   assumes idem: "x \<cdot> x = x"
```
```   475
```
```   476 lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```   477 proof -
```
```   478   have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
```
```   479   also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
```
```   480   also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
```
```   481   finally show ?thesis .
```
```   482 qed
```
```   483
```
```   484 lemmas (in ACf) AC = assoc commute left_commute
```
```   485
```
```   486 lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
```
```   487 proof -
```
```   488   have "x \<cdot> e = x" by (rule ident)
```
```   489   thus ?thesis by (subst commute)
```
```   490 qed
```
```   491
```
```   492 lemma (in ACIf) idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
```
```   493 proof -
```
```   494   have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
```
```   495   also have "\<dots> = x \<cdot> y" by(simp add:idem)
```
```   496   finally show ?thesis .
```
```   497 qed
```
```   498
```
```   499 lemmas (in ACIf) ACI = AC idem idem2
```
```   500
```
```   501 text{* Interpretation of locales: *}
```
```   502
```
```   503 interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"]
```
```   504 by(auto intro: ACf.intro ACe_axioms.intro add_assoc add_commute)
```
```   505
```
```   506 interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"]
```
```   507   apply -
```
```   508    apply (fast intro: ACf.intro mult_assoc mult_commute)
```
```   509   apply (fastsimp intro: ACe_axioms.intro mult_assoc mult_commute)
```
```   510   done
```
```   511
```
```   512
```
```   513 subsubsection{*From @{term foldSet} to @{term fold}*}
```
```   514
```
```   515 lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
```
```   516 by (auto simp add: less_Suc_eq)
```
```   517
```
```   518 lemma insert_image_inj_on_eq:
```
```   519      "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A;
```
```   520         inj_on h {i. i < Suc m}|]
```
```   521       ==> A = h ` {i. i < m}"
```
```   522 apply (auto simp add: image_less_Suc inj_on_def)
```
```   523 apply (blast intro: less_trans)
```
```   524 done
```
```   525
```
```   526 lemma insert_inj_onE:
```
```   527   assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A"
```
```   528       and inj_on: "inj_on h {i::nat. i<n}"
```
```   529   shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
```
```   530 proof (cases n)
```
```   531   case 0 thus ?thesis using aA by auto
```
```   532 next
```
```   533   case (Suc m)
```
```   534   have nSuc: "n = Suc m" .
```
```   535   have mlessn: "m<n" by (simp add: nSuc)
```
```   536   from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
```
```   537   let ?hm = "swap k m h"
```
```   538   have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn
```
```   539     by (simp add: inj_on_swap_iff inj_on)
```
```   540   show ?thesis
```
```   541   proof (intro exI conjI)
```
```   542     show "inj_on ?hm {i. i < m}" using inj_hm
```
```   543       by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
```
```   544     show "m<n" by (rule mlessn)
```
```   545     show "A = ?hm ` {i. i < m}"
```
```   546     proof (rule insert_image_inj_on_eq)
```
```   547       show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
```
```   548       show "?hm m \<notin> A" by (simp add: swap_def hkeq anot)
```
```   549       show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
```
```   550 	using aA hkeq nSuc klessn
```
```   551 	by (auto simp add: swap_def image_less_Suc fun_upd_image
```
```   552 			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
```
```   553     qed
```
```   554   qed
```
```   555 qed
```
```   556
```
```   557 lemma (in ACf) foldSet_determ_aux:
```
```   558   "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n};
```
```   559                 (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
```
```   560    \<Longrightarrow> x' = x"
```
```   561 proof (induct n rule: less_induct)
```
```   562   case (less n)
```
```   563     have IH: "!!m h A x x'.
```
```   564                \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m};
```
```   565                 (A,x) \<in> foldSet f g z; (A, x') \<in> foldSet f g z\<rbrakk> \<Longrightarrow> x' = x" .
```
```   566     have Afoldx: "(A,x) \<in> foldSet f g z" and Afoldx': "(A,x') \<in> foldSet f g z"
```
```   567      and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" .
```
```   568     show ?case
```
```   569     proof (rule foldSet.cases [OF Afoldx])
```
```   570       assume "(A, x) = ({}, z)"
```
```   571       with Afoldx' show "x' = x" by blast
```
```   572     next
```
```   573       fix B b u
```
```   574       assume "(A,x) = (insert b B, g b \<cdot> u)" and notinB: "b \<notin> B"
```
```   575          and Bu: "(B,u) \<in> foldSet f g z"
```
```   576       hence AbB: "A = insert b B" and x: "x = g b \<cdot> u" by auto
```
```   577       show "x'=x"
```
```   578       proof (rule foldSet.cases [OF Afoldx'])
```
```   579         assume "(A, x') = ({}, z)"
```
```   580         with AbB show "x' = x" by blast
```
```   581       next
```
```   582 	fix C c v
```
```   583 	assume "(A,x') = (insert c C, g c \<cdot> v)" and notinC: "c \<notin> C"
```
```   584 	   and Cv: "(C,v) \<in> foldSet f g z"
```
```   585 	hence AcC: "A = insert c C" and x': "x' = g c \<cdot> v" by auto
```
```   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: "(?D,d) \<in> foldSet f g z"
```
```   610 	    using finite_imp_foldSet by iprover
```
```   611 	  moreover have cinB: "c \<in> B" using B by auto
```
```   612 	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
```
```   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 "(C, g b \<cdot> d) \<in> foldSet f g z" 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   "(A,x) : foldSet f g z ==> (A,y) : foldSet f g z ==> 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: "(A, y) : foldSet f g z ==> 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     ((insert x A, v) : foldSet f g z) =
```
```   643     (EX y. (A, y) : foldSet f g z & 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: Finites, simp)
```
```   706   apply (simp add: left_commute [of x])
```
```   707   done
```
```   708
```
```   709 lemma (in ACf) fold_nest_Un_Int:
```
```   710   "finite A ==> finite B
```
```   711     ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
```
```   712   apply (induct set: Finites, simp)
```
```   713   apply (simp add: fold_commute Int_insert_left insert_absorb)
```
```   714   done
```
```   715
```
```   716 lemma (in ACf) fold_nest_Un_disjoint:
```
```   717   "finite A ==> finite B ==> A Int B = {}
```
```   718     ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
```
```   719   by (simp add: fold_nest_Un_Int)
```
```   720
```
```   721 lemma (in ACf) fold_reindex:
```
```   722 assumes fin: "finite A"
```
```   723 shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
```
```   724 using fin apply induct
```
```   725  apply simp
```
```   726 apply simp
```
```   727 done
```
```   728
```
```   729 lemma (in ACe) fold_Un_Int:
```
```   730   "finite A ==> finite B ==>
```
```   731     fold f g e A \<cdot> fold f g e B =
```
```   732     fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
```
```   733   apply (induct set: Finites, simp)
```
```   734   apply (simp add: AC insert_absorb Int_insert_left)
```
```   735   done
```
```   736
```
```   737 corollary (in ACe) fold_Un_disjoint:
```
```   738   "finite A ==> finite B ==> A Int B = {} ==>
```
```   739     fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
```
```   740   by (simp add: fold_Un_Int)
```
```   741
```
```   742 lemma (in ACe) fold_UN_disjoint:
```
```   743   "\<lbrakk> finite I; ALL i:I. finite (A i);
```
```   744      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
```
```   745    \<Longrightarrow> fold f g e (UNION I A) =
```
```   746        fold f (%i. fold f g e (A i)) e I"
```
```   747   apply (induct set: Finites, simp, atomize)
```
```   748   apply (subgoal_tac "ALL i:F. x \<noteq> i")
```
```   749    prefer 2 apply blast
```
```   750   apply (subgoal_tac "A x Int UNION F A = {}")
```
```   751    prefer 2 apply blast
```
```   752   apply (simp add: fold_Un_disjoint)
```
```   753   done
```
```   754
```
```   755 text{*Fusion theorem, as described in
```
```   756 Graham Hutton's paper,
```
```   757 A Tutorial on the Universality and Expressiveness of Fold,
```
```   758 JFP 9:4 (355-372), 1999.*}
```
```   759 lemma (in ACf) fold_fusion:
```
```   760       includes ACf g
```
```   761       shows
```
```   762 	"finite A ==>
```
```   763 	 (!!x y. h (g x y) = f x (h y)) ==>
```
```   764          h (fold g j w A) = fold f j (h w) A"
```
```   765   by (induct set: Finites, simp_all)
```
```   766
```
```   767 lemma (in ACf) fold_cong:
```
```   768   "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
```
```   769   apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
```
```   770    apply simp
```
```   771   apply (erule finite_induct, simp)
```
```   772   apply (simp add: subset_insert_iff, clarify)
```
```   773   apply (subgoal_tac "finite C")
```
```   774    prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
```
```   775   apply (subgoal_tac "C = insert x (C - {x})")
```
```   776    prefer 2 apply blast
```
```   777   apply (erule ssubst)
```
```   778   apply (drule spec)
```
```   779   apply (erule (1) notE impE)
```
```   780   apply (simp add: Ball_def del: insert_Diff_single)
```
```   781   done
```
```   782
```
```   783 lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   784   fold f (%x. fold f (g x) e (B x)) e A =
```
```   785   fold f (split g) e (SIGMA x:A. B x)"
```
```   786 apply (subst Sigma_def)
```
```   787 apply (subst fold_UN_disjoint, assumption, simp)
```
```   788  apply blast
```
```   789 apply (erule fold_cong)
```
```   790 apply (subst fold_UN_disjoint, simp, simp)
```
```   791  apply blast
```
```   792 apply simp
```
```   793 done
```
```   794
```
```   795 lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
```
```   796    fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
```
```   797 apply (erule finite_induct, simp)
```
```   798 apply (simp add:AC)
```
```   799 done
```
```   800
```
```   801
```
```   802 subsection {* Generalized summation over a set *}
```
```   803
```
```   804 constdefs
```
```   805   setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
```
```   806   "setsum f A == if finite A then fold (op +) f 0 A else 0"
```
```   807
```
```   808 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```   809 written @{text"\<Sum>x\<in>A. e"}. *}
```
```   810
```
```   811 syntax
```
```   812   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```   813 syntax (xsymbols)
```
```   814   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   815 syntax (HTML output)
```
```   816   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   817
```
```   818 translations -- {* Beware of argument permutation! *}
```
```   819   "SUM i:A. b" == "setsum (%i. b) A"
```
```   820   "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
```
```   821
```
```   822 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```   823  @{text"\<Sum>x|P. e"}. *}
```
```   824
```
```   825 syntax
```
```   826   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   827 syntax (xsymbols)
```
```   828   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   829 syntax (HTML output)
```
```   830   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   831
```
```   832 translations
```
```   833   "SUM x|P. t" => "setsum (%x. t) {x. P}"
```
```   834   "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
```
```   835
```
```   836 text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
```
```   837
```
```   838 syntax
```
```   839   "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
```
```   840
```
```   841 parse_translation {*
```
```   842   let
```
```   843     fun Setsum_tr [A] = Syntax.const "setsum" \$ Abs ("", dummyT, Bound 0) \$ A
```
```   844   in [("_Setsum", Setsum_tr)] end;
```
```   845 *}
```
```   846
```
```   847 print_translation {*
```
```   848 let
```
```   849   fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" \$ A
```
```   850     | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) \$ Abs(y,Ty,P)] =
```
```   851        if x<>y then raise Match
```
```   852        else let val x' = Syntax.mark_bound x
```
```   853                 val t' = subst_bound(x',t)
```
```   854                 val P' = subst_bound(x',P)
```
```   855             in Syntax.const "_qsetsum" \$ Syntax.mark_bound x \$ P' \$ t' end
```
```   856 in
```
```   857 [("setsum", setsum_tr')]
```
```   858 end
```
```   859 *}
```
```   860
```
```   861 lemma setsum_empty [simp]: "setsum f {} = 0"
```
```   862   by (simp add: setsum_def)
```
```   863
```
```   864 lemma setsum_insert [simp]:
```
```   865     "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   866   by (simp add: setsum_def)
```
```   867
```
```   868 lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
```
```   869   by (simp add: setsum_def)
```
```   870
```
```   871 lemma setsum_reindex:
```
```   872      "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   873 by(auto simp add: setsum_def AC_add.fold_reindex dest!:finite_imageD)
```
```   874
```
```   875 lemma setsum_reindex_id:
```
```   876      "inj_on f B ==> setsum f B = setsum id (f ` B)"
```
```   877 by (auto simp add: setsum_reindex)
```
```   878
```
```   879 lemma setsum_cong:
```
```   880   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   881 by(fastsimp simp: setsum_def intro: AC_add.fold_cong)
```
```   882
```
```   883 lemma strong_setsum_cong[cong]:
```
```   884   "A = B ==> (!!x. x:B =simp=> f x = g x)
```
```   885    ==> setsum (%x. f x) A = setsum (%x. g x) B"
```
```   886 by(fastsimp simp: simp_implies_def setsum_def intro: AC_add.fold_cong)
```
```   887
```
```   888 lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
```
```   889   by (rule setsum_cong[OF refl], auto);
```
```   890
```
```   891 lemma setsum_reindex_cong:
```
```   892      "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
```
```   893       ==> setsum h B = setsum g A"
```
```   894   by (simp add: setsum_reindex cong: setsum_cong)
```
```   895
```
```   896 lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
```
```   897 apply (clarsimp simp: setsum_def)
```
```   898 apply (erule finite_induct, auto)
```
```   899 done
```
```   900
```
```   901 lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
```
```   902 by(simp add:setsum_cong)
```
```   903
```
```   904 lemma setsum_Un_Int: "finite A ==> finite B ==>
```
```   905   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   906   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   907 by(simp add: setsum_def AC_add.fold_Un_Int [symmetric])
```
```   908
```
```   909 lemma setsum_Un_disjoint: "finite A ==> finite B
```
```   910   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   911 by (subst setsum_Un_Int [symmetric], auto)
```
```   912
```
```   913 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   914   the lhs need not be, since UNION I A could still be finite.*)
```
```   915 lemma setsum_UN_disjoint:
```
```   916     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```   917         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```   918       setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   919 by(simp add: setsum_def AC_add.fold_UN_disjoint cong: setsum_cong)
```
```   920
```
```   921 text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
```
```   922 directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
```
```   923 lemma setsum_Union_disjoint:
```
```   924   "[| (ALL A:C. finite A);
```
```   925       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```   926    ==> setsum f (Union C) = setsum (setsum f) C"
```
```   927 apply (cases "finite C")
```
```   928  prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
```
```   929   apply (frule setsum_UN_disjoint [of C id f])
```
```   930  apply (unfold Union_def id_def, assumption+)
```
```   931 done
```
```   932
```
```   933 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```   934   the rhs need not be, since SIGMA A B could still be finite.*)
```
```   935 lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   936     (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```   937 by(simp add:setsum_def AC_add.fold_Sigma split_def cong:setsum_cong)
```
```   938
```
```   939 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```   940 lemma setsum_cartesian_product:
```
```   941    "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
```
```   942 apply (cases "finite A")
```
```   943  apply (cases "finite B")
```
```   944   apply (simp add: setsum_Sigma)
```
```   945  apply (cases "A={}", simp)
```
```   946  apply (simp)
```
```   947 apply (auto simp add: setsum_def
```
```   948             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   949 done
```
```   950
```
```   951 lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```   952 by(simp add:setsum_def AC_add.fold_distrib)
```
```   953
```
```   954
```
```   955 subsubsection {* Properties in more restricted classes of structures *}
```
```   956
```
```   957 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   958   apply (case_tac "finite A")
```
```   959    prefer 2 apply (simp add: setsum_def)
```
```   960   apply (erule rev_mp)
```
```   961   apply (erule finite_induct, auto)
```
```   962   done
```
```   963
```
```   964 lemma setsum_eq_0_iff [simp]:
```
```   965     "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   966   by (induct set: Finites) auto
```
```   967
```
```   968 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   969     (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   970   -- {* For the natural numbers, we have subtraction. *}
```
```   971   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
```
```   972
```
```   973 lemma setsum_Un: "finite A ==> finite B ==>
```
```   974     (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   975       setsum f A + setsum f B - setsum f (A Int B)"
```
```   976   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
```
```   977
```
```   978 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   979     (if a:A then setsum f A - f a else setsum f A)"
```
```   980   apply (case_tac "finite A")
```
```   981    prefer 2 apply (simp add: setsum_def)
```
```   982   apply (erule finite_induct)
```
```   983    apply (auto simp add: insert_Diff_if)
```
```   984   apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   985   done
```
```   986
```
```   987 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   988   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   989   (if a:A then setsum f A - f a else setsum f A)"
```
```   990   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   991
```
```   992 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)"
```
```   993   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))"])
```
```   994   apply (auto simp add: insert_Diff_if add_ac)
```
```   995   done
```
```   996
```
```   997 (* By Jeremy Siek: *)
```
```   998
```
```   999 lemma setsum_diff_nat:
```
```  1000   assumes finB: "finite B"
```
```  1001   shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```  1002 using finB
```
```  1003 proof (induct)
```
```  1004   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```  1005 next
```
```  1006   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```  1007     and xFinA: "insert x F \<subseteq> A"
```
```  1008     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```  1009   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```  1010   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```  1011     by (simp add: setsum_diff1_nat)
```
```  1012   from xFinA have "F \<subseteq> A" by simp
```
```  1013   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```  1014   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```  1015     by simp
```
```  1016   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```  1017   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```  1018     by simp
```
```  1019   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```  1020   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```  1021     by simp
```
```  1022   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```  1023 qed
```
```  1024
```
```  1025 lemma setsum_diff:
```
```  1026   assumes le: "finite A" "B \<subseteq> A"
```
```  1027   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```  1028 proof -
```
```  1029   from le have finiteB: "finite B" using finite_subset by auto
```
```  1030   show ?thesis using finiteB le
```
```  1031     proof (induct)
```
```  1032       case empty
```
```  1033       thus ?case by auto
```
```  1034     next
```
```  1035       case (insert x F)
```
```  1036       thus ?case using le finiteB
```
```  1037 	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```  1038     qed
```
```  1039   qed
```
```  1040
```
```  1041 lemma setsum_mono:
```
```  1042   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
```
```  1043   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```  1044 proof (cases "finite K")
```
```  1045   case True
```
```  1046   thus ?thesis using le
```
```  1047   proof (induct)
```
```  1048     case empty
```
```  1049     thus ?case by simp
```
```  1050   next
```
```  1051     case insert
```
```  1052     thus ?case using add_mono
```
```  1053       by force
```
```  1054   qed
```
```  1055 next
```
```  1056   case False
```
```  1057   thus ?thesis
```
```  1058     by (simp add: setsum_def)
```
```  1059 qed
```
```  1060
```
```  1061 lemma setsum_strict_mono:
```
```  1062 fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```  1063 assumes fin_ne: "finite A"  "A \<noteq> {}"
```
```  1064 shows "(!!x. x:A \<Longrightarrow> f x < g x) \<Longrightarrow> setsum f A < setsum g A"
```
```  1065 using fin_ne
```
```  1066 proof (induct rule: finite_ne_induct)
```
```  1067   case singleton thus ?case by simp
```
```  1068 next
```
```  1069   case insert thus ?case by (auto simp: add_strict_mono)
```
```  1070 qed
```
```  1071
```
```  1072 lemma setsum_negf:
```
```  1073  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```  1074 proof (cases "finite A")
```
```  1075   case True thus ?thesis by (induct set: Finites, auto)
```
```  1076 next
```
```  1077   case False thus ?thesis by (simp add: setsum_def)
```
```  1078 qed
```
```  1079
```
```  1080 lemma setsum_subtractf:
```
```  1081  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```  1082   setsum f A - setsum g A"
```
```  1083 proof (cases "finite A")
```
```  1084   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
```
```  1085 next
```
```  1086   case False thus ?thesis by (simp add: setsum_def)
```
```  1087 qed
```
```  1088
```
```  1089 lemma setsum_nonneg:
```
```  1090 assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```  1091 shows "0 \<le> setsum f A"
```
```  1092 proof (cases "finite A")
```
```  1093   case True thus ?thesis using nn
```
```  1094   apply (induct set: Finites, auto)
```
```  1095   apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
```
```  1096   apply (blast intro: add_mono)
```
```  1097   done
```
```  1098 next
```
```  1099   case False thus ?thesis by (simp add: setsum_def)
```
```  1100 qed
```
```  1101
```
```  1102 lemma setsum_nonpos:
```
```  1103 assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
```
```  1104 shows "setsum f A \<le> 0"
```
```  1105 proof (cases "finite A")
```
```  1106   case True thus ?thesis using np
```
```  1107   apply (induct set: Finites, auto)
```
```  1108   apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
```
```  1109   apply (blast intro: add_mono)
```
```  1110   done
```
```  1111 next
```
```  1112   case False thus ?thesis by (simp add: setsum_def)
```
```  1113 qed
```
```  1114
```
```  1115 lemma setsum_mono2:
```
```  1116 fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
```
```  1117 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```  1118 shows "setsum f A \<le> setsum f B"
```
```  1119 proof -
```
```  1120   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```  1121     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```  1122   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```  1123     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```  1124   also have "A \<union> (B-A) = B" using sub by blast
```
```  1125   finally show ?thesis .
```
```  1126 qed
```
```  1127
```
```  1128 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```  1129     ALL x: B - A.
```
```  1130       0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
```
```  1131         setsum f A <= setsum f B"
```
```  1132   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```  1133   apply (erule ssubst)
```
```  1134   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```  1135   apply simp
```
```  1136   apply (rule add_left_mono)
```
```  1137   apply (erule setsum_nonneg)
```
```  1138   apply (subst setsum_Un_disjoint [THEN sym])
```
```  1139   apply (erule finite_subset, assumption)
```
```  1140   apply (rule finite_subset)
```
```  1141   prefer 2
```
```  1142   apply assumption
```
```  1143   apply auto
```
```  1144   apply (rule setsum_cong)
```
```  1145   apply auto
```
```  1146 done
```
```  1147
```
```  1148 (* FIXME: this is distributitivty, name as such! *)
```
```  1149 (* suggested name: setsum_right_distrib (CB) *)
```
```  1150
```
```  1151 lemma setsum_mult:
```
```  1152   fixes f :: "'a => ('b::semiring_0_cancel)"
```
```  1153   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```  1154 proof (cases "finite A")
```
```  1155   case True
```
```  1156   thus ?thesis
```
```  1157   proof (induct)
```
```  1158     case empty thus ?case by simp
```
```  1159   next
```
```  1160     case (insert x A) thus ?case by (simp add: right_distrib)
```
```  1161   qed
```
```  1162 next
```
```  1163   case False thus ?thesis by (simp add: setsum_def)
```
```  1164 qed
```
```  1165
```
```  1166 lemma setsum_left_distrib:
```
```  1167   "setsum f A * (r::'a::semiring_0_cancel) = (\<Sum>n\<in>A. f n * r)"
```
```  1168 proof (cases "finite A")
```
```  1169   case True
```
```  1170   then show ?thesis
```
```  1171   proof induct
```
```  1172     case empty thus ?case by simp
```
```  1173   next
```
```  1174     case (insert x A) thus ?case by (simp add: left_distrib)
```
```  1175   qed
```
```  1176 next
```
```  1177   case False thus ?thesis by (simp add: setsum_def)
```
```  1178 qed
```
```  1179
```
```  1180 lemma setsum_divide_distrib:
```
```  1181   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```  1182 proof (cases "finite A")
```
```  1183   case True
```
```  1184   then show ?thesis
```
```  1185   proof induct
```
```  1186     case empty thus ?case by simp
```
```  1187   next
```
```  1188     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```  1189   qed
```
```  1190 next
```
```  1191   case False thus ?thesis by (simp add: setsum_def)
```
```  1192 qed
```
```  1193
```
```  1194 lemma setsum_abs[iff]:
```
```  1195   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1196   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```  1197 proof (cases "finite A")
```
```  1198   case True
```
```  1199   thus ?thesis
```
```  1200   proof (induct)
```
```  1201     case empty thus ?case by simp
```
```  1202   next
```
```  1203     case (insert x A)
```
```  1204     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```  1205   qed
```
```  1206 next
```
```  1207   case False thus ?thesis by (simp add: setsum_def)
```
```  1208 qed
```
```  1209
```
```  1210 lemma setsum_abs_ge_zero[iff]:
```
```  1211   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1212   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```  1213 proof (cases "finite A")
```
```  1214   case True
```
```  1215   thus ?thesis
```
```  1216   proof (induct)
```
```  1217     case empty thus ?case by simp
```
```  1218   next
```
```  1219     case (insert x A) thus ?case by (auto intro: order_trans)
```
```  1220   qed
```
```  1221 next
```
```  1222   case False thus ?thesis by (simp add: setsum_def)
```
```  1223 qed
```
```  1224
```
```  1225 lemma abs_setsum_abs[simp]:
```
```  1226   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1227   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```  1228 proof (cases "finite A")
```
```  1229   case True
```
```  1230   thus ?thesis
```
```  1231   proof (induct)
```
```  1232     case empty thus ?case by simp
```
```  1233   next
```
```  1234     case (insert a A)
```
```  1235     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
```
```  1236     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```  1237     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```  1238       by (simp del: abs_of_nonneg)
```
```  1239     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```  1240     finally show ?case .
```
```  1241   qed
```
```  1242 next
```
```  1243   case False thus ?thesis by (simp add: setsum_def)
```
```  1244 qed
```
```  1245
```
```  1246
```
```  1247 text {* Commuting outer and inner summation *}
```
```  1248
```
```  1249 lemma swap_inj_on:
```
```  1250   "inj_on (%(i, j). (j, i)) (A \<times> B)"
```
```  1251   by (unfold inj_on_def) fast
```
```  1252
```
```  1253 lemma swap_product:
```
```  1254   "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
```
```  1255   by (simp add: split_def image_def) blast
```
```  1256
```
```  1257 lemma setsum_commute:
```
```  1258   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
```
```  1259 proof (simp add: setsum_cartesian_product)
```
```  1260   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
```
```  1261     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
```
```  1262     (is "?s = _")
```
```  1263     apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
```
```  1264     apply (simp add: split_def)
```
```  1265     done
```
```  1266   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
```
```  1267     (is "_ = ?t")
```
```  1268     apply (simp add: swap_product)
```
```  1269     done
```
```  1270   finally show "?s = ?t" .
```
```  1271 qed
```
```  1272
```
```  1273
```
```  1274 subsection {* Generalized product over a set *}
```
```  1275
```
```  1276 constdefs
```
```  1277   setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
```
```  1278   "setprod f A == if finite A then fold (op *) f 1 A else 1"
```
```  1279
```
```  1280 syntax
```
```  1281   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```  1282 syntax (xsymbols)
```
```  1283   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1284 syntax (HTML output)
```
```  1285   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1286
```
```  1287 translations -- {* Beware of argument permutation! *}
```
```  1288   "PROD i:A. b" == "setprod (%i. b) A"
```
```  1289   "\<Prod>i\<in>A. b" == "setprod (%i. b) A"
```
```  1290
```
```  1291 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```  1292  @{text"\<Prod>x|P. e"}. *}
```
```  1293
```
```  1294 syntax
```
```  1295   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```  1296 syntax (xsymbols)
```
```  1297   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1298 syntax (HTML output)
```
```  1299   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```  1300
```
```  1301 translations
```
```  1302   "PROD x|P. t" => "setprod (%x. t) {x. P}"
```
```  1303   "\<Prod>x|P. t" => "setprod (%x. t) {x. P}"
```
```  1304
```
```  1305 text{* Finally we abbreviate @{term"\<Prod>x\<in>A. x"} by @{text"\<Prod>A"}. *}
```
```  1306
```
```  1307 syntax
```
```  1308   "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
```
```  1309
```
```  1310 parse_translation {*
```
```  1311   let
```
```  1312     fun Setprod_tr [A] = Syntax.const "setprod" \$ Abs ("", dummyT, Bound 0) \$ A
```
```  1313   in [("_Setprod", Setprod_tr)] end;
```
```  1314 *}
```
```  1315 print_translation {*
```
```  1316 let fun setprod_tr' [Abs(x,Tx,t), A] =
```
```  1317     if t = Bound 0 then Syntax.const "_Setprod" \$ A else raise Match
```
```  1318 in
```
```  1319 [("setprod", setprod_tr')]
```
```  1320 end
```
```  1321 *}
```
```  1322
```
```  1323
```
```  1324 lemma setprod_empty [simp]: "setprod f {} = 1"
```
```  1325   by (auto simp add: setprod_def)
```
```  1326
```
```  1327 lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
```
```  1328     setprod f (insert a A) = f a * setprod f A"
```
```  1329 by (simp add: setprod_def)
```
```  1330
```
```  1331 lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
```
```  1332   by (simp add: setprod_def)
```
```  1333
```
```  1334 lemma setprod_reindex:
```
```  1335      "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```  1336 by(auto simp: setprod_def AC_mult.fold_reindex dest!:finite_imageD)
```
```  1337
```
```  1338 lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```  1339 by (auto simp add: setprod_reindex)
```
```  1340
```
```  1341 lemma setprod_cong:
```
```  1342   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```  1343 by(fastsimp simp: setprod_def intro: AC_mult.fold_cong)
```
```  1344
```
```  1345 lemma strong_setprod_cong:
```
```  1346   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
```
```  1347 by(fastsimp simp: simp_implies_def setprod_def intro: AC_mult.fold_cong)
```
```  1348
```
```  1349 lemma setprod_reindex_cong: "inj_on f A ==>
```
```  1350     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```  1351   by (frule setprod_reindex, simp)
```
```  1352
```
```  1353
```
```  1354 lemma setprod_1: "setprod (%i. 1) A = 1"
```
```  1355   apply (case_tac "finite A")
```
```  1356   apply (erule finite_induct, auto simp add: mult_ac)
```
```  1357   done
```
```  1358
```
```  1359 lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
```
```  1360   apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
```
```  1361   apply (erule ssubst, rule setprod_1)
```
```  1362   apply (rule setprod_cong, auto)
```
```  1363   done
```
```  1364
```
```  1365 lemma setprod_Un_Int: "finite A ==> finite B
```
```  1366     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```  1367 by(simp add: setprod_def AC_mult.fold_Un_Int[symmetric])
```
```  1368
```
```  1369 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```  1370   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```  1371 by (subst setprod_Un_Int [symmetric], auto)
```
```  1372
```
```  1373 lemma setprod_UN_disjoint:
```
```  1374     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1375         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1376       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```  1377 by(simp add: setprod_def AC_mult.fold_UN_disjoint cong: setprod_cong)
```
```  1378
```
```  1379 lemma setprod_Union_disjoint:
```
```  1380   "[| (ALL A:C. finite A);
```
```  1381       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```  1382    ==> setprod f (Union C) = setprod (setprod f) C"
```
```  1383 apply (cases "finite C")
```
```  1384  prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
```
```  1385   apply (frule setprod_UN_disjoint [of C id f])
```
```  1386  apply (unfold Union_def id_def, assumption+)
```
```  1387 done
```
```  1388
```
```  1389 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1390     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
```
```  1391     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```  1392 by(simp add:setprod_def AC_mult.fold_Sigma split_def cong:setprod_cong)
```
```  1393
```
```  1394 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1395 lemma setprod_cartesian_product:
```
```  1396      "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
```
```  1397 apply (cases "finite A")
```
```  1398  apply (cases "finite B")
```
```  1399   apply (simp add: setprod_Sigma)
```
```  1400  apply (cases "A={}", simp)
```
```  1401  apply (simp add: setprod_1)
```
```  1402 apply (auto simp add: setprod_def
```
```  1403             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1404 done
```
```  1405
```
```  1406 lemma setprod_timesf:
```
```  1407      "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
```
```  1408 by(simp add:setprod_def AC_mult.fold_distrib)
```
```  1409
```
```  1410
```
```  1411 subsubsection {* Properties in more restricted classes of structures *}
```
```  1412
```
```  1413 lemma setprod_eq_1_iff [simp]:
```
```  1414     "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
```
```  1415   by (induct set: Finites) auto
```
```  1416
```
```  1417 lemma setprod_zero:
```
```  1418      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
```
```  1419   apply (induct set: Finites, force, clarsimp)
```
```  1420   apply (erule disjE, auto)
```
```  1421   done
```
```  1422
```
```  1423 lemma setprod_nonneg [rule_format]:
```
```  1424      "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1425   apply (case_tac "finite A")
```
```  1426   apply (induct set: Finites, force, clarsimp)
```
```  1427   apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
```
```  1428   apply (rule mult_mono, assumption+)
```
```  1429   apply (auto simp add: setprod_def)
```
```  1430   done
```
```  1431
```
```  1432 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
```
```  1433      --> 0 < setprod f A"
```
```  1434   apply (case_tac "finite A")
```
```  1435   apply (induct set: Finites, force, clarsimp)
```
```  1436   apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
```
```  1437   apply (rule mult_strict_mono, assumption+)
```
```  1438   apply (auto simp add: setprod_def)
```
```  1439   done
```
```  1440
```
```  1441 lemma setprod_nonzero [rule_format]:
```
```  1442     "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
```
```  1443       finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
```
```  1444   apply (erule finite_induct, auto)
```
```  1445   done
```
```  1446
```
```  1447 lemma setprod_zero_eq:
```
```  1448     "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
```
```  1449      finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
```
```  1450   apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
```
```  1451   done
```
```  1452
```
```  1453 lemma setprod_nonzero_field:
```
```  1454     "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
```
```  1455   apply (rule setprod_nonzero, auto)
```
```  1456   done
```
```  1457
```
```  1458 lemma setprod_zero_eq_field:
```
```  1459     "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
```
```  1460   apply (rule setprod_zero_eq, auto)
```
```  1461   done
```
```  1462
```
```  1463 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1464     (setprod f (A Un B) :: 'a ::{field})
```
```  1465       = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1466   apply (subst setprod_Un_Int [symmetric], auto)
```
```  1467   apply (subgoal_tac "finite (A Int B)")
```
```  1468   apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
```
```  1469   apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
```
```  1470   done
```
```  1471
```
```  1472 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1473     (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1474       (if a:A then setprod f A / f a else setprod f A)"
```
```  1475   apply (erule finite_induct)
```
```  1476    apply (auto simp add: insert_Diff_if)
```
```  1477   apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
```
```  1478   apply (erule ssubst)
```
```  1479   apply (subst times_divide_eq_right [THEN sym])
```
```  1480   apply (auto simp add: mult_ac times_divide_eq_right divide_self)
```
```  1481   done
```
```  1482
```
```  1483 lemma setprod_inversef: "finite A ==>
```
```  1484     ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
```
```  1485       setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1486   apply (erule finite_induct)
```
```  1487   apply (simp, simp)
```
```  1488   done
```
```  1489
```
```  1490 lemma setprod_dividef:
```
```  1491      "[|finite A;
```
```  1492         \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
```
```  1493       ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1494   apply (subgoal_tac
```
```  1495          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1496   apply (erule ssubst)
```
```  1497   apply (subst divide_inverse)
```
```  1498   apply (subst setprod_timesf)
```
```  1499   apply (subst setprod_inversef, assumption+, rule refl)
```
```  1500   apply (rule setprod_cong, rule refl)
```
```  1501   apply (subst divide_inverse, auto)
```
```  1502   done
```
```  1503
```
```  1504 subsection {* Finite cardinality *}
```
```  1505
```
```  1506 text {* This definition, although traditional, is ugly to work with:
```
```  1507 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
```
```  1508 But now that we have @{text setsum} things are easy:
```
```  1509 *}
```
```  1510
```
```  1511 constdefs
```
```  1512   card :: "'a set => nat"
```
```  1513   "card A == setsum (%x. 1::nat) A"
```
```  1514
```
```  1515 lemma card_empty [simp]: "card {} = 0"
```
```  1516   by (simp add: card_def)
```
```  1517
```
```  1518 lemma card_infinite [simp]: "~ finite A ==> card A = 0"
```
```  1519   by (simp add: card_def)
```
```  1520
```
```  1521 lemma card_eq_setsum: "card A = setsum (%x. 1) A"
```
```  1522 by (simp add: card_def)
```
```  1523
```
```  1524 lemma card_insert_disjoint [simp]:
```
```  1525   "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
```
```  1526 by(simp add: card_def)
```
```  1527
```
```  1528 lemma card_insert_if:
```
```  1529     "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
```
```  1530   by (simp add: insert_absorb)
```
```  1531
```
```  1532 lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
```
```  1533   apply auto
```
```  1534   apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
```
```  1535   done
```
```  1536
```
```  1537 lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
```
```  1538 by auto
```
```  1539
```
```  1540 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
```
```  1541 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1542 apply(simp del:insert_Diff_single)
```
```  1543 done
```
```  1544
```
```  1545 lemma card_Diff_singleton:
```
```  1546     "finite A ==> x: A ==> card (A - {x}) = card A - 1"
```
```  1547   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1548
```
```  1549 lemma card_Diff_singleton_if:
```
```  1550     "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
```
```  1551   by (simp add: card_Diff_singleton)
```
```  1552
```
```  1553 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1554   by (simp add: card_insert_if card_Suc_Diff1)
```
```  1555
```
```  1556 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1557   by (simp add: card_insert_if)
```
```  1558
```
```  1559 lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
```
```  1560 by (simp add: card_def setsum_mono2)
```
```  1561
```
```  1562 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1563   apply (induct set: Finites, simp, clarify)
```
```  1564   apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1565    prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1566   apply (drule_tac x = "A - {x}" in spec)
```
```  1567   apply (simp add: card_Diff_singleton_if split add: split_if_asm)
```
```  1568   apply (case_tac "card A", auto)
```
```  1569   done
```
```  1570
```
```  1571 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1572   apply (simp add: psubset_def linorder_not_le [symmetric])
```
```  1573   apply (blast dest: card_seteq)
```
```  1574   done
```
```  1575
```
```  1576 lemma card_Un_Int: "finite A ==> finite B
```
```  1577     ==> card A + card B = card (A Un B) + card (A Int B)"
```
```  1578 by(simp add:card_def setsum_Un_Int)
```
```  1579
```
```  1580 lemma card_Un_disjoint: "finite A ==> finite B
```
```  1581     ==> A Int B = {} ==> card (A Un B) = card A + card B"
```
```  1582   by (simp add: card_Un_Int)
```
```  1583
```
```  1584 lemma card_Diff_subset:
```
```  1585   "finite B ==> B <= A ==> card (A - B) = card A - card B"
```
```  1586 by(simp add:card_def setsum_diff_nat)
```
```  1587
```
```  1588 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1589   apply (rule Suc_less_SucD)
```
```  1590   apply (simp add: card_Suc_Diff1)
```
```  1591   done
```
```  1592
```
```  1593 lemma card_Diff2_less:
```
```  1594     "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1595   apply (case_tac "x = y")
```
```  1596    apply (simp add: card_Diff1_less)
```
```  1597   apply (rule less_trans)
```
```  1598    prefer 2 apply (auto intro!: card_Diff1_less)
```
```  1599   done
```
```  1600
```
```  1601 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1602   apply (case_tac "x : A")
```
```  1603    apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1604   done
```
```  1605
```
```  1606 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1607 by (erule psubsetI, blast)
```
```  1608
```
```  1609 lemma insert_partition:
```
```  1610   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1611   \<Longrightarrow> x \<inter> \<Union> F = {}"
```
```  1612 by auto
```
```  1613
```
```  1614 (* main cardinality theorem *)
```
```  1615 lemma card_partition [rule_format]:
```
```  1616      "finite C ==>
```
```  1617         finite (\<Union> C) -->
```
```  1618         (\<forall>c\<in>C. card c = k) -->
```
```  1619         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  1620         k * card(C) = card (\<Union> C)"
```
```  1621 apply (erule finite_induct, simp)
```
```  1622 apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition
```
```  1623        finite_subset [of _ "\<Union> (insert x F)"])
```
```  1624 done
```
```  1625
```
```  1626
```
```  1627 lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
```
```  1628 apply (cases "finite A")
```
```  1629 apply (erule finite_induct)
```
```  1630 apply (auto simp add: ring_distrib add_ac)
```
```  1631 done
```
```  1632
```
```  1633
```
```  1634 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::recpower)) = y^(card A)"
```
```  1635   apply (erule finite_induct)
```
```  1636   apply (auto simp add: power_Suc)
```
```  1637   done
```
```  1638
```
```  1639 lemma setsum_bounded:
```
```  1640   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{comm_semiring_1_cancel, pordered_ab_semigroup_add})"
```
```  1641   shows "setsum f A \<le> of_nat(card A) * K"
```
```  1642 proof (cases "finite A")
```
```  1643   case True
```
```  1644   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```  1645 next
```
```  1646   case False thus ?thesis by (simp add: setsum_def)
```
```  1647 qed
```
```  1648
```
```  1649
```
```  1650 subsubsection {* Cardinality of unions *}
```
```  1651
```
```  1652 lemma of_nat_id[simp]: "(of_nat n :: nat) = n"
```
```  1653 by(induct n, auto)
```
```  1654
```
```  1655 lemma card_UN_disjoint:
```
```  1656     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1657         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1658       card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```  1659   apply (simp add: card_def del: setsum_constant)
```
```  1660   apply (subgoal_tac
```
```  1661            "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
```
```  1662   apply (simp add: setsum_UN_disjoint del: setsum_constant)
```
```  1663   apply (simp cong: setsum_cong)
```
```  1664   done
```
```  1665
```
```  1666 lemma card_Union_disjoint:
```
```  1667   "finite C ==> (ALL A:C. finite A) ==>
```
```  1668         (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
```
```  1669       card (Union C) = setsum card C"
```
```  1670   apply (frule card_UN_disjoint [of C id])
```
```  1671   apply (unfold Union_def id_def, assumption+)
```
```  1672   done
```
```  1673
```
```  1674 subsubsection {* Cardinality of image *}
```
```  1675
```
```  1676 text{*The image of a finite set can be expressed using @{term fold}.*}
```
```  1677 lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
```
```  1678   apply (erule finite_induct, simp)
```
```  1679   apply (subst ACf.fold_insert)
```
```  1680   apply (auto simp add: ACf_def)
```
```  1681   done
```
```  1682
```
```  1683 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
```
```  1684   apply (induct set: Finites, simp)
```
```  1685   apply (simp add: le_SucI finite_imageI card_insert_if)
```
```  1686   done
```
```  1687
```
```  1688 lemma card_image: "inj_on f A ==> card (f ` A) = card A"
```
```  1689 by(simp add:card_def setsum_reindex o_def del:setsum_constant)
```
```  1690
```
```  1691 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  1692   by (simp add: card_seteq card_image)
```
```  1693
```
```  1694 lemma eq_card_imp_inj_on:
```
```  1695   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
```
```  1696 apply (induct rule:finite_induct, simp)
```
```  1697 apply(frule card_image_le[where f = f])
```
```  1698 apply(simp add:card_insert_if split:if_splits)
```
```  1699 done
```
```  1700
```
```  1701 lemma inj_on_iff_eq_card:
```
```  1702   "finite A ==> inj_on f A = (card(f ` A) = card A)"
```
```  1703 by(blast intro: card_image eq_card_imp_inj_on)
```
```  1704
```
```  1705
```
```  1706 lemma card_inj_on_le:
```
```  1707     "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
```
```  1708 apply (subgoal_tac "finite A")
```
```  1709  apply (force intro: card_mono simp add: card_image [symmetric])
```
```  1710 apply (blast intro: finite_imageD dest: finite_subset)
```
```  1711 done
```
```  1712
```
```  1713 lemma card_bij_eq:
```
```  1714     "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  1715        finite A; finite B |] ==> card A = card B"
```
```  1716   by (auto intro: le_anti_sym card_inj_on_le)
```
```  1717
```
```  1718
```
```  1719 subsubsection {* Cardinality of products *}
```
```  1720
```
```  1721 (*
```
```  1722 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1723   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1724   by auto
```
```  1725 *)
```
```  1726
```
```  1727 lemma card_SigmaI [simp]:
```
```  1728   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```  1729   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1730 by(simp add:card_def setsum_Sigma del:setsum_constant)
```
```  1731
```
```  1732 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```  1733 apply (cases "finite A")
```
```  1734 apply (cases "finite B")
```
```  1735 apply (auto simp add: card_eq_0_iff
```
```  1736             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1737 done
```
```  1738
```
```  1739 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```  1740 by (simp add: card_cartesian_product)
```
```  1741
```
```  1742
```
```  1743
```
```  1744 subsubsection {* Cardinality of the Powerset *}
```
```  1745
```
```  1746 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
```
```  1747   apply (induct set: Finites)
```
```  1748    apply (simp_all add: Pow_insert)
```
```  1749   apply (subst card_Un_disjoint, blast)
```
```  1750     apply (blast intro: finite_imageI, blast)
```
```  1751   apply (subgoal_tac "inj_on (insert x) (Pow F)")
```
```  1752    apply (simp add: card_image Pow_insert)
```
```  1753   apply (unfold inj_on_def)
```
```  1754   apply (blast elim!: equalityE)
```
```  1755   done
```
```  1756
```
```  1757 text {* Relates to equivalence classes.  Based on a theorem of
```
```  1758 F. Kammüller's.  *}
```
```  1759
```
```  1760 lemma dvd_partition:
```
```  1761   "finite (Union C) ==>
```
```  1762     ALL c : C. k dvd card c ==>
```
```  1763     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
```
```  1764   k dvd card (Union C)"
```
```  1765 apply(frule finite_UnionD)
```
```  1766 apply(rotate_tac -1)
```
```  1767   apply (induct set: Finites, simp_all, clarify)
```
```  1768   apply (subst card_Un_disjoint)
```
```  1769   apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
```
```  1770   done
```
```  1771
```
```  1772
```
```  1773 subsection{* A fold functional for non-empty sets *}
```
```  1774
```
```  1775 text{* Does not require start value. *}
```
```  1776
```
```  1777 consts
```
```  1778   fold1Set :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
```
```  1779
```
```  1780 inductive "fold1Set f"
```
```  1781 intros
```
```  1782   fold1Set_insertI [intro]:
```
```  1783    "\<lbrakk> (A,x) \<in> foldSet f id a; a \<notin> A \<rbrakk> \<Longrightarrow> (insert a A, x) \<in> fold1Set f"
```
```  1784
```
```  1785 constdefs
```
```  1786   fold1 :: "('a => 'a => 'a) => 'a set => 'a"
```
```  1787   "fold1 f A == THE x. (A, x) : fold1Set f"
```
```  1788
```
```  1789 lemma fold1Set_nonempty:
```
```  1790  "(A, x) : fold1Set f \<Longrightarrow> A \<noteq> {}"
```
```  1791 by(erule fold1Set.cases, simp_all)
```
```  1792
```
```  1793
```
```  1794 inductive_cases empty_fold1SetE [elim!]: "({}, x) : fold1Set f"
```
```  1795
```
```  1796 inductive_cases insert_fold1SetE [elim!]: "(insert a X, x) : fold1Set f"
```
```  1797
```
```  1798
```
```  1799 lemma fold1Set_sing [iff]: "(({a},b) : fold1Set f) = (a = b)"
```
```  1800   by (blast intro: foldSet.intros elim: foldSet.cases)
```
```  1801
```
```  1802 lemma fold1_singleton[simp]: "fold1 f {a} = a"
```
```  1803   by (unfold fold1_def) blast
```
```  1804
```
```  1805 lemma finite_nonempty_imp_fold1Set:
```
```  1806   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : fold1Set f"
```
```  1807 apply (induct A rule: finite_induct)
```
```  1808 apply (auto dest: finite_imp_foldSet [of _ f id])
```
```  1809 done
```
```  1810
```
```  1811 text{*First, some lemmas about @{term foldSet}.*}
```
```  1812
```
```  1813 lemma (in ACf) foldSet_insert_swap:
```
```  1814 assumes fold: "(A,y) \<in> foldSet f id b"
```
```  1815 shows "b \<notin> A \<Longrightarrow> (insert b A, z \<cdot> y) \<in> foldSet f id z"
```
```  1816 using fold
```
```  1817 proof (induct rule: foldSet.induct)
```
```  1818   case emptyI thus ?case by (force simp add: fold_insert_aux commute)
```
```  1819 next
```
```  1820   case (insertI A x y)
```
```  1821     have "(insert x (insert b A), x \<cdot> (z \<cdot> y)) \<in> foldSet f (\<lambda>u. u) z"
```
```  1822       using insertI by force  --{*how does @{term id} get unfolded?*}
```
```  1823     thus ?case by (simp add: insert_commute AC)
```
```  1824 qed
```
```  1825
```
```  1826 lemma (in ACf) foldSet_permute_diff:
```
```  1827 assumes fold: "(A,x) \<in> foldSet f id b"
```
```  1828 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> (insert b (A-{a}), x) \<in> foldSet f id a"
```
```  1829 using fold
```
```  1830 proof (induct rule: foldSet.induct)
```
```  1831   case emptyI thus ?case by simp
```
```  1832 next
```
```  1833   case (insertI A x y)
```
```  1834   have "a = x \<or> a \<in> A" using insertI by simp
```
```  1835   thus ?case
```
```  1836   proof
```
```  1837     assume "a = x"
```
```  1838     with insertI show ?thesis
```
```  1839       by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap)
```
```  1840   next
```
```  1841     assume ainA: "a \<in> A"
```
```  1842     hence "(insert x (insert b (A - {a})), x \<cdot> y) \<in> foldSet f id a"
```
```  1843       using insertI by (force simp: id_def)
```
```  1844     moreover
```
```  1845     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
```
```  1846       using ainA insertI by blast
```
```  1847     ultimately show ?thesis by (simp add: id_def)
```
```  1848   qed
```
```  1849 qed
```
```  1850
```
```  1851 lemma (in ACf) fold1_eq_fold:
```
```  1852      "[|finite A; a \<notin> A|] ==> fold1 f (insert a A) = fold f id a A"
```
```  1853 apply (simp add: fold1_def fold_def)
```
```  1854 apply (rule the_equality)
```
```  1855 apply (best intro: foldSet_determ theI dest: finite_imp_foldSet [of _ f id])
```
```  1856 apply (rule sym, clarify)
```
```  1857 apply (case_tac "Aa=A")
```
```  1858  apply (best intro: the_equality foldSet_determ)
```
```  1859 apply (subgoal_tac "(A,x) \<in> foldSet f id a")
```
```  1860  apply (best intro: the_equality foldSet_determ)
```
```  1861 apply (subgoal_tac "insert aa (Aa - {a}) = A")
```
```  1862  prefer 2 apply (blast elim: equalityE)
```
```  1863 apply (auto dest: foldSet_permute_diff [where a=a])
```
```  1864 done
```
```  1865
```
```  1866 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
```
```  1867 apply safe
```
```  1868 apply simp
```
```  1869 apply (drule_tac x=x in spec)
```
```  1870 apply (drule_tac x="A-{x}" in spec, auto)
```
```  1871 done
```
```  1872
```
```  1873 lemma (in ACf) fold1_insert:
```
```  1874   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
```
```  1875   shows "fold1 f (insert x A) = f x (fold1 f A)"
```
```  1876 proof -
```
```  1877   from nonempty obtain a A' where "A = insert a A' & a ~: A'"
```
```  1878     by (auto simp add: nonempty_iff)
```
```  1879   with A show ?thesis
```
```  1880     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
```
```  1881 qed
```
```  1882
```
```  1883 lemma (in ACIf) fold1_insert_idem [simp]:
```
```  1884   assumes nonempty: "A \<noteq> {}" and A: "finite A"
```
```  1885   shows "fold1 f (insert x A) = f x (fold1 f A)"
```
```  1886 proof -
```
```  1887   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
```
```  1888     by (auto simp add: nonempty_iff)
```
```  1889   show ?thesis
```
```  1890   proof cases
```
```  1891     assume "a = x"
```
```  1892     thus ?thesis
```
```  1893     proof cases
```
```  1894       assume "A' = {}"
```
```  1895       with prems show ?thesis by (simp add: idem)
```
```  1896     next
```
```  1897       assume "A' \<noteq> {}"
```
```  1898       with prems show ?thesis
```
```  1899 	by (simp add: fold1_insert assoc [symmetric] idem)
```
```  1900     qed
```
```  1901   next
```
```  1902     assume "a \<noteq> x"
```
```  1903     with prems show ?thesis
```
```  1904       by (simp add: insert_commute fold1_eq_fold fold_insert_idem)
```
```  1905   qed
```
```  1906 qed
```
```  1907
```
```  1908
```
```  1909 text{* Now the recursion rules for definitions: *}
```
```  1910
```
```  1911 lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
```
```  1912 by(simp add:fold1_singleton)
```
```  1913
```
```  1914 lemma (in ACf) fold1_insert_def:
```
```  1915   "\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
```
```  1916 by(simp add:fold1_insert)
```
```  1917
```
```  1918 lemma (in ACIf) fold1_insert_idem_def:
```
```  1919   "\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
```
```  1920 by(simp add:fold1_insert_idem)
```
```  1921
```
```  1922 subsubsection{* Determinacy for @{term fold1Set} *}
```
```  1923
```
```  1924 text{*Not actually used!!*}
```
```  1925
```
```  1926 lemma (in ACf) foldSet_permute:
```
```  1927   "[|(insert a A, x) \<in> foldSet f id b; a \<notin> A; b \<notin> A|]
```
```  1928    ==> (insert b A, x) \<in> foldSet f id a"
```
```  1929 apply (case_tac "a=b")
```
```  1930 apply (auto dest: foldSet_permute_diff)
```
```  1931 done
```
```  1932
```
```  1933 lemma (in ACf) fold1Set_determ:
```
```  1934   "(A, x) \<in> fold1Set f ==> (A, y) \<in> fold1Set f ==> y = x"
```
```  1935 proof (clarify elim!: fold1Set.cases)
```
```  1936   fix A x B y a b
```
```  1937   assume Ax: "(A, x) \<in> foldSet f id a"
```
```  1938   assume By: "(B, y) \<in> foldSet f id b"
```
```  1939   assume anotA:  "a \<notin> A"
```
```  1940   assume bnotB:  "b \<notin> B"
```
```  1941   assume eq: "insert a A = insert b B"
```
```  1942   show "y=x"
```
```  1943   proof cases
```
```  1944     assume same: "a=b"
```
```  1945     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
```
```  1946     thus ?thesis using Ax By same by (blast intro: foldSet_determ)
```
```  1947   next
```
```  1948     assume diff: "a\<noteq>b"
```
```  1949     let ?D = "B - {a}"
```
```  1950     have B: "B = insert a ?D" and A: "A = insert b ?D"
```
```  1951      and aB: "a \<in> B" and bA: "b \<in> A"
```
```  1952       using eq anotA bnotB diff by (blast elim!:equalityE)+
```
```  1953     with aB bnotB By
```
```  1954     have "(insert b ?D, y) \<in> foldSet f id a"
```
```  1955       by (auto intro: foldSet_permute simp add: insert_absorb)
```
```  1956     moreover
```
```  1957     have "(insert b ?D, x) \<in> foldSet f id a"
```
```  1958       by (simp add: A [symmetric] Ax)
```
```  1959     ultimately show ?thesis by (blast intro: foldSet_determ)
```
```  1960   qed
```
```  1961 qed
```
```  1962
```
```  1963 lemma (in ACf) fold1Set_equality: "(A, y) : fold1Set f ==> fold1 f A = y"
```
```  1964   by (unfold fold1_def) (blast intro: fold1Set_determ)
```
```  1965
```
```  1966 declare
```
```  1967   empty_foldSetE [rule del]   foldSet.intros [rule del]
```
```  1968   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
```
```  1969   -- {* No more proves involve these relations. *}
```
```  1970
```
```  1971 subsubsection{* Semi-Lattices *}
```
```  1972
```
```  1973 locale ACIfSL = ACIf +
```
```  1974   fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
```
```  1975   assumes below_def: "(x \<sqsubseteq> y) = (x\<cdot>y = x)"
```
```  1976
```
```  1977 locale ACIfSLlin = ACIfSL +
```
```  1978   assumes lin: "x\<cdot>y \<in> {x,y}"
```
```  1979
```
```  1980 lemma (in ACIfSL) below_refl[simp]: "x \<sqsubseteq> x"
```
```  1981 by(simp add: below_def idem)
```
```  1982
```
```  1983 lemma (in ACIfSL) below_f_conv[simp]: "x \<sqsubseteq> y \<cdot> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
```
```  1984 proof
```
```  1985   assume "x \<sqsubseteq> y \<cdot> z"
```
```  1986   hence xyzx: "x \<cdot> (y \<cdot> z) = x"  by(simp add: below_def)
```
```  1987   have "x \<cdot> y = x"
```
```  1988   proof -
```
```  1989     have "x \<cdot> y = (x \<cdot> (y \<cdot> z)) \<cdot> y" by(rule subst[OF xyzx], rule refl)
```
```  1990     also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  1991     also have "\<dots> = x" by(rule xyzx)
```
```  1992     finally show ?thesis .
```
```  1993   qed
```
```  1994   moreover have "x \<cdot> z = x"
```
```  1995   proof -
```
```  1996     have "x \<cdot> z = (x \<cdot> (y \<cdot> z)) \<cdot> z" by(rule subst[OF xyzx], rule refl)
```
```  1997     also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  1998     also have "\<dots> = x" by(rule xyzx)
```
```  1999     finally show ?thesis .
```
```  2000   qed
```
```  2001   ultimately show "x \<sqsubseteq> y \<and> x \<sqsubseteq> z" by(simp add: below_def)
```
```  2002 next
```
```  2003   assume a: "x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
```
```  2004   hence y: "x \<cdot> y = x" and z: "x \<cdot> z = x" by(simp_all add: below_def)
```
```  2005   have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by(simp add:assoc)
```
```  2006   also have "x \<cdot> y = x" using a by(simp_all add: below_def)
```
```  2007   also have "x \<cdot> z = x" using a by(simp_all add: below_def)
```
```  2008   finally show "x \<sqsubseteq> y \<cdot> z" by(simp_all add: below_def)
```
```  2009 qed
```
```  2010
```
```  2011 lemma (in ACIfSLlin) above_f_conv:
```
```  2012  "x \<cdot> y \<sqsubseteq> z = (x \<sqsubseteq> z \<or> y \<sqsubseteq> z)"
```
```  2013 proof
```
```  2014   assume a: "x \<cdot> y \<sqsubseteq> z"
```
```  2015   have "x \<cdot> y = x \<or> x \<cdot> y = y" using lin[of x y] by simp
```
```  2016   thus "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
```
```  2017   proof
```
```  2018     assume "x \<cdot> y = x" hence "x \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
```
```  2019   next
```
```  2020     assume "x \<cdot> y = y" hence "y \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
```
```  2021   qed
```
```  2022 next
```
```  2023   assume "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
```
```  2024   thus "x \<cdot> y \<sqsubseteq> z"
```
```  2025   proof
```
```  2026     assume a: "x \<sqsubseteq> z"
```
```  2027     have "(x \<cdot> y) \<cdot> z = (x \<cdot> z) \<cdot> y" by(simp add:ACI)
```
```  2028     also have "x \<cdot> z = x" using a by(simp add:below_def)
```
```  2029     finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
```
```  2030   next
```
```  2031     assume a: "y \<sqsubseteq> z"
```
```  2032     have "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  2033     also have "y \<cdot> z = y" using a by(simp add:below_def)
```
```  2034     finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
```
```  2035   qed
```
```  2036 qed
```
```  2037
```
```  2038
```
```  2039 subsubsection{* Lemmas about @{text fold1} *}
```
```  2040
```
```  2041 lemma (in ACf) fold1_Un:
```
```  2042 assumes A: "finite A" "A \<noteq> {}"
```
```  2043 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
```
```  2044        fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
```
```  2045 using A
```
```  2046 proof(induct rule:finite_ne_induct)
```
```  2047   case singleton thus ?case by(simp add:fold1_insert)
```
```  2048 next
```
```  2049   case insert thus ?case by (simp add:fold1_insert assoc)
```
```  2050 qed
```
```  2051
```
```  2052 lemma (in ACIf) fold1_Un2:
```
```  2053 assumes A: "finite A" "A \<noteq> {}"
```
```  2054 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
```
```  2055        fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
```
```  2056 using A
```
```  2057 proof(induct rule:finite_ne_induct)
```
```  2058   case singleton thus ?case by(simp add:fold1_insert_idem)
```
```  2059 next
```
```  2060   case insert thus ?case by (simp add:fold1_insert_idem assoc)
```
```  2061 qed
```
```  2062
```
```  2063 lemma (in ACf) fold1_in:
```
```  2064   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}"
```
```  2065   shows "fold1 f A \<in> A"
```
```  2066 using A
```
```  2067 proof (induct rule:finite_ne_induct)
```
```  2068   case singleton thus ?case by simp
```
```  2069 next
```
```  2070   case insert thus ?case using elem by (force simp add:fold1_insert)
```
```  2071 qed
```
```  2072
```
```  2073 lemma (in ACIfSL) below_fold1_iff:
```
```  2074 assumes A: "finite A" "A \<noteq> {}"
```
```  2075 shows "x \<sqsubseteq> fold1 f A = (\<forall>a\<in>A. x \<sqsubseteq> a)"
```
```  2076 using A
```
```  2077 by(induct rule:finite_ne_induct) simp_all
```
```  2078
```
```  2079 lemma (in ACIfSL) fold1_belowI:
```
```  2080 assumes A: "finite A" "A \<noteq> {}"
```
```  2081 shows "a \<in> A \<Longrightarrow> fold1 f A \<sqsubseteq> a"
```
```  2082 using A
```
```  2083 proof (induct rule:finite_ne_induct)
```
```  2084   case singleton thus ?case by simp
```
```  2085 next
```
```  2086   case (insert x F)
```
```  2087   from insert(5) have "a = x \<or> a \<in> F" by simp
```
```  2088   thus ?case
```
```  2089   proof
```
```  2090     assume "a = x" thus ?thesis using insert by(simp add:below_def ACI)
```
```  2091   next
```
```  2092     assume "a \<in> F"
```
```  2093     hence bel: "fold1 f F \<sqsubseteq> a" by(rule insert)
```
```  2094     have "fold1 f (insert x F) \<cdot> a = x \<cdot> (fold1 f F \<cdot> a)"
```
```  2095       using insert by(simp add:below_def ACI)
```
```  2096     also have "fold1 f F \<cdot> a = fold1 f F"
```
```  2097       using bel  by(simp add:below_def ACI)
```
```  2098     also have "x \<cdot> \<dots> = fold1 f (insert x F)"
```
```  2099       using insert by(simp add:below_def ACI)
```
```  2100     finally show ?thesis  by(simp add:below_def)
```
```  2101   qed
```
```  2102 qed
```
```  2103
```
```  2104 lemma (in ACIfSLlin) fold1_below_iff:
```
```  2105 assumes A: "finite A" "A \<noteq> {}"
```
```  2106 shows "fold1 f A \<sqsubseteq> x = (\<exists>a\<in>A. a \<sqsubseteq> x)"
```
```  2107 using A
```
```  2108 by(induct rule:finite_ne_induct)(simp_all add:above_f_conv)
```
```  2109
```
```  2110
```
```  2111 subsubsection{* Lattices *}
```
```  2112
```
```  2113 locale Lattice = lattice +
```
```  2114   fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
```
```  2115   and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```  2116   defines "Inf == fold1 inf"  and "Sup == fold1 sup"
```
```  2117
```
```  2118 locale Distrib_Lattice = distrib_lattice + Lattice
```
```  2119
```
```  2120 text{* Lattices are semilattices *}
```
```  2121
```
```  2122 lemma (in Lattice) ACf_inf: "ACf inf"
```
```  2123 by(blast intro: ACf.intro inf_commute inf_assoc)
```
```  2124
```
```  2125 lemma (in Lattice) ACf_sup: "ACf sup"
```
```  2126 by(blast intro: ACf.intro sup_commute sup_assoc)
```
```  2127
```
```  2128 lemma (in Lattice) ACIf_inf: "ACIf inf"
```
```  2129 apply(rule ACIf.intro)
```
```  2130 apply(rule ACf_inf)
```
```  2131 apply(rule ACIf_axioms.intro)
```
```  2132 apply(rule inf_idem)
```
```  2133 done
```
```  2134
```
```  2135 lemma (in Lattice) ACIf_sup: "ACIf sup"
```
```  2136 apply(rule ACIf.intro)
```
```  2137 apply(rule ACf_sup)
```
```  2138 apply(rule ACIf_axioms.intro)
```
```  2139 apply(rule sup_idem)
```
```  2140 done
```
```  2141
```
```  2142 lemma (in Lattice) ACIfSL_inf: "ACIfSL inf (op \<sqsubseteq>)"
```
```  2143 apply(rule ACIfSL.intro)
```
```  2144 apply(rule ACf_inf)
```
```  2145 apply(rule ACIf.axioms[OF ACIf_inf])
```
```  2146 apply(rule ACIfSL_axioms.intro)
```
```  2147 apply(rule iffI)
```
```  2148  apply(blast intro: antisym inf_le1 inf_le2 inf_least refl)
```
```  2149 apply(erule subst)
```
```  2150 apply(rule inf_le2)
```
```  2151 done
```
```  2152
```
```  2153 lemma (in Lattice) ACIfSL_sup: "ACIfSL sup (%x y. y \<sqsubseteq> x)"
```
```  2154 apply(rule ACIfSL.intro)
```
```  2155 apply(rule ACf_sup)
```
```  2156 apply(rule ACIf.axioms[OF ACIf_sup])
```
```  2157 apply(rule ACIfSL_axioms.intro)
```
```  2158 apply(rule iffI)
```
```  2159  apply(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
```
```  2160 apply(erule subst)
```
```  2161 apply(rule sup_ge2)
```
```  2162 done
```
```  2163
```
```  2164
```
```  2165 subsubsection{* Fold laws in lattices *}
```
```  2166
```
```  2167 lemma (in Lattice) Inf_le_Sup[simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Squnion>A"
```
```  2168 apply(unfold Sup_def Inf_def)
```
```  2169 apply(subgoal_tac "EX a. a:A")
```
```  2170 prefer 2 apply blast
```
```  2171 apply(erule exE)
```
```  2172 apply(rule trans)
```
```  2173 apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_inf])
```
```  2174 apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup])
```
```  2175 done
```
```  2176
```
```  2177 lemma (in Lattice) sup_Inf_absorb[simp]:
```
```  2178   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<squnion> \<Sqinter>A) = a"
```
```  2179 apply(subst sup_commute)
```
```  2180 apply(simp add:Inf_def sup_absorb ACIfSL.fold1_belowI[OF ACIfSL_inf])
```
```  2181 done
```
```  2182
```
```  2183 lemma (in Lattice) inf_Sup_absorb[simp]:
```
```  2184   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<sqinter> \<Squnion>A) = a"
```
```  2185 by(simp add:Sup_def inf_absorb ACIfSL.fold1_belowI[OF ACIfSL_sup])
```
```  2186
```
```  2187
```
```  2188 lemma (in Distrib_Lattice) sup_Inf1_distrib:
```
```  2189 assumes A: "finite A" "A \<noteq> {}"
```
```  2190 shows "(x \<squnion> \<Sqinter>A) = \<Sqinter>{x \<squnion> a|a. a \<in> A}"
```
```  2191 using A
```
```  2192 proof (induct rule: finite_ne_induct)
```
```  2193   case singleton thus ?case by(simp add:Inf_def)
```
```  2194 next
```
```  2195   case (insert y A)
```
```  2196   have fin: "finite {x \<squnion> a |a. a \<in> A}"
```
```  2197     by(fast intro: finite_surj[where f = "%a. x \<squnion> a", OF insert(1)])
```
```  2198   have "x \<squnion> \<Sqinter> (insert y A) = x \<squnion> (y \<sqinter> \<Sqinter> A)"
```
```  2199     using insert by(simp add:ACf.fold1_insert_def[OF ACf_inf Inf_def])
```
```  2200   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> \<Sqinter> A)" by(rule sup_inf_distrib1)
```
```  2201   also have "x \<squnion> \<Sqinter> A = \<Sqinter>{x \<squnion> a|a. a \<in> A}" using insert by simp
```
```  2202   also have "(x \<squnion> y) \<sqinter> \<dots> = \<Sqinter> (insert (x \<squnion> y) {x \<squnion> a |a. a \<in> A})"
```
```  2203     using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def fin])
```
```  2204   also have "insert (x\<squnion>y) {x\<squnion>a |a. a \<in> A} = {x\<squnion>a |a. a \<in> insert y A}"
```
```  2205     by blast
```
```  2206   finally show ?case .
```
```  2207 qed
```
```  2208
```
```  2209 lemma (in Distrib_Lattice) sup_Inf2_distrib:
```
```  2210 assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  2211 shows "(\<Sqinter>A \<squnion> \<Sqinter>B) = \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
```
```  2212 using A
```
```  2213 proof (induct rule: finite_ne_induct)
```
```  2214   case singleton thus ?case
```
```  2215     by(simp add: sup_Inf1_distrib[OF B] fold1_singleton_def[OF Inf_def])
```
```  2216 next
```
```  2217   case (insert x A)
```
```  2218   have finB: "finite {x \<squnion> b |b. b \<in> B}"
```
```  2219     by(fast intro: finite_surj[where f = "%b. x \<squnion> b", OF B(1)])
```
```  2220   have finAB: "finite {a \<squnion> b |a b. a \<in> A \<and> b \<in> B}"
```
```  2221   proof -
```
```  2222     have "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<squnion> b})"
```
```  2223       by blast
```
```  2224     thus ?thesis by(simp add: insert(1) B(1))
```
```  2225   qed
```
```  2226   have ne: "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  2227   have "\<Sqinter>(insert x A) \<squnion> \<Sqinter>B = (x \<sqinter> \<Sqinter>A) \<squnion> \<Sqinter>B"
```
```  2228     using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def])
```
```  2229   also have "\<dots> = (x \<squnion> \<Sqinter>B) \<sqinter> (\<Sqinter>A \<squnion> \<Sqinter>B)" by(rule sup_inf_distrib2)
```
```  2230   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}"
```
```  2231     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  2232   also have "\<dots> = \<Sqinter>({x\<squnion>b |b. b \<in> B} \<union> {a\<squnion>b |a b. a \<in> A \<and> b \<in> B})"
```
```  2233     (is "_ = \<Sqinter>?M")
```
```  2234     using B insert
```
```  2235     by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne])
```
```  2236   also have "?M = {a \<squnion> b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  2237     by blast
```
```  2238   finally show ?case .
```
```  2239 qed
```
```  2240
```
```  2241
```
```  2242 subsection{*Min and Max*}
```
```  2243
```
```  2244 text{* As an application of @{text fold1} we define the minimal and
```
```  2245 maximal element of a (non-empty) set over a linear order. *}
```
```  2246
```
```  2247 constdefs
```
```  2248   Min :: "('a::linorder)set => 'a"
```
```  2249   "Min  ==  fold1 min"
```
```  2250
```
```  2251   Max :: "('a::linorder)set => 'a"
```
```  2252   "Max  ==  fold1 max"
```
```  2253
```
```  2254
```
```  2255 text{* Before we can do anything, we need to show that @{text min} and
```
```  2256 @{text max} are ACI and the ordering is linear: *}
```
```  2257
```
```  2258 interpretation min: ACf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
```
```  2259 apply(rule ACf.intro)
```
```  2260 apply(auto simp:min_def)
```
```  2261 done
```
```  2262
```
```  2263 interpretation min: ACIf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
```
```  2264 apply(rule ACIf_axioms.intro)
```
```  2265 apply(auto simp:min_def)
```
```  2266 done
```
```  2267
```
```  2268 interpretation max: ACf ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
```
```  2269 apply(rule ACf.intro)
```
```  2270 apply(auto simp:max_def)
```
```  2271 done
```
```  2272
```
```  2273 interpretation max: ACIf ["max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
```
```  2274 apply(rule ACIf_axioms.intro)
```
```  2275 apply(auto simp:max_def)
```
```  2276 done
```
```  2277
```
```  2278 interpretation min:
```
```  2279   ACIfSL ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>"]
```
```  2280 apply(rule ACIfSL_axioms.intro)
```
```  2281 apply(auto simp:min_def)
```
```  2282 done
```
```  2283
```
```  2284 interpretation min:
```
```  2285   ACIfSLlin ["min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>"]
```
```  2286 apply(rule ACIfSLlin_axioms.intro)
```
```  2287 apply(auto simp:min_def)
```
```  2288 done
```
```  2289
```
```  2290 interpretation max:
```
```  2291   ACIfSL ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x"]
```
```  2292 apply(rule ACIfSL_axioms.intro)
```
```  2293 apply(auto simp:max_def)
```
```  2294 done
```
```  2295
```
```  2296 interpretation max:
```
```  2297   ACIfSLlin ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x"]
```
```  2298 apply(rule ACIfSLlin_axioms.intro)
```
```  2299 apply(auto simp:max_def)
```
```  2300 done
```
```  2301
```
```  2302 interpretation min_max:
```
```  2303   Lattice ["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
```
```  2304 apply -
```
```  2305 apply(rule Min_def)
```
```  2306 apply(rule Max_def)
```
```  2307 done
```
```  2308
```
```  2309
```
```  2310 interpretation min_max:
```
```  2311   Distrib_Lattice ["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
```
```  2312 .
```
```  2313
```
```  2314 text{* Now we instantiate the recursion equations and declare them
```
```  2315 simplification rules: *}
```
```  2316
```
```  2317 (* Making Min or Max a defined parameter of a locale, suitably
```
```  2318   extending ACIf, could make the following interpretations more automatic. *)
```
```  2319
```
```  2320 lemmas Min_singleton = fold1_singleton_def [OF Min_def]
```
```  2321 lemmas Max_singleton = fold1_singleton_def [OF Max_def]
```
```  2322 lemmas Min_insert = min.fold1_insert_idem_def [OF Min_def]
```
```  2323 lemmas Max_insert = max.fold1_insert_idem_def [OF Max_def]
```
```  2324
```
```  2325 declare Min_singleton [simp]  Max_singleton [simp]
```
```  2326 declare Min_insert [simp]  Max_insert [simp]
```
```  2327
```
```  2328
```
```  2329 text{* Now we instantiate some @{text fold1} properties: *}
```
```  2330
```
```  2331 lemma Min_in [simp]:
```
```  2332   shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A"
```
```  2333 using min.fold1_in
```
```  2334 by(fastsimp simp: Min_def min_def)
```
```  2335
```
```  2336 lemma Max_in [simp]:
```
```  2337   shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A"
```
```  2338 using max.fold1_in
```
```  2339 by(fastsimp simp: Max_def max_def)
```
```  2340
```
```  2341 lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<le> x"
```
```  2342 by(simp add: Min_def min.fold1_belowI)
```
```  2343
```
```  2344 lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<le> Max A"
```
```  2345 by(simp add: Max_def max.fold1_belowI)
```
```  2346
```
```  2347 lemma Min_ge_iff[simp]:
```
```  2348   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
```
```  2349 by(simp add: Min_def min.below_fold1_iff)
```
```  2350
```
```  2351 lemma Max_le_iff[simp]:
```
```  2352   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
```
```  2353 by(simp add: Max_def max.below_fold1_iff)
```
```  2354
```
```  2355 lemma Min_le_iff:
```
```  2356   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Min A \<le> x) = (\<exists>a\<in>A. a \<le> x)"
```
```  2357 by(simp add: Min_def min.fold1_below_iff)
```
```  2358
```
```  2359 lemma Max_ge_iff:
```
```  2360   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Max A) = (\<exists>a\<in>A. x \<le> a)"
```
```  2361 by(simp add: Max_def max.fold1_below_iff)
```
```  2362
```
```  2363 subsection {* Properties of axclass @{text finite} *}
```
```  2364
```
```  2365 text{* Many of these are by Brian Huffman. *}
```
```  2366
```
```  2367 lemma finite_set: "finite (A::'a::finite set)"
```
```  2368 by (rule finite_subset [OF subset_UNIV finite])
```
```  2369
```
```  2370
```
```  2371 instance unit :: finite
```
```  2372 proof
```
```  2373   have "finite {()}" by simp
```
```  2374   also have "{()} = UNIV" by auto
```
```  2375   finally show "finite (UNIV :: unit set)" .
```
```  2376 qed
```
```  2377
```
```  2378 instance bool :: finite
```
```  2379 proof
```
```  2380   have "finite {True, False}" by simp
```
```  2381   also have "{True, False} = UNIV" by auto
```
```  2382   finally show "finite (UNIV :: bool set)" .
```
```  2383 qed
```
```  2384
```
```  2385
```
```  2386 instance * :: (finite, finite) finite
```
```  2387 proof
```
```  2388   show "finite (UNIV :: ('a \<times> 'b) set)"
```
```  2389   proof (rule finite_Prod_UNIV)
```
```  2390     show "finite (UNIV :: 'a set)" by (rule finite)
```
```  2391     show "finite (UNIV :: 'b set)" by (rule finite)
```
```  2392   qed
```
```  2393 qed
```
```  2394
```
```  2395 instance "+" :: (finite, finite) finite
```
```  2396 proof
```
```  2397   have a: "finite (UNIV :: 'a set)" by (rule finite)
```
```  2398   have b: "finite (UNIV :: 'b set)" by (rule finite)
```
```  2399   from a b have "finite ((UNIV :: 'a set) <+> (UNIV :: 'b set))"
```
```  2400     by (rule finite_Plus)
```
```  2401   thus "finite (UNIV :: ('a + 'b) set)" by simp
```
```  2402 qed
```
```  2403
```
```  2404
```
```  2405 instance set :: (finite) finite
```
```  2406 proof
```
```  2407   have "finite (UNIV :: 'a set)" by (rule finite)
```
```  2408   hence "finite (Pow (UNIV :: 'a set))"
```
```  2409     by (rule finite_Pow_iff [THEN iffD2])
```
```  2410   thus "finite (UNIV :: 'a set set)" by simp
```
```  2411 qed
```
```  2412
```
```  2413 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```  2414 by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
```
```  2415
```
```  2416 instance fun :: (finite, finite) finite
```
```  2417 proof
```
```  2418   show "finite (UNIV :: ('a => 'b) set)"
```
```  2419   proof (rule finite_imageD)
```
```  2420     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```  2421     show "finite (range ?graph)" by (rule finite_set)
```
```  2422     show "inj ?graph" by (rule inj_graph)
```
```  2423   qed
```
```  2424 qed
```
```  2425
```
```  2426 end
```