src/HOL/Finite_Set.thy
 author ballarin Mon Apr 25 17:58:41 2005 +0200 (2005-04-25) changeset 15837 7a567dcd4cda parent 15791 446ec11266be child 16550 e14b89d6ef13 permissions -rw-r--r--
Subsumption of locale interpretations.
```     1 (*  Title:      HOL/Finite_Set.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
```
```     4                 Additions by Jeremy Avigad in Feb 2004
```
```     5 *)
```
```     6
```
```     7 header {* Finite sets *}
```
```     8
```
```     9 theory Finite_Set
```
```    10 imports Divides 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_Un [iff]: "finite (F Un G) = (finite F & finite G)"
```
```   181   by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
```
```   182
```
```   183 lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
```
```   184   -- {* The converse obviously fails. *}
```
```   185   by (blast intro: finite_subset)
```
```   186
```
```   187 lemma finite_insert [simp]: "finite (insert a A) = finite A"
```
```   188   apply (subst insert_is_Un)
```
```   189   apply (simp only: finite_Un, blast)
```
```   190   done
```
```   191
```
```   192 lemma finite_Union[simp, intro]:
```
```   193  "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
```
```   194 by (induct rule:finite_induct) simp_all
```
```   195
```
```   196 lemma finite_empty_induct:
```
```   197   "finite A ==>
```
```   198   P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
```
```   199 proof -
```
```   200   assume "finite A"
```
```   201     and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
```
```   202   have "P (A - A)"
```
```   203   proof -
```
```   204     fix c b :: "'a set"
```
```   205     presume c: "finite c" and b: "finite b"
```
```   206       and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
```
```   207     from c show "c \<subseteq> b ==> P (b - c)"
```
```   208     proof induct
```
```   209       case empty
```
```   210       from P1 show ?case by simp
```
```   211     next
```
```   212       case (insert x F)
```
```   213       have "P (b - F - {x})"
```
```   214       proof (rule P2)
```
```   215         from _ b show "finite (b - F)" by (rule finite_subset) blast
```
```   216         from insert show "x \<in> b - F" by simp
```
```   217         from insert show "P (b - F)" by simp
```
```   218       qed
```
```   219       also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
```
```   220       finally show ?case .
```
```   221     qed
```
```   222   next
```
```   223     show "A \<subseteq> A" ..
```
```   224   qed
```
```   225   thus "P {}" by simp
```
```   226 qed
```
```   227
```
```   228 lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
```
```   229   by (rule Diff_subset [THEN finite_subset])
```
```   230
```
```   231 lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
```
```   232   apply (subst Diff_insert)
```
```   233   apply (case_tac "a : A - B")
```
```   234    apply (rule finite_insert [symmetric, THEN trans])
```
```   235    apply (subst insert_Diff, simp_all)
```
```   236   done
```
```   237
```
```   238
```
```   239 text {* Image and Inverse Image over Finite Sets *}
```
```   240
```
```   241 lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
```
```   242   -- {* The image of a finite set is finite. *}
```
```   243   by (induct set: Finites) simp_all
```
```   244
```
```   245 lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
```
```   246   apply (frule finite_imageI)
```
```   247   apply (erule finite_subset, assumption)
```
```   248   done
```
```   249
```
```   250 lemma finite_range_imageI:
```
```   251     "finite (range g) ==> finite (range (%x. f (g x)))"
```
```   252   apply (drule finite_imageI, simp)
```
```   253   done
```
```   254
```
```   255 lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
```
```   256 proof -
```
```   257   have aux: "!!A. finite (A - {}) = finite A" by simp
```
```   258   fix B :: "'a set"
```
```   259   assume "finite B"
```
```   260   thus "!!A. f`A = B ==> inj_on f A ==> finite A"
```
```   261     apply induct
```
```   262      apply simp
```
```   263     apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
```
```   264      apply clarify
```
```   265      apply (simp (no_asm_use) add: inj_on_def)
```
```   266      apply (blast dest!: aux [THEN iffD1], atomize)
```
```   267     apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
```
```   268     apply (frule subsetD [OF equalityD2 insertI1], clarify)
```
```   269     apply (rule_tac x = xa in bexI)
```
```   270      apply (simp_all add: inj_on_image_set_diff)
```
```   271     done
```
```   272 qed (rule refl)
```
```   273
```
```   274
```
```   275 lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
```
```   276   -- {* The inverse image of a singleton under an injective function
```
```   277          is included in a singleton. *}
```
```   278   apply (auto simp add: inj_on_def)
```
```   279   apply (blast intro: the_equality [symmetric])
```
```   280   done
```
```   281
```
```   282 lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
```
```   283   -- {* The inverse image of a finite set under an injective function
```
```   284          is finite. *}
```
```   285   apply (induct set: Finites, simp_all)
```
```   286   apply (subst vimage_insert)
```
```   287   apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
```
```   288   done
```
```   289
```
```   290
```
```   291 text {* The finite UNION of finite sets *}
```
```   292
```
```   293 lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
```
```   294   by (induct set: Finites) simp_all
```
```   295
```
```   296 text {*
```
```   297   Strengthen RHS to
```
```   298   @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
```
```   299
```
```   300   We'd need to prove
```
```   301   @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
```
```   302   by induction. *}
```
```   303
```
```   304 lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
```
```   305   by (blast intro: finite_UN_I finite_subset)
```
```   306
```
```   307
```
```   308 text {* Sigma of finite sets *}
```
```   309
```
```   310 lemma finite_SigmaI [simp]:
```
```   311     "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   312   by (unfold Sigma_def) (blast intro!: finite_UN_I)
```
```   313
```
```   314 lemma finite_cartesian_product: "[| finite A; finite B |] ==>
```
```   315     finite (A <*> B)"
```
```   316   by (rule finite_SigmaI)
```
```   317
```
```   318 lemma finite_Prod_UNIV:
```
```   319     "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
```
```   320   apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
```
```   321    apply (erule ssubst)
```
```   322    apply (erule finite_SigmaI, auto)
```
```   323   done
```
```   324
```
```   325 lemma finite_cartesian_productD1:
```
```   326      "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
```
```   327 apply (auto simp add: finite_conv_nat_seg_image)
```
```   328 apply (drule_tac x=n in spec)
```
```   329 apply (drule_tac x="fst o f" in spec)
```
```   330 apply (auto simp add: o_def)
```
```   331  prefer 2 apply (force dest!: equalityD2)
```
```   332 apply (drule equalityD1)
```
```   333 apply (rename_tac y x)
```
```   334 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   335  prefer 2 apply force
```
```   336 apply clarify
```
```   337 apply (rule_tac x=k in image_eqI, auto)
```
```   338 done
```
```   339
```
```   340 lemma finite_cartesian_productD2:
```
```   341      "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
```
```   342 apply (auto simp add: finite_conv_nat_seg_image)
```
```   343 apply (drule_tac x=n in spec)
```
```   344 apply (drule_tac x="snd o f" in spec)
```
```   345 apply (auto simp add: o_def)
```
```   346  prefer 2 apply (force dest!: equalityD2)
```
```   347 apply (drule equalityD1)
```
```   348 apply (rename_tac x y)
```
```   349 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   350  prefer 2 apply force
```
```   351 apply clarify
```
```   352 apply (rule_tac x=k in image_eqI, auto)
```
```   353 done
```
```   354
```
```   355
```
```   356 instance unit :: finite
```
```   357 proof
```
```   358   have "finite {()}" by simp
```
```   359   also have "{()} = UNIV" by auto
```
```   360   finally show "finite (UNIV :: unit set)" .
```
```   361 qed
```
```   362
```
```   363 instance * :: (finite, finite) finite
```
```   364 proof
```
```   365   show "finite (UNIV :: ('a \<times> 'b) set)"
```
```   366   proof (rule finite_Prod_UNIV)
```
```   367     show "finite (UNIV :: 'a set)" by (rule finite)
```
```   368     show "finite (UNIV :: 'b set)" by (rule finite)
```
```   369   qed
```
```   370 qed
```
```   371
```
```   372
```
```   373 text {* The powerset of a finite set *}
```
```   374
```
```   375 lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
```
```   376 proof
```
```   377   assume "finite (Pow A)"
```
```   378   with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
```
```   379   thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   380 next
```
```   381   assume "finite A"
```
```   382   thus "finite (Pow A)"
```
```   383     by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
```
```   384 qed
```
```   385
```
```   386
```
```   387 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   388 by(blast intro: finite_subset[OF subset_Pow_Union])
```
```   389
```
```   390
```
```   391 lemma finite_converse [iff]: "finite (r^-1) = finite r"
```
```   392   apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
```
```   393    apply simp
```
```   394    apply (rule iffI)
```
```   395     apply (erule finite_imageD [unfolded inj_on_def])
```
```   396     apply (simp split add: split_split)
```
```   397    apply (erule finite_imageI)
```
```   398   apply (simp add: converse_def image_def, auto)
```
```   399   apply (rule bexI)
```
```   400    prefer 2 apply assumption
```
```   401   apply simp
```
```   402   done
```
```   403
```
```   404
```
```   405 text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
```
```   406 Ehmety) *}
```
```   407
```
```   408 lemma finite_Field: "finite r ==> finite (Field r)"
```
```   409   -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
```
```   410   apply (induct set: Finites)
```
```   411    apply (auto simp add: Field_def Domain_insert Range_insert)
```
```   412   done
```
```   413
```
```   414 lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
```
```   415   apply clarify
```
```   416   apply (erule trancl_induct)
```
```   417    apply (auto simp add: Field_def)
```
```   418   done
```
```   419
```
```   420 lemma finite_trancl: "finite (r^+) = finite r"
```
```   421   apply auto
```
```   422    prefer 2
```
```   423    apply (rule trancl_subset_Field2 [THEN finite_subset])
```
```   424    apply (rule finite_SigmaI)
```
```   425     prefer 3
```
```   426     apply (blast intro: r_into_trancl' finite_subset)
```
```   427    apply (auto simp add: finite_Field)
```
```   428   done
```
```   429
```
```   430
```
```   431 subsection {* A fold functional for finite sets *}
```
```   432
```
```   433 text {* The intended behaviour is
```
```   434 @{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
```
```   435 if @{text f} is associative-commutative. For an application of @{text fold}
```
```   436 se the definitions of sums and products over finite sets.
```
```   437 *}
```
```   438
```
```   439 consts
```
```   440   foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
```
```   441
```
```   442 inductive "foldSet f g z"
```
```   443 intros
```
```   444 emptyI [intro]: "({}, z) : foldSet f g z"
```
```   445 insertI [intro]:
```
```   446      "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
```
```   447       \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
```
```   448
```
```   449 inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
```
```   450
```
```   451 constdefs
```
```   452   fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
```
```   453   "fold f g z A == THE x. (A, x) : foldSet f g z"
```
```   454
```
```   455 text{*A tempting alternative for the definiens is
```
```   456 @{term "if finite A then THE x. (A, x) : foldSet f g e else e"}.
```
```   457 It allows the removal of finiteness assumptions from the theorems
```
```   458 @{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
```
```   459 The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
```
```   460
```
```   461
```
```   462 lemma Diff1_foldSet:
```
```   463   "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
```
```   464 by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
```
```   465
```
```   466 lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
```
```   467   by (induct set: foldSet) auto
```
```   468
```
```   469 lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
```
```   470   by (induct set: Finites) auto
```
```   471
```
```   472
```
```   473 subsubsection {* Commutative monoids *}
```
```   474
```
```   475 locale ACf =
```
```   476   fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
```
```   477   assumes commute: "x \<cdot> y = y \<cdot> x"
```
```   478     and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
```
```   479
```
```   480 locale ACe = ACf +
```
```   481   fixes e :: 'a
```
```   482   assumes ident [simp]: "x \<cdot> e = x"
```
```   483
```
```   484 locale ACIf = ACf +
```
```   485   assumes idem: "x \<cdot> x = x"
```
```   486
```
```   487 lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```   488 proof -
```
```   489   have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
```
```   490   also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
```
```   491   also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
```
```   492   finally show ?thesis .
```
```   493 qed
```
```   494
```
```   495 lemmas (in ACf) AC = assoc commute left_commute
```
```   496
```
```   497 lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
```
```   498 proof -
```
```   499   have "x \<cdot> e = x" by (rule ident)
```
```   500   thus ?thesis by (subst commute)
```
```   501 qed
```
```   502
```
```   503 lemma (in ACIf) idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
```
```   504 proof -
```
```   505   have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
```
```   506   also have "\<dots> = x \<cdot> y" by(simp add:idem)
```
```   507   finally show ?thesis .
```
```   508 qed
```
```   509
```
```   510 lemmas (in ACIf) ACI = AC idem idem2
```
```   511
```
```   512 text{* Interpretation of locales: *}
```
```   513
```
```   514 interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"]
```
```   515 by(auto intro: ACf.intro ACe_axioms.intro add_assoc add_commute)
```
```   516
```
```   517 interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"]
```
```   518   apply -
```
```   519    apply (fast intro: ACf.intro mult_assoc mult_commute)
```
```   520   apply (fastsimp intro: ACe_axioms.intro mult_assoc mult_commute)
```
```   521   done
```
```   522
```
```   523
```
```   524 subsubsection{*From @{term foldSet} to @{term fold}*}
```
```   525
```
```   526 lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
```
```   527 by (auto simp add: less_Suc_eq)
```
```   528
```
```   529 lemma insert_image_inj_on_eq:
```
```   530      "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A;
```
```   531         inj_on h {i. i < Suc m}|]
```
```   532       ==> A = h ` {i. i < m}"
```
```   533 apply (auto simp add: image_less_Suc inj_on_def)
```
```   534 apply (blast intro: less_trans)
```
```   535 done
```
```   536
```
```   537 lemma insert_inj_onE:
```
```   538   assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A"
```
```   539       and inj_on: "inj_on h {i::nat. i<n}"
```
```   540   shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
```
```   541 proof (cases n)
```
```   542   case 0 thus ?thesis using aA by auto
```
```   543 next
```
```   544   case (Suc m)
```
```   545   have nSuc: "n = Suc m" .
```
```   546   have mlessn: "m<n" by (simp add: nSuc)
```
```   547   from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
```
```   548   let ?hm = "swap k m h"
```
```   549   have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn
```
```   550     by (simp add: inj_on_swap_iff inj_on)
```
```   551   show ?thesis
```
```   552   proof (intro exI conjI)
```
```   553     show "inj_on ?hm {i. i < m}" using inj_hm
```
```   554       by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
```
```   555     show "m<n" by (rule mlessn)
```
```   556     show "A = ?hm ` {i. i < m}"
```
```   557     proof (rule insert_image_inj_on_eq)
```
```   558       show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
```
```   559       show "?hm m \<notin> A" by (simp add: swap_def hkeq anot)
```
```   560       show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
```
```   561 	using aA hkeq nSuc klessn
```
```   562 	by (auto simp add: swap_def image_less_Suc fun_upd_image
```
```   563 			   less_Suc_eq inj_on_image_set_diff [OF inj_on])
```
```   564     qed
```
```   565   qed
```
```   566 qed
```
```   567
```
```   568 lemma (in ACf) foldSet_determ_aux:
```
```   569   "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n};
```
```   570                 (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
```
```   571    \<Longrightarrow> x' = x"
```
```   572 proof (induct n rule: less_induct)
```
```   573   case (less n)
```
```   574     have IH: "!!m h A x x'.
```
```   575                \<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m};
```
```   576                 (A,x) \<in> foldSet f g z; (A, x') \<in> foldSet f g z\<rbrakk> \<Longrightarrow> x' = x" .
```
```   577     have Afoldx: "(A,x) \<in> foldSet f g z" and Afoldx': "(A,x') \<in> foldSet f g z"
```
```   578      and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" .
```
```   579     show ?case
```
```   580     proof (rule foldSet.cases [OF Afoldx])
```
```   581       assume "(A, x) = ({}, z)"
```
```   582       with Afoldx' show "x' = x" by blast
```
```   583     next
```
```   584       fix B b u
```
```   585       assume "(A,x) = (insert b B, g b \<cdot> u)" and notinB: "b \<notin> B"
```
```   586          and Bu: "(B,u) \<in> foldSet f g z"
```
```   587       hence AbB: "A = insert b B" and x: "x = g b \<cdot> u" by auto
```
```   588       show "x'=x"
```
```   589       proof (rule foldSet.cases [OF Afoldx'])
```
```   590         assume "(A, x') = ({}, z)"
```
```   591         with AbB show "x' = x" by blast
```
```   592       next
```
```   593 	fix C c v
```
```   594 	assume "(A,x') = (insert c C, g c \<cdot> v)" and notinC: "c \<notin> C"
```
```   595 	   and Cv: "(C,v) \<in> foldSet f g z"
```
```   596 	hence AcC: "A = insert c C" and x': "x' = g c \<cdot> v" by auto
```
```   597 	from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
```
```   598         from insert_inj_onE [OF Beq notinB injh]
```
```   599         obtain hB mB where inj_onB: "inj_on hB {i. i < mB}"
```
```   600                      and Beq: "B = hB ` {i. i < mB}"
```
```   601                      and lessB: "mB < n" by auto
```
```   602 	from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
```
```   603         from insert_inj_onE [OF Ceq notinC injh]
```
```   604         obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
```
```   605                        and Ceq: "C = hC ` {i. i < mC}"
```
```   606                        and lessC: "mC < n" by auto
```
```   607 	show "x'=x"
```
```   608 	proof cases
```
```   609           assume "b=c"
```
```   610 	  then moreover have "B = C" using AbB AcC notinB notinC by auto
```
```   611 	  ultimately show ?thesis  using Bu Cv x x' IH[OF lessC Ceq inj_onC]
```
```   612             by auto
```
```   613 	next
```
```   614 	  assume diff: "b \<noteq> c"
```
```   615 	  let ?D = "B - {c}"
```
```   616 	  have B: "B = insert c ?D" and C: "C = insert b ?D"
```
```   617 	    using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
```
```   618 	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
```
```   619 	  with AbB have "finite ?D" by simp
```
```   620 	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g z"
```
```   621 	    using finite_imp_foldSet by rules
```
```   622 	  moreover have cinB: "c \<in> B" using B by auto
```
```   623 	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
```
```   624 	    by(rule Diff1_foldSet)
```
```   625 	  hence "g c \<cdot> d = u" by (rule IH [OF lessB Beq inj_onB Bu])
```
```   626           moreover have "g b \<cdot> d = v"
```
```   627 	  proof (rule IH[OF lessC Ceq inj_onC Cv])
```
```   628 	    show "(C, g b \<cdot> d) \<in> foldSet f g z" using C notinB Dfoldd
```
```   629 	      by fastsimp
```
```   630 	  qed
```
```   631 	  ultimately show ?thesis using x x' by (auto simp: AC)
```
```   632 	qed
```
```   633       qed
```
```   634     qed
```
```   635   qed
```
```   636
```
```   637
```
```   638 lemma (in ACf) foldSet_determ:
```
```   639   "(A,x) : foldSet f g z ==> (A,y) : foldSet f g z ==> y = x"
```
```   640 apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on])
```
```   641 apply (blast intro: foldSet_determ_aux [rule_format])
```
```   642 done
```
```   643
```
```   644 lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y"
```
```   645   by (unfold fold_def) (blast intro: foldSet_determ)
```
```   646
```
```   647 text{* The base case for @{text fold}: *}
```
```   648
```
```   649 lemma fold_empty [simp]: "fold f g z {} = z"
```
```   650   by (unfold fold_def) blast
```
```   651
```
```   652 lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
```
```   653     ((insert x A, v) : foldSet f g z) =
```
```   654     (EX y. (A, y) : foldSet f g z & v = f (g x) y)"
```
```   655   apply auto
```
```   656   apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
```
```   657    apply (fastsimp dest: foldSet_imp_finite)
```
```   658   apply (blast intro: foldSet_determ)
```
```   659   done
```
```   660
```
```   661 text{* The recursion equation for @{text fold}: *}
```
```   662
```
```   663 lemma (in ACf) fold_insert[simp]:
```
```   664     "finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
```
```   665   apply (unfold fold_def)
```
```   666   apply (simp add: fold_insert_aux)
```
```   667   apply (rule the_equality)
```
```   668   apply (auto intro: finite_imp_foldSet
```
```   669     cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
```
```   670   done
```
```   671
```
```   672 lemma (in ACf) fold_rec:
```
```   673 assumes fin: "finite A" and a: "a:A"
```
```   674 shows "fold f g z A = f (g a) (fold f g z (A - {a}))"
```
```   675 proof-
```
```   676   have A: "A = insert a (A - {a})" using a by blast
```
```   677   hence "fold f g z A = fold f g z (insert a (A - {a}))" by simp
```
```   678   also have "\<dots> = f (g a) (fold f g z (A - {a}))"
```
```   679     by(rule fold_insert) (simp add:fin)+
```
```   680   finally show ?thesis .
```
```   681 qed
```
```   682
```
```   683
```
```   684 text{* A simplified version for idempotent functions: *}
```
```   685
```
```   686 lemma (in ACIf) fold_insert_idem:
```
```   687 assumes finA: "finite A"
```
```   688 shows "fold f g z (insert a A) = g a \<cdot> fold f g z A"
```
```   689 proof cases
```
```   690   assume "a \<in> A"
```
```   691   then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
```
```   692     by(blast dest: mk_disjoint_insert)
```
```   693   show ?thesis
```
```   694   proof -
```
```   695     from finA A have finB: "finite B" by(blast intro: finite_subset)
```
```   696     have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
```
```   697     also have "\<dots> = (g a) \<cdot> (fold f g z B)"
```
```   698       using finB disj by simp
```
```   699     also have "\<dots> = g a \<cdot> fold f g z A"
```
```   700       using A finB disj by(simp add:idem assoc[symmetric])
```
```   701     finally show ?thesis .
```
```   702   qed
```
```   703 next
```
```   704   assume "a \<notin> A"
```
```   705   with finA show ?thesis by simp
```
```   706 qed
```
```   707
```
```   708 lemma (in ACIf) foldI_conv_id:
```
```   709   "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
```
```   710 by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert)
```
```   711
```
```   712 subsubsection{*Lemmas about @{text fold}*}
```
```   713
```
```   714 lemma (in ACf) fold_commute:
```
```   715   "finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
```
```   716   apply (induct set: Finites, simp)
```
```   717   apply (simp add: left_commute [of x])
```
```   718   done
```
```   719
```
```   720 lemma (in ACf) fold_nest_Un_Int:
```
```   721   "finite A ==> finite B
```
```   722     ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
```
```   723   apply (induct set: Finites, simp)
```
```   724   apply (simp add: fold_commute Int_insert_left insert_absorb)
```
```   725   done
```
```   726
```
```   727 lemma (in ACf) fold_nest_Un_disjoint:
```
```   728   "finite A ==> finite B ==> A Int B = {}
```
```   729     ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
```
```   730   by (simp add: fold_nest_Un_Int)
```
```   731
```
```   732 lemma (in ACf) fold_reindex:
```
```   733 assumes fin: "finite A"
```
```   734 shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
```
```   735 using fin apply induct
```
```   736  apply simp
```
```   737 apply simp
```
```   738 done
```
```   739
```
```   740 lemma (in ACe) fold_Un_Int:
```
```   741   "finite A ==> finite B ==>
```
```   742     fold f g e A \<cdot> fold f g e B =
```
```   743     fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
```
```   744   apply (induct set: Finites, simp)
```
```   745   apply (simp add: AC insert_absorb Int_insert_left)
```
```   746   done
```
```   747
```
```   748 corollary (in ACe) fold_Un_disjoint:
```
```   749   "finite A ==> finite B ==> A Int B = {} ==>
```
```   750     fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
```
```   751   by (simp add: fold_Un_Int)
```
```   752
```
```   753 lemma (in ACe) fold_UN_disjoint:
```
```   754   "\<lbrakk> finite I; ALL i:I. finite (A i);
```
```   755      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
```
```   756    \<Longrightarrow> fold f g e (UNION I A) =
```
```   757        fold f (%i. fold f g e (A i)) e I"
```
```   758   apply (induct set: Finites, simp, atomize)
```
```   759   apply (subgoal_tac "ALL i:F. x \<noteq> i")
```
```   760    prefer 2 apply blast
```
```   761   apply (subgoal_tac "A x Int UNION F A = {}")
```
```   762    prefer 2 apply blast
```
```   763   apply (simp add: fold_Un_disjoint)
```
```   764   done
```
```   765
```
```   766 text{*Fusion theorem, as described in
```
```   767 Graham Hutton's paper,
```
```   768 A Tutorial on the Universality and Expressiveness of Fold,
```
```   769 JFP 9:4 (355-372), 1999.*}
```
```   770 lemma (in ACf) fold_fusion:
```
```   771       includes ACf g
```
```   772       shows
```
```   773 	"finite A ==>
```
```   774 	 (!!x y. h (g x y) = f x (h y)) ==>
```
```   775          h (fold g j w A) = fold f j (h w) A"
```
```   776   by (induct set: Finites, simp_all)
```
```   777
```
```   778 lemma (in ACf) fold_cong:
```
```   779   "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
```
```   780   apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
```
```   781    apply simp
```
```   782   apply (erule finite_induct, simp)
```
```   783   apply (simp add: subset_insert_iff, clarify)
```
```   784   apply (subgoal_tac "finite C")
```
```   785    prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
```
```   786   apply (subgoal_tac "C = insert x (C - {x})")
```
```   787    prefer 2 apply blast
```
```   788   apply (erule ssubst)
```
```   789   apply (drule spec)
```
```   790   apply (erule (1) notE impE)
```
```   791   apply (simp add: Ball_def del: insert_Diff_single)
```
```   792   done
```
```   793
```
```   794 lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   795   fold f (%x. fold f (g x) e (B x)) e A =
```
```   796   fold f (split g) e (SIGMA x:A. B x)"
```
```   797 apply (subst Sigma_def)
```
```   798 apply (subst fold_UN_disjoint, assumption, simp)
```
```   799  apply blast
```
```   800 apply (erule fold_cong)
```
```   801 apply (subst fold_UN_disjoint, simp, simp)
```
```   802  apply blast
```
```   803 apply simp
```
```   804 done
```
```   805
```
```   806 lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
```
```   807    fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
```
```   808 apply (erule finite_induct, simp)
```
```   809 apply (simp add:AC)
```
```   810 done
```
```   811
```
```   812
```
```   813 subsection {* Generalized summation over a set *}
```
```   814
```
```   815 constdefs
```
```   816   setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
```
```   817   "setsum f A == if finite A then fold (op +) f 0 A else 0"
```
```   818
```
```   819 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```   820 written @{text"\<Sum>x\<in>A. e"}. *}
```
```   821
```
```   822 syntax
```
```   823   "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```   824 syntax (xsymbols)
```
```   825   "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   826 syntax (HTML output)
```
```   827   "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   828
```
```   829 translations -- {* Beware of argument permutation! *}
```
```   830   "SUM i:A. b" == "setsum (%i. b) A"
```
```   831   "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
```
```   832
```
```   833 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```   834  @{text"\<Sum>x|P. e"}. *}
```
```   835
```
```   836 syntax
```
```   837   "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   838 syntax (xsymbols)
```
```   839   "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   840 syntax (HTML output)
```
```   841   "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   842
```
```   843 translations
```
```   844   "SUM x|P. t" => "setsum (%x. t) {x. P}"
```
```   845   "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
```
```   846
```
```   847 text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
```
```   848
```
```   849 syntax
```
```   850   "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
```
```   851
```
```   852 parse_translation {*
```
```   853   let
```
```   854     fun Setsum_tr [A] = Syntax.const "setsum" \$ Abs ("", dummyT, Bound 0) \$ A
```
```   855   in [("_Setsum", Setsum_tr)] end;
```
```   856 *}
```
```   857
```
```   858 print_translation {*
```
```   859 let
```
```   860   fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" \$ A
```
```   861     | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) \$ Abs(y,Ty,P)] =
```
```   862        if x<>y then raise Match
```
```   863        else let val x' = Syntax.mark_bound x
```
```   864                 val t' = subst_bound(x',t)
```
```   865                 val P' = subst_bound(x',P)
```
```   866             in Syntax.const "_qsetsum" \$ Syntax.mark_bound x \$ P' \$ t' end
```
```   867 in
```
```   868 [("setsum", setsum_tr')]
```
```   869 end
```
```   870 *}
```
```   871
```
```   872 lemma setsum_empty [simp]: "setsum f {} = 0"
```
```   873   by (simp add: setsum_def)
```
```   874
```
```   875 lemma setsum_insert [simp]:
```
```   876     "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   877   by (simp add: setsum_def)
```
```   878
```
```   879 lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
```
```   880   by (simp add: setsum_def)
```
```   881
```
```   882 lemma setsum_reindex:
```
```   883      "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   884 by(auto simp add: setsum_def AC_add.fold_reindex dest!:finite_imageD)
```
```   885
```
```   886 lemma setsum_reindex_id:
```
```   887      "inj_on f B ==> setsum f B = setsum id (f ` B)"
```
```   888 by (auto simp add: setsum_reindex)
```
```   889
```
```   890 lemma setsum_cong:
```
```   891   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   892 by(fastsimp simp: setsum_def intro: AC_add.fold_cong)
```
```   893
```
```   894 lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
```
```   895   by (rule setsum_cong[OF refl], auto);
```
```   896
```
```   897 lemma setsum_reindex_cong:
```
```   898      "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
```
```   899       ==> setsum h B = setsum g A"
```
```   900   by (simp add: setsum_reindex cong: setsum_cong)
```
```   901
```
```   902 lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
```
```   903 apply (clarsimp simp: setsum_def)
```
```   904 apply (erule finite_induct, auto)
```
```   905 done
```
```   906
```
```   907 lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
```
```   908 by(simp add:setsum_cong)
```
```   909
```
```   910 lemma setsum_Un_Int: "finite A ==> finite B ==>
```
```   911   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   912   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   913 by(simp add: setsum_def AC_add.fold_Un_Int [symmetric])
```
```   914
```
```   915 lemma setsum_Un_disjoint: "finite A ==> finite B
```
```   916   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   917 by (subst setsum_Un_Int [symmetric], auto)
```
```   918
```
```   919 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   920   the lhs need not be, since UNION I A could still be finite.*)
```
```   921 lemma setsum_UN_disjoint:
```
```   922     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```   923         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```   924       setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   925 by(simp add: setsum_def AC_add.fold_UN_disjoint cong: setsum_cong)
```
```   926
```
```   927 text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
```
```   928 directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
```
```   929 lemma setsum_Union_disjoint:
```
```   930   "[| (ALL A:C. finite A);
```
```   931       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```   932    ==> setsum f (Union C) = setsum (setsum f) C"
```
```   933 apply (cases "finite C")
```
```   934  prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
```
```   935   apply (frule setsum_UN_disjoint [of C id f])
```
```   936  apply (unfold Union_def id_def, assumption+)
```
```   937 done
```
```   938
```
```   939 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```   940   the rhs need not be, since SIGMA A B could still be finite.*)
```
```   941 lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   942     (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
```
```   943     (\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
```
```   944 by(simp add:setsum_def AC_add.fold_Sigma split_def cong:setsum_cong)
```
```   945
```
```   946 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```   947 lemma setsum_cartesian_product:
```
```   948    "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>z\<in>A <*> B. f (fst z) (snd z))"
```
```   949 apply (cases "finite A")
```
```   950  apply (cases "finite B")
```
```   951   apply (simp add: setsum_Sigma)
```
```   952  apply (cases "A={}", simp)
```
```   953  apply (simp)
```
```   954 apply (auto simp add: setsum_def
```
```   955             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   956 done
```
```   957
```
```   958 lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```   959 by(simp add:setsum_def AC_add.fold_distrib)
```
```   960
```
```   961
```
```   962 subsubsection {* Properties in more restricted classes of structures *}
```
```   963
```
```   964 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   965   apply (case_tac "finite A")
```
```   966    prefer 2 apply (simp add: setsum_def)
```
```   967   apply (erule rev_mp)
```
```   968   apply (erule finite_induct, auto)
```
```   969   done
```
```   970
```
```   971 lemma setsum_eq_0_iff [simp]:
```
```   972     "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   973   by (induct set: Finites) auto
```
```   974
```
```   975 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   976     (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   977   -- {* For the natural numbers, we have subtraction. *}
```
```   978   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
```
```   979
```
```   980 lemma setsum_Un: "finite A ==> finite B ==>
```
```   981     (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   982       setsum f A + setsum f B - setsum f (A Int B)"
```
```   983   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
```
```   984
```
```   985 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   986     (if a:A then setsum f A - f a else setsum f A)"
```
```   987   apply (case_tac "finite A")
```
```   988    prefer 2 apply (simp add: setsum_def)
```
```   989   apply (erule finite_induct)
```
```   990    apply (auto simp add: insert_Diff_if)
```
```   991   apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   992   done
```
```   993
```
```   994 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   995   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   996   (if a:A then setsum f A - f a else setsum f A)"
```
```   997   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   998
```
```   999 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)"
```
```  1000   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))"])
```
```  1001   apply (auto simp add: insert_Diff_if add_ac)
```
```  1002   done
```
```  1003
```
```  1004 (* By Jeremy Siek: *)
```
```  1005
```
```  1006 lemma setsum_diff_nat:
```
```  1007   assumes finB: "finite B"
```
```  1008   shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```  1009 using finB
```
```  1010 proof (induct)
```
```  1011   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```  1012 next
```
```  1013   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```  1014     and xFinA: "insert x F \<subseteq> A"
```
```  1015     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```  1016   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```  1017   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```  1018     by (simp add: setsum_diff1_nat)
```
```  1019   from xFinA have "F \<subseteq> A" by simp
```
```  1020   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```  1021   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```  1022     by simp
```
```  1023   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```  1024   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```  1025     by simp
```
```  1026   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```  1027   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```  1028     by simp
```
```  1029   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```  1030 qed
```
```  1031
```
```  1032 lemma setsum_diff:
```
```  1033   assumes le: "finite A" "B \<subseteq> A"
```
```  1034   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```  1035 proof -
```
```  1036   from le have finiteB: "finite B" using finite_subset by auto
```
```  1037   show ?thesis using finiteB le
```
```  1038     proof (induct)
```
```  1039       case empty
```
```  1040       thus ?case by auto
```
```  1041     next
```
```  1042       case (insert x F)
```
```  1043       thus ?case using le finiteB
```
```  1044 	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```  1045     qed
```
```  1046   qed
```
```  1047
```
```  1048 lemma setsum_mono:
```
```  1049   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
```
```  1050   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```  1051 proof (cases "finite K")
```
```  1052   case True
```
```  1053   thus ?thesis using le
```
```  1054   proof (induct)
```
```  1055     case empty
```
```  1056     thus ?case by simp
```
```  1057   next
```
```  1058     case insert
```
```  1059     thus ?case using add_mono
```
```  1060       by force
```
```  1061   qed
```
```  1062 next
```
```  1063   case False
```
```  1064   thus ?thesis
```
```  1065     by (simp add: setsum_def)
```
```  1066 qed
```
```  1067
```
```  1068 lemma setsum_strict_mono:
```
```  1069 fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```  1070 assumes fin_ne: "finite A"  "A \<noteq> {}"
```
```  1071 shows "(!!x. x:A \<Longrightarrow> f x < g x) \<Longrightarrow> setsum f A < setsum g A"
```
```  1072 using fin_ne
```
```  1073 proof (induct rule: finite_ne_induct)
```
```  1074   case singleton thus ?case by simp
```
```  1075 next
```
```  1076   case insert thus ?case by (auto simp: add_strict_mono)
```
```  1077 qed
```
```  1078
```
```  1079 lemma setsum_negf:
```
```  1080  "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```  1081 proof (cases "finite A")
```
```  1082   case True thus ?thesis by (induct set: Finites, auto)
```
```  1083 next
```
```  1084   case False thus ?thesis by (simp add: setsum_def)
```
```  1085 qed
```
```  1086
```
```  1087 lemma setsum_subtractf:
```
```  1088  "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```  1089   setsum f A - setsum g A"
```
```  1090 proof (cases "finite A")
```
```  1091   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
```
```  1092 next
```
```  1093   case False thus ?thesis by (simp add: setsum_def)
```
```  1094 qed
```
```  1095
```
```  1096 lemma setsum_nonneg:
```
```  1097 assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```  1098 shows "0 \<le> setsum f A"
```
```  1099 proof (cases "finite A")
```
```  1100   case True thus ?thesis using nn
```
```  1101   apply (induct set: Finites, auto)
```
```  1102   apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
```
```  1103   apply (blast intro: add_mono)
```
```  1104   done
```
```  1105 next
```
```  1106   case False thus ?thesis by (simp add: setsum_def)
```
```  1107 qed
```
```  1108
```
```  1109 lemma setsum_nonpos:
```
```  1110 assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
```
```  1111 shows "setsum f A \<le> 0"
```
```  1112 proof (cases "finite A")
```
```  1113   case True thus ?thesis using np
```
```  1114   apply (induct set: Finites, auto)
```
```  1115   apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
```
```  1116   apply (blast intro: add_mono)
```
```  1117   done
```
```  1118 next
```
```  1119   case False thus ?thesis by (simp add: setsum_def)
```
```  1120 qed
```
```  1121
```
```  1122 lemma setsum_mono2:
```
```  1123 fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
```
```  1124 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```  1125 shows "setsum f A \<le> setsum f B"
```
```  1126 proof -
```
```  1127   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```  1128     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```  1129   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```  1130     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```  1131   also have "A \<union> (B-A) = B" using sub by blast
```
```  1132   finally show ?thesis .
```
```  1133 qed
```
```  1134
```
```  1135 (* FIXME: this is distributitivty, name as such! *)
```
```  1136
```
```  1137 lemma setsum_mult:
```
```  1138   fixes f :: "'a => ('b::semiring_0_cancel)"
```
```  1139   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```  1140 proof (cases "finite A")
```
```  1141   case True
```
```  1142   thus ?thesis
```
```  1143   proof (induct)
```
```  1144     case empty thus ?case by simp
```
```  1145   next
```
```  1146     case (insert x A) thus ?case by (simp add: right_distrib)
```
```  1147   qed
```
```  1148 next
```
```  1149   case False thus ?thesis by (simp add: setsum_def)
```
```  1150 qed
```
```  1151
```
```  1152 lemma setsum_abs[iff]:
```
```  1153   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1154   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```  1155 proof (cases "finite A")
```
```  1156   case True
```
```  1157   thus ?thesis
```
```  1158   proof (induct)
```
```  1159     case empty thus ?case by simp
```
```  1160   next
```
```  1161     case (insert x A)
```
```  1162     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```  1163   qed
```
```  1164 next
```
```  1165   case False thus ?thesis by (simp add: setsum_def)
```
```  1166 qed
```
```  1167
```
```  1168 lemma setsum_abs_ge_zero[iff]:
```
```  1169   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1170   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```  1171 proof (cases "finite A")
```
```  1172   case True
```
```  1173   thus ?thesis
```
```  1174   proof (induct)
```
```  1175     case empty thus ?case by simp
```
```  1176   next
```
```  1177     case (insert x A) thus ?case by (auto intro: order_trans)
```
```  1178   qed
```
```  1179 next
```
```  1180   case False thus ?thesis by (simp add: setsum_def)
```
```  1181 qed
```
```  1182
```
```  1183 lemma abs_setsum_abs[simp]:
```
```  1184   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1185   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```  1186 proof (cases "finite A")
```
```  1187   case True
```
```  1188   thus ?thesis
```
```  1189   proof (induct)
```
```  1190     case empty thus ?case by simp
```
```  1191   next
```
```  1192     case (insert a A)
```
```  1193     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
```
```  1194     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```  1195     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>" by simp
```
```  1196     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```  1197     finally show ?case .
```
```  1198   qed
```
```  1199 next
```
```  1200   case False thus ?thesis by (simp add: setsum_def)
```
```  1201 qed
```
```  1202
```
```  1203
```
```  1204 subsection {* Generalized product over a set *}
```
```  1205
```
```  1206 constdefs
```
```  1207   setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
```
```  1208   "setprod f A == if finite A then fold (op *) f 1 A else 1"
```
```  1209
```
```  1210 syntax
```
```  1211   "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_:_. _)" [0, 51, 10] 10)
```
```  1212
```
```  1213 syntax (xsymbols)
```
```  1214   "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1215 syntax (HTML output)
```
```  1216   "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1217 translations
```
```  1218   "\<Prod>i:A. b" == "setprod (%i. b) A"  -- {* Beware of argument permutation! *}
```
```  1219
```
```  1220 syntax
```
```  1221   "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
```
```  1222
```
```  1223 parse_translation {*
```
```  1224   let
```
```  1225     fun Setprod_tr [A] = Syntax.const "setprod" \$ Abs ("", dummyT, Bound 0) \$ A
```
```  1226   in [("_Setprod", Setprod_tr)] end;
```
```  1227 *}
```
```  1228 print_translation {*
```
```  1229 let fun setprod_tr' [Abs(x,Tx,t), A] =
```
```  1230     if t = Bound 0 then Syntax.const "_Setprod" \$ A else raise Match
```
```  1231 in
```
```  1232 [("setprod", setprod_tr')]
```
```  1233 end
```
```  1234 *}
```
```  1235
```
```  1236
```
```  1237 lemma setprod_empty [simp]: "setprod f {} = 1"
```
```  1238   by (auto simp add: setprod_def)
```
```  1239
```
```  1240 lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
```
```  1241     setprod f (insert a A) = f a * setprod f A"
```
```  1242 by (simp add: setprod_def)
```
```  1243
```
```  1244 lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
```
```  1245   by (simp add: setprod_def)
```
```  1246
```
```  1247 lemma setprod_reindex:
```
```  1248      "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```  1249 by(auto simp: setprod_def AC_mult.fold_reindex dest!:finite_imageD)
```
```  1250
```
```  1251 lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```  1252 by (auto simp add: setprod_reindex)
```
```  1253
```
```  1254 lemma setprod_cong:
```
```  1255   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```  1256 by(fastsimp simp: setprod_def intro: AC_mult.fold_cong)
```
```  1257
```
```  1258 lemma setprod_reindex_cong: "inj_on f A ==>
```
```  1259     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```  1260   by (frule setprod_reindex, simp)
```
```  1261
```
```  1262
```
```  1263 lemma setprod_1: "setprod (%i. 1) A = 1"
```
```  1264   apply (case_tac "finite A")
```
```  1265   apply (erule finite_induct, auto simp add: mult_ac)
```
```  1266   done
```
```  1267
```
```  1268 lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
```
```  1269   apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
```
```  1270   apply (erule ssubst, rule setprod_1)
```
```  1271   apply (rule setprod_cong, auto)
```
```  1272   done
```
```  1273
```
```  1274 lemma setprod_Un_Int: "finite A ==> finite B
```
```  1275     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```  1276 by(simp add: setprod_def AC_mult.fold_Un_Int[symmetric])
```
```  1277
```
```  1278 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```  1279   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```  1280 by (subst setprod_Un_Int [symmetric], auto)
```
```  1281
```
```  1282 lemma setprod_UN_disjoint:
```
```  1283     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1284         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1285       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```  1286 by(simp add: setprod_def AC_mult.fold_UN_disjoint cong: setprod_cong)
```
```  1287
```
```  1288 lemma setprod_Union_disjoint:
```
```  1289   "[| (ALL A:C. finite A);
```
```  1290       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```  1291    ==> setprod f (Union C) = setprod (setprod f) C"
```
```  1292 apply (cases "finite C")
```
```  1293  prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
```
```  1294   apply (frule setprod_UN_disjoint [of C id f])
```
```  1295  apply (unfold Union_def id_def, assumption+)
```
```  1296 done
```
```  1297
```
```  1298 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1299     (\<Prod>x:A. (\<Prod>y: B x. f x y)) =
```
```  1300     (\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))"
```
```  1301 by(simp add:setprod_def AC_mult.fold_Sigma split_def cong:setprod_cong)
```
```  1302
```
```  1303 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1304 lemma setprod_cartesian_product:
```
```  1305      "(\<Prod>x:A. (\<Prod>y: B. f x y)) = (\<Prod>z:(A <*> B). f (fst z) (snd z))"
```
```  1306 apply (cases "finite A")
```
```  1307  apply (cases "finite B")
```
```  1308   apply (simp add: setprod_Sigma)
```
```  1309  apply (cases "A={}", simp)
```
```  1310  apply (simp add: setprod_1)
```
```  1311 apply (auto simp add: setprod_def
```
```  1312             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1313 done
```
```  1314
```
```  1315 lemma setprod_timesf:
```
```  1316      "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
```
```  1317 by(simp add:setprod_def AC_mult.fold_distrib)
```
```  1318
```
```  1319
```
```  1320 subsubsection {* Properties in more restricted classes of structures *}
```
```  1321
```
```  1322 lemma setprod_eq_1_iff [simp]:
```
```  1323     "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
```
```  1324   by (induct set: Finites) auto
```
```  1325
```
```  1326 lemma setprod_zero:
```
```  1327      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
```
```  1328   apply (induct set: Finites, force, clarsimp)
```
```  1329   apply (erule disjE, auto)
```
```  1330   done
```
```  1331
```
```  1332 lemma setprod_nonneg [rule_format]:
```
```  1333      "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1334   apply (case_tac "finite A")
```
```  1335   apply (induct set: Finites, force, clarsimp)
```
```  1336   apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
```
```  1337   apply (rule mult_mono, assumption+)
```
```  1338   apply (auto simp add: setprod_def)
```
```  1339   done
```
```  1340
```
```  1341 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
```
```  1342      --> 0 < setprod f A"
```
```  1343   apply (case_tac "finite A")
```
```  1344   apply (induct set: Finites, force, clarsimp)
```
```  1345   apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
```
```  1346   apply (rule mult_strict_mono, assumption+)
```
```  1347   apply (auto simp add: setprod_def)
```
```  1348   done
```
```  1349
```
```  1350 lemma setprod_nonzero [rule_format]:
```
```  1351     "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
```
```  1352       finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
```
```  1353   apply (erule finite_induct, auto)
```
```  1354   done
```
```  1355
```
```  1356 lemma setprod_zero_eq:
```
```  1357     "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
```
```  1358      finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
```
```  1359   apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
```
```  1360   done
```
```  1361
```
```  1362 lemma setprod_nonzero_field:
```
```  1363     "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
```
```  1364   apply (rule setprod_nonzero, auto)
```
```  1365   done
```
```  1366
```
```  1367 lemma setprod_zero_eq_field:
```
```  1368     "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
```
```  1369   apply (rule setprod_zero_eq, auto)
```
```  1370   done
```
```  1371
```
```  1372 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1373     (setprod f (A Un B) :: 'a ::{field})
```
```  1374       = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1375   apply (subst setprod_Un_Int [symmetric], auto)
```
```  1376   apply (subgoal_tac "finite (A Int B)")
```
```  1377   apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
```
```  1378   apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
```
```  1379   done
```
```  1380
```
```  1381 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1382     (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1383       (if a:A then setprod f A / f a else setprod f A)"
```
```  1384   apply (erule finite_induct)
```
```  1385    apply (auto simp add: insert_Diff_if)
```
```  1386   apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
```
```  1387   apply (erule ssubst)
```
```  1388   apply (subst times_divide_eq_right [THEN sym])
```
```  1389   apply (auto simp add: mult_ac times_divide_eq_right divide_self)
```
```  1390   done
```
```  1391
```
```  1392 lemma setprod_inversef: "finite A ==>
```
```  1393     ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
```
```  1394       setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1395   apply (erule finite_induct)
```
```  1396   apply (simp, simp)
```
```  1397   done
```
```  1398
```
```  1399 lemma setprod_dividef:
```
```  1400      "[|finite A;
```
```  1401         \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
```
```  1402       ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1403   apply (subgoal_tac
```
```  1404          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1405   apply (erule ssubst)
```
```  1406   apply (subst divide_inverse)
```
```  1407   apply (subst setprod_timesf)
```
```  1408   apply (subst setprod_inversef, assumption+, rule refl)
```
```  1409   apply (rule setprod_cong, rule refl)
```
```  1410   apply (subst divide_inverse, auto)
```
```  1411   done
```
```  1412
```
```  1413 subsection {* Finite cardinality *}
```
```  1414
```
```  1415 text {* This definition, although traditional, is ugly to work with:
```
```  1416 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
```
```  1417 But now that we have @{text setsum} things are easy:
```
```  1418 *}
```
```  1419
```
```  1420 constdefs
```
```  1421   card :: "'a set => nat"
```
```  1422   "card A == setsum (%x. 1::nat) A"
```
```  1423
```
```  1424 lemma card_empty [simp]: "card {} = 0"
```
```  1425   by (simp add: card_def)
```
```  1426
```
```  1427 lemma card_infinite [simp]: "~ finite A ==> card A = 0"
```
```  1428   by (simp add: card_def)
```
```  1429
```
```  1430 lemma card_eq_setsum: "card A = setsum (%x. 1) A"
```
```  1431 by (simp add: card_def)
```
```  1432
```
```  1433 lemma card_insert_disjoint [simp]:
```
```  1434   "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
```
```  1435 by(simp add: card_def)
```
```  1436
```
```  1437 lemma card_insert_if:
```
```  1438     "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
```
```  1439   by (simp add: insert_absorb)
```
```  1440
```
```  1441 lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
```
```  1442   apply auto
```
```  1443   apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
```
```  1444   done
```
```  1445
```
```  1446 lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
```
```  1447 by auto
```
```  1448
```
```  1449 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
```
```  1450 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1451 apply(simp del:insert_Diff_single)
```
```  1452 done
```
```  1453
```
```  1454 lemma card_Diff_singleton:
```
```  1455     "finite A ==> x: A ==> card (A - {x}) = card A - 1"
```
```  1456   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1457
```
```  1458 lemma card_Diff_singleton_if:
```
```  1459     "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
```
```  1460   by (simp add: card_Diff_singleton)
```
```  1461
```
```  1462 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1463   by (simp add: card_insert_if card_Suc_Diff1)
```
```  1464
```
```  1465 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1466   by (simp add: card_insert_if)
```
```  1467
```
```  1468 lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
```
```  1469 by (simp add: card_def setsum_mono2)
```
```  1470
```
```  1471 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1472   apply (induct set: Finites, simp, clarify)
```
```  1473   apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1474    prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1475   apply (drule_tac x = "A - {x}" in spec)
```
```  1476   apply (simp add: card_Diff_singleton_if split add: split_if_asm)
```
```  1477   apply (case_tac "card A", auto)
```
```  1478   done
```
```  1479
```
```  1480 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1481   apply (simp add: psubset_def linorder_not_le [symmetric])
```
```  1482   apply (blast dest: card_seteq)
```
```  1483   done
```
```  1484
```
```  1485 lemma card_Un_Int: "finite A ==> finite B
```
```  1486     ==> card A + card B = card (A Un B) + card (A Int B)"
```
```  1487 by(simp add:card_def setsum_Un_Int)
```
```  1488
```
```  1489 lemma card_Un_disjoint: "finite A ==> finite B
```
```  1490     ==> A Int B = {} ==> card (A Un B) = card A + card B"
```
```  1491   by (simp add: card_Un_Int)
```
```  1492
```
```  1493 lemma card_Diff_subset:
```
```  1494   "finite B ==> B <= A ==> card (A - B) = card A - card B"
```
```  1495 by(simp add:card_def setsum_diff_nat)
```
```  1496
```
```  1497 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1498   apply (rule Suc_less_SucD)
```
```  1499   apply (simp add: card_Suc_Diff1)
```
```  1500   done
```
```  1501
```
```  1502 lemma card_Diff2_less:
```
```  1503     "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1504   apply (case_tac "x = y")
```
```  1505    apply (simp add: card_Diff1_less)
```
```  1506   apply (rule less_trans)
```
```  1507    prefer 2 apply (auto intro!: card_Diff1_less)
```
```  1508   done
```
```  1509
```
```  1510 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1511   apply (case_tac "x : A")
```
```  1512    apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1513   done
```
```  1514
```
```  1515 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1516 by (erule psubsetI, blast)
```
```  1517
```
```  1518 lemma insert_partition:
```
```  1519   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1520   \<Longrightarrow> x \<inter> \<Union> F = {}"
```
```  1521 by auto
```
```  1522
```
```  1523 (* main cardinality theorem *)
```
```  1524 lemma card_partition [rule_format]:
```
```  1525      "finite C ==>
```
```  1526         finite (\<Union> C) -->
```
```  1527         (\<forall>c\<in>C. card c = k) -->
```
```  1528         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  1529         k * card(C) = card (\<Union> C)"
```
```  1530 apply (erule finite_induct, simp)
```
```  1531 apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition
```
```  1532        finite_subset [of _ "\<Union> (insert x F)"])
```
```  1533 done
```
```  1534
```
```  1535
```
```  1536 lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
```
```  1537 apply (cases "finite A")
```
```  1538 apply (erule finite_induct)
```
```  1539 apply (auto simp add: ring_distrib add_ac)
```
```  1540 done
```
```  1541
```
```  1542
```
```  1543 lemma setprod_constant: "finite A ==> (\<Prod>x: A. (y::'a::recpower)) = y^(card A)"
```
```  1544   apply (erule finite_induct)
```
```  1545   apply (auto simp add: power_Suc)
```
```  1546   done
```
```  1547
```
```  1548 lemma setsum_bounded:
```
```  1549   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{comm_semiring_1_cancel, pordered_ab_semigroup_add})"
```
```  1550   shows "setsum f A \<le> of_nat(card A) * K"
```
```  1551 proof (cases "finite A")
```
```  1552   case True
```
```  1553   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```  1554 next
```
```  1555   case False thus ?thesis by (simp add: setsum_def)
```
```  1556 qed
```
```  1557
```
```  1558
```
```  1559 subsubsection {* Cardinality of unions *}
```
```  1560
```
```  1561 lemma of_nat_id[simp]: "(of_nat n :: nat) = n"
```
```  1562 by(induct n, auto)
```
```  1563
```
```  1564 lemma card_UN_disjoint:
```
```  1565     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1566         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1567       card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```  1568   apply (simp add: card_def del: setsum_constant)
```
```  1569   apply (subgoal_tac
```
```  1570            "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
```
```  1571   apply (simp add: setsum_UN_disjoint del: setsum_constant)
```
```  1572   apply (simp cong: setsum_cong)
```
```  1573   done
```
```  1574
```
```  1575 lemma card_Union_disjoint:
```
```  1576   "finite C ==> (ALL A:C. finite A) ==>
```
```  1577         (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
```
```  1578       card (Union C) = setsum card C"
```
```  1579   apply (frule card_UN_disjoint [of C id])
```
```  1580   apply (unfold Union_def id_def, assumption+)
```
```  1581   done
```
```  1582
```
```  1583 subsubsection {* Cardinality of image *}
```
```  1584
```
```  1585 text{*The image of a finite set can be expressed using @{term fold}.*}
```
```  1586 lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
```
```  1587   apply (erule finite_induct, simp)
```
```  1588   apply (subst ACf.fold_insert)
```
```  1589   apply (auto simp add: ACf_def)
```
```  1590   done
```
```  1591
```
```  1592 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
```
```  1593   apply (induct set: Finites, simp)
```
```  1594   apply (simp add: le_SucI finite_imageI card_insert_if)
```
```  1595   done
```
```  1596
```
```  1597 lemma card_image: "inj_on f A ==> card (f ` A) = card A"
```
```  1598 by(simp add:card_def setsum_reindex o_def del:setsum_constant)
```
```  1599
```
```  1600 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  1601   by (simp add: card_seteq card_image)
```
```  1602
```
```  1603 lemma eq_card_imp_inj_on:
```
```  1604   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
```
```  1605 apply (induct rule:finite_induct, simp)
```
```  1606 apply(frule card_image_le[where f = f])
```
```  1607 apply(simp add:card_insert_if split:if_splits)
```
```  1608 done
```
```  1609
```
```  1610 lemma inj_on_iff_eq_card:
```
```  1611   "finite A ==> inj_on f A = (card(f ` A) = card A)"
```
```  1612 by(blast intro: card_image eq_card_imp_inj_on)
```
```  1613
```
```  1614
```
```  1615 lemma card_inj_on_le:
```
```  1616     "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
```
```  1617 apply (subgoal_tac "finite A")
```
```  1618  apply (force intro: card_mono simp add: card_image [symmetric])
```
```  1619 apply (blast intro: finite_imageD dest: finite_subset)
```
```  1620 done
```
```  1621
```
```  1622 lemma card_bij_eq:
```
```  1623     "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  1624        finite A; finite B |] ==> card A = card B"
```
```  1625   by (auto intro: le_anti_sym card_inj_on_le)
```
```  1626
```
```  1627
```
```  1628 subsubsection {* Cardinality of products *}
```
```  1629
```
```  1630 (*
```
```  1631 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1632   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1633   by auto
```
```  1634 *)
```
```  1635
```
```  1636 lemma card_SigmaI [simp]:
```
```  1637   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```  1638   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1639 by(simp add:card_def setsum_Sigma del:setsum_constant)
```
```  1640
```
```  1641 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```  1642 apply (cases "finite A")
```
```  1643 apply (cases "finite B")
```
```  1644 apply (auto simp add: card_eq_0_iff
```
```  1645             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1646 done
```
```  1647
```
```  1648 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```  1649 by (simp add: card_cartesian_product)
```
```  1650
```
```  1651
```
```  1652
```
```  1653 subsubsection {* Cardinality of the Powerset *}
```
```  1654
```
```  1655 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
```
```  1656   apply (induct set: Finites)
```
```  1657    apply (simp_all add: Pow_insert)
```
```  1658   apply (subst card_Un_disjoint, blast)
```
```  1659     apply (blast intro: finite_imageI, blast)
```
```  1660   apply (subgoal_tac "inj_on (insert x) (Pow F)")
```
```  1661    apply (simp add: card_image Pow_insert)
```
```  1662   apply (unfold inj_on_def)
```
```  1663   apply (blast elim!: equalityE)
```
```  1664   done
```
```  1665
```
```  1666 text {* Relates to equivalence classes.  Based on a theorem of
```
```  1667 F. Kammüller's.  *}
```
```  1668
```
```  1669 lemma dvd_partition:
```
```  1670   "finite (Union C) ==>
```
```  1671     ALL c : C. k dvd card c ==>
```
```  1672     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
```
```  1673   k dvd card (Union C)"
```
```  1674 apply(frule finite_UnionD)
```
```  1675 apply(rotate_tac -1)
```
```  1676   apply (induct set: Finites, simp_all, clarify)
```
```  1677   apply (subst card_Un_disjoint)
```
```  1678   apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
```
```  1679   done
```
```  1680
```
```  1681
```
```  1682 subsubsection {* Theorems about @{text "choose"} *}
```
```  1683
```
```  1684 text {*
```
```  1685   \medskip Basic theorem about @{text "choose"}.  By Florian
```
```  1686   Kamm\"uller, tidied by LCP.
```
```  1687 *}
```
```  1688
```
```  1689 lemma card_s_0_eq_empty:
```
```  1690     "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
```
```  1691   apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
```
```  1692   apply (simp cong add: rev_conj_cong)
```
```  1693   done
```
```  1694
```
```  1695 lemma choose_deconstruct: "finite M ==> x \<notin> M
```
```  1696   ==> {s. s <= insert x M & card(s) = Suc k}
```
```  1697        = {s. s <= M & card(s) = Suc k} Un
```
```  1698          {s. EX t. t <= M & card(t) = k & s = insert x t}"
```
```  1699   apply safe
```
```  1700    apply (auto intro: finite_subset [THEN card_insert_disjoint])
```
```  1701   apply (drule_tac x = "xa - {x}" in spec)
```
```  1702   apply (subgoal_tac "x \<notin> xa", auto)
```
```  1703   apply (erule rev_mp, subst card_Diff_singleton)
```
```  1704   apply (auto intro: finite_subset)
```
```  1705   done
```
```  1706
```
```  1707 text{*There are as many subsets of @{term A} having cardinality @{term k}
```
```  1708  as there are sets obtained from the former by inserting a fixed element
```
```  1709  @{term x} into each.*}
```
```  1710 lemma constr_bij:
```
```  1711    "[|finite A; x \<notin> A|] ==>
```
```  1712     card {B. EX C. C <= A & card(C) = k & B = insert x C} =
```
```  1713     card {B. B <= A & card(B) = k}"
```
```  1714   apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
```
```  1715        apply (auto elim!: equalityE simp add: inj_on_def)
```
```  1716     apply (subst Diff_insert0, auto)
```
```  1717    txt {* finiteness of the two sets *}
```
```  1718    apply (rule_tac [2] B = "Pow (A)" in finite_subset)
```
```  1719    apply (rule_tac B = "Pow (insert x A)" in finite_subset)
```
```  1720    apply fast+
```
```  1721   done
```
```  1722
```
```  1723 text {*
```
```  1724   Main theorem: combinatorial statement about number of subsets of a set.
```
```  1725 *}
```
```  1726
```
```  1727 lemma n_sub_lemma:
```
```  1728   "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
```
```  1729   apply (induct k)
```
```  1730    apply (simp add: card_s_0_eq_empty, atomize)
```
```  1731   apply (rotate_tac -1, erule finite_induct)
```
```  1732    apply (simp_all (no_asm_simp) cong add: conj_cong
```
```  1733      add: card_s_0_eq_empty choose_deconstruct)
```
```  1734   apply (subst card_Un_disjoint)
```
```  1735      prefer 4 apply (force simp add: constr_bij)
```
```  1736     prefer 3 apply force
```
```  1737    prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
```
```  1738      finite_subset [of _ "Pow (insert x F)", standard])
```
```  1739   apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
```
```  1740   done
```
```  1741
```
```  1742 theorem n_subsets:
```
```  1743     "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
```
```  1744   by (simp add: n_sub_lemma)
```
```  1745
```
```  1746
```
```  1747 subsection{* A fold functional for non-empty sets *}
```
```  1748
```
```  1749 text{* Does not require start value. *}
```
```  1750
```
```  1751 consts
```
```  1752   fold1Set :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
```
```  1753
```
```  1754 inductive "fold1Set f"
```
```  1755 intros
```
```  1756   fold1Set_insertI [intro]:
```
```  1757    "\<lbrakk> (A,x) \<in> foldSet f id a; a \<notin> A \<rbrakk> \<Longrightarrow> (insert a A, x) \<in> fold1Set f"
```
```  1758
```
```  1759 constdefs
```
```  1760   fold1 :: "('a => 'a => 'a) => 'a set => 'a"
```
```  1761   "fold1 f A == THE x. (A, x) : fold1Set f"
```
```  1762
```
```  1763 lemma fold1Set_nonempty:
```
```  1764  "(A, x) : fold1Set f \<Longrightarrow> A \<noteq> {}"
```
```  1765 by(erule fold1Set.cases, simp_all)
```
```  1766
```
```  1767
```
```  1768 inductive_cases empty_fold1SetE [elim!]: "({}, x) : fold1Set f"
```
```  1769
```
```  1770 inductive_cases insert_fold1SetE [elim!]: "(insert a X, x) : fold1Set f"
```
```  1771
```
```  1772
```
```  1773 lemma fold1Set_sing [iff]: "(({a},b) : fold1Set f) = (a = b)"
```
```  1774   by (blast intro: foldSet.intros elim: foldSet.cases)
```
```  1775
```
```  1776 lemma fold1_singleton[simp]: "fold1 f {a} = a"
```
```  1777   by (unfold fold1_def) blast
```
```  1778
```
```  1779 lemma finite_nonempty_imp_fold1Set:
```
```  1780   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : fold1Set f"
```
```  1781 apply (induct A rule: finite_induct)
```
```  1782 apply (auto dest: finite_imp_foldSet [of _ f id])
```
```  1783 done
```
```  1784
```
```  1785 text{*First, some lemmas about @{term foldSet}.*}
```
```  1786
```
```  1787 lemma (in ACf) foldSet_insert_swap:
```
```  1788 assumes fold: "(A,y) \<in> foldSet f id b"
```
```  1789 shows "b \<notin> A \<Longrightarrow> (insert b A, z \<cdot> y) \<in> foldSet f id z"
```
```  1790 using fold
```
```  1791 proof (induct rule: foldSet.induct)
```
```  1792   case emptyI thus ?case by (force simp add: fold_insert_aux commute)
```
```  1793 next
```
```  1794   case (insertI A x y)
```
```  1795     have "(insert x (insert b A), x \<cdot> (z \<cdot> y)) \<in> foldSet f (\<lambda>u. u) z"
```
```  1796       using insertI by force  --{*how does @{term id} get unfolded?*}
```
```  1797     thus ?case by (simp add: insert_commute AC)
```
```  1798 qed
```
```  1799
```
```  1800 lemma (in ACf) foldSet_permute_diff:
```
```  1801 assumes fold: "(A,x) \<in> foldSet f id b"
```
```  1802 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> (insert b (A-{a}), x) \<in> foldSet f id a"
```
```  1803 using fold
```
```  1804 proof (induct rule: foldSet.induct)
```
```  1805   case emptyI thus ?case by simp
```
```  1806 next
```
```  1807   case (insertI A x y)
```
```  1808   have "a = x \<or> a \<in> A" using insertI by simp
```
```  1809   thus ?case
```
```  1810   proof
```
```  1811     assume "a = x"
```
```  1812     with insertI show ?thesis
```
```  1813       by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap)
```
```  1814   next
```
```  1815     assume ainA: "a \<in> A"
```
```  1816     hence "(insert x (insert b (A - {a})), x \<cdot> y) \<in> foldSet f id a"
```
```  1817       using insertI by (force simp: id_def)
```
```  1818     moreover
```
```  1819     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
```
```  1820       using ainA insertI by blast
```
```  1821     ultimately show ?thesis by (simp add: id_def)
```
```  1822   qed
```
```  1823 qed
```
```  1824
```
```  1825 lemma (in ACf) fold1_eq_fold:
```
```  1826      "[|finite A; a \<notin> A|] ==> fold1 f (insert a A) = fold f id a A"
```
```  1827 apply (simp add: fold1_def fold_def)
```
```  1828 apply (rule the_equality)
```
```  1829 apply (best intro: foldSet_determ theI dest: finite_imp_foldSet [of _ f id])
```
```  1830 apply (rule sym, clarify)
```
```  1831 apply (case_tac "Aa=A")
```
```  1832  apply (best intro: the_equality foldSet_determ)
```
```  1833 apply (subgoal_tac "(A,x) \<in> foldSet f id a")
```
```  1834  apply (best intro: the_equality foldSet_determ)
```
```  1835 apply (subgoal_tac "insert aa (Aa - {a}) = A")
```
```  1836  prefer 2 apply (blast elim: equalityE)
```
```  1837 apply (auto dest: foldSet_permute_diff [where a=a])
```
```  1838 done
```
```  1839
```
```  1840 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
```
```  1841 apply safe
```
```  1842 apply simp
```
```  1843 apply (drule_tac x=x in spec)
```
```  1844 apply (drule_tac x="A-{x}" in spec, auto)
```
```  1845 done
```
```  1846
```
```  1847 lemma (in ACf) fold1_insert:
```
```  1848   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
```
```  1849   shows "fold1 f (insert x A) = f x (fold1 f A)"
```
```  1850 proof -
```
```  1851   from nonempty obtain a A' where "A = insert a A' & a ~: A'"
```
```  1852     by (auto simp add: nonempty_iff)
```
```  1853   with A show ?thesis
```
```  1854     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
```
```  1855 qed
```
```  1856
```
```  1857 lemma (in ACIf) fold1_insert_idem [simp]:
```
```  1858   assumes nonempty: "A \<noteq> {}" and A: "finite A"
```
```  1859   shows "fold1 f (insert x A) = f x (fold1 f A)"
```
```  1860 proof -
```
```  1861   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
```
```  1862     by (auto simp add: nonempty_iff)
```
```  1863   show ?thesis
```
```  1864   proof cases
```
```  1865     assume "a = x"
```
```  1866     thus ?thesis
```
```  1867     proof cases
```
```  1868       assume "A' = {}"
```
```  1869       with prems show ?thesis by (simp add: idem)
```
```  1870     next
```
```  1871       assume "A' \<noteq> {}"
```
```  1872       with prems show ?thesis
```
```  1873 	by (simp add: fold1_insert assoc [symmetric] idem)
```
```  1874     qed
```
```  1875   next
```
```  1876     assume "a \<noteq> x"
```
```  1877     with prems show ?thesis
```
```  1878       by (simp add: insert_commute fold1_eq_fold fold_insert_idem)
```
```  1879   qed
```
```  1880 qed
```
```  1881
```
```  1882
```
```  1883 text{* Now the recursion rules for definitions: *}
```
```  1884
```
```  1885 lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
```
```  1886 by(simp add:fold1_singleton)
```
```  1887
```
```  1888 lemma (in ACf) fold1_insert_def:
```
```  1889   "\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
```
```  1890 by(simp add:fold1_insert)
```
```  1891
```
```  1892 lemma (in ACIf) fold1_insert_idem_def:
```
```  1893   "\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
```
```  1894 by(simp add:fold1_insert_idem)
```
```  1895
```
```  1896 subsubsection{* Determinacy for @{term fold1Set} *}
```
```  1897
```
```  1898 text{*Not actually used!!*}
```
```  1899
```
```  1900 lemma (in ACf) foldSet_permute:
```
```  1901   "[|(insert a A, x) \<in> foldSet f id b; a \<notin> A; b \<notin> A|]
```
```  1902    ==> (insert b A, x) \<in> foldSet f id a"
```
```  1903 apply (case_tac "a=b")
```
```  1904 apply (auto dest: foldSet_permute_diff)
```
```  1905 done
```
```  1906
```
```  1907 lemma (in ACf) fold1Set_determ:
```
```  1908   "(A, x) \<in> fold1Set f ==> (A, y) \<in> fold1Set f ==> y = x"
```
```  1909 proof (clarify elim!: fold1Set.cases)
```
```  1910   fix A x B y a b
```
```  1911   assume Ax: "(A, x) \<in> foldSet f id a"
```
```  1912   assume By: "(B, y) \<in> foldSet f id b"
```
```  1913   assume anotA:  "a \<notin> A"
```
```  1914   assume bnotB:  "b \<notin> B"
```
```  1915   assume eq: "insert a A = insert b B"
```
```  1916   show "y=x"
```
```  1917   proof cases
```
```  1918     assume same: "a=b"
```
```  1919     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
```
```  1920     thus ?thesis using Ax By same by (blast intro: foldSet_determ)
```
```  1921   next
```
```  1922     assume diff: "a\<noteq>b"
```
```  1923     let ?D = "B - {a}"
```
```  1924     have B: "B = insert a ?D" and A: "A = insert b ?D"
```
```  1925      and aB: "a \<in> B" and bA: "b \<in> A"
```
```  1926       using eq anotA bnotB diff by (blast elim!:equalityE)+
```
```  1927     with aB bnotB By
```
```  1928     have "(insert b ?D, y) \<in> foldSet f id a"
```
```  1929       by (auto intro: foldSet_permute simp add: insert_absorb)
```
```  1930     moreover
```
```  1931     have "(insert b ?D, x) \<in> foldSet f id a"
```
```  1932       by (simp add: A [symmetric] Ax)
```
```  1933     ultimately show ?thesis by (blast intro: foldSet_determ)
```
```  1934   qed
```
```  1935 qed
```
```  1936
```
```  1937 lemma (in ACf) fold1Set_equality: "(A, y) : fold1Set f ==> fold1 f A = y"
```
```  1938   by (unfold fold1_def) (blast intro: fold1Set_determ)
```
```  1939
```
```  1940 declare
```
```  1941   empty_foldSetE [rule del]   foldSet.intros [rule del]
```
```  1942   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
```
```  1943   -- {* No more proves involve these relations. *}
```
```  1944
```
```  1945 subsubsection{* Semi-Lattices *}
```
```  1946
```
```  1947 locale ACIfSL = ACIf +
```
```  1948   fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
```
```  1949   assumes below_def: "(x \<sqsubseteq> y) = (x\<cdot>y = x)"
```
```  1950
```
```  1951 locale ACIfSLlin = ACIfSL +
```
```  1952   assumes lin: "x\<cdot>y \<in> {x,y}"
```
```  1953
```
```  1954 lemma (in ACIfSL) below_refl[simp]: "x \<sqsubseteq> x"
```
```  1955 by(simp add: below_def idem)
```
```  1956
```
```  1957 lemma (in ACIfSL) below_f_conv[simp]: "x \<sqsubseteq> y \<cdot> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
```
```  1958 proof
```
```  1959   assume "x \<sqsubseteq> y \<cdot> z"
```
```  1960   hence xyzx: "x \<cdot> (y \<cdot> z) = x"  by(simp add: below_def)
```
```  1961   have "x \<cdot> y = x"
```
```  1962   proof -
```
```  1963     have "x \<cdot> y = (x \<cdot> (y \<cdot> z)) \<cdot> y" by(rule subst[OF xyzx], rule refl)
```
```  1964     also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  1965     also have "\<dots> = x" by(rule xyzx)
```
```  1966     finally show ?thesis .
```
```  1967   qed
```
```  1968   moreover have "x \<cdot> z = x"
```
```  1969   proof -
```
```  1970     have "x \<cdot> z = (x \<cdot> (y \<cdot> z)) \<cdot> z" by(rule subst[OF xyzx], rule refl)
```
```  1971     also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  1972     also have "\<dots> = x" by(rule xyzx)
```
```  1973     finally show ?thesis .
```
```  1974   qed
```
```  1975   ultimately show "x \<sqsubseteq> y \<and> x \<sqsubseteq> z" by(simp add: below_def)
```
```  1976 next
```
```  1977   assume a: "x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
```
```  1978   hence y: "x \<cdot> y = x" and z: "x \<cdot> z = x" by(simp_all add: below_def)
```
```  1979   have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by(simp add:assoc)
```
```  1980   also have "x \<cdot> y = x" using a by(simp_all add: below_def)
```
```  1981   also have "x \<cdot> z = x" using a by(simp_all add: below_def)
```
```  1982   finally show "x \<sqsubseteq> y \<cdot> z" by(simp_all add: below_def)
```
```  1983 qed
```
```  1984
```
```  1985 lemma (in ACIfSLlin) above_f_conv:
```
```  1986  "x \<cdot> y \<sqsubseteq> z = (x \<sqsubseteq> z \<or> y \<sqsubseteq> z)"
```
```  1987 proof
```
```  1988   assume a: "x \<cdot> y \<sqsubseteq> z"
```
```  1989   have "x \<cdot> y = x \<or> x \<cdot> y = y" using lin[of x y] by simp
```
```  1990   thus "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
```
```  1991   proof
```
```  1992     assume "x \<cdot> y = x" hence "x \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
```
```  1993   next
```
```  1994     assume "x \<cdot> y = y" hence "y \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
```
```  1995   qed
```
```  1996 next
```
```  1997   assume "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
```
```  1998   thus "x \<cdot> y \<sqsubseteq> z"
```
```  1999   proof
```
```  2000     assume a: "x \<sqsubseteq> z"
```
```  2001     have "(x \<cdot> y) \<cdot> z = (x \<cdot> z) \<cdot> y" by(simp add:ACI)
```
```  2002     also have "x \<cdot> z = x" using a by(simp add:below_def)
```
```  2003     finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
```
```  2004   next
```
```  2005     assume a: "y \<sqsubseteq> z"
```
```  2006     have "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  2007     also have "y \<cdot> z = y" using a by(simp add:below_def)
```
```  2008     finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
```
```  2009   qed
```
```  2010 qed
```
```  2011
```
```  2012
```
```  2013 subsubsection{* Lemmas about @{text fold1} *}
```
```  2014
```
```  2015 lemma (in ACf) fold1_Un:
```
```  2016 assumes A: "finite A" "A \<noteq> {}"
```
```  2017 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
```
```  2018        fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
```
```  2019 using A
```
```  2020 proof(induct rule:finite_ne_induct)
```
```  2021   case singleton thus ?case by(simp add:fold1_insert)
```
```  2022 next
```
```  2023   case insert thus ?case by (simp add:fold1_insert assoc)
```
```  2024 qed
```
```  2025
```
```  2026 lemma (in ACIf) fold1_Un2:
```
```  2027 assumes A: "finite A" "A \<noteq> {}"
```
```  2028 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
```
```  2029        fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
```
```  2030 using A
```
```  2031 proof(induct rule:finite_ne_induct)
```
```  2032   case singleton thus ?case by(simp add:fold1_insert_idem)
```
```  2033 next
```
```  2034   case insert thus ?case by (simp add:fold1_insert_idem assoc)
```
```  2035 qed
```
```  2036
```
```  2037 lemma (in ACf) fold1_in:
```
```  2038   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}"
```
```  2039   shows "fold1 f A \<in> A"
```
```  2040 using A
```
```  2041 proof (induct rule:finite_ne_induct)
```
```  2042   case singleton thus ?case by simp
```
```  2043 next
```
```  2044   case insert thus ?case using elem by (force simp add:fold1_insert)
```
```  2045 qed
```
```  2046
```
```  2047 lemma (in ACIfSL) below_fold1_iff:
```
```  2048 assumes A: "finite A" "A \<noteq> {}"
```
```  2049 shows "x \<sqsubseteq> fold1 f A = (\<forall>a\<in>A. x \<sqsubseteq> a)"
```
```  2050 using A
```
```  2051 by(induct rule:finite_ne_induct) simp_all
```
```  2052
```
```  2053 lemma (in ACIfSL) fold1_belowI:
```
```  2054 assumes A: "finite A" "A \<noteq> {}"
```
```  2055 shows "a \<in> A \<Longrightarrow> fold1 f A \<sqsubseteq> a"
```
```  2056 using A
```
```  2057 proof (induct rule:finite_ne_induct)
```
```  2058   case singleton thus ?case by simp
```
```  2059 next
```
```  2060   case (insert x F)
```
```  2061   from insert(5) have "a = x \<or> a \<in> F" by simp
```
```  2062   thus ?case
```
```  2063   proof
```
```  2064     assume "a = x" thus ?thesis using insert by(simp add:below_def ACI)
```
```  2065   next
```
```  2066     assume "a \<in> F"
```
```  2067     hence bel: "fold1 f F \<sqsubseteq> a" by(rule insert)
```
```  2068     have "fold1 f (insert x F) \<cdot> a = x \<cdot> (fold1 f F \<cdot> a)"
```
```  2069       using insert by(simp add:below_def ACI)
```
```  2070     also have "fold1 f F \<cdot> a = fold1 f F"
```
```  2071       using bel  by(simp add:below_def ACI)
```
```  2072     also have "x \<cdot> \<dots> = fold1 f (insert x F)"
```
```  2073       using insert by(simp add:below_def ACI)
```
```  2074     finally show ?thesis  by(simp add:below_def)
```
```  2075   qed
```
```  2076 qed
```
```  2077
```
```  2078 lemma (in ACIfSLlin) fold1_below_iff:
```
```  2079 assumes A: "finite A" "A \<noteq> {}"
```
```  2080 shows "fold1 f A \<sqsubseteq> x = (\<exists>a\<in>A. a \<sqsubseteq> x)"
```
```  2081 using A
```
```  2082 by(induct rule:finite_ne_induct)(simp_all add:above_f_conv)
```
```  2083
```
```  2084
```
```  2085 subsubsection{* Lattices *}
```
```  2086
```
```  2087 locale Lattice = lattice +
```
```  2088   fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
```
```  2089   and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```  2090   defines "Inf == fold1 inf"  and "Sup == fold1 sup"
```
```  2091
```
```  2092 locale Distrib_Lattice = distrib_lattice + Lattice
```
```  2093
```
```  2094 text{* Lattices are semilattices *}
```
```  2095
```
```  2096 lemma (in Lattice) ACf_inf: "ACf inf"
```
```  2097 by(blast intro: ACf.intro inf_commute inf_assoc)
```
```  2098
```
```  2099 lemma (in Lattice) ACf_sup: "ACf sup"
```
```  2100 by(blast intro: ACf.intro sup_commute sup_assoc)
```
```  2101
```
```  2102 lemma (in Lattice) ACIf_inf: "ACIf inf"
```
```  2103 apply(rule ACIf.intro)
```
```  2104 apply(rule ACf_inf)
```
```  2105 apply(rule ACIf_axioms.intro)
```
```  2106 apply(rule inf_idem)
```
```  2107 done
```
```  2108
```
```  2109 lemma (in Lattice) ACIf_sup: "ACIf sup"
```
```  2110 apply(rule ACIf.intro)
```
```  2111 apply(rule ACf_sup)
```
```  2112 apply(rule ACIf_axioms.intro)
```
```  2113 apply(rule sup_idem)
```
```  2114 done
```
```  2115
```
```  2116 lemma (in Lattice) ACIfSL_inf: "ACIfSL inf (op \<sqsubseteq>)"
```
```  2117 apply(rule ACIfSL.intro)
```
```  2118 apply(rule ACf_inf)
```
```  2119 apply(rule ACIf.axioms[OF ACIf_inf])
```
```  2120 apply(rule ACIfSL_axioms.intro)
```
```  2121 apply(rule iffI)
```
```  2122  apply(blast intro: antisym inf_le1 inf_le2 inf_least refl)
```
```  2123 apply(erule subst)
```
```  2124 apply(rule inf_le2)
```
```  2125 done
```
```  2126
```
```  2127 lemma (in Lattice) ACIfSL_sup: "ACIfSL sup (%x y. y \<sqsubseteq> x)"
```
```  2128 apply(rule ACIfSL.intro)
```
```  2129 apply(rule ACf_sup)
```
```  2130 apply(rule ACIf.axioms[OF ACIf_sup])
```
```  2131 apply(rule ACIfSL_axioms.intro)
```
```  2132 apply(rule iffI)
```
```  2133  apply(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
```
```  2134 apply(erule subst)
```
```  2135 apply(rule sup_ge2)
```
```  2136 done
```
```  2137
```
```  2138
```
```  2139 subsubsection{* Fold laws in lattices *}
```
```  2140
```
```  2141 lemma (in Lattice) Inf_le_Sup[simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Squnion>A"
```
```  2142 apply(unfold Sup_def Inf_def)
```
```  2143 apply(subgoal_tac "EX a. a:A")
```
```  2144 prefer 2 apply blast
```
```  2145 apply(erule exE)
```
```  2146 apply(rule trans)
```
```  2147 apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_inf])
```
```  2148 apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup])
```
```  2149 done
```
```  2150
```
```  2151 lemma (in Lattice) sup_Inf_absorb[simp]:
```
```  2152   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<squnion> \<Sqinter>A) = a"
```
```  2153 apply(subst sup_commute)
```
```  2154 apply(simp add:Inf_def sup_absorb ACIfSL.fold1_belowI[OF ACIfSL_inf])
```
```  2155 done
```
```  2156
```
```  2157 lemma (in Lattice) inf_Sup_absorb[simp]:
```
```  2158   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<sqinter> \<Squnion>A) = a"
```
```  2159 by(simp add:Sup_def inf_absorb ACIfSL.fold1_belowI[OF ACIfSL_sup])
```
```  2160
```
```  2161
```
```  2162 lemma (in Distrib_Lattice) sup_Inf1_distrib:
```
```  2163 assumes A: "finite A" "A \<noteq> {}"
```
```  2164 shows "(x \<squnion> \<Sqinter>A) = \<Sqinter>{x \<squnion> a|a. a \<in> A}"
```
```  2165 using A
```
```  2166 proof (induct rule: finite_ne_induct)
```
```  2167   case singleton thus ?case by(simp add:Inf_def)
```
```  2168 next
```
```  2169   case (insert y A)
```
```  2170   have fin: "finite {x \<squnion> a |a. a \<in> A}"
```
```  2171     by(fast intro: finite_surj[where f = "%a. x \<squnion> a", OF insert(1)])
```
```  2172   have "x \<squnion> \<Sqinter> (insert y A) = x \<squnion> (y \<sqinter> \<Sqinter> A)"
```
```  2173     using insert by(simp add:ACf.fold1_insert_def[OF ACf_inf Inf_def])
```
```  2174   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> \<Sqinter> A)" by(rule sup_inf_distrib1)
```
```  2175   also have "x \<squnion> \<Sqinter> A = \<Sqinter>{x \<squnion> a|a. a \<in> A}" using insert by simp
```
```  2176   also have "(x \<squnion> y) \<sqinter> \<dots> = \<Sqinter> (insert (x \<squnion> y) {x \<squnion> a |a. a \<in> A})"
```
```  2177     using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def fin])
```
```  2178   also have "insert (x\<squnion>y) {x\<squnion>a |a. a \<in> A} = {x\<squnion>a |a. a \<in> insert y A}"
```
```  2179     by blast
```
```  2180   finally show ?case .
```
```  2181 qed
```
```  2182
```
```  2183 lemma (in Distrib_Lattice) sup_Inf2_distrib:
```
```  2184 assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  2185 shows "(\<Sqinter>A \<squnion> \<Sqinter>B) = \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
```
```  2186 using A
```
```  2187 proof (induct rule: finite_ne_induct)
```
```  2188   case singleton thus ?case
```
```  2189     by(simp add: sup_Inf1_distrib[OF B] fold1_singleton_def[OF Inf_def])
```
```  2190 next
```
```  2191   case (insert x A)
```
```  2192   have finB: "finite {x \<squnion> b |b. b \<in> B}"
```
```  2193     by(fast intro: finite_surj[where f = "%b. x \<squnion> b", OF B(1)])
```
```  2194   have finAB: "finite {a \<squnion> b |a b. a \<in> A \<and> b \<in> B}"
```
```  2195   proof -
```
```  2196     have "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<squnion> b})"
```
```  2197       by blast
```
```  2198     thus ?thesis by(simp add: insert(1) B(1))
```
```  2199   qed
```
```  2200   have ne: "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  2201   have "\<Sqinter>(insert x A) \<squnion> \<Sqinter>B = (x \<sqinter> \<Sqinter>A) \<squnion> \<Sqinter>B"
```
```  2202     using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def])
```
```  2203   also have "\<dots> = (x \<squnion> \<Sqinter>B) \<sqinter> (\<Sqinter>A \<squnion> \<Sqinter>B)" by(rule sup_inf_distrib2)
```
```  2204   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}"
```
```  2205     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  2206   also have "\<dots> = \<Sqinter>({x\<squnion>b |b. b \<in> B} \<union> {a\<squnion>b |a b. a \<in> A \<and> b \<in> B})"
```
```  2207     (is "_ = \<Sqinter>?M")
```
```  2208     using B insert
```
```  2209     by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne])
```
```  2210   also have "?M = {a \<squnion> b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  2211     by blast
```
```  2212   finally show ?case .
```
```  2213 qed
```
```  2214
```
```  2215
```
```  2216 subsection{*Min and Max*}
```
```  2217
```
```  2218 text{* As an application of @{text fold1} we define the minimal and
```
```  2219 maximal element of a (non-empty) set over a linear order. *}
```
```  2220
```
```  2221 constdefs
```
```  2222   Min :: "('a::linorder)set => 'a"
```
```  2223   "Min  ==  fold1 min"
```
```  2224
```
```  2225   Max :: "('a::linorder)set => 'a"
```
```  2226   "Max  ==  fold1 max"
```
```  2227
```
```  2228
```
```  2229 text{* Before we can do anything, we need to show that @{text min} and
```
```  2230 @{text max} are ACI and the ordering is linear: *}
```
```  2231
```
```  2232 interpretation min: ACf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
```
```  2233 apply(rule ACf.intro)
```
```  2234 apply(auto simp:min_def)
```
```  2235 done
```
```  2236
```
```  2237 interpretation min: ACIf ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
```
```  2238 apply(rule ACIf_axioms.intro)
```
```  2239 apply(auto simp:min_def)
```
```  2240 done
```
```  2241
```
```  2242 interpretation max: ACf ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
```
```  2243 apply(rule ACf.intro)
```
```  2244 apply(auto simp:max_def)
```
```  2245 done
```
```  2246
```
```  2247 interpretation max: ACIf ["max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
```
```  2248 apply(rule ACIf_axioms.intro)
```
```  2249 apply(auto simp:max_def)
```
```  2250 done
```
```  2251
```
```  2252 interpretation min:
```
```  2253   ACIfSL ["min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>"]
```
```  2254 apply(rule ACIfSL_axioms.intro)
```
```  2255 apply(auto simp:min_def)
```
```  2256 done
```
```  2257
```
```  2258 interpretation min:
```
```  2259   ACIfSLlin ["min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "op \<le>"]
```
```  2260 apply(rule ACIfSLlin_axioms.intro)
```
```  2261 apply(auto simp:min_def)
```
```  2262 done
```
```  2263
```
```  2264 interpretation max:
```
```  2265   ACIfSL ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x"]
```
```  2266 apply(rule ACIfSL_axioms.intro)
```
```  2267 apply(auto simp:max_def)
```
```  2268 done
```
```  2269
```
```  2270 interpretation max:
```
```  2271   ACIfSLlin ["max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "%x y. y\<le>x"]
```
```  2272 apply(rule ACIfSLlin_axioms.intro)
```
```  2273 apply(auto simp:max_def)
```
```  2274 done
```
```  2275
```
```  2276 interpretation min_max:
```
```  2277   Lattice ["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
```
```  2278 apply -
```
```  2279 apply(rule Min_def)
```
```  2280 apply(rule Max_def)
```
```  2281 done
```
```  2282
```
```  2283
```
```  2284 interpretation min_max:
```
```  2285   Distrib_Lattice ["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max" "Min" "Max"]
```
```  2286 .
```
```  2287
```
```  2288 text{* Now we instantiate the recursion equations and declare them
```
```  2289 simplification rules: *}
```
```  2290
```
```  2291 (* Making Min (resp. Max) a defined parameter of a locale suitably
```
```  2292   extending ACIf could make the following interpretations more automatic. *)
```
```  2293
```
```  2294 declare
```
```  2295   fold1_singleton_def[OF Min_def, simp]
```
```  2296   min.fold1_insert_idem_def[OF Min_def, simp]
```
```  2297   fold1_singleton_def[OF Max_def, simp]
```
```  2298   max.fold1_insert_idem_def[OF Max_def, simp]
```
```  2299
```
```  2300 text{* Now we instantiate some @{text fold1} properties: *}
```
```  2301
```
```  2302 lemma Min_in [simp]:
```
```  2303   shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A"
```
```  2304 using min.fold1_in
```
```  2305 by(fastsimp simp: Min_def min_def)
```
```  2306
```
```  2307 lemma Max_in [simp]:
```
```  2308   shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A"
```
```  2309 using max.fold1_in
```
```  2310 by(fastsimp simp: Max_def max_def)
```
```  2311
```
```  2312 lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<le> x"
```
```  2313 by(simp add: Min_def min.fold1_belowI)
```
```  2314
```
```  2315 lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<le> Max A"
```
```  2316 by(simp add: Max_def max.fold1_belowI)
```
```  2317
```
```  2318 lemma Min_ge_iff[simp]:
```
```  2319   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
```
```  2320 by(simp add: Min_def min.below_fold1_iff)
```
```  2321
```
```  2322 lemma Max_le_iff[simp]:
```
```  2323   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
```
```  2324 by(simp add: Max_def max.below_fold1_iff)
```
```  2325
```
```  2326 lemma Min_le_iff:
```
```  2327   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Min A \<le> x) = (\<exists>a\<in>A. a \<le> x)"
```
```  2328 by(simp add: Min_def min.fold1_below_iff)
```
```  2329
```
```  2330 lemma Max_ge_iff:
```
```  2331   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Max A) = (\<exists>a\<in>A. x \<le> a)"
```
```  2332 by(simp add: Max_def max.fold1_below_iff)
```
```  2333
```
```  2334 end
```