src/HOL/Finite_Set.thy
 author nipkow Tue Feb 08 09:46:00 2005 +0100 (2005-02-08) changeset 15505 c929e1cbef88 parent 15504 5bc81e50f2c5 child 15506 864238c95b56 permissions -rw-r--r--
Fixed lattice defns
```     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
```
```    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:
```
```   104 assumes fin: "finite A" shows "\<exists> (n::nat) f. A = f ` {i::nat. i<n}"
```
```   105 using fin
```
```   106 proof induct
```
```   107   case empty
```
```   108   show ?case
```
```   109   proof show "\<exists>f. {} = f ` {i::nat. i < 0}" by(simp add:image_def) qed
```
```   110 next
```
```   111   case (insert a A)
```
```   112   from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" by blast
```
```   113   hence "insert a A = (%i. if i<n then f i else a) ` {i. i < n+1}"
```
```   114     by (auto simp add:image_def Ball_def)
```
```   115   thus ?case by blast
```
```   116 qed
```
```   117
```
```   118 lemma nat_seg_image_imp_finite:
```
```   119   "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
```
```   120 proof (induct n)
```
```   121   case 0 thus ?case by simp
```
```   122 next
```
```   123   case (Suc n)
```
```   124   let ?B = "f ` {i. i < n}"
```
```   125   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```   126   show ?case
```
```   127   proof cases
```
```   128     assume "\<exists>k<n. f n = f k"
```
```   129     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   130     thus ?thesis using finB by simp
```
```   131   next
```
```   132     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   133     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   134     thus ?thesis using finB by simp
```
```   135   qed
```
```   136 qed
```
```   137
```
```   138 lemma finite_conv_nat_seg_image:
```
```   139   "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
```
```   140 by(blast intro: finite_imp_nat_seg_image nat_seg_image_imp_finite)
```
```   141
```
```   142 subsubsection{* Finiteness and set theoretic constructions *}
```
```   143
```
```   144 lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
```
```   145   -- {* The union of two finite sets is finite. *}
```
```   146   by (induct set: Finites) simp_all
```
```   147
```
```   148 lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
```
```   149   -- {* Every subset of a finite set is finite. *}
```
```   150 proof -
```
```   151   assume "finite B"
```
```   152   thus "!!A. A \<subseteq> B ==> finite A"
```
```   153   proof induct
```
```   154     case empty
```
```   155     thus ?case by simp
```
```   156   next
```
```   157     case (insert x F A)
```
```   158     have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
```
```   159     show "finite A"
```
```   160     proof cases
```
```   161       assume x: "x \<in> A"
```
```   162       with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   163       with r have "finite (A - {x})" .
```
```   164       hence "finite (insert x (A - {x}))" ..
```
```   165       also have "insert x (A - {x}) = A" by (rule insert_Diff)
```
```   166       finally show ?thesis .
```
```   167     next
```
```   168       show "A \<subseteq> F ==> ?thesis" .
```
```   169       assume "x \<notin> A"
```
```   170       with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   171     qed
```
```   172   qed
```
```   173 qed
```
```   174
```
```   175 lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
```
```   176   by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
```
```   177
```
```   178 lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
```
```   179   -- {* The converse obviously fails. *}
```
```   180   by (blast intro: finite_subset)
```
```   181
```
```   182 lemma finite_insert [simp]: "finite (insert a A) = finite A"
```
```   183   apply (subst insert_is_Un)
```
```   184   apply (simp only: finite_Un, blast)
```
```   185   done
```
```   186
```
```   187 lemma finite_Union[simp, intro]:
```
```   188  "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
```
```   189 by (induct rule:finite_induct) simp_all
```
```   190
```
```   191 lemma finite_empty_induct:
```
```   192   "finite A ==>
```
```   193   P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
```
```   194 proof -
```
```   195   assume "finite A"
```
```   196     and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
```
```   197   have "P (A - A)"
```
```   198   proof -
```
```   199     fix c b :: "'a set"
```
```   200     presume c: "finite c" and b: "finite b"
```
```   201       and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
```
```   202     from c show "c \<subseteq> b ==> P (b - c)"
```
```   203     proof induct
```
```   204       case empty
```
```   205       from P1 show ?case by simp
```
```   206     next
```
```   207       case (insert x F)
```
```   208       have "P (b - F - {x})"
```
```   209       proof (rule P2)
```
```   210         from _ b show "finite (b - F)" by (rule finite_subset) blast
```
```   211         from insert show "x \<in> b - F" by simp
```
```   212         from insert show "P (b - F)" by simp
```
```   213       qed
```
```   214       also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
```
```   215       finally show ?case .
```
```   216     qed
```
```   217   next
```
```   218     show "A \<subseteq> A" ..
```
```   219   qed
```
```   220   thus "P {}" by simp
```
```   221 qed
```
```   222
```
```   223 lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
```
```   224   by (rule Diff_subset [THEN finite_subset])
```
```   225
```
```   226 lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
```
```   227   apply (subst Diff_insert)
```
```   228   apply (case_tac "a : A - B")
```
```   229    apply (rule finite_insert [symmetric, THEN trans])
```
```   230    apply (subst insert_Diff, simp_all)
```
```   231   done
```
```   232
```
```   233
```
```   234 text {* Image and Inverse Image over Finite Sets *}
```
```   235
```
```   236 lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
```
```   237   -- {* The image of a finite set is finite. *}
```
```   238   by (induct set: Finites) simp_all
```
```   239
```
```   240 lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
```
```   241   apply (frule finite_imageI)
```
```   242   apply (erule finite_subset, assumption)
```
```   243   done
```
```   244
```
```   245 lemma finite_range_imageI:
```
```   246     "finite (range g) ==> finite (range (%x. f (g x)))"
```
```   247   apply (drule finite_imageI, simp)
```
```   248   done
```
```   249
```
```   250 lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
```
```   251 proof -
```
```   252   have aux: "!!A. finite (A - {}) = finite A" by simp
```
```   253   fix B :: "'a set"
```
```   254   assume "finite B"
```
```   255   thus "!!A. f`A = B ==> inj_on f A ==> finite A"
```
```   256     apply induct
```
```   257      apply simp
```
```   258     apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
```
```   259      apply clarify
```
```   260      apply (simp (no_asm_use) add: inj_on_def)
```
```   261      apply (blast dest!: aux [THEN iffD1], atomize)
```
```   262     apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
```
```   263     apply (frule subsetD [OF equalityD2 insertI1], clarify)
```
```   264     apply (rule_tac x = xa in bexI)
```
```   265      apply (simp_all add: inj_on_image_set_diff)
```
```   266     done
```
```   267 qed (rule refl)
```
```   268
```
```   269
```
```   270 lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
```
```   271   -- {* The inverse image of a singleton under an injective function
```
```   272          is included in a singleton. *}
```
```   273   apply (auto simp add: inj_on_def)
```
```   274   apply (blast intro: the_equality [symmetric])
```
```   275   done
```
```   276
```
```   277 lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
```
```   278   -- {* The inverse image of a finite set under an injective function
```
```   279          is finite. *}
```
```   280   apply (induct set: Finites, simp_all)
```
```   281   apply (subst vimage_insert)
```
```   282   apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
```
```   283   done
```
```   284
```
```   285
```
```   286 text {* The finite UNION of finite sets *}
```
```   287
```
```   288 lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
```
```   289   by (induct set: Finites) simp_all
```
```   290
```
```   291 text {*
```
```   292   Strengthen RHS to
```
```   293   @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
```
```   294
```
```   295   We'd need to prove
```
```   296   @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
```
```   297   by induction. *}
```
```   298
```
```   299 lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
```
```   300   by (blast intro: finite_UN_I finite_subset)
```
```   301
```
```   302
```
```   303 text {* Sigma of finite sets *}
```
```   304
```
```   305 lemma finite_SigmaI [simp]:
```
```   306     "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   307   by (unfold Sigma_def) (blast intro!: finite_UN_I)
```
```   308
```
```   309 lemma finite_cartesian_product: "[| finite A; finite B |] ==>
```
```   310     finite (A <*> B)"
```
```   311   by (rule finite_SigmaI)
```
```   312
```
```   313 lemma finite_Prod_UNIV:
```
```   314     "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
```
```   315   apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
```
```   316    apply (erule ssubst)
```
```   317    apply (erule finite_SigmaI, auto)
```
```   318   done
```
```   319
```
```   320 lemma finite_cartesian_productD1:
```
```   321      "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
```
```   322 apply (auto simp add: finite_conv_nat_seg_image)
```
```   323 apply (drule_tac x=n in spec)
```
```   324 apply (drule_tac x="fst o f" in spec)
```
```   325 apply (auto simp add: o_def)
```
```   326  prefer 2 apply (force dest!: equalityD2)
```
```   327 apply (drule equalityD1)
```
```   328 apply (rename_tac y x)
```
```   329 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   330  prefer 2 apply force
```
```   331 apply clarify
```
```   332 apply (rule_tac x=k in image_eqI, auto)
```
```   333 done
```
```   334
```
```   335 lemma finite_cartesian_productD2:
```
```   336      "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
```
```   337 apply (auto simp add: finite_conv_nat_seg_image)
```
```   338 apply (drule_tac x=n in spec)
```
```   339 apply (drule_tac x="snd o f" in spec)
```
```   340 apply (auto simp add: o_def)
```
```   341  prefer 2 apply (force dest!: equalityD2)
```
```   342 apply (drule equalityD1)
```
```   343 apply (rename_tac x y)
```
```   344 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   345  prefer 2 apply force
```
```   346 apply clarify
```
```   347 apply (rule_tac x=k in image_eqI, auto)
```
```   348 done
```
```   349
```
```   350
```
```   351 instance unit :: finite
```
```   352 proof
```
```   353   have "finite {()}" by simp
```
```   354   also have "{()} = UNIV" by auto
```
```   355   finally show "finite (UNIV :: unit set)" .
```
```   356 qed
```
```   357
```
```   358 instance * :: (finite, finite) finite
```
```   359 proof
```
```   360   show "finite (UNIV :: ('a \<times> 'b) set)"
```
```   361   proof (rule finite_Prod_UNIV)
```
```   362     show "finite (UNIV :: 'a set)" by (rule finite)
```
```   363     show "finite (UNIV :: 'b set)" by (rule finite)
```
```   364   qed
```
```   365 qed
```
```   366
```
```   367
```
```   368 text {* The powerset of a finite set *}
```
```   369
```
```   370 lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
```
```   371 proof
```
```   372   assume "finite (Pow A)"
```
```   373   with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
```
```   374   thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   375 next
```
```   376   assume "finite A"
```
```   377   thus "finite (Pow A)"
```
```   378     by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
```
```   379 qed
```
```   380
```
```   381
```
```   382 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   383 by(blast intro: finite_subset[OF subset_Pow_Union])
```
```   384
```
```   385
```
```   386 lemma finite_converse [iff]: "finite (r^-1) = finite r"
```
```   387   apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
```
```   388    apply simp
```
```   389    apply (rule iffI)
```
```   390     apply (erule finite_imageD [unfolded inj_on_def])
```
```   391     apply (simp split add: split_split)
```
```   392    apply (erule finite_imageI)
```
```   393   apply (simp add: converse_def image_def, auto)
```
```   394   apply (rule bexI)
```
```   395    prefer 2 apply assumption
```
```   396   apply simp
```
```   397   done
```
```   398
```
```   399
```
```   400 text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
```
```   401 Ehmety) *}
```
```   402
```
```   403 lemma finite_Field: "finite r ==> finite (Field r)"
```
```   404   -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
```
```   405   apply (induct set: Finites)
```
```   406    apply (auto simp add: Field_def Domain_insert Range_insert)
```
```   407   done
```
```   408
```
```   409 lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
```
```   410   apply clarify
```
```   411   apply (erule trancl_induct)
```
```   412    apply (auto simp add: Field_def)
```
```   413   done
```
```   414
```
```   415 lemma finite_trancl: "finite (r^+) = finite r"
```
```   416   apply auto
```
```   417    prefer 2
```
```   418    apply (rule trancl_subset_Field2 [THEN finite_subset])
```
```   419    apply (rule finite_SigmaI)
```
```   420     prefer 3
```
```   421     apply (blast intro: r_into_trancl' finite_subset)
```
```   422    apply (auto simp add: finite_Field)
```
```   423   done
```
```   424
```
```   425
```
```   426 subsection {* A fold functional for finite sets *}
```
```   427
```
```   428 text {* The intended behaviour is
```
```   429 @{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
```
```   430 if @{text f} is associative-commutative. For an application of @{text fold}
```
```   431 se the definitions of sums and products over finite sets.
```
```   432 *}
```
```   433
```
```   434 consts
```
```   435   foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
```
```   436
```
```   437 inductive "foldSet f g z"
```
```   438 intros
```
```   439 emptyI [intro]: "({}, z) : foldSet f g z"
```
```   440 insertI [intro]: "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
```
```   441  \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
```
```   442
```
```   443 inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
```
```   444
```
```   445 constdefs
```
```   446   fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
```
```   447   "fold f g z A == THE x. (A, x) : foldSet f g z"
```
```   448
```
```   449 text{*A tempting alternative for the definiens is
```
```   450 @{term "if finite A then THE x. (A, x) : foldSet f g e else e"}.
```
```   451 It allows the removal of finiteness assumptions from the theorems
```
```   452 @{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
```
```   453 The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
```
```   454
```
```   455
```
```   456 lemma Diff1_foldSet:
```
```   457   "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
```
```   458 by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
```
```   459
```
```   460 lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
```
```   461   by (induct set: foldSet) auto
```
```   462
```
```   463 lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
```
```   464   by (induct set: Finites) auto
```
```   465
```
```   466
```
```   467 subsubsection {* Commutative monoids *}
```
```   468
```
```   469 locale ACf =
```
```   470   fixes f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
```
```   471   assumes commute: "x \<cdot> y = y \<cdot> x"
```
```   472     and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
```
```   473
```
```   474 locale ACe = ACf +
```
```   475   fixes e :: 'a
```
```   476   assumes ident [simp]: "x \<cdot> e = x"
```
```   477
```
```   478 locale ACIf = ACf +
```
```   479   assumes idem: "x \<cdot> x = x"
```
```   480
```
```   481 lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
```
```   482 proof -
```
```   483   have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
```
```   484   also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
```
```   485   also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
```
```   486   finally show ?thesis .
```
```   487 qed
```
```   488
```
```   489 lemmas (in ACf) AC = assoc commute left_commute
```
```   490
```
```   491 lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
```
```   492 proof -
```
```   493   have "x \<cdot> e = x" by (rule ident)
```
```   494   thus ?thesis by (subst commute)
```
```   495 qed
```
```   496
```
```   497 lemma (in ACIf) idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
```
```   498 proof -
```
```   499   have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
```
```   500   also have "\<dots> = x \<cdot> y" by(simp add:idem)
```
```   501   finally show ?thesis .
```
```   502 qed
```
```   503
```
```   504 lemmas (in ACIf) ACI = AC idem idem2
```
```   505
```
```   506 text{* Instantiation of locales: *}
```
```   507
```
```   508 lemma ACf_add: "ACf (op + :: 'a::comm_monoid_add \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```   509 by(fastsimp intro: ACf.intro add_assoc add_commute)
```
```   510
```
```   511 lemma ACe_add: "ACe (op +) (0::'a::comm_monoid_add)"
```
```   512 by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_add)
```
```   513
```
```   514
```
```   515 lemma ACf_mult: "ACf (op * :: 'a::comm_monoid_mult \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```   516 by(fast intro: ACf.intro mult_assoc ab_semigroup_mult.mult_commute)
```
```   517
```
```   518 lemma ACe_mult: "ACe (op *) (1::'a::comm_monoid_mult)"
```
```   519 by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_mult)
```
```   520
```
```   521
```
```   522 subsubsection{*From @{term foldSet} to @{term fold}*}
```
```   523
```
```   524 (* only used in the next lemma, but in there twice *)
```
```   525 lemma card_lemma: assumes A1: "A = insert b B" and notinB: "b \<notin> B" and
```
```   526   card: "A = h`{i. i<Suc n}" and new: "\<not>(EX k<n. h n = h k)"
```
```   527 shows "EX h. B = h`{i. i<n}" (is "EX h. ?P h")
```
```   528 proof
```
```   529   let ?h = "%i. if h i = b then h n else h i"
```
```   530   show "B = ?h`{i. i<n}" (is "_ = ?r")
```
```   531   proof
```
```   532     show "B \<subseteq> ?r"
```
```   533     proof
```
```   534       fix u assume "u \<in> B"
```
```   535       hence uinA: "u \<in> A" and unotb: "u \<noteq> b" using A1 notinB by blast+
```
```   536       then obtain i\<^isub>u where below: "i\<^isub>u < Suc n" and [simp]: "u = h i\<^isub>u"
```
```   537 	using card by(auto simp:image_def)
```
```   538       show "u \<in> ?r"
```
```   539       proof cases
```
```   540 	assume "i\<^isub>u < n"
```
```   541 	thus ?thesis using unotb by(fastsimp)
```
```   542       next
```
```   543 	assume "\<not> i\<^isub>u < n"
```
```   544 	with below have [simp]: "i\<^isub>u = n" by arith
```
```   545 	obtain i\<^isub>k where i\<^isub>k: "i\<^isub>k < Suc n" and [simp]: "b = h i\<^isub>k"
```
```   546 	  using A1 card by blast
```
```   547 	have "i\<^isub>k < n"
```
```   548 	proof (rule ccontr)
```
```   549 	  assume "\<not> i\<^isub>k < n"
```
```   550 	  hence "i\<^isub>k = n" using i\<^isub>k by arith
```
```   551 	  thus False using unotb by simp
```
```   552 	qed
```
```   553 	thus ?thesis by(auto simp add:image_def)
```
```   554       qed
```
```   555     qed
```
```   556   next
```
```   557     show "?r \<subseteq> B"
```
```   558     proof
```
```   559       fix u assume "u \<in> ?r"
```
```   560       then obtain i\<^isub>u where below: "i\<^isub>u < n" and
```
```   561         or: "b = h i\<^isub>u \<and> u = h n \<or> h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
```
```   562 	by(auto simp:image_def)
```
```   563       from or show "u \<in> B"
```
```   564       proof
```
```   565 	assume [simp]: "b = h i\<^isub>u \<and> u = h n"
```
```   566 	have "u \<in> A" using card by auto
```
```   567         moreover have "u \<noteq> b" using new below by auto
```
```   568 	ultimately show "u \<in> B" using A1 by blast
```
```   569       next
```
```   570 	assume "h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
```
```   571 	moreover hence "u \<in> A" using card below by auto
```
```   572 	ultimately show "u \<in> B" using A1 by blast
```
```   573       qed
```
```   574     qed
```
```   575   qed
```
```   576 qed
```
```   577
```
```   578 lemma (in ACf) foldSet_determ_aux:
```
```   579   "!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
```
```   580    \<Longrightarrow> x' = x"
```
```   581 proof (induct n)
```
```   582   case 0 thus ?case by auto
```
```   583 next
```
```   584   case (Suc n)
```
```   585   have IH: "!!A x x' h. \<lbrakk>A = h`{i::nat. i<n}; (A,x) \<in> foldSet f g z; (A,x') \<in> foldSet f g z\<rbrakk>
```
```   586            \<Longrightarrow> x' = x" and card: "A = h`{i. i<Suc n}"
```
```   587   and Afoldx: "(A, x) \<in> foldSet f g z" and Afoldy: "(A,x') \<in> foldSet f g z" .
```
```   588   show ?case
```
```   589   proof cases
```
```   590     assume "EX k<n. h n = h k"
```
```   591       --{*@{term h} is not injective, so the cardinality has not increased*}
```
```   592     hence card': "A = h ` {i. i < n}"
```
```   593       using card by (auto simp:image_def less_Suc_eq)
```
```   594     show ?thesis by(rule IH[OF card' Afoldx Afoldy])
```
```   595   next
```
```   596     assume new: "\<not>(EX k<n. h n = h k)"
```
```   597     show ?thesis
```
```   598     proof (rule foldSet.cases[OF Afoldx])
```
```   599       assume "(A, x) = ({}, z)"  --{*fold of a singleton set*}
```
```   600       thus "x' = x" using Afoldy by (auto)
```
```   601     next
```
```   602       fix B b y
```
```   603       assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
```
```   604 	and y: "(B,y) \<in> foldSet f g z" and notinB: "b \<notin> B"
```
```   605       hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
```
```   606       show ?thesis
```
```   607       proof (rule foldSet.cases[OF Afoldy])
```
```   608 	assume "(A,x') = ({}, z)"
```
```   609 	thus ?thesis using A1 by auto
```
```   610       next
```
```   611 	fix C c r
```
```   612 	assume eq2: "(A,x') = (insert c C, g c \<cdot> r)"
```
```   613 	  and r: "(C,r) \<in> foldSet f g z" and notinC: "c \<notin> C"
```
```   614 	hence A2: "A = insert c C" and x': "x' = g c \<cdot> r" by auto
```
```   615 	obtain hB where lessB: "B = hB ` {i. i<n}"
```
```   616 	  using card_lemma[OF A1 notinB card new] by auto
```
```   617 	obtain hC where lessC: "C = hC ` {i. i<n}"
```
```   618 	  using card_lemma[OF A2 notinC card new] by auto
```
```   619 	show ?thesis
```
```   620 	proof cases
```
```   621 	  assume "b = c"
```
```   622 	  then moreover have "B = C" using A1 A2 notinB notinC by auto
```
```   623 	  ultimately show ?thesis using IH[OF lessB] y r x x' by auto
```
```   624 	next
```
```   625 	  assume diff: "b \<noteq> c"
```
```   626 	  let ?D = "B - {c}"
```
```   627 	  have B: "B = insert c ?D" and C: "C = insert b ?D"
```
```   628 	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
```
```   629 	  have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
```
```   630 	  with A1 have "finite ?D" by simp
```
```   631 	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g z"
```
```   632 	    using finite_imp_foldSet by rules
```
```   633 	  moreover have cinB: "c \<in> B" using B by(auto)
```
```   634 	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
```
```   635 	    by(rule Diff1_foldSet)
```
```   636 	  hence "g c \<cdot> d = y" by(rule IH[OF lessB y])
```
```   637           moreover have "g b \<cdot> d = r"
```
```   638 	  proof (rule IH[OF lessC r])
```
```   639 	    show "(C,g b \<cdot> d) \<in> foldSet f g z" using C notinB Dfoldd
```
```   640 	      by fastsimp
```
```   641 	  qed
```
```   642 	  ultimately show ?thesis using x x' by(auto simp:AC)
```
```   643 	qed
```
```   644       qed
```
```   645     qed
```
```   646   qed
```
```   647 qed
```
```   648
```
```   649 (* The same proof, but using card
```
```   650 lemma (in ACf) foldSet_determ_aux:
```
```   651   "!!A x x'. \<lbrakk> card A < n; (A,x) : foldSet f g e; (A,x') : foldSet f g e \<rbrakk>
```
```   652    \<Longrightarrow> x' = x"
```
```   653 proof (induct n)
```
```   654   case 0 thus ?case by simp
```
```   655 next
```
```   656   case (Suc n)
```
```   657   have IH: "!!A x x'. \<lbrakk>card A < n; (A,x) \<in> foldSet f g e; (A,x') \<in> foldSet f g e\<rbrakk>
```
```   658            \<Longrightarrow> x' = x" and card: "card A < Suc n"
```
```   659   and Afoldx: "(A, x) \<in> foldSet f g e" and Afoldy: "(A,x') \<in> foldSet f g e" .
```
```   660   from card have "card A < n \<or> card A = n" by arith
```
```   661   thus ?case
```
```   662   proof
```
```   663     assume less: "card A < n"
```
```   664     show ?thesis by(rule IH[OF less Afoldx Afoldy])
```
```   665   next
```
```   666     assume cardA: "card A = n"
```
```   667     show ?thesis
```
```   668     proof (rule foldSet.cases[OF Afoldx])
```
```   669       assume "(A, x) = ({}, e)"
```
```   670       thus "x' = x" using Afoldy by (auto)
```
```   671     next
```
```   672       fix B b y
```
```   673       assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
```
```   674 	and y: "(B,y) \<in> foldSet f g e" and notinB: "b \<notin> B"
```
```   675       hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
```
```   676       show ?thesis
```
```   677       proof (rule foldSet.cases[OF Afoldy])
```
```   678 	assume "(A,x') = ({}, e)"
```
```   679 	thus ?thesis using A1 by auto
```
```   680       next
```
```   681 	fix C c z
```
```   682 	assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
```
```   683 	  and z: "(C,z) \<in> foldSet f g e" and notinC: "c \<notin> C"
```
```   684 	hence A2: "A = insert c C" and x': "x' = g c \<cdot> z" by auto
```
```   685 	have finA: "finite A" by(rule foldSet_imp_finite[OF Afoldx])
```
```   686 	with cardA A1 notinB have less: "card B < n" by simp
```
```   687 	show ?thesis
```
```   688 	proof cases
```
```   689 	  assume "b = c"
```
```   690 	  then moreover have "B = C" using A1 A2 notinB notinC by auto
```
```   691 	  ultimately show ?thesis using IH[OF less] y z x x' by auto
```
```   692 	next
```
```   693 	  assume diff: "b \<noteq> c"
```
```   694 	  let ?D = "B - {c}"
```
```   695 	  have B: "B = insert c ?D" and C: "C = insert b ?D"
```
```   696 	    using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
```
```   697 	  have "finite ?D" using finA A1 by simp
```
```   698 	  then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
```
```   699 	    using finite_imp_foldSet by rules
```
```   700 	  moreover have cinB: "c \<in> B" using B by(auto)
```
```   701 	  ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
```
```   702 	    by(rule Diff1_foldSet)
```
```   703 	  hence "g c \<cdot> d = y" by(rule IH[OF less y])
```
```   704           moreover have "g b \<cdot> d = z"
```
```   705 	  proof (rule IH[OF _ z])
```
```   706 	    show "card C < n" using C cardA A1 notinB finA cinB
```
```   707 	      by(auto simp:card_Diff1_less)
```
```   708 	  next
```
```   709 	    show "(C,g b \<cdot> d) \<in> foldSet f g e" using C notinB Dfoldd
```
```   710 	      by fastsimp
```
```   711 	  qed
```
```   712 	  ultimately show ?thesis using x x' by(auto simp:AC)
```
```   713 	qed
```
```   714       qed
```
```   715     qed
```
```   716   qed
```
```   717 qed
```
```   718 *)
```
```   719
```
```   720 lemma (in ACf) foldSet_determ:
```
```   721   "(A, x) : foldSet f g z ==> (A, y) : foldSet f g z ==> y = x"
```
```   722 apply(frule foldSet_imp_finite)
```
```   723 apply(simp add:finite_conv_nat_seg_image)
```
```   724 apply(blast intro: foldSet_determ_aux [rule_format])
```
```   725 done
```
```   726
```
```   727 lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y"
```
```   728   by (unfold fold_def) (blast intro: foldSet_determ)
```
```   729
```
```   730 text{* The base case for @{text fold}: *}
```
```   731
```
```   732 lemma fold_empty [simp]: "fold f g z {} = z"
```
```   733   by (unfold fold_def) blast
```
```   734
```
```   735 lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
```
```   736     ((insert x A, v) : foldSet f g z) =
```
```   737     (EX y. (A, y) : foldSet f g z & v = f (g x) y)"
```
```   738   apply auto
```
```   739   apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
```
```   740    apply (fastsimp dest: foldSet_imp_finite)
```
```   741   apply (blast intro: foldSet_determ)
```
```   742   done
```
```   743
```
```   744 text{* The recursion equation for @{text fold}: *}
```
```   745
```
```   746 lemma (in ACf) fold_insert[simp]:
```
```   747     "finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
```
```   748   apply (unfold fold_def)
```
```   749   apply (simp add: fold_insert_aux)
```
```   750   apply (rule the_equality)
```
```   751   apply (auto intro: finite_imp_foldSet
```
```   752     cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
```
```   753   done
```
```   754
```
```   755 declare
```
```   756   empty_foldSetE [rule del]  foldSet.intros [rule del]
```
```   757   -- {* Delete rules to do with @{text foldSet} relation. *}
```
```   758
```
```   759 text{* A simplified version for idempotent functions: *}
```
```   760
```
```   761 lemma (in ACIf) fold_insert2:
```
```   762 assumes finA: "finite A"
```
```   763 shows "fold (op \<cdot>) g z (insert a A) = g a \<cdot> fold f g z A"
```
```   764 proof cases
```
```   765   assume "a \<in> A"
```
```   766   then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
```
```   767     by(blast dest: mk_disjoint_insert)
```
```   768   show ?thesis
```
```   769   proof -
```
```   770     from finA A have finB: "finite B" by(blast intro: finite_subset)
```
```   771     have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
```
```   772     also have "\<dots> = (g a) \<cdot> (fold f g z B)"
```
```   773       using finB disj by(simp)
```
```   774     also have "\<dots> = g a \<cdot> fold f g z A"
```
```   775       using A finB disj by(simp add:idem assoc[symmetric])
```
```   776     finally show ?thesis .
```
```   777   qed
```
```   778 next
```
```   779   assume "a \<notin> A"
```
```   780   with finA show ?thesis by simp
```
```   781 qed
```
```   782
```
```   783 lemma (in ACIf) foldI_conv_id:
```
```   784   "finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
```
```   785 by(erule finite_induct)(simp_all add: fold_insert2 del: fold_insert)
```
```   786
```
```   787 subsubsection{*Lemmas about @{text fold}*}
```
```   788
```
```   789 lemma (in ACf) fold_commute:
```
```   790   "finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
```
```   791   apply (induct set: Finites, simp)
```
```   792   apply (simp add: left_commute [of x])
```
```   793   done
```
```   794
```
```   795 lemma (in ACf) fold_nest_Un_Int:
```
```   796   "finite A ==> finite B
```
```   797     ==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
```
```   798   apply (induct set: Finites, simp)
```
```   799   apply (simp add: fold_commute Int_insert_left insert_absorb)
```
```   800   done
```
```   801
```
```   802 lemma (in ACf) fold_nest_Un_disjoint:
```
```   803   "finite A ==> finite B ==> A Int B = {}
```
```   804     ==> fold f g z (A Un B) = fold f g (fold f g z B) A"
```
```   805   by (simp add: fold_nest_Un_Int)
```
```   806
```
```   807 lemma (in ACf) fold_reindex:
```
```   808 assumes fin: "finite A"
```
```   809 shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
```
```   810 using fin apply (induct)
```
```   811  apply simp
```
```   812 apply simp
```
```   813 done
```
```   814
```
```   815 lemma (in ACe) fold_Un_Int:
```
```   816   "finite A ==> finite B ==>
```
```   817     fold f g e A \<cdot> fold f g e B =
```
```   818     fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
```
```   819   apply (induct set: Finites, simp)
```
```   820   apply (simp add: AC insert_absorb Int_insert_left)
```
```   821   done
```
```   822
```
```   823 corollary (in ACe) fold_Un_disjoint:
```
```   824   "finite A ==> finite B ==> A Int B = {} ==>
```
```   825     fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
```
```   826   by (simp add: fold_Un_Int)
```
```   827
```
```   828 lemma (in ACe) fold_UN_disjoint:
```
```   829   "\<lbrakk> finite I; ALL i:I. finite (A i);
```
```   830      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
```
```   831    \<Longrightarrow> fold f g e (UNION I A) =
```
```   832        fold f (%i. fold f g e (A i)) e I"
```
```   833   apply (induct set: Finites, simp, atomize)
```
```   834   apply (subgoal_tac "ALL i:F. x \<noteq> i")
```
```   835    prefer 2 apply blast
```
```   836   apply (subgoal_tac "A x Int UNION F A = {}")
```
```   837    prefer 2 apply blast
```
```   838   apply (simp add: fold_Un_disjoint)
```
```   839   done
```
```   840
```
```   841 lemma (in ACf) fold_cong:
```
```   842   "finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
```
```   843   apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
```
```   844    apply simp
```
```   845   apply (erule finite_induct, simp)
```
```   846   apply (simp add: subset_insert_iff, clarify)
```
```   847   apply (subgoal_tac "finite C")
```
```   848    prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
```
```   849   apply (subgoal_tac "C = insert x (C - {x})")
```
```   850    prefer 2 apply blast
```
```   851   apply (erule ssubst)
```
```   852   apply (drule spec)
```
```   853   apply (erule (1) notE impE)
```
```   854   apply (simp add: Ball_def del: insert_Diff_single)
```
```   855   done
```
```   856
```
```   857 lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   858   fold f (%x. fold f (g x) e (B x)) e A =
```
```   859   fold f (split g) e (SIGMA x:A. B x)"
```
```   860 apply (subst Sigma_def)
```
```   861 apply (subst fold_UN_disjoint)
```
```   862    apply assumption
```
```   863   apply simp
```
```   864  apply blast
```
```   865 apply (erule fold_cong)
```
```   866 apply (subst fold_UN_disjoint)
```
```   867    apply simp
```
```   868   apply simp
```
```   869  apply blast
```
```   870 apply (simp)
```
```   871 done
```
```   872
```
```   873 lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
```
```   874    fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
```
```   875 apply (erule finite_induct)
```
```   876  apply simp
```
```   877 apply (simp add:AC)
```
```   878 done
```
```   879
```
```   880
```
```   881 subsection {* Generalized summation over a set *}
```
```   882
```
```   883 constdefs
```
```   884   setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
```
```   885   "setsum f A == if finite A then fold (op +) f 0 A else 0"
```
```   886
```
```   887 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```   888 written @{text"\<Sum>x\<in>A. e"}. *}
```
```   889
```
```   890 syntax
```
```   891   "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```   892 syntax (xsymbols)
```
```   893   "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   894 syntax (HTML output)
```
```   895   "_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   896
```
```   897 translations -- {* Beware of argument permutation! *}
```
```   898   "SUM i:A. b" == "setsum (%i. b) A"
```
```   899   "\<Sum>i\<in>A. b" == "setsum (%i. b) A"
```
```   900
```
```   901 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```   902  @{text"\<Sum>x|P. e"}. *}
```
```   903
```
```   904 syntax
```
```   905   "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   906 syntax (xsymbols)
```
```   907   "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   908 syntax (HTML output)
```
```   909   "_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   910
```
```   911 translations
```
```   912   "SUM x|P. t" => "setsum (%x. t) {x. P}"
```
```   913   "\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
```
```   914
```
```   915 text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
```
```   916
```
```   917 syntax
```
```   918   "_Setsum" :: "'a set => 'a::comm_monoid_mult"  ("\<Sum>_" [1000] 999)
```
```   919
```
```   920 parse_translation {*
```
```   921   let
```
```   922     fun Setsum_tr [A] = Syntax.const "setsum" \$ Abs ("", dummyT, Bound 0) \$ A
```
```   923   in [("_Setsum", Setsum_tr)] end;
```
```   924 *}
```
```   925
```
```   926 print_translation {*
```
```   927 let
```
```   928   fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" \$ A
```
```   929     | setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) \$ Abs(y,Ty,P)] =
```
```   930        if x<>y then raise Match
```
```   931        else let val x' = Syntax.mark_bound x
```
```   932                 val t' = subst_bound(x',t)
```
```   933                 val P' = subst_bound(x',P)
```
```   934             in Syntax.const "_qsetsum" \$ Syntax.mark_bound x \$ P' \$ t' end
```
```   935 in
```
```   936 [("setsum", setsum_tr')]
```
```   937 end
```
```   938 *}
```
```   939
```
```   940 lemma setsum_empty [simp]: "setsum f {} = 0"
```
```   941   by (simp add: setsum_def)
```
```   942
```
```   943 lemma setsum_insert [simp]:
```
```   944     "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   945   by (simp add: setsum_def ACf.fold_insert [OF ACf_add])
```
```   946
```
```   947 lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
```
```   948   by (simp add: setsum_def)
```
```   949
```
```   950 lemma setsum_reindex:
```
```   951      "inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   952 by(auto simp add: setsum_def ACf.fold_reindex[OF ACf_add] dest!:finite_imageD)
```
```   953
```
```   954 lemma setsum_reindex_id:
```
```   955      "inj_on f B ==> setsum f B = setsum id (f ` B)"
```
```   956 by (auto simp add: setsum_reindex)
```
```   957
```
```   958 lemma setsum_cong:
```
```   959   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   960 by(fastsimp simp: setsum_def intro: ACf.fold_cong[OF ACf_add])
```
```   961
```
```   962 lemma setsum_reindex_cong:
```
```   963      "[|inj_on f A; B = f ` A; !!a. g a = h (f a)|]
```
```   964       ==> setsum h B = setsum g A"
```
```   965   by (simp add: setsum_reindex cong: setsum_cong)
```
```   966
```
```   967 lemma setsum_0: "setsum (%i. 0) A = 0"
```
```   968 apply (clarsimp simp: setsum_def)
```
```   969 apply (erule finite_induct, auto simp:ACf.fold_insert [OF ACf_add])
```
```   970 done
```
```   971
```
```   972 lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0"
```
```   973   apply (subgoal_tac "setsum f F = setsum (%x. 0) F")
```
```   974   apply (erule ssubst, rule setsum_0)
```
```   975   apply (rule setsum_cong, auto)
```
```   976   done
```
```   977
```
```   978 lemma setsum_Un_Int: "finite A ==> finite B ==>
```
```   979   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   980   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   981 by(simp add: setsum_def ACe.fold_Un_Int[OF ACe_add,symmetric])
```
```   982
```
```   983 lemma setsum_Un_disjoint: "finite A ==> finite B
```
```   984   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   985 by (subst setsum_Un_Int [symmetric], auto)
```
```   986
```
```   987 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   988   the lhs need not be, since UNION I A could still be finite.*)
```
```   989 lemma setsum_UN_disjoint:
```
```   990     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```   991         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```   992       setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   993 by(simp add: setsum_def ACe.fold_UN_disjoint[OF ACe_add] cong: setsum_cong)
```
```   994
```
```   995 text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
```
```   996 directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
```
```   997 lemma setsum_Union_disjoint:
```
```   998   "[| (ALL A:C. finite A);
```
```   999       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```  1000    ==> setsum f (Union C) = setsum (setsum f) C"
```
```  1001 apply (cases "finite C")
```
```  1002  prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
```
```  1003   apply (frule setsum_UN_disjoint [of C id f])
```
```  1004  apply (unfold Union_def id_def, assumption+)
```
```  1005 done
```
```  1006
```
```  1007 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```  1008   the rhs need not be, since SIGMA A B could still be finite.*)
```
```  1009 lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1010     (\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
```
```  1011     (\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
```
```  1012 by(simp add:setsum_def ACe.fold_Sigma[OF ACe_add] split_def cong:setsum_cong)
```
```  1013
```
```  1014 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1015 lemma setsum_cartesian_product:
```
```  1016    "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>z\<in>A <*> B. f (fst z) (snd z))"
```
```  1017 apply (cases "finite A")
```
```  1018  apply (cases "finite B")
```
```  1019   apply (simp add: setsum_Sigma)
```
```  1020  apply (cases "A={}", simp)
```
```  1021  apply (simp add: setsum_0)
```
```  1022 apply (auto simp add: setsum_def
```
```  1023             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1024 done
```
```  1025
```
```  1026 lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```  1027 by(simp add:setsum_def ACe.fold_distrib[OF ACe_add])
```
```  1028
```
```  1029
```
```  1030 subsubsection {* Properties in more restricted classes of structures *}
```
```  1031
```
```  1032 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```  1033   apply (case_tac "finite A")
```
```  1034    prefer 2 apply (simp add: setsum_def)
```
```  1035   apply (erule rev_mp)
```
```  1036   apply (erule finite_induct, auto)
```
```  1037   done
```
```  1038
```
```  1039 lemma setsum_eq_0_iff [simp]:
```
```  1040     "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```  1041   by (induct set: Finites) auto
```
```  1042
```
```  1043 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```  1044     (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```  1045   -- {* For the natural numbers, we have subtraction. *}
```
```  1046   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
```
```  1047
```
```  1048 lemma setsum_Un: "finite A ==> finite B ==>
```
```  1049     (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```  1050       setsum f A + setsum f B - setsum f (A Int B)"
```
```  1051   by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
```
```  1052
```
```  1053 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```  1054     (if a:A then setsum f A - f a else setsum f A)"
```
```  1055   apply (case_tac "finite A")
```
```  1056    prefer 2 apply (simp add: setsum_def)
```
```  1057   apply (erule finite_induct)
```
```  1058    apply (auto simp add: insert_Diff_if)
```
```  1059   apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```  1060   done
```
```  1061
```
```  1062 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```  1063   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```  1064   (if a:A then setsum f A - f a else setsum f A)"
```
```  1065   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```  1066
```
```  1067 (* By Jeremy Siek: *)
```
```  1068
```
```  1069 lemma setsum_diff_nat:
```
```  1070   assumes finB: "finite B"
```
```  1071   shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```  1072 using finB
```
```  1073 proof (induct)
```
```  1074   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```  1075 next
```
```  1076   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```  1077     and xFinA: "insert x F \<subseteq> A"
```
```  1078     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```  1079   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```  1080   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```  1081     by (simp add: setsum_diff1_nat)
```
```  1082   from xFinA have "F \<subseteq> A" by simp
```
```  1083   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```  1084   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```  1085     by simp
```
```  1086   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```  1087   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```  1088     by simp
```
```  1089   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```  1090   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```  1091     by simp
```
```  1092   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```  1093 qed
```
```  1094
```
```  1095 lemma setsum_diff:
```
```  1096   assumes le: "finite A" "B \<subseteq> A"
```
```  1097   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```  1098 proof -
```
```  1099   from le have finiteB: "finite B" using finite_subset by auto
```
```  1100   show ?thesis using finiteB le
```
```  1101     proof (induct)
```
```  1102       case empty
```
```  1103       thus ?case by auto
```
```  1104     next
```
```  1105       case (insert x F)
```
```  1106       thus ?case using le finiteB
```
```  1107 	by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```  1108     qed
```
```  1109   qed
```
```  1110
```
```  1111 lemma setsum_mono:
```
```  1112   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
```
```  1113   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```  1114 proof (cases "finite K")
```
```  1115   case True
```
```  1116   thus ?thesis using le
```
```  1117   proof (induct)
```
```  1118     case empty
```
```  1119     thus ?case by simp
```
```  1120   next
```
```  1121     case insert
```
```  1122     thus ?case using add_mono
```
```  1123       by force
```
```  1124   qed
```
```  1125 next
```
```  1126   case False
```
```  1127   thus ?thesis
```
```  1128     by (simp add: setsum_def)
```
```  1129 qed
```
```  1130
```
```  1131 lemma setsum_mono2_nat:
```
```  1132   assumes fin: "finite B" and sub: "A \<subseteq> B"
```
```  1133 shows "setsum f A \<le> (setsum f B :: nat)"
```
```  1134 proof -
```
```  1135   have "setsum f A \<le> setsum f A + setsum f (B-A)" by arith
```
```  1136   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```  1137     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```  1138   also have "A \<union> (B-A) = B" using sub by blast
```
```  1139   finally show ?thesis .
```
```  1140 qed
```
```  1141
```
```  1142 lemma setsum_negf: "finite A ==> setsum (%x. - (f x)::'a::ab_group_add) A =
```
```  1143   - setsum f A"
```
```  1144   by (induct set: Finites, auto)
```
```  1145
```
```  1146 lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```  1147   setsum f A - setsum g A"
```
```  1148   by (simp add: diff_minus setsum_addf setsum_negf)
```
```  1149
```
```  1150 lemma setsum_nonneg: "[| finite A;
```
```  1151     \<forall>x \<in> A. (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) \<le> f x |] ==>
```
```  1152     0 \<le> setsum f A";
```
```  1153   apply (induct set: Finites, auto)
```
```  1154   apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
```
```  1155   apply (blast intro: add_mono)
```
```  1156   done
```
```  1157
```
```  1158 lemma setsum_nonpos: "[| finite A;
```
```  1159     \<forall>x \<in> A. f x \<le> (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) |] ==>
```
```  1160     setsum f A \<le> 0";
```
```  1161   apply (induct set: Finites, auto)
```
```  1162   apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
```
```  1163   apply (blast intro: add_mono)
```
```  1164   done
```
```  1165
```
```  1166 lemma setsum_mult:
```
```  1167   fixes f :: "'a => ('b::semiring_0_cancel)"
```
```  1168   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```  1169 proof (cases "finite A")
```
```  1170   case True
```
```  1171   thus ?thesis
```
```  1172   proof (induct)
```
```  1173     case empty thus ?case by simp
```
```  1174   next
```
```  1175     case (insert x A) thus ?case by (simp add: right_distrib)
```
```  1176   qed
```
```  1177 next
```
```  1178   case False thus ?thesis by (simp add: setsum_def)
```
```  1179 qed
```
```  1180
```
```  1181 lemma setsum_abs:
```
```  1182   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1183   assumes fin: "finite A"
```
```  1184   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```  1185 using fin
```
```  1186 proof (induct)
```
```  1187   case empty thus ?case by simp
```
```  1188 next
```
```  1189   case (insert x A)
```
```  1190   thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```  1191 qed
```
```  1192
```
```  1193 lemma setsum_abs_ge_zero:
```
```  1194   fixes f :: "'a => ('b::lordered_ab_group_abs)"
```
```  1195   assumes fin: "finite A"
```
```  1196   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```  1197 using fin
```
```  1198 proof (induct)
```
```  1199   case empty thus ?case by simp
```
```  1200 next
```
```  1201   case (insert x A) thus ?case by (auto intro: order_trans)
```
```  1202 qed
```
```  1203
```
```  1204
```
```  1205 subsection {* Generalized product over a set *}
```
```  1206
```
```  1207 constdefs
```
```  1208   setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
```
```  1209   "setprod f A == if finite A then fold (op *) f 1 A else 1"
```
```  1210
```
```  1211 syntax
```
```  1212   "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_:_. _)" [0, 51, 10] 10)
```
```  1213
```
```  1214 syntax (xsymbols)
```
```  1215   "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1216 syntax (HTML output)
```
```  1217   "_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```  1218 translations
```
```  1219   "\<Prod>i:A. b" == "setprod (%i. b) A"  -- {* Beware of argument permutation! *}
```
```  1220
```
```  1221 syntax
```
```  1222   "_Setprod" :: "'a set => 'a::comm_monoid_mult"  ("\<Prod>_" [1000] 999)
```
```  1223
```
```  1224 parse_translation {*
```
```  1225   let
```
```  1226     fun Setprod_tr [A] = Syntax.const "setprod" \$ Abs ("", dummyT, Bound 0) \$ A
```
```  1227   in [("_Setprod", Setprod_tr)] end;
```
```  1228 *}
```
```  1229 print_translation {*
```
```  1230 let fun setprod_tr' [Abs(x,Tx,t), A] =
```
```  1231     if t = Bound 0 then Syntax.const "_Setprod" \$ A else raise Match
```
```  1232 in
```
```  1233 [("setprod", setprod_tr')]
```
```  1234 end
```
```  1235 *}
```
```  1236
```
```  1237
```
```  1238 lemma setprod_empty [simp]: "setprod f {} = 1"
```
```  1239   by (auto simp add: setprod_def)
```
```  1240
```
```  1241 lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
```
```  1242     setprod f (insert a A) = f a * setprod f A"
```
```  1243 by (simp add: setprod_def ACf.fold_insert [OF ACf_mult])
```
```  1244
```
```  1245 lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
```
```  1246   by (simp add: setprod_def)
```
```  1247
```
```  1248 lemma setprod_reindex:
```
```  1249      "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```  1250 by(auto simp: setprod_def ACf.fold_reindex[OF ACf_mult] dest!:finite_imageD)
```
```  1251
```
```  1252 lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```  1253 by (auto simp add: setprod_reindex)
```
```  1254
```
```  1255 lemma setprod_cong:
```
```  1256   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```  1257 by(fastsimp simp: setprod_def intro: ACf.fold_cong[OF ACf_mult])
```
```  1258
```
```  1259 lemma setprod_reindex_cong: "inj_on f A ==>
```
```  1260     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```  1261   by (frule setprod_reindex, simp)
```
```  1262
```
```  1263
```
```  1264 lemma setprod_1: "setprod (%i. 1) A = 1"
```
```  1265   apply (case_tac "finite A")
```
```  1266   apply (erule finite_induct, auto simp add: mult_ac)
```
```  1267   done
```
```  1268
```
```  1269 lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
```
```  1270   apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
```
```  1271   apply (erule ssubst, rule setprod_1)
```
```  1272   apply (rule setprod_cong, auto)
```
```  1273   done
```
```  1274
```
```  1275 lemma setprod_Un_Int: "finite A ==> finite B
```
```  1276     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```  1277 by(simp add: setprod_def ACe.fold_Un_Int[OF ACe_mult,symmetric])
```
```  1278
```
```  1279 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```  1280   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```  1281 by (subst setprod_Un_Int [symmetric], auto)
```
```  1282
```
```  1283 lemma setprod_UN_disjoint:
```
```  1284     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1285         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1286       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```  1287 by(simp add: setprod_def ACe.fold_UN_disjoint[OF ACe_mult] cong: setprod_cong)
```
```  1288
```
```  1289 lemma setprod_Union_disjoint:
```
```  1290   "[| (ALL A:C. finite A);
```
```  1291       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```  1292    ==> setprod f (Union C) = setprod (setprod f) C"
```
```  1293 apply (cases "finite C")
```
```  1294  prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
```
```  1295   apply (frule setprod_UN_disjoint [of C id f])
```
```  1296  apply (unfold Union_def id_def, assumption+)
```
```  1297 done
```
```  1298
```
```  1299 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1300     (\<Prod>x:A. (\<Prod>y: B x. f x y)) =
```
```  1301     (\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))"
```
```  1302 by(simp add:setprod_def ACe.fold_Sigma[OF ACe_mult] split_def cong:setprod_cong)
```
```  1303
```
```  1304 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1305 lemma setprod_cartesian_product:
```
```  1306      "(\<Prod>x:A. (\<Prod>y: B. f x y)) = (\<Prod>z:(A <*> B). f (fst z) (snd z))"
```
```  1307 apply (cases "finite A")
```
```  1308  apply (cases "finite B")
```
```  1309   apply (simp add: setprod_Sigma)
```
```  1310  apply (cases "A={}", simp)
```
```  1311  apply (simp add: setprod_1)
```
```  1312 apply (auto simp add: setprod_def
```
```  1313             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1314 done
```
```  1315
```
```  1316 lemma setprod_timesf:
```
```  1317      "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
```
```  1318 by(simp add:setprod_def ACe.fold_distrib[OF ACe_mult])
```
```  1319
```
```  1320
```
```  1321 subsubsection {* Properties in more restricted classes of structures *}
```
```  1322
```
```  1323 lemma setprod_eq_1_iff [simp]:
```
```  1324     "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
```
```  1325   by (induct set: Finites) auto
```
```  1326
```
```  1327 lemma setprod_zero:
```
```  1328      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
```
```  1329   apply (induct set: Finites, force, clarsimp)
```
```  1330   apply (erule disjE, auto)
```
```  1331   done
```
```  1332
```
```  1333 lemma setprod_nonneg [rule_format]:
```
```  1334      "(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1335   apply (case_tac "finite A")
```
```  1336   apply (induct set: Finites, force, clarsimp)
```
```  1337   apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
```
```  1338   apply (rule mult_mono, assumption+)
```
```  1339   apply (auto simp add: setprod_def)
```
```  1340   done
```
```  1341
```
```  1342 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
```
```  1343      --> 0 < setprod f A"
```
```  1344   apply (case_tac "finite A")
```
```  1345   apply (induct set: Finites, force, clarsimp)
```
```  1346   apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
```
```  1347   apply (rule mult_strict_mono, assumption+)
```
```  1348   apply (auto simp add: setprod_def)
```
```  1349   done
```
```  1350
```
```  1351 lemma setprod_nonzero [rule_format]:
```
```  1352     "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
```
```  1353       finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
```
```  1354   apply (erule finite_induct, auto)
```
```  1355   done
```
```  1356
```
```  1357 lemma setprod_zero_eq:
```
```  1358     "(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
```
```  1359      finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
```
```  1360   apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
```
```  1361   done
```
```  1362
```
```  1363 lemma setprod_nonzero_field:
```
```  1364     "finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
```
```  1365   apply (rule setprod_nonzero, auto)
```
```  1366   done
```
```  1367
```
```  1368 lemma setprod_zero_eq_field:
```
```  1369     "finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
```
```  1370   apply (rule setprod_zero_eq, auto)
```
```  1371   done
```
```  1372
```
```  1373 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1374     (setprod f (A Un B) :: 'a ::{field})
```
```  1375       = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1376   apply (subst setprod_Un_Int [symmetric], auto)
```
```  1377   apply (subgoal_tac "finite (A Int B)")
```
```  1378   apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
```
```  1379   apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
```
```  1380   done
```
```  1381
```
```  1382 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1383     (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1384       (if a:A then setprod f A / f a else setprod f A)"
```
```  1385   apply (erule finite_induct)
```
```  1386    apply (auto simp add: insert_Diff_if)
```
```  1387   apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
```
```  1388   apply (erule ssubst)
```
```  1389   apply (subst times_divide_eq_right [THEN sym])
```
```  1390   apply (auto simp add: mult_ac times_divide_eq_right divide_self)
```
```  1391   done
```
```  1392
```
```  1393 lemma setprod_inversef: "finite A ==>
```
```  1394     ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
```
```  1395       setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1396   apply (erule finite_induct)
```
```  1397   apply (simp, simp)
```
```  1398   done
```
```  1399
```
```  1400 lemma setprod_dividef:
```
```  1401      "[|finite A;
```
```  1402         \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
```
```  1403       ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1404   apply (subgoal_tac
```
```  1405          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1406   apply (erule ssubst)
```
```  1407   apply (subst divide_inverse)
```
```  1408   apply (subst setprod_timesf)
```
```  1409   apply (subst setprod_inversef, assumption+, rule refl)
```
```  1410   apply (rule setprod_cong, rule refl)
```
```  1411   apply (subst divide_inverse, auto)
```
```  1412   done
```
```  1413
```
```  1414 subsection {* Finite cardinality *}
```
```  1415
```
```  1416 text {* This definition, although traditional, is ugly to work with:
```
```  1417 @{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
```
```  1418 But now that we have @{text setsum} things are easy:
```
```  1419 *}
```
```  1420
```
```  1421 constdefs
```
```  1422   card :: "'a set => nat"
```
```  1423   "card A == setsum (%x. 1::nat) A"
```
```  1424
```
```  1425 lemma card_empty [simp]: "card {} = 0"
```
```  1426   by (simp add: card_def)
```
```  1427
```
```  1428 lemma card_infinite [simp]: "~ finite A ==> card A = 0"
```
```  1429   by (simp add: card_def)
```
```  1430
```
```  1431 lemma card_eq_setsum: "card A = setsum (%x. 1) A"
```
```  1432 by (simp add: card_def)
```
```  1433
```
```  1434 lemma card_insert_disjoint [simp]:
```
```  1435   "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
```
```  1436 by(simp add: card_def ACf.fold_insert[OF ACf_add])
```
```  1437
```
```  1438 lemma card_insert_if:
```
```  1439     "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
```
```  1440   by (simp add: insert_absorb)
```
```  1441
```
```  1442 lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
```
```  1443   apply auto
```
```  1444   apply (drule_tac a = x in mk_disjoint_insert, clarify)
```
```  1445   apply (auto)
```
```  1446   done
```
```  1447
```
```  1448 lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
```
```  1449 by auto
```
```  1450
```
```  1451 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
```
```  1452 apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
```
```  1453 apply(simp del:insert_Diff_single)
```
```  1454 done
```
```  1455
```
```  1456 lemma card_Diff_singleton:
```
```  1457     "finite A ==> x: A ==> card (A - {x}) = card A - 1"
```
```  1458   by (simp add: card_Suc_Diff1 [symmetric])
```
```  1459
```
```  1460 lemma card_Diff_singleton_if:
```
```  1461     "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
```
```  1462   by (simp add: card_Diff_singleton)
```
```  1463
```
```  1464 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
```
```  1465   by (simp add: card_insert_if card_Suc_Diff1)
```
```  1466
```
```  1467 lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
```
```  1468   by (simp add: card_insert_if)
```
```  1469
```
```  1470 lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
```
```  1471 by (simp add: card_def setsum_mono2_nat)
```
```  1472
```
```  1473 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
```
```  1474   apply (induct set: Finites, simp, clarify)
```
```  1475   apply (subgoal_tac "finite A & A - {x} <= F")
```
```  1476    prefer 2 apply (blast intro: finite_subset, atomize)
```
```  1477   apply (drule_tac x = "A - {x}" in spec)
```
```  1478   apply (simp add: card_Diff_singleton_if split add: split_if_asm)
```
```  1479   apply (case_tac "card A", auto)
```
```  1480   done
```
```  1481
```
```  1482 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
```
```  1483   apply (simp add: psubset_def linorder_not_le [symmetric])
```
```  1484   apply (blast dest: card_seteq)
```
```  1485   done
```
```  1486
```
```  1487 lemma card_Un_Int: "finite A ==> finite B
```
```  1488     ==> card A + card B = card (A Un B) + card (A Int B)"
```
```  1489 by(simp add:card_def setsum_Un_Int)
```
```  1490
```
```  1491 lemma card_Un_disjoint: "finite A ==> finite B
```
```  1492     ==> A Int B = {} ==> card (A Un B) = card A + card B"
```
```  1493   by (simp add: card_Un_Int)
```
```  1494
```
```  1495 lemma card_Diff_subset:
```
```  1496   "finite B ==> B <= A ==> card (A - B) = card A - card B"
```
```  1497 by(simp add:card_def setsum_diff_nat)
```
```  1498
```
```  1499 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
```
```  1500   apply (rule Suc_less_SucD)
```
```  1501   apply (simp add: card_Suc_Diff1)
```
```  1502   done
```
```  1503
```
```  1504 lemma card_Diff2_less:
```
```  1505     "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
```
```  1506   apply (case_tac "x = y")
```
```  1507    apply (simp add: card_Diff1_less)
```
```  1508   apply (rule less_trans)
```
```  1509    prefer 2 apply (auto intro!: card_Diff1_less)
```
```  1510   done
```
```  1511
```
```  1512 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
```
```  1513   apply (case_tac "x : A")
```
```  1514    apply (simp_all add: card_Diff1_less less_imp_le)
```
```  1515   done
```
```  1516
```
```  1517 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
```
```  1518 by (erule psubsetI, blast)
```
```  1519
```
```  1520 lemma insert_partition:
```
```  1521   "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
```
```  1522   \<Longrightarrow> x \<inter> \<Union> F = {}"
```
```  1523 by auto
```
```  1524
```
```  1525 (* main cardinality theorem *)
```
```  1526 lemma card_partition [rule_format]:
```
```  1527      "finite C ==>
```
```  1528         finite (\<Union> C) -->
```
```  1529         (\<forall>c\<in>C. card c = k) -->
```
```  1530         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
```
```  1531         k * card(C) = card (\<Union> C)"
```
```  1532 apply (erule finite_induct, simp)
```
```  1533 apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition
```
```  1534        finite_subset [of _ "\<Union> (insert x F)"])
```
```  1535 done
```
```  1536
```
```  1537
```
```  1538 lemma setsum_constant_nat: "(\<Sum>x\<in>A. y) = (card A) * y"
```
```  1539   -- {* Generalized to any @{text comm_semiring_1_cancel} in
```
```  1540         @{text IntDef} as @{text setsum_constant}. *}
```
```  1541 apply (cases "finite A")
```
```  1542 apply (erule finite_induct, auto)
```
```  1543 done
```
```  1544
```
```  1545 lemma setprod_constant: "finite A ==> (\<Prod>x: A. (y::'a::recpower)) = y^(card A)"
```
```  1546   apply (erule finite_induct)
```
```  1547   apply (auto simp add: power_Suc)
```
```  1548   done
```
```  1549
```
```  1550
```
```  1551 subsubsection {* Cardinality of unions *}
```
```  1552
```
```  1553 lemma card_UN_disjoint:
```
```  1554     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1555         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1556       card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```  1557   apply (simp add: card_def)
```
```  1558   apply (subgoal_tac
```
```  1559            "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
```
```  1560   apply (simp add: setsum_UN_disjoint)
```
```  1561   apply (simp add: setsum_constant_nat cong: setsum_cong)
```
```  1562   done
```
```  1563
```
```  1564 lemma card_Union_disjoint:
```
```  1565   "finite C ==> (ALL A:C. finite A) ==>
```
```  1566         (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
```
```  1567       card (Union C) = setsum card C"
```
```  1568   apply (frule card_UN_disjoint [of C id])
```
```  1569   apply (unfold Union_def id_def, assumption+)
```
```  1570   done
```
```  1571
```
```  1572 subsubsection {* Cardinality of image *}
```
```  1573
```
```  1574 text{*The image of a finite set can be expressed using @{term fold}.*}
```
```  1575 lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
```
```  1576   apply (erule finite_induct, simp)
```
```  1577   apply (subst ACf.fold_insert)
```
```  1578   apply (auto simp add: ACf_def)
```
```  1579   done
```
```  1580
```
```  1581 lemma card_image_le: "finite A ==> card (f ` A) <= card A"
```
```  1582   apply (induct set: Finites, simp)
```
```  1583   apply (simp add: le_SucI finite_imageI card_insert_if)
```
```  1584   done
```
```  1585
```
```  1586 lemma card_image: "inj_on f A ==> card (f ` A) = card A"
```
```  1587 by(simp add:card_def setsum_reindex o_def)
```
```  1588
```
```  1589 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
```
```  1590   by (simp add: card_seteq card_image)
```
```  1591
```
```  1592 lemma eq_card_imp_inj_on:
```
```  1593   "[| finite A; card(f ` A) = card A |] ==> inj_on f A"
```
```  1594 apply(induct rule:finite_induct)
```
```  1595  apply simp
```
```  1596 apply(frule card_image_le[where f = f])
```
```  1597 apply(simp add:card_insert_if split:if_splits)
```
```  1598 done
```
```  1599
```
```  1600 lemma inj_on_iff_eq_card:
```
```  1601   "finite A ==> inj_on f A = (card(f ` A) = card A)"
```
```  1602 by(blast intro: card_image eq_card_imp_inj_on)
```
```  1603
```
```  1604
```
```  1605 lemma card_inj_on_le:
```
```  1606     "[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
```
```  1607 apply (subgoal_tac "finite A")
```
```  1608  apply (force intro: card_mono simp add: card_image [symmetric])
```
```  1609 apply (blast intro: finite_imageD dest: finite_subset)
```
```  1610 done
```
```  1611
```
```  1612 lemma card_bij_eq:
```
```  1613     "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
```
```  1614        finite A; finite B |] ==> card A = card B"
```
```  1615   by (auto intro: le_anti_sym card_inj_on_le)
```
```  1616
```
```  1617
```
```  1618 subsubsection {* Cardinality of products *}
```
```  1619
```
```  1620 (*
```
```  1621 lemma SigmaI_insert: "y \<notin> A ==>
```
```  1622   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```  1623   by auto
```
```  1624 *)
```
```  1625
```
```  1626 lemma card_SigmaI [simp]:
```
```  1627   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```  1628   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```  1629 by(simp add:card_def setsum_Sigma)
```
```  1630
```
```  1631 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```  1632 apply (cases "finite A")
```
```  1633 apply (cases "finite B")
```
```  1634   apply (simp add: setsum_constant_nat)
```
```  1635 apply (auto simp add: card_eq_0_iff
```
```  1636             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1637 done
```
```  1638
```
```  1639 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```  1640 by (simp add: card_cartesian_product)
```
```  1641
```
```  1642
```
```  1643
```
```  1644 subsubsection {* Cardinality of the Powerset *}
```
```  1645
```
```  1646 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"  (* FIXME numeral 2 (!?) *)
```
```  1647   apply (induct set: Finites)
```
```  1648    apply (simp_all add: Pow_insert)
```
```  1649   apply (subst card_Un_disjoint, blast)
```
```  1650     apply (blast intro: finite_imageI, blast)
```
```  1651   apply (subgoal_tac "inj_on (insert x) (Pow F)")
```
```  1652    apply (simp add: card_image Pow_insert)
```
```  1653   apply (unfold inj_on_def)
```
```  1654   apply (blast elim!: equalityE)
```
```  1655   done
```
```  1656
```
```  1657 text {* Relates to equivalence classes.  Based on a theorem of
```
```  1658 F. Kammüller's.  *}
```
```  1659
```
```  1660 lemma dvd_partition:
```
```  1661   "finite (Union C) ==>
```
```  1662     ALL c : C. k dvd card c ==>
```
```  1663     (ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
```
```  1664   k dvd card (Union C)"
```
```  1665 apply(frule finite_UnionD)
```
```  1666 apply(rotate_tac -1)
```
```  1667   apply (induct set: Finites, simp_all, clarify)
```
```  1668   apply (subst card_Un_disjoint)
```
```  1669   apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
```
```  1670   done
```
```  1671
```
```  1672
```
```  1673 subsubsection {* Theorems about @{text "choose"} *}
```
```  1674
```
```  1675 text {*
```
```  1676   \medskip Basic theorem about @{text "choose"}.  By Florian
```
```  1677   Kamm\"uller, tidied by LCP.
```
```  1678 *}
```
```  1679
```
```  1680 lemma card_s_0_eq_empty:
```
```  1681     "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
```
```  1682   apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
```
```  1683   apply (simp cong add: rev_conj_cong)
```
```  1684   done
```
```  1685
```
```  1686 lemma choose_deconstruct: "finite M ==> x \<notin> M
```
```  1687   ==> {s. s <= insert x M & card(s) = Suc k}
```
```  1688        = {s. s <= M & card(s) = Suc k} Un
```
```  1689          {s. EX t. t <= M & card(t) = k & s = insert x t}"
```
```  1690   apply safe
```
```  1691    apply (auto intro: finite_subset [THEN card_insert_disjoint])
```
```  1692   apply (drule_tac x = "xa - {x}" in spec)
```
```  1693   apply (subgoal_tac "x \<notin> xa", auto)
```
```  1694   apply (erule rev_mp, subst card_Diff_singleton)
```
```  1695   apply (auto intro: finite_subset)
```
```  1696   done
```
```  1697
```
```  1698 text{*There are as many subsets of @{term A} having cardinality @{term k}
```
```  1699  as there are sets obtained from the former by inserting a fixed element
```
```  1700  @{term x} into each.*}
```
```  1701 lemma constr_bij:
```
```  1702    "[|finite A; x \<notin> A|] ==>
```
```  1703     card {B. EX C. C <= A & card(C) = k & B = insert x C} =
```
```  1704     card {B. B <= A & card(B) = k}"
```
```  1705   apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
```
```  1706        apply (auto elim!: equalityE simp add: inj_on_def)
```
```  1707     apply (subst Diff_insert0, auto)
```
```  1708    txt {* finiteness of the two sets *}
```
```  1709    apply (rule_tac [2] B = "Pow (A)" in finite_subset)
```
```  1710    apply (rule_tac B = "Pow (insert x A)" in finite_subset)
```
```  1711    apply fast+
```
```  1712   done
```
```  1713
```
```  1714 text {*
```
```  1715   Main theorem: combinatorial statement about number of subsets of a set.
```
```  1716 *}
```
```  1717
```
```  1718 lemma n_sub_lemma:
```
```  1719   "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
```
```  1720   apply (induct k)
```
```  1721    apply (simp add: card_s_0_eq_empty, atomize)
```
```  1722   apply (rotate_tac -1, erule finite_induct)
```
```  1723    apply (simp_all (no_asm_simp) cong add: conj_cong
```
```  1724      add: card_s_0_eq_empty choose_deconstruct)
```
```  1725   apply (subst card_Un_disjoint)
```
```  1726      prefer 4 apply (force simp add: constr_bij)
```
```  1727     prefer 3 apply force
```
```  1728    prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
```
```  1729      finite_subset [of _ "Pow (insert x F)", standard])
```
```  1730   apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
```
```  1731   done
```
```  1732
```
```  1733 theorem n_subsets:
```
```  1734     "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
```
```  1735   by (simp add: n_sub_lemma)
```
```  1736
```
```  1737
```
```  1738 subsection{* A fold functional for non-empty sets *}
```
```  1739
```
```  1740 text{* Does not require start value. *}
```
```  1741
```
```  1742 consts
```
```  1743   foldSet1 :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
```
```  1744
```
```  1745 inductive "foldSet1 f"
```
```  1746 intros
```
```  1747 foldSet1_singletonI [intro]: "({a}, a) : foldSet1 f"
```
```  1748 foldSet1_insertI [intro]:
```
```  1749  "\<lbrakk> (A, x) : foldSet1 f; a \<notin> A; A \<noteq> {} \<rbrakk>
```
```  1750   \<Longrightarrow> (insert a A, f a x) : foldSet1 f"
```
```  1751
```
```  1752 constdefs
```
```  1753   fold1 :: "('a => 'a => 'a) => 'a set => 'a"
```
```  1754   "fold1 f A == THE x. (A, x) : foldSet1 f"
```
```  1755
```
```  1756 lemma foldSet1_nonempty:
```
```  1757  "(A, x) : foldSet1 f \<Longrightarrow> A \<noteq> {}"
```
```  1758 by(erule foldSet1.cases, simp_all)
```
```  1759
```
```  1760
```
```  1761 inductive_cases empty_foldSet1E [elim!]: "({}, x) : foldSet1 f"
```
```  1762
```
```  1763 lemma foldSet1_sing[iff]: "(({a},b) : foldSet1 f) = (a = b)"
```
```  1764 apply(rule iffI)
```
```  1765  prefer 2 apply fast
```
```  1766 apply (erule foldSet1.cases)
```
```  1767  apply blast
```
```  1768 apply (erule foldSet1.cases)
```
```  1769  apply blast
```
```  1770 apply blast
```
```  1771 done
```
```  1772
```
```  1773 lemma Diff1_foldSet1:
```
```  1774   "(A - {x}, y) : foldSet1 f ==> x: A ==> (A, f x y) : foldSet1 f"
```
```  1775 by (erule insert_Diff [THEN subst], rule foldSet1.intros,
```
```  1776     auto dest!:foldSet1_nonempty)
```
```  1777
```
```  1778 lemma foldSet1_imp_finite: "(A, x) : foldSet1 f ==> finite A"
```
```  1779   by (induct set: foldSet1) auto
```
```  1780
```
```  1781 lemma finite_nonempty_imp_foldSet1:
```
```  1782   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : foldSet1 f"
```
```  1783   by (induct set: Finites) auto
```
```  1784
```
```  1785 lemma (in ACf) foldSet1_determ_aux:
```
```  1786   "!!A x y. \<lbrakk> card A < n; (A, x) : foldSet1 f; (A, y) : foldSet1 f \<rbrakk> \<Longrightarrow> y = x"
```
```  1787 proof (induct n)
```
```  1788   case 0 thus ?case by simp
```
```  1789 next
```
```  1790   case (Suc n)
```
```  1791   have IH: "!!A x y. \<lbrakk>card A < n; (A, x) \<in> foldSet1 f; (A, y) \<in> foldSet1 f\<rbrakk>
```
```  1792            \<Longrightarrow> y = x" and card: "card A < Suc n"
```
```  1793   and Afoldx: "(A, x) \<in> foldSet1 f" and Afoldy: "(A, y) \<in> foldSet1 f" .
```
```  1794   from card have "card A < n \<or> card A = n" by arith
```
```  1795   thus ?case
```
```  1796   proof
```
```  1797     assume less: "card A < n"
```
```  1798     show ?thesis by(rule IH[OF less Afoldx Afoldy])
```
```  1799   next
```
```  1800     assume cardA: "card A = n"
```
```  1801     show ?thesis
```
```  1802     proof (rule foldSet1.cases[OF Afoldx])
```
```  1803       fix a assume "(A, x) = ({a}, a)"
```
```  1804       thus "y = x" using Afoldy by (simp add:foldSet1_sing)
```
```  1805     next
```
```  1806       fix Ax ax x'
```
```  1807       assume eq1: "(A, x) = (insert ax Ax, ax \<cdot> x')"
```
```  1808 	and x': "(Ax, x') \<in> foldSet1 f" and notinx: "ax \<notin> Ax"
```
```  1809 	and Axnon: "Ax \<noteq> {}"
```
```  1810       hence A1: "A = insert ax Ax" and x: "x = ax \<cdot> x'" by auto
```
```  1811       show ?thesis
```
```  1812       proof (rule foldSet1.cases[OF Afoldy])
```
```  1813 	fix ay assume "(A, y) = ({ay}, ay)"
```
```  1814 	thus ?thesis using eq1 x' Axnon notinx
```
```  1815 	  by (fastsimp simp:foldSet1_sing)
```
```  1816       next
```
```  1817 	fix Ay ay y'
```
```  1818 	assume eq2: "(A, y) = (insert ay Ay, ay \<cdot> y')"
```
```  1819 	  and y': "(Ay, y') \<in> foldSet1 f" and notiny: "ay \<notin> Ay"
```
```  1820 	  and Aynon: "Ay \<noteq> {}"
```
```  1821 	hence A2: "A = insert ay Ay" and y: "y = ay \<cdot> y'" by auto
```
```  1822 	have finA: "finite A" by(rule foldSet1_imp_finite[OF Afoldx])
```
```  1823 	with cardA A1 notinx have less: "card Ax < n" by simp
```
```  1824 	show ?thesis
```
```  1825 	proof cases
```
```  1826 	  assume "ax = ay"
```
```  1827 	  then moreover have "Ax = Ay" using A1 A2 notinx notiny by auto
```
```  1828 	  ultimately show ?thesis using IH[OF less x'] y' eq1 eq2 by auto
```
```  1829 	next
```
```  1830 	  assume diff: "ax \<noteq> ay"
```
```  1831 	  let ?B = "Ax - {ay}"
```
```  1832 	  have Ax: "Ax = insert ay ?B" and Ay: "Ay = insert ax ?B"
```
```  1833 	    using A1 A2 notinx notiny diff by(blast elim!:equalityE)+
```
```  1834 	  show ?thesis
```
```  1835 	  proof cases
```
```  1836 	    assume "?B = {}"
```
```  1837 	    with Ax Ay show ?thesis using x' y' x y by(simp add:commute)
```
```  1838 	  next
```
```  1839 	    assume Bnon: "?B \<noteq> {}"
```
```  1840 	    moreover have "finite ?B" using finA A1 by simp
```
```  1841 	    ultimately obtain b where Bfoldb: "(?B,b) \<in> foldSet1 f"
```
```  1842 	      using finite_nonempty_imp_foldSet1 by(blast)
```
```  1843 	    moreover have ayinAx: "ay \<in> Ax" using Ax by(auto)
```
```  1844 	    ultimately have "(Ax,ay\<cdot>b) \<in> foldSet1 f" by(rule Diff1_foldSet1)
```
```  1845 	    hence "ay\<cdot>b = x'" by(rule IH[OF less x'])
```
```  1846             moreover have "ax\<cdot>b = y'"
```
```  1847 	    proof (rule IH[OF _ y'])
```
```  1848 	      show "card Ay < n" using Ay cardA A1 notinx finA ayinAx
```
```  1849 		by(auto simp:card_Diff1_less)
```
```  1850 	    next
```
```  1851 	      show "(Ay,ax\<cdot>b) \<in> foldSet1 f" using Ay notinx Bfoldb Bnon
```
```  1852 		by fastsimp
```
```  1853 	    qed
```
```  1854 	    ultimately show ?thesis using x y by(auto simp:AC)
```
```  1855 	  qed
```
```  1856 	qed
```
```  1857       qed
```
```  1858     qed
```
```  1859   qed
```
```  1860 qed
```
```  1861
```
```  1862
```
```  1863 lemma (in ACf) foldSet1_determ:
```
```  1864   "(A, x) : foldSet1 f ==> (A, y) : foldSet1 f ==> y = x"
```
```  1865 by (blast intro: foldSet1_determ_aux [rule_format])
```
```  1866
```
```  1867 lemma (in ACf) foldSet1_equality: "(A, y) : foldSet1 f ==> fold1 f A = y"
```
```  1868   by (unfold fold1_def) (blast intro: foldSet1_determ)
```
```  1869
```
```  1870 lemma fold1_singleton[simp]: "fold1 f {a} = a"
```
```  1871   by (unfold fold1_def) blast
```
```  1872
```
```  1873 lemma (in ACf) foldSet1_insert_aux: "x \<notin> A ==> A \<noteq> {} \<Longrightarrow>
```
```  1874     ((insert x A, v) : foldSet1 f) =
```
```  1875     (EX y. (A, y) : foldSet1 f & v = f x y)"
```
```  1876 apply auto
```
```  1877 apply (rule_tac A1 = A and f1 = f in finite_nonempty_imp_foldSet1 [THEN exE])
```
```  1878   apply (fastsimp dest: foldSet1_imp_finite)
```
```  1879  apply blast
```
```  1880 apply (blast intro: foldSet1_determ)
```
```  1881 done
```
```  1882
```
```  1883 lemma (in ACf) fold1_insert:
```
```  1884   "finite A ==> x \<notin> A ==> A \<noteq> {} \<Longrightarrow> fold1 f (insert x A) = f x (fold1 f A)"
```
```  1885 apply (unfold fold1_def)
```
```  1886 apply (simp add: foldSet1_insert_aux)
```
```  1887 apply (rule the_equality)
```
```  1888 apply (auto intro: finite_nonempty_imp_foldSet1
```
```  1889     cong add: conj_cong simp add: fold1_def [symmetric] foldSet1_equality)
```
```  1890 done
```
```  1891
```
```  1892 lemma (in ACIf) fold1_insert2[simp]:
```
```  1893 assumes finA: "finite A" and nonA: "A \<noteq> {}"
```
```  1894 shows "fold1 f (insert a A) = f a (fold1 f A)"
```
```  1895 proof cases
```
```  1896   assume "a \<in> A"
```
```  1897   then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
```
```  1898     by(blast dest: mk_disjoint_insert)
```
```  1899   show ?thesis
```
```  1900   proof cases
```
```  1901     assume "B = {}"
```
```  1902     thus ?thesis using A by(simp add:idem fold1_singleton)
```
```  1903   next
```
```  1904     assume nonB: "B \<noteq> {}"
```
```  1905     from finA A have finB: "finite B" by(blast intro: finite_subset)
```
```  1906     have "fold1 f (insert a A) = fold1 f (insert a B)" using A by simp
```
```  1907     also have "\<dots> = f a (fold1 f B)"
```
```  1908       using finB nonB disj by(simp add: fold1_insert)
```
```  1909     also have "\<dots> = f a (fold1 f A)"
```
```  1910       using A finB nonB disj by(simp add:idem fold1_insert assoc[symmetric])
```
```  1911     finally show ?thesis .
```
```  1912   qed
```
```  1913 next
```
```  1914   assume "a \<notin> A"
```
```  1915   with finA nonA show ?thesis by(simp add:fold1_insert)
```
```  1916 qed
```
```  1917
```
```  1918 text{* Now the recursion rules for definitions: *}
```
```  1919
```
```  1920 lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
```
```  1921 by(simp add:fold1_singleton)
```
```  1922
```
```  1923 lemma (in ACf) fold1_insert_def:
```
```  1924   "\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
```
```  1925 by(simp add:fold1_insert)
```
```  1926
```
```  1927 lemma (in ACIf) fold1_insert2_def:
```
```  1928   "\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
```
```  1929 by(simp add:fold1_insert2)
```
```  1930
```
```  1931
```
```  1932 subsubsection{* Semi-Lattices *}
```
```  1933
```
```  1934 locale ACIfSL = ACIf +
```
```  1935   fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
```
```  1936   assumes below_def: "(x \<sqsubseteq> y) = (x\<cdot>y = x)"
```
```  1937
```
```  1938 locale ACIfSLlin = ACIfSL +
```
```  1939   assumes lin: "x\<cdot>y \<in> {x,y}"
```
```  1940
```
```  1941 lemma (in ACIfSL) below_refl[simp]: "x \<sqsubseteq> x"
```
```  1942 by(simp add: below_def idem)
```
```  1943
```
```  1944 lemma (in ACIfSL) below_f_conv[simp]: "x \<sqsubseteq> y \<cdot> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
```
```  1945 proof
```
```  1946   assume "x \<sqsubseteq> y \<cdot> z"
```
```  1947   hence xyzx: "x \<cdot> (y \<cdot> z) = x"  by(simp add: below_def)
```
```  1948   have "x \<cdot> y = x"
```
```  1949   proof -
```
```  1950     have "x \<cdot> y = (x \<cdot> (y \<cdot> z)) \<cdot> y" by(rule subst[OF xyzx], rule refl)
```
```  1951     also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  1952     also have "\<dots> = x" by(rule xyzx)
```
```  1953     finally show ?thesis .
```
```  1954   qed
```
```  1955   moreover have "x \<cdot> z = x"
```
```  1956   proof -
```
```  1957     have "x \<cdot> z = (x \<cdot> (y \<cdot> z)) \<cdot> z" by(rule subst[OF xyzx], rule refl)
```
```  1958     also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  1959     also have "\<dots> = x" by(rule xyzx)
```
```  1960     finally show ?thesis .
```
```  1961   qed
```
```  1962   ultimately show "x \<sqsubseteq> y \<and> x \<sqsubseteq> z" by(simp add: below_def)
```
```  1963 next
```
```  1964   assume a: "x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
```
```  1965   hence y: "x \<cdot> y = x" and z: "x \<cdot> z = x" by(simp_all add: below_def)
```
```  1966   have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by(simp add:assoc)
```
```  1967   also have "x \<cdot> y = x" using a by(simp_all add: below_def)
```
```  1968   also have "x \<cdot> z = x" using a by(simp_all add: below_def)
```
```  1969   finally show "x \<sqsubseteq> y \<cdot> z" by(simp_all add: below_def)
```
```  1970 qed
```
```  1971
```
```  1972 lemma (in ACIfSLlin) above_f_conv:
```
```  1973  "x \<cdot> y \<sqsubseteq> z = (x \<sqsubseteq> z \<or> y \<sqsubseteq> z)"
```
```  1974 proof
```
```  1975   assume a: "x \<cdot> y \<sqsubseteq> z"
```
```  1976   have "x \<cdot> y = x \<or> x \<cdot> y = y" using lin[of x y] by simp
```
```  1977   thus "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
```
```  1978   proof
```
```  1979     assume "x \<cdot> y = x" hence "x \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
```
```  1980   next
```
```  1981     assume "x \<cdot> y = y" hence "y \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis ..
```
```  1982   qed
```
```  1983 next
```
```  1984   assume "x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
```
```  1985   thus "x \<cdot> y \<sqsubseteq> z"
```
```  1986   proof
```
```  1987     assume a: "x \<sqsubseteq> z"
```
```  1988     have "(x \<cdot> y) \<cdot> z = (x \<cdot> z) \<cdot> y" by(simp add:ACI)
```
```  1989     also have "x \<cdot> z = x" using a by(simp add:below_def)
```
```  1990     finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
```
```  1991   next
```
```  1992     assume a: "y \<sqsubseteq> z"
```
```  1993     have "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
```
```  1994     also have "y \<cdot> z = y" using a by(simp add:below_def)
```
```  1995     finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def)
```
```  1996   qed
```
```  1997 qed
```
```  1998
```
```  1999
```
```  2000 subsubsection{* Lemmas about @{text fold1} *}
```
```  2001
```
```  2002 lemma (in ACf) fold1_Un:
```
```  2003 assumes A: "finite A" "A \<noteq> {}"
```
```  2004 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
```
```  2005        fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
```
```  2006 using A
```
```  2007 proof(induct rule:finite_ne_induct)
```
```  2008   case singleton thus ?case by(simp add:fold1_insert)
```
```  2009 next
```
```  2010   case insert thus ?case by (simp add:fold1_insert assoc)
```
```  2011 qed
```
```  2012
```
```  2013 lemma (in ACIf) fold1_Un2:
```
```  2014 assumes A: "finite A" "A \<noteq> {}"
```
```  2015 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
```
```  2016        fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
```
```  2017 using A
```
```  2018 proof(induct rule:finite_ne_induct)
```
```  2019   case singleton thus ?case by(simp add:fold1_insert2)
```
```  2020 next
```
```  2021   case insert thus ?case by (simp add:fold1_insert2 assoc)
```
```  2022 qed
```
```  2023
```
```  2024 lemma (in ACf) fold1_in:
```
```  2025   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}"
```
```  2026   shows "fold1 f A \<in> A"
```
```  2027 using A
```
```  2028 proof (induct rule:finite_ne_induct)
```
```  2029   case singleton thus ?case by(simp)
```
```  2030 next
```
```  2031   case insert thus ?case using elem by (force simp add:fold1_insert)
```
```  2032 qed
```
```  2033
```
```  2034 lemma (in ACIfSL) below_fold1_iff:
```
```  2035 assumes A: "finite A" "A \<noteq> {}"
```
```  2036 shows "x \<sqsubseteq> fold1 f A = (\<forall>a\<in>A. x \<sqsubseteq> a)"
```
```  2037 using A
```
```  2038 by(induct rule:finite_ne_induct) simp_all
```
```  2039
```
```  2040 lemma (in ACIfSL) fold1_belowI:
```
```  2041 assumes A: "finite A" "A \<noteq> {}"
```
```  2042 shows "a \<in> A \<Longrightarrow> fold1 f A \<sqsubseteq> a"
```
```  2043 using A
```
```  2044 proof (induct rule:finite_ne_induct)
```
```  2045   case singleton thus ?case by simp
```
```  2046 next
```
```  2047   case (insert x F)
```
```  2048   from insert(4) have "a = x \<or> a \<in> F" by simp
```
```  2049   thus ?case
```
```  2050   proof
```
```  2051     assume "a = x" thus ?thesis using insert by(simp add:below_def ACI)
```
```  2052   next
```
```  2053     assume "a \<in> F"
```
```  2054     hence bel: "fold1 op \<cdot> F \<sqsubseteq> a" by(rule insert)
```
```  2055     have "fold1 op \<cdot> (insert x F) \<cdot> a = x \<cdot> (fold1 op \<cdot> F \<cdot> a)"
```
```  2056       using insert by(simp add:below_def ACI)
```
```  2057     also have "fold1 op \<cdot> F \<cdot> a = fold1 op \<cdot> F"
```
```  2058       using bel  by(simp add:below_def ACI)
```
```  2059     also have "x \<cdot> \<dots> = fold1 op \<cdot> (insert x F)"
```
```  2060       using insert by(simp add:below_def ACI)
```
```  2061     finally show ?thesis  by(simp add:below_def)
```
```  2062   qed
```
```  2063 qed
```
```  2064
```
```  2065 lemma (in ACIfSLlin) fold1_below_iff:
```
```  2066 assumes A: "finite A" "A \<noteq> {}"
```
```  2067 shows "fold1 f A \<sqsubseteq> x = (\<exists>a\<in>A. a \<sqsubseteq> x)"
```
```  2068 using A
```
```  2069 by(induct rule:finite_ne_induct)(simp_all add:above_f_conv)
```
```  2070
```
```  2071 subsubsection{* Lattices *}
```
```  2072
```
```  2073 locale Lattice =
```
```  2074   fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
```
```  2075   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
```
```  2076   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
```
```  2077   and Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
```
```  2078   and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```  2079   assumes refl: "x \<sqsubseteq> x"
```
```  2080   and trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
```
```  2081   and antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
```
```  2082   and inf_le1: "x \<sqinter> y \<sqsubseteq> x" and inf_le2: "x \<sqinter> y \<sqsubseteq> y"
```
```  2083   and inf_least: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
```
```  2084   and sup_ge1: "x \<sqsubseteq> x \<squnion> y" and sup_ge2: "y \<sqsubseteq> x \<squnion> y"
```
```  2085   and sup_greatest: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
```
```  2086   and inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
```
```  2087   and sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
```
```  2088   defines "Inf == fold1 inf"  and "Sup == fold1 sup"
```
```  2089
```
```  2090
```
```  2091 lemma (in Lattice) inf_comm: "(x \<sqinter> y) = (y \<sqinter> x)"
```
```  2092 by(blast intro: antisym inf_le1 inf_le2 inf_least)
```
```  2093
```
```  2094 lemma (in Lattice) sup_comm: "(x \<squnion> y) = (y \<squnion> x)"
```
```  2095 by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest)
```
```  2096
```
```  2097 lemma (in Lattice) inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
```
```  2098 by(blast intro: antisym inf_le1 inf_le2 inf_least trans)
```
```  2099
```
```  2100 lemma (in Lattice) sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
```
```  2101 by(blast intro!: antisym sup_ge1 sup_ge2 intro: sup_greatest trans)
```
```  2102
```
```  2103 lemma (in Lattice) inf_idem[simp]: "x \<sqinter> x = x"
```
```  2104 by(blast intro: antisym inf_le1 inf_le2 inf_least refl)
```
```  2105
```
```  2106 lemma (in Lattice) sup_idem[simp]: "x \<squnion> x = x"
```
```  2107 by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
```
```  2108
```
```  2109 lemma (in Lattice) sup_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
```
```  2110 by(blast intro: antisym sup_ge2 sup_greatest refl)
```
```  2111
```
```  2112 lemma (in Lattice) inf_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
```
```  2113 by(blast intro: antisym inf_le1 inf_least refl)
```
```  2114
```
```  2115 text{* Towards distributivity: if you have one of them, you have them all. *}
```
```  2116
```
```  2117 lemma (in Lattice) distrib_imp1:
```
```  2118 assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```  2119 shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```  2120 proof-
```
```  2121   have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
```
```  2122   also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_comm sup_assoc)
```
```  2123   also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
```
```  2124     by(simp add:inf_sup_absorb inf_comm)
```
```  2125   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
```
```  2126   finally show ?thesis .
```
```  2127 qed
```
```  2128
```
```  2129 lemma (in Lattice) distrib_imp2:
```
```  2130 assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```  2131 shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```  2132 proof-
```
```  2133   have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
```
```  2134   also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_comm inf_assoc)
```
```  2135   also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
```
```  2136     by(simp add:sup_inf_absorb sup_comm)
```
```  2137   also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
```
```  2138   finally show ?thesis .
```
```  2139 qed
```
```  2140
```
```  2141 text{* Lattices are semilattices *}
```
```  2142
```
```  2143 lemma (in Lattice) ACf_inf: "ACf inf"
```
```  2144 by(blast intro: ACf.intro inf_comm inf_assoc)
```
```  2145
```
```  2146 lemma (in Lattice) ACf_sup: "ACf sup"
```
```  2147 by(blast intro: ACf.intro sup_comm sup_assoc)
```
```  2148
```
```  2149 lemma (in Lattice) ACIf_inf: "ACIf inf"
```
```  2150 apply(rule ACIf.intro)
```
```  2151 apply(rule ACf_inf)
```
```  2152 apply(rule ACIf_axioms.intro)
```
```  2153 apply(rule inf_idem)
```
```  2154 done
```
```  2155
```
```  2156 lemma (in Lattice) ACIf_sup: "ACIf sup"
```
```  2157 apply(rule ACIf.intro)
```
```  2158 apply(rule ACf_sup)
```
```  2159 apply(rule ACIf_axioms.intro)
```
```  2160 apply(rule sup_idem)
```
```  2161 done
```
```  2162
```
```  2163 lemma (in Lattice) ACIfSL_inf: "ACIfSL inf (op \<sqsubseteq>)"
```
```  2164 apply(rule ACIfSL.intro)
```
```  2165 apply(rule ACf_inf)
```
```  2166 apply(rule ACIf.axioms[OF ACIf_inf])
```
```  2167 apply(rule ACIfSL_axioms.intro)
```
```  2168 apply(rule iffI)
```
```  2169  apply(blast intro: antisym inf_le1 inf_le2 inf_least refl)
```
```  2170 apply(erule subst)
```
```  2171 apply(rule inf_le2)
```
```  2172 done
```
```  2173
```
```  2174 lemma (in Lattice) ACIfSL_sup: "ACIfSL sup (%x y. y \<sqsubseteq> x)"
```
```  2175 apply(rule ACIfSL.intro)
```
```  2176 apply(rule ACf_sup)
```
```  2177 apply(rule ACIf.axioms[OF ACIf_sup])
```
```  2178 apply(rule ACIfSL_axioms.intro)
```
```  2179 apply(rule iffI)
```
```  2180  apply(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
```
```  2181 apply(erule subst)
```
```  2182 apply(rule sup_ge2)
```
```  2183 done
```
```  2184
```
```  2185 text{* A package of rewrite rules for deciding equivalence wrt ACI: *}
```
```  2186
```
```  2187 lemmas (in Lattice) ACI = ACIf.ACI[OF ACIf_inf] ACIf.ACI[OF ACIf_sup]
```
```  2188
```
```  2189 subsubsection{* Distributive lattices *}
```
```  2190
```
```  2191 locale DistribLattice = Lattice +
```
```  2192   assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```  2193
```
```  2194 lemma (in DistribLattice) sup_inf_distrib2: "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
```
```  2195 by(simp add:ACI sup_inf_distrib1)
```
```  2196
```
```  2197 lemma (in DistribLattice) inf_sup_distrib1: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```  2198 by(rule distrib_imp2[OF sup_inf_distrib1])
```
```  2199
```
```  2200 lemma (in DistribLattice) inf_sup_distrib2: "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
```
```  2201 by(simp add:ACI inf_sup_distrib1)
```
```  2202
```
```  2203 lemmas (in DistribLattice) distrib =
```
```  2204   sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
```
```  2205
```
```  2206
```
```  2207 subsubsection{* Fold laws in lattices *}
```
```  2208
```
```  2209 lemma (in Lattice) Inf_le_Sup: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Squnion>A"
```
```  2210 apply(unfold Sup_def Inf_def)
```
```  2211 apply(subgoal_tac "EX a. a:A")
```
```  2212 prefer 2 apply blast
```
```  2213 apply(erule exE)
```
```  2214 apply(rule trans)
```
```  2215 apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_inf])
```
```  2216 apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup])
```
```  2217 done
```
```  2218
```
```  2219 lemma (in Lattice) sup_Inf_absorb:
```
```  2220   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<squnion> \<Sqinter>A) = a"
```
```  2221 apply(subst sup_comm)
```
```  2222 apply(simp add:Inf_def sup_absorb ACIfSL.fold1_belowI[OF ACIfSL_inf])
```
```  2223 done
```
```  2224
```
```  2225 lemma (in Lattice) inf_Sup_absorb:
```
```  2226   "\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<sqinter> \<Squnion>A) = a"
```
```  2227 by(simp add:Sup_def inf_absorb ACIfSL.fold1_belowI[OF ACIfSL_sup])
```
```  2228
```
```  2229
```
```  2230 lemma (in DistribLattice) sup_Inf1_distrib:
```
```  2231 assumes A: "finite A" "A \<noteq> {}"
```
```  2232 shows "(x \<squnion> \<Sqinter>A) = \<Sqinter>{x \<squnion> a|a. a \<in> A}"
```
```  2233 using A
```
```  2234 proof (induct rule: finite_ne_induct)
```
```  2235   case singleton thus ?case by(simp add:Inf_def)
```
```  2236 next
```
```  2237   case (insert y A)
```
```  2238   have fin: "finite {x \<squnion> a |a. a \<in> A}"
```
```  2239     by(fast intro: finite_surj[where f = "%a. x \<squnion> a", OF insert(0)])
```
```  2240   have "x \<squnion> \<Sqinter> (insert y A) = x \<squnion> (y \<sqinter> \<Sqinter> A)"
```
```  2241     using insert by(simp add:ACf.fold1_insert_def[OF ACf_inf Inf_def])
```
```  2242   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> \<Sqinter> A)" by(rule sup_inf_distrib1)
```
```  2243   also have "x \<squnion> \<Sqinter> A = \<Sqinter>{x \<squnion> a|a. a \<in> A}" using insert by simp
```
```  2244   also have "(x \<squnion> y) \<sqinter> \<dots> = \<Sqinter> (insert (x \<squnion> y) {x \<squnion> a |a. a \<in> A})"
```
```  2245     using insert by(simp add:ACIf.fold1_insert2_def[OF ACIf_inf Inf_def fin])
```
```  2246   also have "insert (x\<squnion>y) {x\<squnion>a |a. a \<in> A} = {x\<squnion>a |a. a \<in> insert y A}"
```
```  2247     by blast
```
```  2248   finally show ?case .
```
```  2249 qed
```
```  2250
```
```  2251 lemma (in DistribLattice) sup_Inf2_distrib:
```
```  2252 assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  2253 shows "(\<Sqinter>A \<squnion> \<Sqinter>B) = \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}"
```
```  2254 using A
```
```  2255 proof (induct rule: finite_ne_induct)
```
```  2256   case singleton thus ?case
```
```  2257     by(simp add: sup_Inf1_distrib[OF B] fold1_singleton_def[OF Inf_def])
```
```  2258 next
```
```  2259   case (insert x A)
```
```  2260   have finB: "finite {x \<squnion> b |b. b \<in> B}"
```
```  2261     by(fast intro: finite_surj[where f = "%b. x \<squnion> b", OF B(0)])
```
```  2262   have finAB: "finite {a \<squnion> b |a b. a \<in> A \<and> b \<in> B}"
```
```  2263   proof -
```
```  2264     have "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<squnion> b})"
```
```  2265       by blast
```
```  2266     thus ?thesis by(simp add: insert(0) B(0))
```
```  2267   qed
```
```  2268   have ne: "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  2269   have "\<Sqinter>(insert x A) \<squnion> \<Sqinter>B = (x \<sqinter> \<Sqinter>A) \<squnion> \<Sqinter>B"
```
```  2270     using insert by(simp add:ACIf.fold1_insert2_def[OF ACIf_inf Inf_def])
```
```  2271   also have "\<dots> = (x \<squnion> \<Sqinter>B) \<sqinter> (\<Sqinter>A \<squnion> \<Sqinter>B)" by(rule sup_inf_distrib2)
```
```  2272   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}"
```
```  2273     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  2274   also have "\<dots> = \<Sqinter>({x\<squnion>b |b. b \<in> B} \<union> {a\<squnion>b |a b. a \<in> A \<and> b \<in> B})"
```
```  2275     (is "_ = \<Sqinter>?M")
```
```  2276     using B insert
```
```  2277     by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne])
```
```  2278   also have "?M = {a \<squnion> b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  2279     by blast
```
```  2280   finally show ?case .
```
```  2281 qed
```
```  2282
```
```  2283
```
```  2284 subsection{*Min and Max*}
```
```  2285
```
```  2286 text{* As an application of @{text fold1} we define the minimal and
```
```  2287 maximal element of a (non-empty) set over a linear order. *}
```
```  2288
```
```  2289 constdefs
```
```  2290   Min :: "('a::linorder)set => 'a"
```
```  2291   "Min  ==  fold1 min"
```
```  2292
```
```  2293   Max :: "('a::linorder)set => 'a"
```
```  2294   "Max  ==  fold1 max"
```
```  2295
```
```  2296
```
```  2297 text{* Before we can do anything, we need to show that @{text min} and
```
```  2298 @{text max} are ACI and the ordering is linear: *}
```
```  2299
```
```  2300 lemma ACf_min: "ACf(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```  2301 apply(rule ACf.intro)
```
```  2302 apply(auto simp:min_def)
```
```  2303 done
```
```  2304
```
```  2305 lemma ACIf_min: "ACIf(min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```  2306 apply(rule ACIf.intro[OF ACf_min])
```
```  2307 apply(rule ACIf_axioms.intro)
```
```  2308 apply(auto simp:min_def)
```
```  2309 done
```
```  2310
```
```  2311 lemma ACf_max: "ACf(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```  2312 apply(rule ACf.intro)
```
```  2313 apply(auto simp:max_def)
```
```  2314 done
```
```  2315
```
```  2316 lemma ACIf_max: "ACIf(max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```  2317 apply(rule ACIf.intro[OF ACf_max])
```
```  2318 apply(rule ACIf_axioms.intro)
```
```  2319 apply(auto simp:max_def)
```
```  2320 done
```
```  2321
```
```  2322 lemma ACIfSL_min: "ACIfSL(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) (op \<le>)"
```
```  2323 apply(rule ACIfSL.intro)
```
```  2324 apply(rule ACf_min)
```
```  2325 apply(rule ACIf.axioms[OF ACIf_min])
```
```  2326 apply(rule ACIfSL_axioms.intro)
```
```  2327 apply(auto simp:min_def)
```
```  2328 done
```
```  2329
```
```  2330 lemma ACIfSLlin_min: "ACIfSLlin(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) (op \<le>)"
```
```  2331 apply(rule ACIfSLlin.intro)
```
```  2332 apply(rule ACf_min)
```
```  2333 apply(rule ACIf.axioms[OF ACIf_min])
```
```  2334 apply(rule ACIfSL.axioms[OF ACIfSL_min])
```
```  2335 apply(rule ACIfSLlin_axioms.intro)
```
```  2336 apply(auto simp:min_def)
```
```  2337 done
```
```  2338
```
```  2339 lemma ACIfSL_max: "ACIfSL(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) (%x y. y\<le>x)"
```
```  2340 apply(rule ACIfSL.intro)
```
```  2341 apply(rule ACf_max)
```
```  2342 apply(rule ACIf.axioms[OF ACIf_max])
```
```  2343 apply(rule ACIfSL_axioms.intro)
```
```  2344 apply(auto simp:max_def)
```
```  2345 done
```
```  2346
```
```  2347 lemma ACIfSLlin_max: "ACIfSLlin(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) (%x y. y\<le>x)"
```
```  2348 apply(rule ACIfSLlin.intro)
```
```  2349 apply(rule ACf_max)
```
```  2350 apply(rule ACIf.axioms[OF ACIf_max])
```
```  2351 apply(rule ACIfSL.axioms[OF ACIfSL_max])
```
```  2352 apply(rule ACIfSLlin_axioms.intro)
```
```  2353 apply(auto simp:max_def)
```
```  2354 done
```
```  2355
```
```  2356 lemma Lattice_min_max: "Lattice (op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) max"
```
```  2357 apply(rule Lattice.intro)
```
```  2358 apply simp
```
```  2359 apply(erule (1) order_trans)
```
```  2360 apply(erule (1) order_antisym)
```
```  2361 apply(simp add:min_def max_def linorder_not_le order_less_imp_le)
```
```  2362 apply(simp add:min_def max_def linorder_not_le order_less_imp_le)
```
```  2363 apply(simp add:min_def max_def linorder_not_le order_less_imp_le)
```
```  2364 apply(simp add:min_def max_def linorder_not_le order_less_imp_le)
```
```  2365 apply(simp add:min_def max_def linorder_not_le order_less_imp_le)
```
```  2366 apply(simp add:min_def max_def linorder_not_le order_less_imp_le)
```
```  2367 apply(rule_tac x=x and y=y in linorder_le_cases)
```
```  2368 apply(rule_tac x=x and y=z in linorder_le_cases)
```
```  2369 apply(rule_tac x=y and y=z in linorder_le_cases)
```
```  2370 apply(simp add:min_def max_def)
```
```  2371 apply(simp add:min_def max_def)
```
```  2372 apply(rule_tac x=y and y=z in linorder_le_cases)
```
```  2373 apply(simp add:min_def max_def)
```
```  2374 apply(simp add:min_def max_def)
```
```  2375 apply(rule_tac x=x and y=z in linorder_le_cases)
```
```  2376 apply(rule_tac x=y and y=z in linorder_le_cases)
```
```  2377 apply(simp add:min_def max_def)
```
```  2378 apply(simp add:min_def max_def)
```
```  2379 apply(rule_tac x=y and y=z in linorder_le_cases)
```
```  2380 apply(simp add:min_def max_def)
```
```  2381 apply(simp add:min_def max_def)
```
```  2382
```
```  2383 apply(rule_tac x=x and y=y in linorder_le_cases)
```
```  2384 apply(rule_tac x=x and y=z in linorder_le_cases)
```
```  2385 apply(rule_tac x=y and y=z in linorder_le_cases)
```
```  2386 apply(simp add:min_def max_def)
```
```  2387 apply(simp add:min_def max_def)
```
```  2388 apply(rule_tac x=y and y=z in linorder_le_cases)
```
```  2389 apply(simp add:min_def max_def)
```
```  2390 apply(simp add:min_def max_def)
```
```  2391 apply(rule_tac x=x and y=z in linorder_le_cases)
```
```  2392 apply(rule_tac x=y and y=z in linorder_le_cases)
```
```  2393 apply(simp add:min_def max_def)
```
```  2394 apply(simp add:min_def max_def)
```
```  2395 apply(rule_tac x=y and y=z in linorder_le_cases)
```
```  2396 apply(simp add:min_def max_def)
```
```  2397 apply(simp add:min_def max_def)
```
```  2398 done
```
```  2399
```
```  2400 text{* Now we instantiate the recursion equations and declare them
```
```  2401 simplification rules: *}
```
```  2402
```
```  2403 declare
```
```  2404   fold1_singleton_def[OF Min_def, simp]
```
```  2405   ACIf.fold1_insert2_def[OF ACIf_min Min_def, simp]
```
```  2406   fold1_singleton_def[OF Max_def, simp]
```
```  2407   ACIf.fold1_insert2_def[OF ACIf_max Max_def, simp]
```
```  2408
```
```  2409 text{* Now we instantiate some @{text fold1} properties: *}
```
```  2410
```
```  2411 lemma Min_in [simp]:
```
```  2412   shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A"
```
```  2413 using ACf.fold1_in[OF ACf_min]
```
```  2414 by(fastsimp simp: Min_def min_def)
```
```  2415
```
```  2416 lemma Max_in [simp]:
```
```  2417   shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A"
```
```  2418 using ACf.fold1_in[OF ACf_max]
```
```  2419 by(fastsimp simp: Max_def max_def)
```
```  2420
```
```  2421 lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<le> x"
```
```  2422 by(simp add: Min_def ACIfSL.fold1_belowI[OF ACIfSL_min])
```
```  2423
```
```  2424 lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<le> Max A"
```
```  2425 by(simp add: Max_def ACIfSL.fold1_belowI[OF ACIfSL_max])
```
```  2426
```
```  2427 lemma Min_ge_iff[simp]:
```
```  2428   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
```
```  2429 by(simp add: Min_def ACIfSL.below_fold1_iff[OF ACIfSL_min])
```
```  2430
```
```  2431 lemma Max_le_iff[simp]:
```
```  2432   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
```
```  2433 by(simp add: Max_def ACIfSL.below_fold1_iff[OF ACIfSL_max])
```
```  2434
```
```  2435 lemma Min_le_iff:
```
```  2436   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Min A \<le> x) = (\<exists>a\<in>A. a \<le> x)"
```
```  2437 by(simp add: Min_def ACIfSLlin.fold1_below_iff[OF ACIfSLlin_min])
```
```  2438
```
```  2439 lemma Max_ge_iff:
```
```  2440   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Max A) = (\<exists>a\<in>A. x \<le> a)"
```
```  2441 by(simp add: Max_def ACIfSLlin.fold1_below_iff[OF ACIfSLlin_max])
```
```  2442
```
```  2443 lemma Min_le_Max:
```
```  2444   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Min A \<le> Max A"
```
```  2445 by(simp add: Min_def Max_def Lattice.Inf_le_Sup[OF Lattice_min_max])
```
```  2446 (* FIXME
```
```  2447 lemma max_Min2_distrib:
```
```  2448   "\<lbrakk> finite A; A \<noteq> {}; finite B; B \<noteq> {} \<rbrakk> \<Longrightarrow>
```
```  2449   max (Min A) (Min B) = Min{ max a b |a b. a \<in> A \<and> b \<in> B}"
```
```  2450 by(simp add: Min_def Max_def Lattice.sup_Inf2_distrib[OF Lattice_min_max])
```
```  2451 *)
```
```  2452 end
```