src/HOL/Finite_Set.thy
 author haftmann Wed Mar 10 16:53:27 2010 +0100 (2010-03-10) changeset 35719 99b6152aedf5 parent 35577 43b93e294522 child 35722 69419a09a7ff permissions -rw-r--r--
split off theory Big_Operators from theory Finite_Set
```     1 (*  Title:      HOL/Finite_Set.thy
```
```     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
```
```     3                 with contributions by Jeremy Avigad
```
```     4 *)
```
```     5
```
```     6 header {* Finite sets *}
```
```     7
```
```     8 theory Finite_Set
```
```     9 imports Power Option
```
```    10 begin
```
```    11
```
```    12 subsection {* Definition and basic properties *}
```
```    13
```
```    14 inductive finite :: "'a set => bool"
```
```    15   where
```
```    16     emptyI [simp, intro!]: "finite {}"
```
```    17   | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
```
```    18
```
```    19 lemma ex_new_if_finite: -- "does not depend on def of finite at all"
```
```    20   assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
```
```    21   shows "\<exists>a::'a. a \<notin> A"
```
```    22 proof -
```
```    23   from assms have "A \<noteq> UNIV" by blast
```
```    24   thus ?thesis by blast
```
```    25 qed
```
```    26
```
```    27 lemma finite_induct [case_names empty insert, induct set: finite]:
```
```    28   "finite F ==>
```
```    29     P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
```
```    30   -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
```
```    31 proof -
```
```    32   assume "P {}" and
```
```    33     insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
```
```    34   assume "finite F"
```
```    35   thus "P F"
```
```    36   proof induct
```
```    37     show "P {}" by fact
```
```    38     fix x F assume F: "finite F" and P: "P F"
```
```    39     show "P (insert x F)"
```
```    40     proof cases
```
```    41       assume "x \<in> F"
```
```    42       hence "insert x F = F" by (rule insert_absorb)
```
```    43       with P show ?thesis by (simp only:)
```
```    44     next
```
```    45       assume "x \<notin> F"
```
```    46       from F this P show ?thesis by (rule insert)
```
```    47     qed
```
```    48   qed
```
```    49 qed
```
```    50
```
```    51 lemma finite_ne_induct[case_names singleton insert, consumes 2]:
```
```    52 assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
```
```    53  \<lbrakk> \<And>x. P{x};
```
```    54    \<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
```
```    55  \<Longrightarrow> P F"
```
```    56 using fin
```
```    57 proof induct
```
```    58   case empty thus ?case by simp
```
```    59 next
```
```    60   case (insert x F)
```
```    61   show ?case
```
```    62   proof cases
```
```    63     assume "F = {}"
```
```    64     thus ?thesis using `P {x}` by simp
```
```    65   next
```
```    66     assume "F \<noteq> {}"
```
```    67     thus ?thesis using insert by blast
```
```    68   qed
```
```    69 qed
```
```    70
```
```    71 lemma finite_subset_induct [consumes 2, case_names empty insert]:
```
```    72   assumes "finite F" and "F \<subseteq> A"
```
```    73     and empty: "P {}"
```
```    74     and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
```
```    75   shows "P F"
```
```    76 proof -
```
```    77   from `finite F` and `F \<subseteq> A`
```
```    78   show ?thesis
```
```    79   proof induct
```
```    80     show "P {}" by fact
```
```    81   next
```
```    82     fix x F
```
```    83     assume "finite F" and "x \<notin> F" and
```
```    84       P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
```
```    85     show "P (insert x F)"
```
```    86     proof (rule insert)
```
```    87       from i show "x \<in> A" by blast
```
```    88       from i have "F \<subseteq> A" by blast
```
```    89       with P show "P F" .
```
```    90       show "finite F" by fact
```
```    91       show "x \<notin> F" by fact
```
```    92     qed
```
```    93   qed
```
```    94 qed
```
```    95
```
```    96
```
```    97 text{* A finite choice principle. Does not need the SOME choice operator. *}
```
```    98 lemma finite_set_choice:
```
```    99   "finite A \<Longrightarrow> ALL x:A. (EX y. P x y) \<Longrightarrow> EX f. ALL x:A. P x (f x)"
```
```   100 proof (induct set: finite)
```
```   101   case empty thus ?case by simp
```
```   102 next
```
```   103   case (insert a A)
```
```   104   then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
```
```   105   show ?case (is "EX f. ?P f")
```
```   106   proof
```
```   107     show "?P(%x. if x = a then b else f x)" using f ab by auto
```
```   108   qed
```
```   109 qed
```
```   110
```
```   111
```
```   112 text{* Finite sets are the images of initial segments of natural numbers: *}
```
```   113
```
```   114 lemma finite_imp_nat_seg_image_inj_on:
```
```   115   assumes fin: "finite A"
```
```   116   shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
```
```   117 using fin
```
```   118 proof induct
```
```   119   case empty
```
```   120   show ?case
```
```   121   proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp
```
```   122   qed
```
```   123 next
```
```   124   case (insert a A)
```
```   125   have notinA: "a \<notin> A" by fact
```
```   126   from insert.hyps obtain n f
```
```   127     where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
```
```   128   hence "insert a A = f(n:=a) ` {i. i < Suc n}"
```
```   129         "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
```
```   130     by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
```
```   131   thus ?case by blast
```
```   132 qed
```
```   133
```
```   134 lemma nat_seg_image_imp_finite:
```
```   135   "!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
```
```   136 proof (induct n)
```
```   137   case 0 thus ?case by simp
```
```   138 next
```
```   139   case (Suc n)
```
```   140   let ?B = "f ` {i. i < n}"
```
```   141   have finB: "finite ?B" by(rule Suc.hyps[OF refl])
```
```   142   show ?case
```
```   143   proof cases
```
```   144     assume "\<exists>k<n. f n = f k"
```
```   145     hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   146     thus ?thesis using finB by simp
```
```   147   next
```
```   148     assume "\<not>(\<exists> k<n. f n = f k)"
```
```   149     hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
```
```   150     thus ?thesis using finB by simp
```
```   151   qed
```
```   152 qed
```
```   153
```
```   154 lemma finite_conv_nat_seg_image:
```
```   155   "finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
```
```   156 by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
```
```   157
```
```   158 lemma finite_imp_inj_to_nat_seg:
```
```   159 assumes "finite A"
```
```   160 shows "EX f n::nat. f`A = {i. i<n} & inj_on f A"
```
```   161 proof -
```
```   162   from finite_imp_nat_seg_image_inj_on[OF `finite A`]
```
```   163   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
```
```   164     by (auto simp:bij_betw_def)
```
```   165   let ?f = "the_inv_into {i. i<n} f"
```
```   166   have "inj_on ?f A & ?f ` A = {i. i<n}"
```
```   167     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
```
```   168   thus ?thesis by blast
```
```   169 qed
```
```   170
```
```   171 lemma finite_Collect_less_nat[iff]: "finite{n::nat. n<k}"
```
```   172 by(fastsimp simp: finite_conv_nat_seg_image)
```
```   173
```
```   174
```
```   175 subsubsection{* Finiteness and set theoretic constructions *}
```
```   176
```
```   177 lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
```
```   178 by (induct set: finite) simp_all
```
```   179
```
```   180 lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
```
```   181   -- {* Every subset of a finite set is finite. *}
```
```   182 proof -
```
```   183   assume "finite B"
```
```   184   thus "!!A. A \<subseteq> B ==> finite A"
```
```   185   proof induct
```
```   186     case empty
```
```   187     thus ?case by simp
```
```   188   next
```
```   189     case (insert x F A)
```
```   190     have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" by fact+
```
```   191     show "finite A"
```
```   192     proof cases
```
```   193       assume x: "x \<in> A"
```
```   194       with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
```
```   195       with r have "finite (A - {x})" .
```
```   196       hence "finite (insert x (A - {x}))" ..
```
```   197       also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
```
```   198       finally show ?thesis .
```
```   199     next
```
```   200       show "A \<subseteq> F ==> ?thesis" by fact
```
```   201       assume "x \<notin> A"
```
```   202       with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
```
```   203     qed
```
```   204   qed
```
```   205 qed
```
```   206
```
```   207 lemma rev_finite_subset: "finite B ==> A \<subseteq> B ==> finite A"
```
```   208 by (rule finite_subset)
```
```   209
```
```   210 lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
```
```   211 by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
```
```   212
```
```   213 lemma finite_Collect_disjI[simp]:
```
```   214   "finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})"
```
```   215 by(simp add:Collect_disj_eq)
```
```   216
```
```   217 lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
```
```   218   -- {* The converse obviously fails. *}
```
```   219 by (blast intro: finite_subset)
```
```   220
```
```   221 lemma finite_Collect_conjI [simp, intro]:
```
```   222   "finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}"
```
```   223   -- {* The converse obviously fails. *}
```
```   224 by(simp add:Collect_conj_eq)
```
```   225
```
```   226 lemma finite_Collect_le_nat[iff]: "finite{n::nat. n<=k}"
```
```   227 by(simp add: le_eq_less_or_eq)
```
```   228
```
```   229 lemma finite_insert [simp]: "finite (insert a A) = finite A"
```
```   230   apply (subst insert_is_Un)
```
```   231   apply (simp only: finite_Un, blast)
```
```   232   done
```
```   233
```
```   234 lemma finite_Union[simp, intro]:
```
```   235  "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
```
```   236 by (induct rule:finite_induct) simp_all
```
```   237
```
```   238 lemma finite_Inter[intro]: "EX A:M. finite(A) \<Longrightarrow> finite(Inter M)"
```
```   239 by (blast intro: Inter_lower finite_subset)
```
```   240
```
```   241 lemma finite_INT[intro]: "EX x:I. finite(A x) \<Longrightarrow> finite(INT x:I. A x)"
```
```   242 by (blast intro: INT_lower finite_subset)
```
```   243
```
```   244 lemma finite_empty_induct:
```
```   245   assumes "finite A"
```
```   246     and "P A"
```
```   247     and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
```
```   248   shows "P {}"
```
```   249 proof -
```
```   250   have "P (A - A)"
```
```   251   proof -
```
```   252     {
```
```   253       fix c b :: "'a set"
```
```   254       assume c: "finite c" and b: "finite b"
```
```   255         and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
```
```   256       have "c \<subseteq> b ==> P (b - c)"
```
```   257         using c
```
```   258       proof induct
```
```   259         case empty
```
```   260         from P1 show ?case by simp
```
```   261       next
```
```   262         case (insert x F)
```
```   263         have "P (b - F - {x})"
```
```   264         proof (rule P2)
```
```   265           from _ b show "finite (b - F)" by (rule finite_subset) blast
```
```   266           from insert show "x \<in> b - F" by simp
```
```   267           from insert show "P (b - F)" by simp
```
```   268         qed
```
```   269         also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
```
```   270         finally show ?case .
```
```   271       qed
```
```   272     }
```
```   273     then show ?thesis by this (simp_all add: assms)
```
```   274   qed
```
```   275   then show ?thesis by simp
```
```   276 qed
```
```   277
```
```   278 lemma finite_Diff [simp]: "finite A ==> finite (A - B)"
```
```   279 by (rule Diff_subset [THEN finite_subset])
```
```   280
```
```   281 lemma finite_Diff2 [simp]:
```
```   282   assumes "finite B" shows "finite (A - B) = finite A"
```
```   283 proof -
```
```   284   have "finite A \<longleftrightarrow> finite((A-B) Un (A Int B))" by(simp add: Un_Diff_Int)
```
```   285   also have "\<dots> \<longleftrightarrow> finite(A-B)" using `finite B` by(simp)
```
```   286   finally show ?thesis ..
```
```   287 qed
```
```   288
```
```   289 lemma finite_compl[simp]:
```
```   290   "finite(A::'a set) \<Longrightarrow> finite(-A) = finite(UNIV::'a set)"
```
```   291 by(simp add:Compl_eq_Diff_UNIV)
```
```   292
```
```   293 lemma finite_Collect_not[simp]:
```
```   294   "finite{x::'a. P x} \<Longrightarrow> finite{x. ~P x} = finite(UNIV::'a set)"
```
```   295 by(simp add:Collect_neg_eq)
```
```   296
```
```   297 lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
```
```   298   apply (subst Diff_insert)
```
```   299   apply (case_tac "a : A - B")
```
```   300    apply (rule finite_insert [symmetric, THEN trans])
```
```   301    apply (subst insert_Diff, simp_all)
```
```   302   done
```
```   303
```
```   304
```
```   305 text {* Image and Inverse Image over Finite Sets *}
```
```   306
```
```   307 lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
```
```   308   -- {* The image of a finite set is finite. *}
```
```   309   by (induct set: finite) simp_all
```
```   310
```
```   311 lemma finite_image_set [simp]:
```
```   312   "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
```
```   313   by (simp add: image_Collect [symmetric])
```
```   314
```
```   315 lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
```
```   316   apply (frule finite_imageI)
```
```   317   apply (erule finite_subset, assumption)
```
```   318   done
```
```   319
```
```   320 lemma finite_range_imageI:
```
```   321     "finite (range g) ==> finite (range (%x. f (g x)))"
```
```   322   apply (drule finite_imageI, simp add: range_composition)
```
```   323   done
```
```   324
```
```   325 lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
```
```   326 proof -
```
```   327   have aux: "!!A. finite (A - {}) = finite A" by simp
```
```   328   fix B :: "'a set"
```
```   329   assume "finite B"
```
```   330   thus "!!A. f`A = B ==> inj_on f A ==> finite A"
```
```   331     apply induct
```
```   332      apply simp
```
```   333     apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
```
```   334      apply clarify
```
```   335      apply (simp (no_asm_use) add: inj_on_def)
```
```   336      apply (blast dest!: aux [THEN iffD1], atomize)
```
```   337     apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
```
```   338     apply (frule subsetD [OF equalityD2 insertI1], clarify)
```
```   339     apply (rule_tac x = xa in bexI)
```
```   340      apply (simp_all add: inj_on_image_set_diff)
```
```   341     done
```
```   342 qed (rule refl)
```
```   343
```
```   344
```
```   345 lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
```
```   346   -- {* The inverse image of a singleton under an injective function
```
```   347          is included in a singleton. *}
```
```   348   apply (auto simp add: inj_on_def)
```
```   349   apply (blast intro: the_equality [symmetric])
```
```   350   done
```
```   351
```
```   352 lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
```
```   353   -- {* The inverse image of a finite set under an injective function
```
```   354          is finite. *}
```
```   355   apply (induct set: finite)
```
```   356    apply simp_all
```
```   357   apply (subst vimage_insert)
```
```   358   apply (simp add: finite_subset [OF inj_vimage_singleton])
```
```   359   done
```
```   360
```
```   361 lemma finite_vimageD:
```
```   362   assumes fin: "finite (h -` F)" and surj: "surj h"
```
```   363   shows "finite F"
```
```   364 proof -
```
```   365   have "finite (h ` (h -` F))" using fin by (rule finite_imageI)
```
```   366   also have "h ` (h -` F) = F" using surj by (rule surj_image_vimage_eq)
```
```   367   finally show "finite F" .
```
```   368 qed
```
```   369
```
```   370 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
```
```   371   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
```
```   372
```
```   373
```
```   374 text {* The finite UNION of finite sets *}
```
```   375
```
```   376 lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
```
```   377   by (induct set: finite) simp_all
```
```   378
```
```   379 text {*
```
```   380   Strengthen RHS to
```
```   381   @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
```
```   382
```
```   383   We'd need to prove
```
```   384   @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
```
```   385   by induction. *}
```
```   386
```
```   387 lemma finite_UN [simp]:
```
```   388   "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
```
```   389 by (blast intro: finite_UN_I finite_subset)
```
```   390
```
```   391 lemma finite_Collect_bex[simp]: "finite A \<Longrightarrow>
```
```   392   finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})"
```
```   393 apply(subgoal_tac "{x. EX y:A. Q x y} = UNION A (%y. {x. Q x y})")
```
```   394  apply auto
```
```   395 done
```
```   396
```
```   397 lemma finite_Collect_bounded_ex[simp]: "finite{y. P y} \<Longrightarrow>
```
```   398   finite{x. EX y. P y & Q x y} = (ALL y. P y \<longrightarrow> finite{x. Q x y})"
```
```   399 apply(subgoal_tac "{x. EX y. P y & Q x y} = UNION {y. P y} (%y. {x. Q x y})")
```
```   400  apply auto
```
```   401 done
```
```   402
```
```   403
```
```   404 lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
```
```   405 by (simp add: Plus_def)
```
```   406
```
```   407 lemma finite_PlusD:
```
```   408   fixes A :: "'a set" and B :: "'b set"
```
```   409   assumes fin: "finite (A <+> B)"
```
```   410   shows "finite A" "finite B"
```
```   411 proof -
```
```   412   have "Inl ` A \<subseteq> A <+> B" by auto
```
```   413   hence "finite (Inl ` A :: ('a + 'b) set)" using fin by(rule finite_subset)
```
```   414   thus "finite A" by(rule finite_imageD)(auto intro: inj_onI)
```
```   415 next
```
```   416   have "Inr ` B \<subseteq> A <+> B" by auto
```
```   417   hence "finite (Inr ` B :: ('a + 'b) set)" using fin by(rule finite_subset)
```
```   418   thus "finite B" by(rule finite_imageD)(auto intro: inj_onI)
```
```   419 qed
```
```   420
```
```   421 lemma finite_Plus_iff[simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
```
```   422 by(auto intro: finite_PlusD finite_Plus)
```
```   423
```
```   424 lemma finite_Plus_UNIV_iff[simp]:
```
```   425   "finite (UNIV :: ('a + 'b) set) =
```
```   426   (finite (UNIV :: 'a set) & finite (UNIV :: 'b set))"
```
```   427 by(subst UNIV_Plus_UNIV[symmetric])(rule finite_Plus_iff)
```
```   428
```
```   429
```
```   430 text {* Sigma of finite sets *}
```
```   431
```
```   432 lemma finite_SigmaI [simp]:
```
```   433     "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
```
```   434   by (unfold Sigma_def) (blast intro!: finite_UN_I)
```
```   435
```
```   436 lemma finite_cartesian_product: "[| finite A; finite B |] ==>
```
```   437     finite (A <*> B)"
```
```   438   by (rule finite_SigmaI)
```
```   439
```
```   440 lemma finite_Prod_UNIV:
```
```   441     "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
```
```   442   apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
```
```   443    apply (erule ssubst)
```
```   444    apply (erule finite_SigmaI, auto)
```
```   445   done
```
```   446
```
```   447 lemma finite_cartesian_productD1:
```
```   448      "[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
```
```   449 apply (auto simp add: finite_conv_nat_seg_image)
```
```   450 apply (drule_tac x=n in spec)
```
```   451 apply (drule_tac x="fst o f" in spec)
```
```   452 apply (auto simp add: o_def)
```
```   453  prefer 2 apply (force dest!: equalityD2)
```
```   454 apply (drule equalityD1)
```
```   455 apply (rename_tac y x)
```
```   456 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   457  prefer 2 apply force
```
```   458 apply clarify
```
```   459 apply (rule_tac x=k in image_eqI, auto)
```
```   460 done
```
```   461
```
```   462 lemma finite_cartesian_productD2:
```
```   463      "[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
```
```   464 apply (auto simp add: finite_conv_nat_seg_image)
```
```   465 apply (drule_tac x=n in spec)
```
```   466 apply (drule_tac x="snd o f" in spec)
```
```   467 apply (auto simp add: o_def)
```
```   468  prefer 2 apply (force dest!: equalityD2)
```
```   469 apply (drule equalityD1)
```
```   470 apply (rename_tac x y)
```
```   471 apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
```
```   472  prefer 2 apply force
```
```   473 apply clarify
```
```   474 apply (rule_tac x=k in image_eqI, auto)
```
```   475 done
```
```   476
```
```   477
```
```   478 text {* The powerset of a finite set *}
```
```   479
```
```   480 lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
```
```   481 proof
```
```   482   assume "finite (Pow A)"
```
```   483   with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
```
```   484   thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
```
```   485 next
```
```   486   assume "finite A"
```
```   487   thus "finite (Pow A)"
```
```   488     by induct (simp_all add: Pow_insert)
```
```   489 qed
```
```   490
```
```   491 lemma finite_Collect_subsets[simp,intro]: "finite A \<Longrightarrow> finite{B. B \<subseteq> A}"
```
```   492 by(simp add: Pow_def[symmetric])
```
```   493
```
```   494
```
```   495 lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
```
```   496 by(blast intro: finite_subset[OF subset_Pow_Union])
```
```   497
```
```   498
```
```   499 lemma finite_subset_image:
```
```   500   assumes "finite B"
```
```   501   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
```
```   502 using assms proof(induct)
```
```   503   case empty thus ?case by simp
```
```   504 next
```
```   505   case insert thus ?case
```
```   506     by (clarsimp simp del: image_insert simp add: image_insert[symmetric])
```
```   507        blast
```
```   508 qed
```
```   509
```
```   510
```
```   511 subsection {* Class @{text finite}  *}
```
```   512
```
```   513 setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}
```
```   514 class finite =
```
```   515   assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
```
```   516 setup {* Sign.parent_path *}
```
```   517 hide const finite
```
```   518
```
```   519 context finite
```
```   520 begin
```
```   521
```
```   522 lemma finite [simp]: "finite (A \<Colon> 'a set)"
```
```   523   by (rule subset_UNIV finite_UNIV finite_subset)+
```
```   524
```
```   525 end
```
```   526
```
```   527 lemma UNIV_unit [noatp]:
```
```   528   "UNIV = {()}" by auto
```
```   529
```
```   530 instance unit :: finite proof
```
```   531 qed (simp add: UNIV_unit)
```
```   532
```
```   533 lemma UNIV_bool [noatp]:
```
```   534   "UNIV = {False, True}" by auto
```
```   535
```
```   536 instance bool :: finite proof
```
```   537 qed (simp add: UNIV_bool)
```
```   538
```
```   539 instance * :: (finite, finite) finite proof
```
```   540 qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
```
```   541
```
```   542 lemma finite_option_UNIV [simp]:
```
```   543   "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
```
```   544   by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
```
```   545
```
```   546 instance option :: (finite) finite proof
```
```   547 qed (simp add: UNIV_option_conv)
```
```   548
```
```   549 lemma inj_graph: "inj (%f. {(x, y). y = f x})"
```
```   550   by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
```
```   551
```
```   552 instance "fun" :: (finite, finite) finite
```
```   553 proof
```
```   554   show "finite (UNIV :: ('a => 'b) set)"
```
```   555   proof (rule finite_imageD)
```
```   556     let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
```
```   557     have "range ?graph \<subseteq> Pow UNIV" by simp
```
```   558     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
```
```   559       by (simp only: finite_Pow_iff finite)
```
```   560     ultimately show "finite (range ?graph)"
```
```   561       by (rule finite_subset)
```
```   562     show "inj ?graph" by (rule inj_graph)
```
```   563   qed
```
```   564 qed
```
```   565
```
```   566 instance "+" :: (finite, finite) finite proof
```
```   567 qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
```
```   568
```
```   569
```
```   570 subsection {* A fold functional for finite sets *}
```
```   571
```
```   572 text {* The intended behaviour is
```
```   573 @{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
```
```   574 if @{text f} is ``left-commutative'':
```
```   575 *}
```
```   576
```
```   577 locale fun_left_comm =
```
```   578   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   579   assumes fun_left_comm: "f x (f y z) = f y (f x z)"
```
```   580 begin
```
```   581
```
```   582 text{* On a functional level it looks much nicer: *}
```
```   583
```
```   584 lemma fun_comp_comm:  "f x \<circ> f y = f y \<circ> f x"
```
```   585 by (simp add: fun_left_comm expand_fun_eq)
```
```   586
```
```   587 end
```
```   588
```
```   589 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
```
```   590 for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
```
```   591   emptyI [intro]: "fold_graph f z {} z" |
```
```   592   insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
```
```   593       \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
```
```   594
```
```   595 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
```
```   596
```
```   597 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
```
```   598 [code del]: "fold f z A = (THE y. fold_graph f z A y)"
```
```   599
```
```   600 text{*A tempting alternative for the definiens is
```
```   601 @{term "if finite A then THE y. fold_graph f z A y else e"}.
```
```   602 It allows the removal of finiteness assumptions from the theorems
```
```   603 @{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
```
```   604 The proofs become ugly. It is not worth the effort. (???) *}
```
```   605
```
```   606
```
```   607 lemma Diff1_fold_graph:
```
```   608   "fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)"
```
```   609 by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto)
```
```   610
```
```   611 lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A"
```
```   612 by (induct set: fold_graph) auto
```
```   613
```
```   614 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
```
```   615 by (induct set: finite) auto
```
```   616
```
```   617
```
```   618 subsubsection{*From @{const fold_graph} to @{term fold}*}
```
```   619
```
```   620 lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
```
```   621   by (auto simp add: less_Suc_eq)
```
```   622
```
```   623 lemma insert_image_inj_on_eq:
```
```   624      "[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A;
```
```   625         inj_on h {i. i < Suc m}|]
```
```   626       ==> A = h ` {i. i < m}"
```
```   627 apply (auto simp add: image_less_Suc inj_on_def)
```
```   628 apply (blast intro: less_trans)
```
```   629 done
```
```   630
```
```   631 lemma insert_inj_onE:
```
```   632   assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A"
```
```   633       and inj_on: "inj_on h {i::nat. i<n}"
```
```   634   shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
```
```   635 proof (cases n)
```
```   636   case 0 thus ?thesis using aA by auto
```
```   637 next
```
```   638   case (Suc m)
```
```   639   have nSuc: "n = Suc m" by fact
```
```   640   have mlessn: "m<n" by (simp add: nSuc)
```
```   641   from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
```
```   642   let ?hm = "Fun.swap k m h"
```
```   643   have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn
```
```   644     by (simp add: inj_on)
```
```   645   show ?thesis
```
```   646   proof (intro exI conjI)
```
```   647     show "inj_on ?hm {i. i < m}" using inj_hm
```
```   648       by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
```
```   649     show "m<n" by (rule mlessn)
```
```   650     show "A = ?hm ` {i. i < m}"
```
```   651     proof (rule insert_image_inj_on_eq)
```
```   652       show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
```
```   653       show "?hm m \<notin> A" by (simp add: swap_def hkeq anot)
```
```   654       show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
```
```   655         using aA hkeq nSuc klessn
```
```   656         by (auto simp add: swap_def image_less_Suc fun_upd_image
```
```   657                            less_Suc_eq inj_on_image_set_diff [OF inj_on])
```
```   658     qed
```
```   659   qed
```
```   660 qed
```
```   661
```
```   662 context fun_left_comm
```
```   663 begin
```
```   664
```
```   665 lemma fold_graph_determ_aux:
```
```   666   "A = h`{i::nat. i<n} \<Longrightarrow> inj_on h {i. i<n}
```
```   667    \<Longrightarrow> fold_graph f z A x \<Longrightarrow> fold_graph f z A x'
```
```   668    \<Longrightarrow> x' = x"
```
```   669 proof (induct n arbitrary: A x x' h rule: less_induct)
```
```   670   case (less n)
```
```   671   have IH: "\<And>m h A x x'. m < n \<Longrightarrow> A = h ` {i. i<m}
```
```   672       \<Longrightarrow> inj_on h {i. i<m} \<Longrightarrow> fold_graph f z A x
```
```   673       \<Longrightarrow> fold_graph f z A x' \<Longrightarrow> x' = x" by fact
```
```   674   have Afoldx: "fold_graph f z A x" and Afoldx': "fold_graph f z A x'"
```
```   675     and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+
```
```   676   show ?case
```
```   677   proof (rule fold_graph.cases [OF Afoldx])
```
```   678     assume "A = {}" and "x = z"
```
```   679     with Afoldx' show "x' = x" by auto
```
```   680   next
```
```   681     fix B b u
```
```   682     assume AbB: "A = insert b B" and x: "x = f b u"
```
```   683       and notinB: "b \<notin> B" and Bu: "fold_graph f z B u"
```
```   684     show "x'=x"
```
```   685     proof (rule fold_graph.cases [OF Afoldx'])
```
```   686       assume "A = {}" and "x' = z"
```
```   687       with AbB show "x' = x" by blast
```
```   688     next
```
```   689       fix C c v
```
```   690       assume AcC: "A = insert c C" and x': "x' = f c v"
```
```   691         and notinC: "c \<notin> C" and Cv: "fold_graph f z C v"
```
```   692       from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
```
```   693       from insert_inj_onE [OF Beq notinB injh]
```
```   694       obtain hB mB where inj_onB: "inj_on hB {i. i < mB}"
```
```   695         and Beq: "B = hB ` {i. i < mB}" and lessB: "mB < n" by auto
```
```   696       from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
```
```   697       from insert_inj_onE [OF Ceq notinC injh]
```
```   698       obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
```
```   699         and Ceq: "C = hC ` {i. i < mC}" and lessC: "mC < n" by auto
```
```   700       show "x'=x"
```
```   701       proof cases
```
```   702         assume "b=c"
```
```   703         then moreover have "B = C" using AbB AcC notinB notinC by auto
```
```   704         ultimately show ?thesis  using Bu Cv x x' IH [OF lessC Ceq inj_onC]
```
```   705           by auto
```
```   706       next
```
```   707         assume diff: "b \<noteq> c"
```
```   708         let ?D = "B - {c}"
```
```   709         have B: "B = insert c ?D" and C: "C = insert b ?D"
```
```   710           using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
```
```   711         have "finite A" by(rule fold_graph_imp_finite [OF Afoldx])
```
```   712         with AbB have "finite ?D" by simp
```
```   713         then obtain d where Dfoldd: "fold_graph f z ?D d"
```
```   714           using finite_imp_fold_graph by iprover
```
```   715         moreover have cinB: "c \<in> B" using B by auto
```
```   716         ultimately have "fold_graph f z B (f c d)" by(rule Diff1_fold_graph)
```
```   717         hence "f c d = u" by (rule IH [OF lessB Beq inj_onB Bu])
```
```   718         moreover have "f b d = v"
```
```   719         proof (rule IH[OF lessC Ceq inj_onC Cv])
```
```   720           show "fold_graph f z C (f b d)" using C notinB Dfoldd by fastsimp
```
```   721         qed
```
```   722         ultimately show ?thesis
```
```   723           using fun_left_comm [of c b] x x' by (auto simp add: o_def)
```
```   724       qed
```
```   725     qed
```
```   726   qed
```
```   727 qed
```
```   728
```
```   729 lemma fold_graph_determ:
```
```   730   "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
```
```   731 apply (frule fold_graph_imp_finite [THEN finite_imp_nat_seg_image_inj_on])
```
```   732 apply (blast intro: fold_graph_determ_aux [rule_format])
```
```   733 done
```
```   734
```
```   735 lemma fold_equality:
```
```   736   "fold_graph f z A y \<Longrightarrow> fold f z A = y"
```
```   737 by (unfold fold_def) (blast intro: fold_graph_determ)
```
```   738
```
```   739 text{* The base case for @{text fold}: *}
```
```   740
```
```   741 lemma (in -) fold_empty [simp]: "fold f z {} = z"
```
```   742 by (unfold fold_def) blast
```
```   743
```
```   744 text{* The various recursion equations for @{const fold}: *}
```
```   745
```
```   746 lemma fold_insert_aux: "x \<notin> A
```
```   747   \<Longrightarrow> fold_graph f z (insert x A) v \<longleftrightarrow>
```
```   748       (\<exists>y. fold_graph f z A y \<and> v = f x y)"
```
```   749 apply auto
```
```   750 apply (rule_tac A1 = A and f1 = f in finite_imp_fold_graph [THEN exE])
```
```   751  apply (fastsimp dest: fold_graph_imp_finite)
```
```   752 apply (blast intro: fold_graph_determ)
```
```   753 done
```
```   754
```
```   755 lemma fold_insert [simp]:
```
```   756   "finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)"
```
```   757 apply (simp add: fold_def fold_insert_aux)
```
```   758 apply (rule the_equality)
```
```   759  apply (auto intro: finite_imp_fold_graph
```
```   760         cong add: conj_cong simp add: fold_def[symmetric] fold_equality)
```
```   761 done
```
```   762
```
```   763 lemma fold_fun_comm:
```
```   764   "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
```
```   765 proof (induct rule: finite_induct)
```
```   766   case empty then show ?case by simp
```
```   767 next
```
```   768   case (insert y A) then show ?case
```
```   769     by (simp add: fun_left_comm[of x])
```
```   770 qed
```
```   771
```
```   772 lemma fold_insert2:
```
```   773   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   774 by (simp add: fold_fun_comm)
```
```   775
```
```   776 lemma fold_rec:
```
```   777 assumes "finite A" and "x \<in> A"
```
```   778 shows "fold f z A = f x (fold f z (A - {x}))"
```
```   779 proof -
```
```   780   have A: "A = insert x (A - {x})" using `x \<in> A` by blast
```
```   781   then have "fold f z A = fold f z (insert x (A - {x}))" by simp
```
```   782   also have "\<dots> = f x (fold f z (A - {x}))"
```
```   783     by (rule fold_insert) (simp add: `finite A`)+
```
```   784   finally show ?thesis .
```
```   785 qed
```
```   786
```
```   787 lemma fold_insert_remove:
```
```   788   assumes "finite A"
```
```   789   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
```
```   790 proof -
```
```   791   from `finite A` have "finite (insert x A)" by auto
```
```   792   moreover have "x \<in> insert x A" by auto
```
```   793   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
```
```   794     by (rule fold_rec)
```
```   795   then show ?thesis by simp
```
```   796 qed
```
```   797
```
```   798 end
```
```   799
```
```   800 text{* A simplified version for idempotent functions: *}
```
```   801
```
```   802 locale fun_left_comm_idem = fun_left_comm +
```
```   803   assumes fun_left_idem: "f x (f x z) = f x z"
```
```   804 begin
```
```   805
```
```   806 text{* The nice version: *}
```
```   807 lemma fun_comp_idem : "f x o f x = f x"
```
```   808 by (simp add: fun_left_idem expand_fun_eq)
```
```   809
```
```   810 lemma fold_insert_idem:
```
```   811   assumes fin: "finite A"
```
```   812   shows "fold f z (insert x A) = f x (fold f z A)"
```
```   813 proof cases
```
```   814   assume "x \<in> A"
```
```   815   then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
```
```   816   then show ?thesis using assms by (simp add:fun_left_idem)
```
```   817 next
```
```   818   assume "x \<notin> A" then show ?thesis using assms by simp
```
```   819 qed
```
```   820
```
```   821 declare fold_insert[simp del] fold_insert_idem[simp]
```
```   822
```
```   823 lemma fold_insert_idem2:
```
```   824   "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
```
```   825 by(simp add:fold_fun_comm)
```
```   826
```
```   827 end
```
```   828
```
```   829 context ab_semigroup_idem_mult
```
```   830 begin
```
```   831
```
```   832 lemma fun_left_comm_idem: "fun_left_comm_idem(op *)"
```
```   833 apply unfold_locales
```
```   834  apply (rule mult_left_commute)
```
```   835 apply (rule mult_left_idem)
```
```   836 done
```
```   837
```
```   838 end
```
```   839
```
```   840 context semilattice_inf
```
```   841 begin
```
```   842
```
```   843 lemma ab_semigroup_idem_mult_inf: "ab_semigroup_idem_mult inf"
```
```   844 proof qed (rule inf_assoc inf_commute inf_idem)+
```
```   845
```
```   846 lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> fold inf b (insert a A) = inf a (fold inf b A)"
```
```   847 by(rule fun_left_comm_idem.fold_insert_idem[OF ab_semigroup_idem_mult.fun_left_comm_idem[OF ab_semigroup_idem_mult_inf]])
```
```   848
```
```   849 lemma inf_le_fold_inf: "finite A \<Longrightarrow> ALL a:A. b \<le> a \<Longrightarrow> inf b c \<le> fold inf c A"
```
```   850 by (induct pred: finite) (auto intro: le_infI1)
```
```   851
```
```   852 lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> fold inf b A \<le> inf a b"
```
```   853 proof(induct arbitrary: a pred:finite)
```
```   854   case empty thus ?case by simp
```
```   855 next
```
```   856   case (insert x A)
```
```   857   show ?case
```
```   858   proof cases
```
```   859     assume "A = {}" thus ?thesis using insert by simp
```
```   860   next
```
```   861     assume "A \<noteq> {}" thus ?thesis using insert by (auto intro: le_infI2)
```
```   862   qed
```
```   863 qed
```
```   864
```
```   865 end
```
```   866
```
```   867 context semilattice_sup
```
```   868 begin
```
```   869
```
```   870 lemma ab_semigroup_idem_mult_sup: "ab_semigroup_idem_mult sup"
```
```   871 by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
```
```   872
```
```   873 lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> fold sup b (insert a A) = sup a (fold sup b A)"
```
```   874 by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)
```
```   875
```
```   876 lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> fold sup c A \<le> sup b c"
```
```   877 by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)
```
```   878
```
```   879 lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> fold sup b A"
```
```   880 by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)
```
```   881
```
```   882 end
```
```   883
```
```   884
```
```   885 subsubsection{* The derived combinator @{text fold_image} *}
```
```   886
```
```   887 definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
```
```   888 where "fold_image f g = fold (%x y. f (g x) y)"
```
```   889
```
```   890 lemma fold_image_empty[simp]: "fold_image f g z {} = z"
```
```   891 by(simp add:fold_image_def)
```
```   892
```
```   893 context ab_semigroup_mult
```
```   894 begin
```
```   895
```
```   896 lemma fold_image_insert[simp]:
```
```   897 assumes "finite A" and "a \<notin> A"
```
```   898 shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
```
```   899 proof -
```
```   900   interpret I: fun_left_comm "%x y. (g x) * y"
```
```   901     by unfold_locales (simp add: mult_ac)
```
```   902   show ?thesis using assms by(simp add:fold_image_def)
```
```   903 qed
```
```   904
```
```   905 (*
```
```   906 lemma fold_commute:
```
```   907   "finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
```
```   908   apply (induct set: finite)
```
```   909    apply simp
```
```   910   apply (simp add: mult_left_commute [of x])
```
```   911   done
```
```   912
```
```   913 lemma fold_nest_Un_Int:
```
```   914   "finite A ==> finite B
```
```   915     ==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
```
```   916   apply (induct set: finite)
```
```   917    apply simp
```
```   918   apply (simp add: fold_commute Int_insert_left insert_absorb)
```
```   919   done
```
```   920
```
```   921 lemma fold_nest_Un_disjoint:
```
```   922   "finite A ==> finite B ==> A Int B = {}
```
```   923     ==> fold times g z (A Un B) = fold times g (fold times g z B) A"
```
```   924   by (simp add: fold_nest_Un_Int)
```
```   925 *)
```
```   926
```
```   927 lemma fold_image_reindex:
```
```   928 assumes fin: "finite A"
```
```   929 shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A"
```
```   930 using fin by induct auto
```
```   931
```
```   932 (*
```
```   933 text{*
```
```   934   Fusion theorem, as described in Graham Hutton's paper,
```
```   935   A Tutorial on the Universality and Expressiveness of Fold,
```
```   936   JFP 9:4 (355-372), 1999.
```
```   937 *}
```
```   938
```
```   939 lemma fold_fusion:
```
```   940   assumes "ab_semigroup_mult g"
```
```   941   assumes fin: "finite A"
```
```   942     and hyp: "\<And>x y. h (g x y) = times x (h y)"
```
```   943   shows "h (fold g j w A) = fold times j (h w) A"
```
```   944 proof -
```
```   945   class_interpret ab_semigroup_mult [g] by fact
```
```   946   show ?thesis using fin hyp by (induct set: finite) simp_all
```
```   947 qed
```
```   948 *)
```
```   949
```
```   950 lemma fold_image_cong:
```
```   951   "finite A \<Longrightarrow>
```
```   952   (!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A"
```
```   953 apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C")
```
```   954  apply simp
```
```   955 apply (erule finite_induct, simp)
```
```   956 apply (simp add: subset_insert_iff, clarify)
```
```   957 apply (subgoal_tac "finite C")
```
```   958  prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
```
```   959 apply (subgoal_tac "C = insert x (C - {x})")
```
```   960  prefer 2 apply blast
```
```   961 apply (erule ssubst)
```
```   962 apply (drule spec)
```
```   963 apply (erule (1) notE impE)
```
```   964 apply (simp add: Ball_def del: insert_Diff_single)
```
```   965 done
```
```   966
```
```   967 end
```
```   968
```
```   969 context comm_monoid_mult
```
```   970 begin
```
```   971
```
```   972 lemma fold_image_Un_Int:
```
```   973   "finite A ==> finite B ==>
```
```   974     fold_image times g 1 A * fold_image times g 1 B =
```
```   975     fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
```
```   976 by (induct set: finite)
```
```   977    (auto simp add: mult_ac insert_absorb Int_insert_left)
```
```   978
```
```   979 corollary fold_Un_disjoint:
```
```   980   "finite A ==> finite B ==> A Int B = {} ==>
```
```   981    fold_image times g 1 (A Un B) =
```
```   982    fold_image times g 1 A * fold_image times g 1 B"
```
```   983 by (simp add: fold_image_Un_Int)
```
```   984
```
```   985 lemma fold_image_UN_disjoint:
```
```   986   "\<lbrakk> finite I; ALL i:I. finite (A i);
```
```   987      ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
```
```   988    \<Longrightarrow> fold_image times g 1 (UNION I A) =
```
```   989        fold_image times (%i. fold_image times g 1 (A i)) 1 I"
```
```   990 apply (induct set: finite, simp, atomize)
```
```   991 apply (subgoal_tac "ALL i:F. x \<noteq> i")
```
```   992  prefer 2 apply blast
```
```   993 apply (subgoal_tac "A x Int UNION F A = {}")
```
```   994  prefer 2 apply blast
```
```   995 apply (simp add: fold_Un_disjoint)
```
```   996 done
```
```   997
```
```   998 lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   999   fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
```
```  1000   fold_image times (split g) 1 (SIGMA x:A. B x)"
```
```  1001 apply (subst Sigma_def)
```
```  1002 apply (subst fold_image_UN_disjoint, assumption, simp)
```
```  1003  apply blast
```
```  1004 apply (erule fold_image_cong)
```
```  1005 apply (subst fold_image_UN_disjoint, simp, simp)
```
```  1006  apply blast
```
```  1007 apply simp
```
```  1008 done
```
```  1009
```
```  1010 lemma fold_image_distrib: "finite A \<Longrightarrow>
```
```  1011    fold_image times (%x. g x * h x) 1 A =
```
```  1012    fold_image times g 1 A *  fold_image times h 1 A"
```
```  1013 by (erule finite_induct) (simp_all add: mult_ac)
```
```  1014
```
```  1015 lemma fold_image_related:
```
```  1016   assumes Re: "R e e"
```
```  1017   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
```
```  1018   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
```
```  1019   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
```
```  1020   using fS by (rule finite_subset_induct) (insert assms, auto)
```
```  1021
```
```  1022 lemma  fold_image_eq_general:
```
```  1023   assumes fS: "finite S"
```
```  1024   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
```
```  1025   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
```
```  1026   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
```
```  1027 proof-
```
```  1028   from h f12 have hS: "h ` S = S'" by auto
```
```  1029   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
```
```  1030     from f12 h H  have "x = y" by auto }
```
```  1031   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
```
```  1032   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
```
```  1033   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
```
```  1034   also have "\<dots> = fold_image (op *) (f2 o h) e S"
```
```  1035     using fold_image_reindex[OF fS hinj, of f2 e] .
```
```  1036   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
```
```  1037     by blast
```
```  1038   finally show ?thesis ..
```
```  1039 qed
```
```  1040
```
```  1041 lemma fold_image_eq_general_inverses:
```
```  1042   assumes fS: "finite S"
```
```  1043   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```  1044   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
```
```  1045   shows "fold_image (op *) f e S = fold_image (op *) g e T"
```
```  1046   (* metis solves it, but not yet available here *)
```
```  1047   apply (rule fold_image_eq_general[OF fS, of T h g f e])
```
```  1048   apply (rule ballI)
```
```  1049   apply (frule kh)
```
```  1050   apply (rule ex1I[])
```
```  1051   apply blast
```
```  1052   apply clarsimp
```
```  1053   apply (drule hk) apply simp
```
```  1054   apply (rule sym)
```
```  1055   apply (erule conjunct1[OF conjunct2[OF hk]])
```
```  1056   apply (rule ballI)
```
```  1057   apply (drule  hk)
```
```  1058   apply blast
```
```  1059   done
```
```  1060
```
```  1061 end
```
```  1062
```
```  1063
```
```  1064 subsection{* A fold functional for non-empty sets *}
```
```  1065
```
```  1066 text{* Does not require start value. *}
```
```  1067
```
```  1068 inductive
```
```  1069   fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
```
```  1070   for f :: "'a => 'a => 'a"
```
```  1071 where
```
```  1072   fold1Set_insertI [intro]:
```
```  1073    "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
```
```  1074
```
```  1075 definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
```
```  1076   "fold1 f A == THE x. fold1Set f A x"
```
```  1077
```
```  1078 lemma fold1Set_nonempty:
```
```  1079   "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
```
```  1080 by(erule fold1Set.cases, simp_all)
```
```  1081
```
```  1082 inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
```
```  1083
```
```  1084 inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
```
```  1085
```
```  1086
```
```  1087 lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
```
```  1088 by (blast elim: fold_graph.cases)
```
```  1089
```
```  1090 lemma fold1_singleton [simp]: "fold1 f {a} = a"
```
```  1091 by (unfold fold1_def) blast
```
```  1092
```
```  1093 lemma finite_nonempty_imp_fold1Set:
```
```  1094   "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
```
```  1095 apply (induct A rule: finite_induct)
```
```  1096 apply (auto dest: finite_imp_fold_graph [of _ f])
```
```  1097 done
```
```  1098
```
```  1099 text{*First, some lemmas about @{const fold_graph}.*}
```
```  1100
```
```  1101 context ab_semigroup_mult
```
```  1102 begin
```
```  1103
```
```  1104 lemma fun_left_comm: "fun_left_comm(op *)"
```
```  1105 by unfold_locales (simp add: mult_ac)
```
```  1106
```
```  1107 lemma fold_graph_insert_swap:
```
```  1108 assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
```
```  1109 shows "fold_graph times z (insert b A) (z * y)"
```
```  1110 proof -
```
```  1111   interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
```
```  1112 from assms show ?thesis
```
```  1113 proof (induct rule: fold_graph.induct)
```
```  1114   case emptyI thus ?case by (force simp add: fold_insert_aux mult_commute)
```
```  1115 next
```
```  1116   case (insertI x A y)
```
```  1117     have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
```
```  1118       using insertI by force  --{*how does @{term id} get unfolded?*}
```
```  1119     thus ?case by (simp add: insert_commute mult_ac)
```
```  1120 qed
```
```  1121 qed
```
```  1122
```
```  1123 lemma fold_graph_permute_diff:
```
```  1124 assumes fold: "fold_graph times b A x"
```
```  1125 shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
```
```  1126 using fold
```
```  1127 proof (induct rule: fold_graph.induct)
```
```  1128   case emptyI thus ?case by simp
```
```  1129 next
```
```  1130   case (insertI x A y)
```
```  1131   have "a = x \<or> a \<in> A" using insertI by simp
```
```  1132   thus ?case
```
```  1133   proof
```
```  1134     assume "a = x"
```
```  1135     with insertI show ?thesis
```
```  1136       by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
```
```  1137   next
```
```  1138     assume ainA: "a \<in> A"
```
```  1139     hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
```
```  1140       using insertI by force
```
```  1141     moreover
```
```  1142     have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
```
```  1143       using ainA insertI by blast
```
```  1144     ultimately show ?thesis by simp
```
```  1145   qed
```
```  1146 qed
```
```  1147
```
```  1148 lemma fold1_eq_fold:
```
```  1149 assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
```
```  1150 proof -
```
```  1151   interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
```
```  1152   from assms show ?thesis
```
```  1153 apply (simp add: fold1_def fold_def)
```
```  1154 apply (rule the_equality)
```
```  1155 apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
```
```  1156 apply (rule sym, clarify)
```
```  1157 apply (case_tac "Aa=A")
```
```  1158  apply (best intro: fold_graph_determ)
```
```  1159 apply (subgoal_tac "fold_graph times a A x")
```
```  1160  apply (best intro: fold_graph_determ)
```
```  1161 apply (subgoal_tac "insert aa (Aa - {a}) = A")
```
```  1162  prefer 2 apply (blast elim: equalityE)
```
```  1163 apply (auto dest: fold_graph_permute_diff [where a=a])
```
```  1164 done
```
```  1165 qed
```
```  1166
```
```  1167 lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
```
```  1168 apply safe
```
```  1169  apply simp
```
```  1170  apply (drule_tac x=x in spec)
```
```  1171  apply (drule_tac x="A-{x}" in spec, auto)
```
```  1172 done
```
```  1173
```
```  1174 lemma fold1_insert:
```
```  1175   assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
```
```  1176   shows "fold1 times (insert x A) = x * fold1 times A"
```
```  1177 proof -
```
```  1178   interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
```
```  1179   from nonempty obtain a A' where "A = insert a A' & a ~: A'"
```
```  1180     by (auto simp add: nonempty_iff)
```
```  1181   with A show ?thesis
```
```  1182     by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
```
```  1183 qed
```
```  1184
```
```  1185 end
```
```  1186
```
```  1187 context ab_semigroup_idem_mult
```
```  1188 begin
```
```  1189
```
```  1190 lemma fold1_insert_idem [simp]:
```
```  1191   assumes nonempty: "A \<noteq> {}" and A: "finite A"
```
```  1192   shows "fold1 times (insert x A) = x * fold1 times A"
```
```  1193 proof -
```
```  1194   interpret fun_left_comm_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```  1195     by (rule fun_left_comm_idem)
```
```  1196   from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
```
```  1197     by (auto simp add: nonempty_iff)
```
```  1198   show ?thesis
```
```  1199   proof cases
```
```  1200     assume "a = x"
```
```  1201     thus ?thesis
```
```  1202     proof cases
```
```  1203       assume "A' = {}"
```
```  1204       with prems show ?thesis by simp
```
```  1205     next
```
```  1206       assume "A' \<noteq> {}"
```
```  1207       with prems show ?thesis
```
```  1208         by (simp add: fold1_insert mult_assoc [symmetric])
```
```  1209     qed
```
```  1210   next
```
```  1211     assume "a \<noteq> x"
```
```  1212     with prems show ?thesis
```
```  1213       by (simp add: insert_commute fold1_eq_fold)
```
```  1214   qed
```
```  1215 qed
```
```  1216
```
```  1217 lemma hom_fold1_commute:
```
```  1218 assumes hom: "!!x y. h (x * y) = h x * h y"
```
```  1219 and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
```
```  1220 using N proof (induct rule: finite_ne_induct)
```
```  1221   case singleton thus ?case by simp
```
```  1222 next
```
```  1223   case (insert n N)
```
```  1224   then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
```
```  1225   also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
```
```  1226   also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
```
```  1227   also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
```
```  1228     using insert by(simp)
```
```  1229   also have "insert (h n) (h ` N) = h ` insert n N" by simp
```
```  1230   finally show ?case .
```
```  1231 qed
```
```  1232
```
```  1233 lemma fold1_eq_fold_idem:
```
```  1234   assumes "finite A"
```
```  1235   shows "fold1 times (insert a A) = fold times a A"
```
```  1236 proof (cases "a \<in> A")
```
```  1237   case False
```
```  1238   with assms show ?thesis by (simp add: fold1_eq_fold)
```
```  1239 next
```
```  1240   interpret fun_left_comm_idem times by (fact fun_left_comm_idem)
```
```  1241   case True then obtain b B
```
```  1242     where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
```
```  1243   with assms have "finite B" by auto
```
```  1244   then have "fold times a (insert a B) = fold times (a * a) B"
```
```  1245     using `a \<notin> B` by (rule fold_insert2)
```
```  1246   then show ?thesis
```
```  1247     using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
```
```  1248 qed
```
```  1249
```
```  1250 end
```
```  1251
```
```  1252
```
```  1253 text{* Now the recursion rules for definitions: *}
```
```  1254
```
```  1255 lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
```
```  1256 by simp
```
```  1257
```
```  1258 lemma (in ab_semigroup_mult) fold1_insert_def:
```
```  1259   "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
```
```  1260 by (simp add:fold1_insert)
```
```  1261
```
```  1262 lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
```
```  1263   "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
```
```  1264 by simp
```
```  1265
```
```  1266 subsubsection{* Determinacy for @{term fold1Set} *}
```
```  1267
```
```  1268 (*Not actually used!!*)
```
```  1269 (*
```
```  1270 context ab_semigroup_mult
```
```  1271 begin
```
```  1272
```
```  1273 lemma fold_graph_permute:
```
```  1274   "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
```
```  1275    ==> fold_graph times id a (insert b A) x"
```
```  1276 apply (cases "a=b")
```
```  1277 apply (auto dest: fold_graph_permute_diff)
```
```  1278 done
```
```  1279
```
```  1280 lemma fold1Set_determ:
```
```  1281   "fold1Set times A x ==> fold1Set times A y ==> y = x"
```
```  1282 proof (clarify elim!: fold1Set.cases)
```
```  1283   fix A x B y a b
```
```  1284   assume Ax: "fold_graph times id a A x"
```
```  1285   assume By: "fold_graph times id b B y"
```
```  1286   assume anotA:  "a \<notin> A"
```
```  1287   assume bnotB:  "b \<notin> B"
```
```  1288   assume eq: "insert a A = insert b B"
```
```  1289   show "y=x"
```
```  1290   proof cases
```
```  1291     assume same: "a=b"
```
```  1292     hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
```
```  1293     thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
```
```  1294   next
```
```  1295     assume diff: "a\<noteq>b"
```
```  1296     let ?D = "B - {a}"
```
```  1297     have B: "B = insert a ?D" and A: "A = insert b ?D"
```
```  1298      and aB: "a \<in> B" and bA: "b \<in> A"
```
```  1299       using eq anotA bnotB diff by (blast elim!:equalityE)+
```
```  1300     with aB bnotB By
```
```  1301     have "fold_graph times id a (insert b ?D) y"
```
```  1302       by (auto intro: fold_graph_permute simp add: insert_absorb)
```
```  1303     moreover
```
```  1304     have "fold_graph times id a (insert b ?D) x"
```
```  1305       by (simp add: A [symmetric] Ax)
```
```  1306     ultimately show ?thesis by (blast intro: fold_graph_determ)
```
```  1307   qed
```
```  1308 qed
```
```  1309
```
```  1310 lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
```
```  1311   by (unfold fold1_def) (blast intro: fold1Set_determ)
```
```  1312
```
```  1313 end
```
```  1314 *)
```
```  1315
```
```  1316 declare
```
```  1317   empty_fold_graphE [rule del]  fold_graph.intros [rule del]
```
```  1318   empty_fold1SetE [rule del]  insert_fold1SetE [rule del]
```
```  1319   -- {* No more proofs involve these relations. *}
```
```  1320
```
```  1321 subsubsection {* Lemmas about @{text fold1} *}
```
```  1322
```
```  1323 context ab_semigroup_mult
```
```  1324 begin
```
```  1325
```
```  1326 lemma fold1_Un:
```
```  1327 assumes A: "finite A" "A \<noteq> {}"
```
```  1328 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
```
```  1329        fold1 times (A Un B) = fold1 times A * fold1 times B"
```
```  1330 using A by (induct rule: finite_ne_induct)
```
```  1331   (simp_all add: fold1_insert mult_assoc)
```
```  1332
```
```  1333 lemma fold1_in:
```
```  1334   assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
```
```  1335   shows "fold1 times A \<in> A"
```
```  1336 using A
```
```  1337 proof (induct rule:finite_ne_induct)
```
```  1338   case singleton thus ?case by simp
```
```  1339 next
```
```  1340   case insert thus ?case using elem by (force simp add:fold1_insert)
```
```  1341 qed
```
```  1342
```
```  1343 end
```
```  1344
```
```  1345 lemma (in ab_semigroup_idem_mult) fold1_Un2:
```
```  1346 assumes A: "finite A" "A \<noteq> {}"
```
```  1347 shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
```
```  1348        fold1 times (A Un B) = fold1 times A * fold1 times B"
```
```  1349 using A
```
```  1350 proof(induct rule:finite_ne_induct)
```
```  1351   case singleton thus ?case by simp
```
```  1352 next
```
```  1353   case insert thus ?case by (simp add: mult_assoc)
```
```  1354 qed
```
```  1355
```
```  1356
```
```  1357 subsection {* Expressing set operations via @{const fold} *}
```
```  1358
```
```  1359 lemma (in fun_left_comm) fun_left_comm_apply:
```
```  1360   "fun_left_comm (\<lambda>x. f (g x))"
```
```  1361 proof
```
```  1362 qed (simp_all add: fun_left_comm)
```
```  1363
```
```  1364 lemma (in fun_left_comm_idem) fun_left_comm_idem_apply:
```
```  1365   "fun_left_comm_idem (\<lambda>x. f (g x))"
```
```  1366   by (rule fun_left_comm_idem.intro, rule fun_left_comm_apply, unfold_locales)
```
```  1367     (simp_all add: fun_left_idem)
```
```  1368
```
```  1369 lemma fun_left_comm_idem_insert:
```
```  1370   "fun_left_comm_idem insert"
```
```  1371 proof
```
```  1372 qed auto
```
```  1373
```
```  1374 lemma fun_left_comm_idem_remove:
```
```  1375   "fun_left_comm_idem (\<lambda>x A. A - {x})"
```
```  1376 proof
```
```  1377 qed auto
```
```  1378
```
```  1379 lemma (in semilattice_inf) fun_left_comm_idem_inf:
```
```  1380   "fun_left_comm_idem inf"
```
```  1381 proof
```
```  1382 qed (auto simp add: inf_left_commute)
```
```  1383
```
```  1384 lemma (in semilattice_sup) fun_left_comm_idem_sup:
```
```  1385   "fun_left_comm_idem sup"
```
```  1386 proof
```
```  1387 qed (auto simp add: sup_left_commute)
```
```  1388
```
```  1389 lemma union_fold_insert:
```
```  1390   assumes "finite A"
```
```  1391   shows "A \<union> B = fold insert B A"
```
```  1392 proof -
```
```  1393   interpret fun_left_comm_idem insert by (fact fun_left_comm_idem_insert)
```
```  1394   from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
```
```  1395 qed
```
```  1396
```
```  1397 lemma minus_fold_remove:
```
```  1398   assumes "finite A"
```
```  1399   shows "B - A = fold (\<lambda>x A. A - {x}) B A"
```
```  1400 proof -
```
```  1401   interpret fun_left_comm_idem "\<lambda>x A. A - {x}" by (fact fun_left_comm_idem_remove)
```
```  1402   from `finite A` show ?thesis by (induct A arbitrary: B) auto
```
```  1403 qed
```
```  1404
```
```  1405 context complete_lattice
```
```  1406 begin
```
```  1407
```
```  1408 lemma inf_Inf_fold_inf:
```
```  1409   assumes "finite A"
```
```  1410   shows "inf B (Inf A) = fold inf B A"
```
```  1411 proof -
```
```  1412   interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
```
```  1413   from `finite A` show ?thesis by (induct A arbitrary: B)
```
```  1414     (simp_all add: Inf_empty Inf_insert inf_commute fold_fun_comm)
```
```  1415 qed
```
```  1416
```
```  1417 lemma sup_Sup_fold_sup:
```
```  1418   assumes "finite A"
```
```  1419   shows "sup B (Sup A) = fold sup B A"
```
```  1420 proof -
```
```  1421   interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
```
```  1422   from `finite A` show ?thesis by (induct A arbitrary: B)
```
```  1423     (simp_all add: Sup_empty Sup_insert sup_commute fold_fun_comm)
```
```  1424 qed
```
```  1425
```
```  1426 lemma Inf_fold_inf:
```
```  1427   assumes "finite A"
```
```  1428   shows "Inf A = fold inf top A"
```
```  1429   using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
```
```  1430
```
```  1431 lemma Sup_fold_sup:
```
```  1432   assumes "finite A"
```
```  1433   shows "Sup A = fold sup bot A"
```
```  1434   using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
```
```  1435
```
```  1436 lemma inf_INFI_fold_inf:
```
```  1437   assumes "finite A"
```
```  1438   shows "inf B (INFI A f) = fold (\<lambda>A. inf (f A)) B A" (is "?inf = ?fold")
```
```  1439 proof (rule sym)
```
```  1440   interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
```
```  1441   interpret fun_left_comm_idem "\<lambda>A. inf (f A)" by (fact fun_left_comm_idem_apply)
```
```  1442   from `finite A` show "?fold = ?inf"
```
```  1443   by (induct A arbitrary: B)
```
```  1444     (simp_all add: INFI_def Inf_empty Inf_insert inf_left_commute)
```
```  1445 qed
```
```  1446
```
```  1447 lemma sup_SUPR_fold_sup:
```
```  1448   assumes "finite A"
```
```  1449   shows "sup B (SUPR A f) = fold (\<lambda>A. sup (f A)) B A" (is "?sup = ?fold")
```
```  1450 proof (rule sym)
```
```  1451   interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
```
```  1452   interpret fun_left_comm_idem "\<lambda>A. sup (f A)" by (fact fun_left_comm_idem_apply)
```
```  1453   from `finite A` show "?fold = ?sup"
```
```  1454   by (induct A arbitrary: B)
```
```  1455     (simp_all add: SUPR_def Sup_empty Sup_insert sup_left_commute)
```
```  1456 qed
```
```  1457
```
```  1458 lemma INFI_fold_inf:
```
```  1459   assumes "finite A"
```
```  1460   shows "INFI A f = fold (\<lambda>A. inf (f A)) top A"
```
```  1461   using assms inf_INFI_fold_inf [of A top] by simp
```
```  1462
```
```  1463 lemma SUPR_fold_sup:
```
```  1464   assumes "finite A"
```
```  1465   shows "SUPR A f = fold (\<lambda>A. sup (f A)) bot A"
```
```  1466   using assms sup_SUPR_fold_sup [of A bot] by simp
```
```  1467
```
```  1468 end
```
```  1469
```
```  1470
```
```  1471 subsection {* Locales as mini-packages *}
```
```  1472
```
```  1473 locale folding =
```
```  1474   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1475   fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
```
```  1476   assumes commute_comp: "f x \<circ> f y = f y \<circ> f x"
```
```  1477   assumes eq_fold: "F A s = Finite_Set.fold f s A"
```
```  1478 begin
```
```  1479
```
```  1480 lemma fun_left_commute:
```
```  1481   "f x (f y s) = f y (f x s)"
```
```  1482   using commute_comp [of x y] by (simp add: expand_fun_eq)
```
```  1483
```
```  1484 lemma fun_left_comm:
```
```  1485   "fun_left_comm f"
```
```  1486 proof
```
```  1487 qed (fact fun_left_commute)
```
```  1488
```
```  1489 lemma empty [simp]:
```
```  1490   "F {} = id"
```
```  1491   by (simp add: eq_fold expand_fun_eq)
```
```  1492
```
```  1493 lemma insert [simp]:
```
```  1494   assumes "finite A" and "x \<notin> A"
```
```  1495   shows "F (insert x A) = F A \<circ> f x"
```
```  1496 proof -
```
```  1497   interpret fun_left_comm f by (fact fun_left_comm)
```
```  1498   from fold_insert2 assms
```
```  1499   have "\<And>s. Finite_Set.fold f s (insert x A) = Finite_Set.fold f (f x s) A" .
```
```  1500   then show ?thesis by (simp add: eq_fold expand_fun_eq)
```
```  1501 qed
```
```  1502
```
```  1503 lemma remove:
```
```  1504   assumes "finite A" and "x \<in> A"
```
```  1505   shows "F A = F (A - {x}) \<circ> f x"
```
```  1506 proof -
```
```  1507   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
```
```  1508     by (auto dest: mk_disjoint_insert)
```
```  1509   moreover from `finite A` this have "finite B" by simp
```
```  1510   ultimately show ?thesis by simp
```
```  1511 qed
```
```  1512
```
```  1513 lemma insert_remove:
```
```  1514   assumes "finite A"
```
```  1515   shows "F (insert x A) = F (A - {x}) \<circ> f x"
```
```  1516 proof (cases "x \<in> A")
```
```  1517   case True with assms show ?thesis by (simp add: remove insert_absorb)
```
```  1518 next
```
```  1519   case False with assms show ?thesis by simp
```
```  1520 qed
```
```  1521
```
```  1522 lemma commute_comp':
```
```  1523   assumes "finite A"
```
```  1524   shows "f x \<circ> F A = F A \<circ> f x"
```
```  1525 proof (rule ext)
```
```  1526   fix s
```
```  1527   from assms show "(f x \<circ> F A) s = (F A \<circ> f x) s"
```
```  1528     by (induct A arbitrary: s) (simp_all add: fun_left_commute)
```
```  1529 qed
```
```  1530
```
```  1531 lemma fun_left_commute':
```
```  1532   assumes "finite A"
```
```  1533   shows "f x (F A s) = F A (f x s)"
```
```  1534   using commute_comp' assms by (simp add: expand_fun_eq)
```
```  1535
```
```  1536 lemma union:
```
```  1537   assumes "finite A" and "finite B"
```
```  1538   and "A \<inter> B = {}"
```
```  1539   shows "F (A \<union> B) = F A \<circ> F B"
```
```  1540 using `finite A` `A \<inter> B = {}` proof (induct A)
```
```  1541   case empty show ?case by simp
```
```  1542 next
```
```  1543   case (insert x A)
```
```  1544   then have "A \<inter> B = {}" by auto
```
```  1545   with insert(3) have "F (A \<union> B) = F A \<circ> F B" .
```
```  1546   moreover from insert have "x \<notin> B" by simp
```
```  1547   moreover from `finite A` `finite B` have fin: "finite (A \<union> B)" by simp
```
```  1548   moreover from `x \<notin> A` `x \<notin> B` have "x \<notin> A \<union> B" by simp
```
```  1549   ultimately show ?case by (simp add: fun_left_commute')
```
```  1550 qed
```
```  1551
```
```  1552 end
```
```  1553
```
```  1554 locale folding_idem = folding +
```
```  1555   assumes idem_comp: "f x \<circ> f x = f x"
```
```  1556 begin
```
```  1557
```
```  1558 declare insert [simp del]
```
```  1559
```
```  1560 lemma fun_idem:
```
```  1561   "f x (f x s) = f x s"
```
```  1562   using idem_comp [of x] by (simp add: expand_fun_eq)
```
```  1563
```
```  1564 lemma fun_left_comm_idem:
```
```  1565   "fun_left_comm_idem f"
```
```  1566 proof
```
```  1567 qed (fact fun_left_commute fun_idem)+
```
```  1568
```
```  1569 lemma insert_idem [simp]:
```
```  1570   assumes "finite A"
```
```  1571   shows "F (insert x A) = F A \<circ> f x"
```
```  1572 proof -
```
```  1573   interpret fun_left_comm_idem f by (fact fun_left_comm_idem)
```
```  1574   from fold_insert_idem2 assms
```
```  1575   have "\<And>s. Finite_Set.fold f s (insert x A) = Finite_Set.fold f (f x s) A" .
```
```  1576   then show ?thesis by (simp add: eq_fold expand_fun_eq)
```
```  1577 qed
```
```  1578
```
```  1579 lemma union_idem:
```
```  1580   assumes "finite A" and "finite B"
```
```  1581   shows "F (A \<union> B) = F A \<circ> F B"
```
```  1582 using `finite A` proof (induct A)
```
```  1583   case empty show ?case by simp
```
```  1584 next
```
```  1585   case (insert x A)
```
```  1586   from insert(3) have "F (A \<union> B) = F A \<circ> F B" .
```
```  1587   moreover from `finite A` `finite B` have fin: "finite (A \<union> B)" by simp
```
```  1588   ultimately show ?case by (simp add: fun_left_commute')
```
```  1589 qed
```
```  1590
```
```  1591 end
```
```  1592
```
```  1593 end
```