src/HOL/Big_Operators.thy
 author haftmann Fri Jun 11 17:14:02 2010 +0200 (2010-06-11) changeset 37407 61dd8c145da7 parent 36977 71c8973a604b child 39198 f967a16dfcdd permissions -rw-r--r--
declare lex_prod_def [code del]
```     1 (*  Title:      HOL/Big_Operators.thy
```
```     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
```
```     3                 with contributions by Jeremy Avigad
```
```     4 *)
```
```     5
```
```     6 header {* Big operators and finite (non-empty) sets *}
```
```     7
```
```     8 theory Big_Operators
```
```     9 imports Plain
```
```    10 begin
```
```    11
```
```    12 subsection {* Generic monoid operation over a set *}
```
```    13
```
```    14 no_notation times (infixl "*" 70)
```
```    15 no_notation Groups.one ("1")
```
```    16
```
```    17 locale comm_monoid_big = comm_monoid +
```
```    18   fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
```
```    19   assumes F_eq: "F g A = (if finite A then fold_image (op *) g 1 A else 1)"
```
```    20
```
```    21 sublocale comm_monoid_big < folding_image proof
```
```    22 qed (simp add: F_eq)
```
```    23
```
```    24 context comm_monoid_big
```
```    25 begin
```
```    26
```
```    27 lemma infinite [simp]:
```
```    28   "\<not> finite A \<Longrightarrow> F g A = 1"
```
```    29   by (simp add: F_eq)
```
```    30
```
```    31 end
```
```    32
```
```    33 text {* for ad-hoc proofs for @{const fold_image} *}
```
```    34
```
```    35 lemma (in comm_monoid_add) comm_monoid_mult:
```
```    36   "class.comm_monoid_mult (op +) 0"
```
```    37 proof qed (auto intro: add_assoc add_commute)
```
```    38
```
```    39 notation times (infixl "*" 70)
```
```    40 notation Groups.one ("1")
```
```    41
```
```    42
```
```    43 subsection {* Generalized summation over a set *}
```
```    44
```
```    45 definition (in comm_monoid_add) setsum :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
```
```    46   "setsum f A = (if finite A then fold_image (op +) f 0 A else 0)"
```
```    47
```
```    48 sublocale comm_monoid_add < setsum!: comm_monoid_big "op +" 0 setsum proof
```
```    49 qed (fact setsum_def)
```
```    50
```
```    51 abbreviation
```
```    52   Setsum  ("\<Sum>_" [1000] 999) where
```
```    53   "\<Sum>A == setsum (%x. x) A"
```
```    54
```
```    55 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```    56 written @{text"\<Sum>x\<in>A. e"}. *}
```
```    57
```
```    58 syntax
```
```    59   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```    60 syntax (xsymbols)
```
```    61   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```    62 syntax (HTML output)
```
```    63   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```    64
```
```    65 translations -- {* Beware of argument permutation! *}
```
```    66   "SUM i:A. b" == "CONST setsum (%i. b) A"
```
```    67   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
```
```    68
```
```    69 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```    70  @{text"\<Sum>x|P. e"}. *}
```
```    71
```
```    72 syntax
```
```    73   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```    74 syntax (xsymbols)
```
```    75   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```    76 syntax (HTML output)
```
```    77   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```    78
```
```    79 translations
```
```    80   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```    81   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```    82
```
```    83 print_translation {*
```
```    84 let
```
```    85   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```    86         if x <> y then raise Match
```
```    87         else
```
```    88           let
```
```    89             val x' = Syntax.mark_bound x;
```
```    90             val t' = subst_bound (x', t);
```
```    91             val P' = subst_bound (x', P);
```
```    92           in Syntax.const @{syntax_const "_qsetsum"} \$ Syntax.mark_bound x \$ P' \$ t' end
```
```    93     | setsum_tr' _ = raise Match;
```
```    94 in [(@{const_syntax setsum}, setsum_tr')] end
```
```    95 *}
```
```    96
```
```    97 lemma setsum_empty:
```
```    98   "setsum f {} = 0"
```
```    99   by (fact setsum.empty)
```
```   100
```
```   101 lemma setsum_insert:
```
```   102   "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   103   by (fact setsum.insert)
```
```   104
```
```   105 lemma setsum_infinite:
```
```   106   "~ finite A ==> setsum f A = 0"
```
```   107   by (fact setsum.infinite)
```
```   108
```
```   109 lemma (in comm_monoid_add) setsum_reindex:
```
```   110   assumes "inj_on f B" shows "setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   111 proof -
```
```   112   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   113   from assms show ?thesis by (auto simp add: setsum_def fold_image_reindex dest!:finite_imageD)
```
```   114 qed
```
```   115
```
```   116 lemma (in comm_monoid_add) setsum_reindex_id:
```
```   117   "inj_on f B ==> setsum f B = setsum id (f ` B)"
```
```   118   by (simp add: setsum_reindex)
```
```   119
```
```   120 lemma (in comm_monoid_add) setsum_reindex_nonzero:
```
```   121   assumes fS: "finite S"
```
```   122   and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
```
```   123   shows "setsum h (f ` S) = setsum (h o f) S"
```
```   124 using nz
```
```   125 proof(induct rule: finite_induct[OF fS])
```
```   126   case 1 thus ?case by simp
```
```   127 next
```
```   128   case (2 x F)
```
```   129   {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
```
```   130     then obtain y where y: "y \<in> F" "f x = f y" by auto
```
```   131     from "2.hyps" y have xy: "x \<noteq> y" by auto
```
```   132
```
```   133     from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
```
```   134     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
```
```   135     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   136       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   137       using h0
```
```   138       apply simp
```
```   139       apply (rule "2.hyps"(3))
```
```   140       apply (rule_tac y="y" in  "2.prems")
```
```   141       apply simp_all
```
```   142       done
```
```   143     finally have ?case .}
```
```   144   moreover
```
```   145   {assume fxF: "f x \<notin> f ` F"
```
```   146     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)"
```
```   147       using fxF "2.hyps" by simp
```
```   148     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   149       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   150       apply simp
```
```   151       apply (rule cong [OF refl [of "op + (h (f x))"]])
```
```   152       apply (rule "2.hyps"(3))
```
```   153       apply (rule_tac y="y" in  "2.prems")
```
```   154       apply simp_all
```
```   155       done
```
```   156     finally have ?case .}
```
```   157   ultimately show ?case by blast
```
```   158 qed
```
```   159
```
```   160 lemma (in comm_monoid_add) setsum_cong:
```
```   161   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   162   by (cases "finite A") (auto intro: setsum.cong)
```
```   163
```
```   164 lemma (in comm_monoid_add) strong_setsum_cong [cong]:
```
```   165   "A = B ==> (!!x. x:B =simp=> f x = g x)
```
```   166    ==> setsum (%x. f x) A = setsum (%x. g x) B"
```
```   167   by (rule setsum_cong) (simp_all add: simp_implies_def)
```
```   168
```
```   169 lemma (in comm_monoid_add) setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"
```
```   170   by (auto intro: setsum_cong)
```
```   171
```
```   172 lemma (in comm_monoid_add) setsum_reindex_cong:
```
```   173    "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
```
```   174     ==> setsum h B = setsum g A"
```
```   175   by (simp add: setsum_reindex cong: setsum_cong)
```
```   176
```
```   177 lemma (in comm_monoid_add) setsum_0[simp]: "setsum (%i. 0) A = 0"
```
```   178   by (cases "finite A") (erule finite_induct, auto)
```
```   179
```
```   180 lemma (in comm_monoid_add) setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
```
```   181   by (simp add:setsum_cong)
```
```   182
```
```   183 lemma (in comm_monoid_add) setsum_Un_Int: "finite A ==> finite B ==>
```
```   184   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   185   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   186   by (fact setsum.union_inter)
```
```   187
```
```   188 lemma (in comm_monoid_add) setsum_Un_disjoint: "finite A ==> finite B
```
```   189   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   190   by (fact setsum.union_disjoint)
```
```   191
```
```   192 lemma setsum_mono_zero_left:
```
```   193   assumes fT: "finite T" and ST: "S \<subseteq> T"
```
```   194   and z: "\<forall>i \<in> T - S. f i = 0"
```
```   195   shows "setsum f S = setsum f T"
```
```   196 proof-
```
```   197   have eq: "T = S \<union> (T - S)" using ST by blast
```
```   198   have d: "S \<inter> (T - S) = {}" using ST by blast
```
```   199   from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
```
```   200   show ?thesis
```
```   201   by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
```
```   202 qed
```
```   203
```
```   204 lemma setsum_mono_zero_right:
```
```   205   "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. f i = 0 \<Longrightarrow> setsum f T = setsum f S"
```
```   206 by(blast intro!: setsum_mono_zero_left[symmetric])
```
```   207
```
```   208 lemma setsum_mono_zero_cong_left:
```
```   209   assumes fT: "finite T" and ST: "S \<subseteq> T"
```
```   210   and z: "\<forall>i \<in> T - S. g i = 0"
```
```   211   and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
```
```   212   shows "setsum f S = setsum g T"
```
```   213 proof-
```
```   214   have eq: "T = S \<union> (T - S)" using ST by blast
```
```   215   have d: "S \<inter> (T - S) = {}" using ST by blast
```
```   216   from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
```
```   217   show ?thesis
```
```   218     using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
```
```   219 qed
```
```   220
```
```   221 lemma setsum_mono_zero_cong_right:
```
```   222   assumes fT: "finite T" and ST: "S \<subseteq> T"
```
```   223   and z: "\<forall>i \<in> T - S. f i = 0"
```
```   224   and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
```
```   225   shows "setsum f T = setsum g S"
```
```   226 using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto
```
```   227
```
```   228 lemma setsum_delta:
```
```   229   assumes fS: "finite S"
```
```   230   shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
```
```   231 proof-
```
```   232   let ?f = "(\<lambda>k. if k=a then b k else 0)"
```
```   233   {assume a: "a \<notin> S"
```
```   234     hence "\<forall> k\<in> S. ?f k = 0" by simp
```
```   235     hence ?thesis  using a by simp}
```
```   236   moreover
```
```   237   {assume a: "a \<in> S"
```
```   238     let ?A = "S - {a}"
```
```   239     let ?B = "{a}"
```
```   240     have eq: "S = ?A \<union> ?B" using a by blast
```
```   241     have dj: "?A \<inter> ?B = {}" by simp
```
```   242     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   243     have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
```
```   244       using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```   245       by simp
```
```   246     then have ?thesis  using a by simp}
```
```   247   ultimately show ?thesis by blast
```
```   248 qed
```
```   249 lemma setsum_delta':
```
```   250   assumes fS: "finite S" shows
```
```   251   "setsum (\<lambda>k. if a = k then b k else 0) S =
```
```   252      (if a\<in> S then b a else 0)"
```
```   253   using setsum_delta[OF fS, of a b, symmetric]
```
```   254   by (auto intro: setsum_cong)
```
```   255
```
```   256 lemma setsum_restrict_set:
```
```   257   assumes fA: "finite A"
```
```   258   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
```
```   259 proof-
```
```   260   from fA have fab: "finite (A \<inter> B)" by auto
```
```   261   have aba: "A \<inter> B \<subseteq> A" by blast
```
```   262   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
```
```   263   from setsum_mono_zero_left[OF fA aba, of ?g]
```
```   264   show ?thesis by simp
```
```   265 qed
```
```   266
```
```   267 lemma setsum_cases:
```
```   268   assumes fA: "finite A"
```
```   269   shows "setsum (\<lambda>x. if P x then f x else g x) A =
```
```   270          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
```
```   271 proof-
```
```   272   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
```
```   273           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
```
```   274     by blast+
```
```   275   from fA
```
```   276   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
```
```   277   let ?g = "\<lambda>x. if P x then f x else g x"
```
```   278   from setsum_Un_disjoint[OF f a(2), of ?g] a(1)
```
```   279   show ?thesis by simp
```
```   280 qed
```
```   281
```
```   282
```
```   283 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   284   the lhs need not be, since UNION I A could still be finite.*)
```
```   285 lemma (in comm_monoid_add) setsum_UN_disjoint:
```
```   286   assumes "finite I" and "ALL i:I. finite (A i)"
```
```   287     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
```
```   288   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   289 proof -
```
```   290   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   291   from assms show ?thesis by (simp add: setsum_def fold_image_UN_disjoint cong: setsum_cong)
```
```   292 qed
```
```   293
```
```   294 text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
```
```   295 directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
```
```   296 lemma setsum_Union_disjoint:
```
```   297   "[| (ALL A:C. finite A);
```
```   298       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```   299    ==> setsum f (Union C) = setsum (setsum f) C"
```
```   300 apply (cases "finite C")
```
```   301  prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
```
```   302   apply (frule setsum_UN_disjoint [of C id f])
```
```   303  apply (unfold Union_def id_def, assumption+)
```
```   304 done
```
```   305
```
```   306 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```   307   the rhs need not be, since SIGMA A B could still be finite.*)
```
```   308 lemma (in comm_monoid_add) setsum_Sigma:
```
```   309   assumes "finite A" and  "ALL x:A. finite (B x)"
```
```   310   shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```   311 proof -
```
```   312   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   313   from assms show ?thesis by (simp add: setsum_def fold_image_Sigma split_def cong: setsum_cong)
```
```   314 qed
```
```   315
```
```   316 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```   317 lemma setsum_cartesian_product:
```
```   318    "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
```
```   319 apply (cases "finite A")
```
```   320  apply (cases "finite B")
```
```   321   apply (simp add: setsum_Sigma)
```
```   322  apply (cases "A={}", simp)
```
```   323  apply (simp)
```
```   324 apply (auto simp add: setsum_def
```
```   325             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   326 done
```
```   327
```
```   328 lemma (in comm_monoid_add) setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```   329   by (cases "finite A") (simp_all add: setsum.distrib)
```
```   330
```
```   331
```
```   332 subsubsection {* Properties in more restricted classes of structures *}
```
```   333
```
```   334 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   335 apply (case_tac "finite A")
```
```   336  prefer 2 apply (simp add: setsum_def)
```
```   337 apply (erule rev_mp)
```
```   338 apply (erule finite_induct, auto)
```
```   339 done
```
```   340
```
```   341 lemma setsum_eq_0_iff [simp]:
```
```   342     "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   343 by (induct set: finite) auto
```
```   344
```
```   345 lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
```
```   346   (setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
```
```   347 apply(erule finite_induct)
```
```   348 apply (auto simp add:add_is_1)
```
```   349 done
```
```   350
```
```   351 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   352
```
```   353 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   354   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   355   -- {* For the natural numbers, we have subtraction. *}
```
```   356 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   357
```
```   358 lemma setsum_Un: "finite A ==> finite B ==>
```
```   359   (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   360    setsum f A + setsum f B - setsum f (A Int B)"
```
```   361 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   362
```
```   363 lemma (in comm_monoid_add) setsum_eq_general_reverses:
```
```   364   assumes fS: "finite S" and fT: "finite T"
```
```   365   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```   366   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
```
```   367   shows "setsum f S = setsum g T"
```
```   368 proof -
```
```   369   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   370   show ?thesis
```
```   371   apply (simp add: setsum_def fS fT)
```
```   372   apply (rule fold_image_eq_general_inverses)
```
```   373   apply (rule fS)
```
```   374   apply (erule kh)
```
```   375   apply (erule hk)
```
```   376   done
```
```   377 qed
```
```   378
```
```   379 lemma (in comm_monoid_add) setsum_Un_zero:
```
```   380   assumes fS: "finite S" and fT: "finite T"
```
```   381   and I0: "\<forall>x \<in> S\<inter>T. f x = 0"
```
```   382   shows "setsum f (S \<union> T) = setsum f S  + setsum f T"
```
```   383 proof -
```
```   384   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   385   show ?thesis
```
```   386   using fS fT
```
```   387   apply (simp add: setsum_def)
```
```   388   apply (rule fold_image_Un_one)
```
```   389   using I0 by auto
```
```   390 qed
```
```   391
```
```   392 lemma setsum_UNION_zero:
```
```   393   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
```
```   394   and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
```
```   395   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
```
```   396   using fSS f0
```
```   397 proof(induct rule: finite_induct[OF fS])
```
```   398   case 1 thus ?case by simp
```
```   399 next
```
```   400   case (2 T F)
```
```   401   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F"
```
```   402     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
```
```   403   from fTF have fUF: "finite (\<Union>F)" by auto
```
```   404   from "2.prems" TF fTF
```
```   405   show ?case
```
```   406     by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f])
```
```   407 qed
```
```   408
```
```   409 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   410   (if a:A then setsum f A - f a else setsum f A)"
```
```   411 apply (case_tac "finite A")
```
```   412  prefer 2 apply (simp add: setsum_def)
```
```   413 apply (erule finite_induct)
```
```   414  apply (auto simp add: insert_Diff_if)
```
```   415 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   416 done
```
```   417
```
```   418 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   419   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   420   (if a:A then setsum f A - f a else setsum f A)"
```
```   421 by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   422
```
```   423 lemma setsum_diff1'[rule_format]:
```
```   424   "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
```
```   425 apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
```
```   426 apply (auto simp add: insert_Diff_if add_ac)
```
```   427 done
```
```   428
```
```   429 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
```
```   430   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
```
```   431 unfolding setsum_diff1'[OF assms] by auto
```
```   432
```
```   433 (* By Jeremy Siek: *)
```
```   434
```
```   435 lemma setsum_diff_nat:
```
```   436 assumes "finite B" and "B \<subseteq> A"
```
```   437 shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```   438 using assms
```
```   439 proof induct
```
```   440   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```   441 next
```
```   442   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```   443     and xFinA: "insert x F \<subseteq> A"
```
```   444     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```   445   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```   446   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```   447     by (simp add: setsum_diff1_nat)
```
```   448   from xFinA have "F \<subseteq> A" by simp
```
```   449   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```   450   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```   451     by simp
```
```   452   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```   453   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```   454     by simp
```
```   455   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```   456   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```   457     by simp
```
```   458   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```   459 qed
```
```   460
```
```   461 lemma setsum_diff:
```
```   462   assumes le: "finite A" "B \<subseteq> A"
```
```   463   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```   464 proof -
```
```   465   from le have finiteB: "finite B" using finite_subset by auto
```
```   466   show ?thesis using finiteB le
```
```   467   proof induct
```
```   468     case empty
```
```   469     thus ?case by auto
```
```   470   next
```
```   471     case (insert x F)
```
```   472     thus ?case using le finiteB
```
```   473       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```   474   qed
```
```   475 qed
```
```   476
```
```   477 lemma setsum_mono:
```
```   478   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
```
```   479   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   480 proof (cases "finite K")
```
```   481   case True
```
```   482   thus ?thesis using le
```
```   483   proof induct
```
```   484     case empty
```
```   485     thus ?case by simp
```
```   486   next
```
```   487     case insert
```
```   488     thus ?case using add_mono by fastsimp
```
```   489   qed
```
```   490 next
```
```   491   case False
```
```   492   thus ?thesis
```
```   493     by (simp add: setsum_def)
```
```   494 qed
```
```   495
```
```   496 lemma setsum_strict_mono:
```
```   497   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```   498   assumes "finite A"  "A \<noteq> {}"
```
```   499     and "!!x. x:A \<Longrightarrow> f x < g x"
```
```   500   shows "setsum f A < setsum g A"
```
```   501   using prems
```
```   502 proof (induct rule: finite_ne_induct)
```
```   503   case singleton thus ?case by simp
```
```   504 next
```
```   505   case insert thus ?case by (auto simp: add_strict_mono)
```
```   506 qed
```
```   507
```
```   508 lemma setsum_negf:
```
```   509   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```   510 proof (cases "finite A")
```
```   511   case True thus ?thesis by (induct set: finite) auto
```
```   512 next
```
```   513   case False thus ?thesis by (simp add: setsum_def)
```
```   514 qed
```
```   515
```
```   516 lemma setsum_subtractf:
```
```   517   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```   518     setsum f A - setsum g A"
```
```   519 proof (cases "finite A")
```
```   520   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
```
```   521 next
```
```   522   case False thus ?thesis by (simp add: setsum_def)
```
```   523 qed
```
```   524
```
```   525 lemma setsum_nonneg:
```
```   526   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```   527   shows "0 \<le> setsum f A"
```
```   528 proof (cases "finite A")
```
```   529   case True thus ?thesis using nn
```
```   530   proof induct
```
```   531     case empty then show ?case by simp
```
```   532   next
```
```   533     case (insert x F)
```
```   534     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   535     with insert show ?case by simp
```
```   536   qed
```
```   537 next
```
```   538   case False thus ?thesis by (simp add: setsum_def)
```
```   539 qed
```
```   540
```
```   541 lemma setsum_nonpos:
```
```   542   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
```
```   543   shows "setsum f A \<le> 0"
```
```   544 proof (cases "finite A")
```
```   545   case True thus ?thesis using np
```
```   546   proof induct
```
```   547     case empty then show ?case by simp
```
```   548   next
```
```   549     case (insert x F)
```
```   550     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   551     with insert show ?case by simp
```
```   552   qed
```
```   553 next
```
```   554   case False thus ?thesis by (simp add: setsum_def)
```
```   555 qed
```
```   556
```
```   557 lemma setsum_nonneg_leq_bound:
```
```   558   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   559   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   560   shows "f i \<le> B"
```
```   561 proof -
```
```   562   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
```
```   563     using assms by (auto intro!: setsum_nonneg)
```
```   564   moreover
```
```   565   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
```
```   566     using assms by (simp add: setsum_diff1)
```
```   567   ultimately show ?thesis by auto
```
```   568 qed
```
```   569
```
```   570 lemma setsum_nonneg_0:
```
```   571   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   572   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
```
```   573   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
```
```   574   shows "f i = 0"
```
```   575   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
```
```   576
```
```   577 lemma setsum_mono2:
```
```   578 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
```
```   579 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   580 shows "setsum f A \<le> setsum f B"
```
```   581 proof -
```
```   582   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   583     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   584   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```   585     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```   586   also have "A \<union> (B-A) = B" using sub by blast
```
```   587   finally show ?thesis .
```
```   588 qed
```
```   589
```
```   590 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```   591     ALL x: B - A.
```
```   592       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
```
```   593         setsum f A <= setsum f B"
```
```   594   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```   595   apply (erule ssubst)
```
```   596   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```   597   apply simp
```
```   598   apply (rule add_left_mono)
```
```   599   apply (erule setsum_nonneg)
```
```   600   apply (subst setsum_Un_disjoint [THEN sym])
```
```   601   apply (erule finite_subset, assumption)
```
```   602   apply (rule finite_subset)
```
```   603   prefer 2
```
```   604   apply assumption
```
```   605   apply (auto simp add: sup_absorb2)
```
```   606 done
```
```   607
```
```   608 lemma setsum_right_distrib:
```
```   609   fixes f :: "'a => ('b::semiring_0)"
```
```   610   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```   611 proof (cases "finite A")
```
```   612   case True
```
```   613   thus ?thesis
```
```   614   proof induct
```
```   615     case empty thus ?case by simp
```
```   616   next
```
```   617     case (insert x A) thus ?case by (simp add: right_distrib)
```
```   618   qed
```
```   619 next
```
```   620   case False thus ?thesis by (simp add: setsum_def)
```
```   621 qed
```
```   622
```
```   623 lemma setsum_left_distrib:
```
```   624   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```   625 proof (cases "finite A")
```
```   626   case True
```
```   627   then show ?thesis
```
```   628   proof induct
```
```   629     case empty thus ?case by simp
```
```   630   next
```
```   631     case (insert x A) thus ?case by (simp add: left_distrib)
```
```   632   qed
```
```   633 next
```
```   634   case False thus ?thesis by (simp add: setsum_def)
```
```   635 qed
```
```   636
```
```   637 lemma setsum_divide_distrib:
```
```   638   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```   639 proof (cases "finite A")
```
```   640   case True
```
```   641   then show ?thesis
```
```   642   proof induct
```
```   643     case empty thus ?case by simp
```
```   644   next
```
```   645     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```   646   qed
```
```   647 next
```
```   648   case False thus ?thesis by (simp add: setsum_def)
```
```   649 qed
```
```   650
```
```   651 lemma setsum_abs[iff]:
```
```   652   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   653   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```   654 proof (cases "finite A")
```
```   655   case True
```
```   656   thus ?thesis
```
```   657   proof induct
```
```   658     case empty thus ?case by simp
```
```   659   next
```
```   660     case (insert x A)
```
```   661     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   662   qed
```
```   663 next
```
```   664   case False thus ?thesis by (simp add: setsum_def)
```
```   665 qed
```
```   666
```
```   667 lemma setsum_abs_ge_zero[iff]:
```
```   668   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   669   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```   670 proof (cases "finite A")
```
```   671   case True
```
```   672   thus ?thesis
```
```   673   proof induct
```
```   674     case empty thus ?case by simp
```
```   675   next
```
```   676     case (insert x A) thus ?case by auto
```
```   677   qed
```
```   678 next
```
```   679   case False thus ?thesis by (simp add: setsum_def)
```
```   680 qed
```
```   681
```
```   682 lemma abs_setsum_abs[simp]:
```
```   683   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   684   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```   685 proof (cases "finite A")
```
```   686   case True
```
```   687   thus ?thesis
```
```   688   proof induct
```
```   689     case empty thus ?case by simp
```
```   690   next
```
```   691     case (insert a A)
```
```   692     hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
```
```   693     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```   694     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```   695       by (simp del: abs_of_nonneg)
```
```   696     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```   697     finally show ?case .
```
```   698   qed
```
```   699 next
```
```   700   case False thus ?thesis by (simp add: setsum_def)
```
```   701 qed
```
```   702
```
```   703 lemma setsum_Plus:
```
```   704   fixes A :: "'a set" and B :: "'b set"
```
```   705   assumes fin: "finite A" "finite B"
```
```   706   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
```
```   707 proof -
```
```   708   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   709   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
```
```   710     by(auto intro: finite_imageI)
```
```   711   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
```
```   712   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
```
```   713   ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)
```
```   714 qed
```
```   715
```
```   716
```
```   717 text {* Commuting outer and inner summation *}
```
```   718
```
```   719 lemma setsum_commute:
```
```   720   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
```
```   721 proof (simp add: setsum_cartesian_product)
```
```   722   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
```
```   723     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
```
```   724     (is "?s = _")
```
```   725     apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
```
```   726     apply (simp add: split_def)
```
```   727     done
```
```   728   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
```
```   729     (is "_ = ?t")
```
```   730     apply (simp add: swap_product)
```
```   731     done
```
```   732   finally show "?s = ?t" .
```
```   733 qed
```
```   734
```
```   735 lemma setsum_product:
```
```   736   fixes f :: "'a => ('b::semiring_0)"
```
```   737   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   738   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
```
```   739
```
```   740 lemma setsum_mult_setsum_if_inj:
```
```   741 fixes f :: "'a => ('b::semiring_0)"
```
```   742 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
```
```   743   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
```
```   744 by(auto simp: setsum_product setsum_cartesian_product
```
```   745         intro!:  setsum_reindex_cong[symmetric])
```
```   746
```
```   747 lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
```
```   748 apply (cases "finite A")
```
```   749 apply (erule finite_induct)
```
```   750 apply (auto simp add: algebra_simps)
```
```   751 done
```
```   752
```
```   753 lemma setsum_bounded:
```
```   754   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
```
```   755   shows "setsum f A \<le> of_nat(card A) * K"
```
```   756 proof (cases "finite A")
```
```   757   case True
```
```   758   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```   759 next
```
```   760   case False thus ?thesis by (simp add: setsum_def)
```
```   761 qed
```
```   762
```
```   763
```
```   764 subsubsection {* Cardinality as special case of @{const setsum} *}
```
```   765
```
```   766 lemma card_eq_setsum:
```
```   767   "card A = setsum (\<lambda>x. 1) A"
```
```   768   by (simp only: card_def setsum_def)
```
```   769
```
```   770 lemma card_UN_disjoint:
```
```   771   "finite I ==> (ALL i:I. finite (A i)) ==>
```
```   772    (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {})
```
```   773    ==> card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```   774 apply (simp add: card_eq_setsum del: setsum_constant)
```
```   775 apply (subgoal_tac
```
```   776          "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
```
```   777 apply (simp add: setsum_UN_disjoint del: setsum_constant)
```
```   778 apply (simp cong: setsum_cong)
```
```   779 done
```
```   780
```
```   781 lemma card_Union_disjoint:
```
```   782   "finite C ==> (ALL A:C. finite A) ==>
```
```   783    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
```
```   784    ==> card (Union C) = setsum card C"
```
```   785 apply (frule card_UN_disjoint [of C id])
```
```   786 apply (unfold Union_def id_def, assumption+)
```
```   787 done
```
```   788
```
```   789 text{*The image of a finite set can be expressed using @{term fold_image}.*}
```
```   790 lemma image_eq_fold_image:
```
```   791   "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A"
```
```   792 proof (induct rule: finite_induct)
```
```   793   case empty then show ?case by simp
```
```   794 next
```
```   795   interpret ab_semigroup_mult "op Un"
```
```   796     proof qed auto
```
```   797   case insert
```
```   798   then show ?case by simp
```
```   799 qed
```
```   800
```
```   801 subsubsection {* Cardinality of products *}
```
```   802
```
```   803 lemma card_SigmaI [simp]:
```
```   804   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```   805   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```   806 by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
```
```   807
```
```   808 (*
```
```   809 lemma SigmaI_insert: "y \<notin> A ==>
```
```   810   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```   811   by auto
```
```   812 *)
```
```   813
```
```   814 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```   815   by (cases "finite A \<and> finite B")
```
```   816     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   817
```
```   818 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```   819 by (simp add: card_cartesian_product)
```
```   820
```
```   821
```
```   822 subsection {* Generalized product over a set *}
```
```   823
```
```   824 definition (in comm_monoid_mult) setprod :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
```
```   825   "setprod f A = (if finite A then fold_image (op *) f 1 A else 1)"
```
```   826
```
```   827 sublocale comm_monoid_mult < setprod!: comm_monoid_big "op *" 1 setprod proof
```
```   828 qed (fact setprod_def)
```
```   829
```
```   830 abbreviation
```
```   831   Setprod  ("\<Prod>_" [1000] 999) where
```
```   832   "\<Prod>A == setprod (%x. x) A"
```
```   833
```
```   834 syntax
```
```   835   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```   836 syntax (xsymbols)
```
```   837   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```   838 syntax (HTML output)
```
```   839   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```   840
```
```   841 translations -- {* Beware of argument permutation! *}
```
```   842   "PROD i:A. b" == "CONST setprod (%i. b) A"
```
```   843   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
```
```   844
```
```   845 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```   846  @{text"\<Prod>x|P. e"}. *}
```
```   847
```
```   848 syntax
```
```   849   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```   850 syntax (xsymbols)
```
```   851   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```   852 syntax (HTML output)
```
```   853   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```   854
```
```   855 translations
```
```   856   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```   857   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```   858
```
```   859 lemma setprod_empty: "setprod f {} = 1"
```
```   860   by (fact setprod.empty)
```
```   861
```
```   862 lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
```
```   863     setprod f (insert a A) = f a * setprod f A"
```
```   864   by (fact setprod.insert)
```
```   865
```
```   866 lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
```
```   867   by (fact setprod.infinite)
```
```   868
```
```   869 lemma setprod_reindex:
```
```   870    "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```   871 by(auto simp: setprod_def fold_image_reindex dest!:finite_imageD)
```
```   872
```
```   873 lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```   874 by (auto simp add: setprod_reindex)
```
```   875
```
```   876 lemma setprod_cong:
```
```   877   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```   878 by(fastsimp simp: setprod_def intro: fold_image_cong)
```
```   879
```
```   880 lemma strong_setprod_cong[cong]:
```
```   881   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
```
```   882 by(fastsimp simp: simp_implies_def setprod_def intro: fold_image_cong)
```
```   883
```
```   884 lemma setprod_reindex_cong: "inj_on f A ==>
```
```   885     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```   886 by (frule setprod_reindex, simp)
```
```   887
```
```   888 lemma strong_setprod_reindex_cong: assumes i: "inj_on f A"
```
```   889   and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
```
```   890   shows "setprod h B = setprod g A"
```
```   891 proof-
```
```   892     have "setprod h B = setprod (h o f) A"
```
```   893       by (simp add: B setprod_reindex[OF i, of h])
```
```   894     then show ?thesis apply simp
```
```   895       apply (rule setprod_cong)
```
```   896       apply simp
```
```   897       by (simp add: eq)
```
```   898 qed
```
```   899
```
```   900 lemma setprod_Un_one:
```
```   901   assumes fS: "finite S" and fT: "finite T"
```
```   902   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
```
```   903   shows "setprod f (S \<union> T) = setprod f S  * setprod f T"
```
```   904   using fS fT
```
```   905   apply (simp add: setprod_def)
```
```   906   apply (rule fold_image_Un_one)
```
```   907   using I0 by auto
```
```   908
```
```   909
```
```   910 lemma setprod_1: "setprod (%i. 1) A = 1"
```
```   911 apply (case_tac "finite A")
```
```   912 apply (erule finite_induct, auto simp add: mult_ac)
```
```   913 done
```
```   914
```
```   915 lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
```
```   916 apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
```
```   917 apply (erule ssubst, rule setprod_1)
```
```   918 apply (rule setprod_cong, auto)
```
```   919 done
```
```   920
```
```   921 lemma setprod_Un_Int: "finite A ==> finite B
```
```   922     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```   923 by(simp add: setprod_def fold_image_Un_Int[symmetric])
```
```   924
```
```   925 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```   926   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```   927 by (subst setprod_Un_Int [symmetric], auto)
```
```   928
```
```   929 lemma setprod_mono_one_left:
```
```   930   assumes fT: "finite T" and ST: "S \<subseteq> T"
```
```   931   and z: "\<forall>i \<in> T - S. f i = 1"
```
```   932   shows "setprod f S = setprod f T"
```
```   933 proof-
```
```   934   have eq: "T = S \<union> (T - S)" using ST by blast
```
```   935   have d: "S \<inter> (T - S) = {}" using ST by blast
```
```   936   from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
```
```   937   show ?thesis
```
```   938   by (simp add: setprod_Un_disjoint[OF f d, unfolded eq[symmetric]] setprod_1'[OF z])
```
```   939 qed
```
```   940
```
```   941 lemmas setprod_mono_one_right = setprod_mono_one_left [THEN sym]
```
```   942
```
```   943 lemma setprod_delta:
```
```   944   assumes fS: "finite S"
```
```   945   shows "setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   946 proof-
```
```   947   let ?f = "(\<lambda>k. if k=a then b k else 1)"
```
```   948   {assume a: "a \<notin> S"
```
```   949     hence "\<forall> k\<in> S. ?f k = 1" by simp
```
```   950     hence ?thesis  using a by (simp add: setprod_1 cong add: setprod_cong) }
```
```   951   moreover
```
```   952   {assume a: "a \<in> S"
```
```   953     let ?A = "S - {a}"
```
```   954     let ?B = "{a}"
```
```   955     have eq: "S = ?A \<union> ?B" using a by blast
```
```   956     have dj: "?A \<inter> ?B = {}" by simp
```
```   957     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   958     have fA1: "setprod ?f ?A = 1" apply (rule setprod_1') by auto
```
```   959     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```   960       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```   961       by simp
```
```   962     then have ?thesis  using a by (simp add: fA1 cong add: setprod_cong cong del: if_weak_cong)}
```
```   963   ultimately show ?thesis by blast
```
```   964 qed
```
```   965
```
```   966 lemma setprod_delta':
```
```   967   assumes fS: "finite S" shows
```
```   968   "setprod (\<lambda>k. if a = k then b k else 1) S =
```
```   969      (if a\<in> S then b a else 1)"
```
```   970   using setprod_delta[OF fS, of a b, symmetric]
```
```   971   by (auto intro: setprod_cong)
```
```   972
```
```   973
```
```   974 lemma setprod_UN_disjoint:
```
```   975     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```   976         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```   977       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```   978 by(simp add: setprod_def fold_image_UN_disjoint cong: setprod_cong)
```
```   979
```
```   980 lemma setprod_Union_disjoint:
```
```   981   "[| (ALL A:C. finite A);
```
```   982       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```   983    ==> setprod f (Union C) = setprod (setprod f) C"
```
```   984 apply (cases "finite C")
```
```   985  prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
```
```   986   apply (frule setprod_UN_disjoint [of C id f])
```
```   987  apply (unfold Union_def id_def, assumption+)
```
```   988 done
```
```   989
```
```   990 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```   991     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
```
```   992     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```   993 by(simp add:setprod_def fold_image_Sigma split_def cong:setprod_cong)
```
```   994
```
```   995 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```   996 lemma setprod_cartesian_product:
```
```   997      "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
```
```   998 apply (cases "finite A")
```
```   999  apply (cases "finite B")
```
```  1000   apply (simp add: setprod_Sigma)
```
```  1001  apply (cases "A={}", simp)
```
```  1002  apply (simp add: setprod_1)
```
```  1003 apply (auto simp add: setprod_def
```
```  1004             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1005 done
```
```  1006
```
```  1007 lemma setprod_timesf:
```
```  1008      "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
```
```  1009 by(simp add:setprod_def fold_image_distrib)
```
```  1010
```
```  1011
```
```  1012 subsubsection {* Properties in more restricted classes of structures *}
```
```  1013
```
```  1014 lemma setprod_eq_1_iff [simp]:
```
```  1015   "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
```
```  1016 by (induct set: finite) auto
```
```  1017
```
```  1018 lemma setprod_zero:
```
```  1019      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
```
```  1020 apply (induct set: finite, force, clarsimp)
```
```  1021 apply (erule disjE, auto)
```
```  1022 done
```
```  1023
```
```  1024 lemma setprod_nonneg [rule_format]:
```
```  1025    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1026 by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
```
```  1027
```
```  1028 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
```
```  1029   --> 0 < setprod f A"
```
```  1030 by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
```
```  1031
```
```  1032 lemma setprod_zero_iff[simp]: "finite A ==>
```
```  1033   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
```
```  1034   (EX x: A. f x = 0)"
```
```  1035 by (erule finite_induct, auto simp:no_zero_divisors)
```
```  1036
```
```  1037 lemma setprod_pos_nat:
```
```  1038   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
```
```  1039 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1040
```
```  1041 lemma setprod_pos_nat_iff[simp]:
```
```  1042   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
```
```  1043 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1044
```
```  1045 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1046   (setprod f (A Un B) :: 'a ::{field})
```
```  1047    = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1048 by (subst setprod_Un_Int [symmetric], auto)
```
```  1049
```
```  1050 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1051   (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1052   (if a:A then setprod f A / f a else setprod f A)"
```
```  1053   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```  1054
```
```  1055 lemma setprod_inversef:
```
```  1056   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1057   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1058 by (erule finite_induct) auto
```
```  1059
```
```  1060 lemma setprod_dividef:
```
```  1061   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1062   shows "finite A
```
```  1063     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1064 apply (subgoal_tac
```
```  1065          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1066 apply (erule ssubst)
```
```  1067 apply (subst divide_inverse)
```
```  1068 apply (subst setprod_timesf)
```
```  1069 apply (subst setprod_inversef, assumption+, rule refl)
```
```  1070 apply (rule setprod_cong, rule refl)
```
```  1071 apply (subst divide_inverse, auto)
```
```  1072 done
```
```  1073
```
```  1074 lemma setprod_dvd_setprod [rule_format]:
```
```  1075     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
```
```  1076   apply (cases "finite A")
```
```  1077   apply (induct set: finite)
```
```  1078   apply (auto simp add: dvd_def)
```
```  1079   apply (rule_tac x = "k * ka" in exI)
```
```  1080   apply (simp add: algebra_simps)
```
```  1081 done
```
```  1082
```
```  1083 lemma setprod_dvd_setprod_subset:
```
```  1084   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1085   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
```
```  1086   apply (unfold dvd_def, blast)
```
```  1087   apply (subst setprod_Un_disjoint [symmetric])
```
```  1088   apply (auto elim: finite_subset intro: setprod_cong)
```
```  1089 done
```
```  1090
```
```  1091 lemma setprod_dvd_setprod_subset2:
```
```  1092   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow>
```
```  1093       setprod f A dvd setprod g B"
```
```  1094   apply (rule dvd_trans)
```
```  1095   apply (rule setprod_dvd_setprod, erule (1) bspec)
```
```  1096   apply (erule (1) setprod_dvd_setprod_subset)
```
```  1097 done
```
```  1098
```
```  1099 lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow>
```
```  1100     (f i ::'a::comm_semiring_1) dvd setprod f A"
```
```  1101 by (induct set: finite) (auto intro: dvd_mult)
```
```  1102
```
```  1103 lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow>
```
```  1104     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
```
```  1105   apply (cases "finite A")
```
```  1106   apply (induct set: finite)
```
```  1107   apply auto
```
```  1108 done
```
```  1109
```
```  1110 lemma setprod_mono:
```
```  1111   fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
```
```  1112   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
```
```  1113   shows "setprod f A \<le> setprod g A"
```
```  1114 proof (cases "finite A")
```
```  1115   case True
```
```  1116   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
```
```  1117   proof (induct A rule: finite_subset_induct)
```
```  1118     case (insert a F)
```
```  1119     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
```
```  1120       unfolding setprod_insert[OF insert(1,3)]
```
```  1121       using assms[rule_format,OF insert(2)] insert
```
```  1122       by (auto intro: mult_mono mult_nonneg_nonneg)
```
```  1123   qed auto
```
```  1124   thus ?thesis by simp
```
```  1125 qed auto
```
```  1126
```
```  1127 lemma abs_setprod:
```
```  1128   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
```
```  1129   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
```
```  1130 proof (cases "finite A")
```
```  1131   case True thus ?thesis
```
```  1132     by induct (auto simp add: field_simps abs_mult)
```
```  1133 qed auto
```
```  1134
```
```  1135 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
```
```  1136 apply (erule finite_induct)
```
```  1137 apply auto
```
```  1138 done
```
```  1139
```
```  1140 lemma setprod_gen_delta:
```
```  1141   assumes fS: "finite S"
```
```  1142   shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::{comm_monoid_mult}) * c^ (card S - 1) else c^ card S)"
```
```  1143 proof-
```
```  1144   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```  1145   {assume a: "a \<notin> S"
```
```  1146     hence "\<forall> k\<in> S. ?f k = c" by simp
```
```  1147     hence ?thesis  using a setprod_constant[OF fS, of c] by (simp add: setprod_1 cong add: setprod_cong) }
```
```  1148   moreover
```
```  1149   {assume a: "a \<in> S"
```
```  1150     let ?A = "S - {a}"
```
```  1151     let ?B = "{a}"
```
```  1152     have eq: "S = ?A \<union> ?B" using a by blast
```
```  1153     have dj: "?A \<inter> ?B = {}" by simp
```
```  1154     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```  1155     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```  1156       apply (rule setprod_cong) by auto
```
```  1157     have cA: "card ?A = card S - 1" using fS a by auto
```
```  1158     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
```
```  1159     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```  1160       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```  1161       by simp
```
```  1162     then have ?thesis using a cA
```
```  1163       by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
```
```  1164   ultimately show ?thesis by blast
```
```  1165 qed
```
```  1166
```
```  1167
```
```  1168 subsection {* Versions of @{const inf} and @{const sup} on non-empty sets *}
```
```  1169
```
```  1170 no_notation times (infixl "*" 70)
```
```  1171 no_notation Groups.one ("1")
```
```  1172
```
```  1173 locale semilattice_big = semilattice +
```
```  1174   fixes F :: "'a set \<Rightarrow> 'a"
```
```  1175   assumes F_eq: "finite A \<Longrightarrow> F A = fold1 (op *) A"
```
```  1176
```
```  1177 sublocale semilattice_big < folding_one_idem proof
```
```  1178 qed (simp_all add: F_eq)
```
```  1179
```
```  1180 notation times (infixl "*" 70)
```
```  1181 notation Groups.one ("1")
```
```  1182
```
```  1183 context lattice
```
```  1184 begin
```
```  1185
```
```  1186 definition Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900) where
```
```  1187   "Inf_fin = fold1 inf"
```
```  1188
```
```  1189 definition Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900) where
```
```  1190   "Sup_fin = fold1 sup"
```
```  1191
```
```  1192 end
```
```  1193
```
```  1194 sublocale lattice < Inf_fin!: semilattice_big inf Inf_fin proof
```
```  1195 qed (simp add: Inf_fin_def)
```
```  1196
```
```  1197 sublocale lattice < Sup_fin!: semilattice_big sup Sup_fin proof
```
```  1198 qed (simp add: Sup_fin_def)
```
```  1199
```
```  1200 context semilattice_inf
```
```  1201 begin
```
```  1202
```
```  1203 lemma ab_semigroup_idem_mult_inf:
```
```  1204   "class.ab_semigroup_idem_mult inf"
```
```  1205 proof qed (rule inf_assoc inf_commute inf_idem)+
```
```  1206
```
```  1207 lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> fold inf b (insert a A) = inf a (fold inf b A)"
```
```  1208 by(rule fun_left_comm_idem.fold_insert_idem[OF ab_semigroup_idem_mult.fun_left_comm_idem[OF ab_semigroup_idem_mult_inf]])
```
```  1209
```
```  1210 lemma inf_le_fold_inf: "finite A \<Longrightarrow> ALL a:A. b \<le> a \<Longrightarrow> inf b c \<le> fold inf c A"
```
```  1211 by (induct pred: finite) (auto intro: le_infI1)
```
```  1212
```
```  1213 lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> fold inf b A \<le> inf a b"
```
```  1214 proof(induct arbitrary: a pred:finite)
```
```  1215   case empty thus ?case by simp
```
```  1216 next
```
```  1217   case (insert x A)
```
```  1218   show ?case
```
```  1219   proof cases
```
```  1220     assume "A = {}" thus ?thesis using insert by simp
```
```  1221   next
```
```  1222     assume "A \<noteq> {}" thus ?thesis using insert by (auto intro: le_infI2)
```
```  1223   qed
```
```  1224 qed
```
```  1225
```
```  1226 lemma below_fold1_iff:
```
```  1227   assumes "finite A" "A \<noteq> {}"
```
```  1228   shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  1229 proof -
```
```  1230   interpret ab_semigroup_idem_mult inf
```
```  1231     by (rule ab_semigroup_idem_mult_inf)
```
```  1232   show ?thesis using assms by (induct rule: finite_ne_induct) simp_all
```
```  1233 qed
```
```  1234
```
```  1235 lemma fold1_belowI:
```
```  1236   assumes "finite A"
```
```  1237     and "a \<in> A"
```
```  1238   shows "fold1 inf A \<le> a"
```
```  1239 proof -
```
```  1240   from assms have "A \<noteq> {}" by auto
```
```  1241   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```  1242   proof (induct rule: finite_ne_induct)
```
```  1243     case singleton thus ?case by simp
```
```  1244   next
```
```  1245     interpret ab_semigroup_idem_mult inf
```
```  1246       by (rule ab_semigroup_idem_mult_inf)
```
```  1247     case (insert x F)
```
```  1248     from insert(5) have "a = x \<or> a \<in> F" by simp
```
```  1249     thus ?case
```
```  1250     proof
```
```  1251       assume "a = x" thus ?thesis using insert
```
```  1252         by (simp add: mult_ac)
```
```  1253     next
```
```  1254       assume "a \<in> F"
```
```  1255       hence bel: "fold1 inf F \<le> a" by (rule insert)
```
```  1256       have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
```
```  1257         using insert by (simp add: mult_ac)
```
```  1258       also have "inf (fold1 inf F) a = fold1 inf F"
```
```  1259         using bel by (auto intro: antisym)
```
```  1260       also have "inf x \<dots> = fold1 inf (insert x F)"
```
```  1261         using insert by (simp add: mult_ac)
```
```  1262       finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
```
```  1263       moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
```
```  1264       ultimately show ?thesis by simp
```
```  1265     qed
```
```  1266   qed
```
```  1267 qed
```
```  1268
```
```  1269 end
```
```  1270
```
```  1271 context semilattice_sup
```
```  1272 begin
```
```  1273
```
```  1274 lemma ab_semigroup_idem_mult_sup: "class.ab_semigroup_idem_mult sup"
```
```  1275 by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
```
```  1276
```
```  1277 lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> fold sup b (insert a A) = sup a (fold sup b A)"
```
```  1278 by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)
```
```  1279
```
```  1280 lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> fold sup c A \<le> sup b c"
```
```  1281 by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)
```
```  1282
```
```  1283 lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> fold sup b A"
```
```  1284 by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)
```
```  1285
```
```  1286 end
```
```  1287
```
```  1288 context lattice
```
```  1289 begin
```
```  1290
```
```  1291 lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
```
```  1292 apply(unfold Sup_fin_def Inf_fin_def)
```
```  1293 apply(subgoal_tac "EX a. a:A")
```
```  1294 prefer 2 apply blast
```
```  1295 apply(erule exE)
```
```  1296 apply(rule order_trans)
```
```  1297 apply(erule (1) fold1_belowI)
```
```  1298 apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice])
```
```  1299 done
```
```  1300
```
```  1301 lemma sup_Inf_absorb [simp]:
```
```  1302   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
```
```  1303 apply(subst sup_commute)
```
```  1304 apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)
```
```  1305 done
```
```  1306
```
```  1307 lemma inf_Sup_absorb [simp]:
```
```  1308   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
```
```  1309 by (simp add: Sup_fin_def inf_absorb1
```
```  1310   semilattice_inf.fold1_belowI [OF dual_semilattice])
```
```  1311
```
```  1312 end
```
```  1313
```
```  1314 context distrib_lattice
```
```  1315 begin
```
```  1316
```
```  1317 lemma sup_Inf1_distrib:
```
```  1318   assumes "finite A"
```
```  1319     and "A \<noteq> {}"
```
```  1320   shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
```
```  1321 proof -
```
```  1322   interpret ab_semigroup_idem_mult inf
```
```  1323     by (rule ab_semigroup_idem_mult_inf)
```
```  1324   from assms show ?thesis
```
```  1325     by (simp add: Inf_fin_def image_def
```
```  1326       hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])
```
```  1327         (rule arg_cong [where f="fold1 inf"], blast)
```
```  1328 qed
```
```  1329
```
```  1330 lemma sup_Inf2_distrib:
```
```  1331   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1332   shows "sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B) = \<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B}"
```
```  1333 using A proof (induct rule: finite_ne_induct)
```
```  1334   case singleton thus ?case
```
```  1335     by (simp add: sup_Inf1_distrib [OF B] fold1_singleton_def [OF Inf_fin_def])
```
```  1336 next
```
```  1337   interpret ab_semigroup_idem_mult inf
```
```  1338     by (rule ab_semigroup_idem_mult_inf)
```
```  1339   case (insert x A)
```
```  1340   have finB: "finite {sup x b |b. b \<in> B}"
```
```  1341     by(rule finite_surj[where f = "sup x", OF B(1)], auto)
```
```  1342   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1343   proof -
```
```  1344     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
```
```  1345       by blast
```
```  1346     thus ?thesis by(simp add: insert(1) B(1))
```
```  1347   qed
```
```  1348   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1349   have "sup (\<Sqinter>\<^bsub>fin\<^esub>(insert x A)) (\<Sqinter>\<^bsub>fin\<^esub>B) = sup (inf x (\<Sqinter>\<^bsub>fin\<^esub>A)) (\<Sqinter>\<^bsub>fin\<^esub>B)"
```
```  1350     using insert by (simp add: fold1_insert_idem_def [OF Inf_fin_def])
```
```  1351   also have "\<dots> = inf (sup x (\<Sqinter>\<^bsub>fin\<^esub>B)) (sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2)
```
```  1352   also have "\<dots> = inf (\<Sqinter>\<^bsub>fin\<^esub>{sup x b|b. b \<in> B}) (\<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B})"
```
```  1353     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  1354   also have "\<dots> = \<Sqinter>\<^bsub>fin\<^esub>({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
```
```  1355     (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
```
```  1356     using B insert
```
```  1357     by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
```
```  1358   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1359     by blast
```
```  1360   finally show ?case .
```
```  1361 qed
```
```  1362
```
```  1363 lemma inf_Sup1_distrib:
```
```  1364   assumes "finite A" and "A \<noteq> {}"
```
```  1365   shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
```
```  1366 proof -
```
```  1367   interpret ab_semigroup_idem_mult sup
```
```  1368     by (rule ab_semigroup_idem_mult_sup)
```
```  1369   from assms show ?thesis
```
```  1370     by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])
```
```  1371       (rule arg_cong [where f="fold1 sup"], blast)
```
```  1372 qed
```
```  1373
```
```  1374 lemma inf_Sup2_distrib:
```
```  1375   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1376   shows "inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B) = \<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B}"
```
```  1377 using A proof (induct rule: finite_ne_induct)
```
```  1378   case singleton thus ?case
```
```  1379     by(simp add: inf_Sup1_distrib [OF B] fold1_singleton_def [OF Sup_fin_def])
```
```  1380 next
```
```  1381   case (insert x A)
```
```  1382   have finB: "finite {inf x b |b. b \<in> B}"
```
```  1383     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
```
```  1384   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1385   proof -
```
```  1386     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
```
```  1387       by blast
```
```  1388     thus ?thesis by(simp add: insert(1) B(1))
```
```  1389   qed
```
```  1390   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1391   interpret ab_semigroup_idem_mult sup
```
```  1392     by (rule ab_semigroup_idem_mult_sup)
```
```  1393   have "inf (\<Squnion>\<^bsub>fin\<^esub>(insert x A)) (\<Squnion>\<^bsub>fin\<^esub>B) = inf (sup x (\<Squnion>\<^bsub>fin\<^esub>A)) (\<Squnion>\<^bsub>fin\<^esub>B)"
```
```  1394     using insert by (simp add: fold1_insert_idem_def [OF Sup_fin_def])
```
```  1395   also have "\<dots> = sup (inf x (\<Squnion>\<^bsub>fin\<^esub>B)) (inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2)
```
```  1396   also have "\<dots> = sup (\<Squnion>\<^bsub>fin\<^esub>{inf x b|b. b \<in> B}) (\<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B})"
```
```  1397     using insert by(simp add:inf_Sup1_distrib[OF B])
```
```  1398   also have "\<dots> = \<Squnion>\<^bsub>fin\<^esub>({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
```
```  1399     (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
```
```  1400     using B insert
```
```  1401     by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
```
```  1402   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1403     by blast
```
```  1404   finally show ?case .
```
```  1405 qed
```
```  1406
```
```  1407 end
```
```  1408
```
```  1409 context complete_lattice
```
```  1410 begin
```
```  1411
```
```  1412 lemma Inf_fin_Inf:
```
```  1413   assumes "finite A" and "A \<noteq> {}"
```
```  1414   shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
```
```  1415 proof -
```
```  1416   interpret ab_semigroup_idem_mult inf
```
```  1417     by (rule ab_semigroup_idem_mult_inf)
```
```  1418   from `A \<noteq> {}` obtain b B where "A = insert b B" by auto
```
```  1419   moreover with `finite A` have "finite B" by simp
```
```  1420   ultimately show ?thesis
```
```  1421   by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])
```
```  1422     (simp add: Inf_fold_inf)
```
```  1423 qed
```
```  1424
```
```  1425 lemma Sup_fin_Sup:
```
```  1426   assumes "finite A" and "A \<noteq> {}"
```
```  1427   shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
```
```  1428 proof -
```
```  1429   interpret ab_semigroup_idem_mult sup
```
```  1430     by (rule ab_semigroup_idem_mult_sup)
```
```  1431   from `A \<noteq> {}` obtain b B where "A = insert b B" by auto
```
```  1432   moreover with `finite A` have "finite B" by simp
```
```  1433   ultimately show ?thesis
```
```  1434   by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric])
```
```  1435     (simp add: Sup_fold_sup)
```
```  1436 qed
```
```  1437
```
```  1438 end
```
```  1439
```
```  1440
```
```  1441 subsection {* Versions of @{const min} and @{const max} on non-empty sets *}
```
```  1442
```
```  1443 definition (in linorder) Min :: "'a set \<Rightarrow> 'a" where
```
```  1444   "Min = fold1 min"
```
```  1445
```
```  1446 definition (in linorder) Max :: "'a set \<Rightarrow> 'a" where
```
```  1447   "Max = fold1 max"
```
```  1448
```
```  1449 sublocale linorder < Min!: semilattice_big min Min proof
```
```  1450 qed (simp add: Min_def)
```
```  1451
```
```  1452 sublocale linorder < Max!: semilattice_big max Max proof
```
```  1453 qed (simp add: Max_def)
```
```  1454
```
```  1455 context linorder
```
```  1456 begin
```
```  1457
```
```  1458 lemmas Min_singleton = Min.singleton
```
```  1459 lemmas Max_singleton = Max.singleton
```
```  1460
```
```  1461 lemma Min_insert:
```
```  1462   assumes "finite A" and "A \<noteq> {}"
```
```  1463   shows "Min (insert x A) = min x (Min A)"
```
```  1464   using assms by simp
```
```  1465
```
```  1466 lemma Max_insert:
```
```  1467   assumes "finite A" and "A \<noteq> {}"
```
```  1468   shows "Max (insert x A) = max x (Max A)"
```
```  1469   using assms by simp
```
```  1470
```
```  1471 lemma Min_Un:
```
```  1472   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
```
```  1473   shows "Min (A \<union> B) = min (Min A) (Min B)"
```
```  1474   using assms by (rule Min.union_idem)
```
```  1475
```
```  1476 lemma Max_Un:
```
```  1477   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
```
```  1478   shows "Max (A \<union> B) = max (Max A) (Max B)"
```
```  1479   using assms by (rule Max.union_idem)
```
```  1480
```
```  1481 lemma hom_Min_commute:
```
```  1482   assumes "\<And>x y. h (min x y) = min (h x) (h y)"
```
```  1483     and "finite N" and "N \<noteq> {}"
```
```  1484   shows "h (Min N) = Min (h ` N)"
```
```  1485   using assms by (rule Min.hom_commute)
```
```  1486
```
```  1487 lemma hom_Max_commute:
```
```  1488   assumes "\<And>x y. h (max x y) = max (h x) (h y)"
```
```  1489     and "finite N" and "N \<noteq> {}"
```
```  1490   shows "h (Max N) = Max (h ` N)"
```
```  1491   using assms by (rule Max.hom_commute)
```
```  1492
```
```  1493 lemma ab_semigroup_idem_mult_min:
```
```  1494   "class.ab_semigroup_idem_mult min"
```
```  1495   proof qed (auto simp add: min_def)
```
```  1496
```
```  1497 lemma ab_semigroup_idem_mult_max:
```
```  1498   "class.ab_semigroup_idem_mult max"
```
```  1499   proof qed (auto simp add: max_def)
```
```  1500
```
```  1501 lemma max_lattice:
```
```  1502   "class.semilattice_inf (op \<ge>) (op >) max"
```
```  1503   by (fact min_max.dual_semilattice)
```
```  1504
```
```  1505 lemma dual_max:
```
```  1506   "ord.max (op \<ge>) = min"
```
```  1507   by (auto simp add: ord.max_def_raw min_def expand_fun_eq)
```
```  1508
```
```  1509 lemma dual_min:
```
```  1510   "ord.min (op \<ge>) = max"
```
```  1511   by (auto simp add: ord.min_def_raw max_def expand_fun_eq)
```
```  1512
```
```  1513 lemma strict_below_fold1_iff:
```
```  1514   assumes "finite A" and "A \<noteq> {}"
```
```  1515   shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  1516 proof -
```
```  1517   interpret ab_semigroup_idem_mult min
```
```  1518     by (rule ab_semigroup_idem_mult_min)
```
```  1519   from assms show ?thesis
```
```  1520   by (induct rule: finite_ne_induct)
```
```  1521     (simp_all add: fold1_insert)
```
```  1522 qed
```
```  1523
```
```  1524 lemma fold1_below_iff:
```
```  1525   assumes "finite A" and "A \<noteq> {}"
```
```  1526   shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  1527 proof -
```
```  1528   interpret ab_semigroup_idem_mult min
```
```  1529     by (rule ab_semigroup_idem_mult_min)
```
```  1530   from assms show ?thesis
```
```  1531   by (induct rule: finite_ne_induct)
```
```  1532     (simp_all add: fold1_insert min_le_iff_disj)
```
```  1533 qed
```
```  1534
```
```  1535 lemma fold1_strict_below_iff:
```
```  1536   assumes "finite A" and "A \<noteq> {}"
```
```  1537   shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  1538 proof -
```
```  1539   interpret ab_semigroup_idem_mult min
```
```  1540     by (rule ab_semigroup_idem_mult_min)
```
```  1541   from assms show ?thesis
```
```  1542   by (induct rule: finite_ne_induct)
```
```  1543     (simp_all add: fold1_insert min_less_iff_disj)
```
```  1544 qed
```
```  1545
```
```  1546 lemma fold1_antimono:
```
```  1547   assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
```
```  1548   shows "fold1 min B \<le> fold1 min A"
```
```  1549 proof cases
```
```  1550   assume "A = B" thus ?thesis by simp
```
```  1551 next
```
```  1552   interpret ab_semigroup_idem_mult min
```
```  1553     by (rule ab_semigroup_idem_mult_min)
```
```  1554   assume "A \<noteq> B"
```
```  1555   have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
```
```  1556   have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
```
```  1557   also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
```
```  1558   proof -
```
```  1559     have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
```
```  1560     moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *)
```
```  1561     moreover have "(B-A) \<noteq> {}" using prems by blast
```
```  1562     moreover have "A Int (B-A) = {}" using prems by blast
```
```  1563     ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
```
```  1564   qed
```
```  1565   also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
```
```  1566   finally show ?thesis .
```
```  1567 qed
```
```  1568
```
```  1569 lemma Min_in [simp]:
```
```  1570   assumes "finite A" and "A \<noteq> {}"
```
```  1571   shows "Min A \<in> A"
```
```  1572 proof -
```
```  1573   interpret ab_semigroup_idem_mult min
```
```  1574     by (rule ab_semigroup_idem_mult_min)
```
```  1575   from assms fold1_in show ?thesis by (fastsimp simp: Min_def min_def)
```
```  1576 qed
```
```  1577
```
```  1578 lemma Max_in [simp]:
```
```  1579   assumes "finite A" and "A \<noteq> {}"
```
```  1580   shows "Max A \<in> A"
```
```  1581 proof -
```
```  1582   interpret ab_semigroup_idem_mult max
```
```  1583     by (rule ab_semigroup_idem_mult_max)
```
```  1584   from assms fold1_in [of A] show ?thesis by (fastsimp simp: Max_def max_def)
```
```  1585 qed
```
```  1586
```
```  1587 lemma Min_le [simp]:
```
```  1588   assumes "finite A" and "x \<in> A"
```
```  1589   shows "Min A \<le> x"
```
```  1590   using assms by (simp add: Min_def min_max.fold1_belowI)
```
```  1591
```
```  1592 lemma Max_ge [simp]:
```
```  1593   assumes "finite A" and "x \<in> A"
```
```  1594   shows "x \<le> Max A"
```
```  1595 proof -
```
```  1596   interpret semilattice_inf "op \<ge>" "op >" max
```
```  1597     by (rule max_lattice)
```
```  1598   from assms show ?thesis by (simp add: Max_def fold1_belowI)
```
```  1599 qed
```
```  1600
```
```  1601 lemma Min_ge_iff [simp, no_atp]:
```
```  1602   assumes "finite A" and "A \<noteq> {}"
```
```  1603   shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  1604   using assms by (simp add: Min_def min_max.below_fold1_iff)
```
```  1605
```
```  1606 lemma Max_le_iff [simp, no_atp]:
```
```  1607   assumes "finite A" and "A \<noteq> {}"
```
```  1608   shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
```
```  1609 proof -
```
```  1610   interpret semilattice_inf "op \<ge>" "op >" max
```
```  1611     by (rule max_lattice)
```
```  1612   from assms show ?thesis by (simp add: Max_def below_fold1_iff)
```
```  1613 qed
```
```  1614
```
```  1615 lemma Min_gr_iff [simp, no_atp]:
```
```  1616   assumes "finite A" and "A \<noteq> {}"
```
```  1617   shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  1618   using assms by (simp add: Min_def strict_below_fold1_iff)
```
```  1619
```
```  1620 lemma Max_less_iff [simp, no_atp]:
```
```  1621   assumes "finite A" and "A \<noteq> {}"
```
```  1622   shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
```
```  1623 proof -
```
```  1624   interpret dual: linorder "op \<ge>" "op >"
```
```  1625     by (rule dual_linorder)
```
```  1626   from assms show ?thesis
```
```  1627     by (simp add: Max_def dual.strict_below_fold1_iff [folded dual.dual_max])
```
```  1628 qed
```
```  1629
```
```  1630 lemma Min_le_iff [no_atp]:
```
```  1631   assumes "finite A" and "A \<noteq> {}"
```
```  1632   shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  1633   using assms by (simp add: Min_def fold1_below_iff)
```
```  1634
```
```  1635 lemma Max_ge_iff [no_atp]:
```
```  1636   assumes "finite A" and "A \<noteq> {}"
```
```  1637   shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
```
```  1638 proof -
```
```  1639   interpret dual: linorder "op \<ge>" "op >"
```
```  1640     by (rule dual_linorder)
```
```  1641   from assms show ?thesis
```
```  1642     by (simp add: Max_def dual.fold1_below_iff [folded dual.dual_max])
```
```  1643 qed
```
```  1644
```
```  1645 lemma Min_less_iff [no_atp]:
```
```  1646   assumes "finite A" and "A \<noteq> {}"
```
```  1647   shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  1648   using assms by (simp add: Min_def fold1_strict_below_iff)
```
```  1649
```
```  1650 lemma Max_gr_iff [no_atp]:
```
```  1651   assumes "finite A" and "A \<noteq> {}"
```
```  1652   shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
```
```  1653 proof -
```
```  1654   interpret dual: linorder "op \<ge>" "op >"
```
```  1655     by (rule dual_linorder)
```
```  1656   from assms show ?thesis
```
```  1657     by (simp add: Max_def dual.fold1_strict_below_iff [folded dual.dual_max])
```
```  1658 qed
```
```  1659
```
```  1660 lemma Min_eqI:
```
```  1661   assumes "finite A"
```
```  1662   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
```
```  1663     and "x \<in> A"
```
```  1664   shows "Min A = x"
```
```  1665 proof (rule antisym)
```
```  1666   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1667   with assms show "Min A \<ge> x" by simp
```
```  1668 next
```
```  1669   from assms show "x \<ge> Min A" by simp
```
```  1670 qed
```
```  1671
```
```  1672 lemma Max_eqI:
```
```  1673   assumes "finite A"
```
```  1674   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
```
```  1675     and "x \<in> A"
```
```  1676   shows "Max A = x"
```
```  1677 proof (rule antisym)
```
```  1678   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1679   with assms show "Max A \<le> x" by simp
```
```  1680 next
```
```  1681   from assms show "x \<le> Max A" by simp
```
```  1682 qed
```
```  1683
```
```  1684 lemma Min_antimono:
```
```  1685   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  1686   shows "Min N \<le> Min M"
```
```  1687   using assms by (simp add: Min_def fold1_antimono)
```
```  1688
```
```  1689 lemma Max_mono:
```
```  1690   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  1691   shows "Max M \<le> Max N"
```
```  1692 proof -
```
```  1693   interpret dual: linorder "op \<ge>" "op >"
```
```  1694     by (rule dual_linorder)
```
```  1695   from assms show ?thesis
```
```  1696     by (simp add: Max_def dual.fold1_antimono [folded dual.dual_max])
```
```  1697 qed
```
```  1698
```
```  1699 lemma finite_linorder_max_induct[consumes 1, case_names empty insert]:
```
```  1700  assumes fin: "finite A"
```
```  1701  and   empty: "P {}"
```
```  1702  and  insert: "(!!b A. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))"
```
```  1703  shows "P A"
```
```  1704 using fin empty insert
```
```  1705 proof (induct rule: finite_psubset_induct)
```
```  1706   case (psubset A)
```
```  1707   have IH: "\<And>B. \<lbrakk>B < A; P {}; (\<And>A b. \<lbrakk>finite A; \<forall>a\<in>A. a<b; P A\<rbrakk> \<Longrightarrow> P (insert b A))\<rbrakk> \<Longrightarrow> P B" by fact
```
```  1708   have fin: "finite A" by fact
```
```  1709   have empty: "P {}" by fact
```
```  1710   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
```
```  1711   show "P A"
```
```  1712   proof (cases "A = {}")
```
```  1713     assume "A = {}"
```
```  1714     then show "P A" using `P {}` by simp
```
```  1715   next
```
```  1716     let ?B = "A - {Max A}"
```
```  1717     let ?A = "insert (Max A) ?B"
```
```  1718     have "finite ?B" using `finite A` by simp
```
```  1719     assume "A \<noteq> {}"
```
```  1720     with `finite A` have "Max A : A" by auto
```
```  1721     then have A: "?A = A" using insert_Diff_single insert_absorb by auto
```
```  1722     then have "P ?B" using `P {}` step IH[of ?B] by blast
```
```  1723     moreover
```
```  1724     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastsimp
```
```  1725     ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastsimp
```
```  1726   qed
```
```  1727 qed
```
```  1728
```
```  1729 lemma finite_linorder_min_induct[consumes 1, case_names empty insert]:
```
```  1730  "\<lbrakk>finite A; P {}; \<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"
```
```  1731 by(rule linorder.finite_linorder_max_induct[OF dual_linorder])
```
```  1732
```
```  1733 end
```
```  1734
```
```  1735 context linordered_ab_semigroup_add
```
```  1736 begin
```
```  1737
```
```  1738 lemma add_Min_commute:
```
```  1739   fixes k
```
```  1740   assumes "finite N" and "N \<noteq> {}"
```
```  1741   shows "k + Min N = Min {k + m | m. m \<in> N}"
```
```  1742 proof -
```
```  1743   have "\<And>x y. k + min x y = min (k + x) (k + y)"
```
```  1744     by (simp add: min_def not_le)
```
```  1745       (blast intro: antisym less_imp_le add_left_mono)
```
```  1746   with assms show ?thesis
```
```  1747     using hom_Min_commute [of "plus k" N]
```
```  1748     by simp (blast intro: arg_cong [where f = Min])
```
```  1749 qed
```
```  1750
```
```  1751 lemma add_Max_commute:
```
```  1752   fixes k
```
```  1753   assumes "finite N" and "N \<noteq> {}"
```
```  1754   shows "k + Max N = Max {k + m | m. m \<in> N}"
```
```  1755 proof -
```
```  1756   have "\<And>x y. k + max x y = max (k + x) (k + y)"
```
```  1757     by (simp add: max_def not_le)
```
```  1758       (blast intro: antisym less_imp_le add_left_mono)
```
```  1759   with assms show ?thesis
```
```  1760     using hom_Max_commute [of "plus k" N]
```
```  1761     by simp (blast intro: arg_cong [where f = Max])
```
```  1762 qed
```
```  1763
```
```  1764 end
```
```  1765
```
```  1766 context linordered_ab_group_add
```
```  1767 begin
```
```  1768
```
```  1769 lemma minus_Max_eq_Min [simp]:
```
```  1770   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Max S) = Min (uminus ` S)"
```
```  1771   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
```
```  1772
```
```  1773 lemma minus_Min_eq_Max [simp]:
```
```  1774   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Min S) = Max (uminus ` S)"
```
```  1775   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
```
```  1776
```
```  1777 end
```
```  1778
```
```  1779 end
```