src/HOL/Big_Operators.thy
 author krauss Fri Sep 09 00:22:18 2011 +0200 (2011-09-09) changeset 44845 5e51075cbd97 parent 42986 11fd8c04ea24 child 44890 22f665a2e91c permissions -rw-r--r--
added syntactic classes for "inf" and "sup"
```     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 lemma F_cong:
```
```    32   assumes "A = B" "\<And>x. x \<in> B \<Longrightarrow> h x = g x"
```
```    33   shows "F h A = F g B"
```
```    34 proof cases
```
```    35   assume "finite A"
```
```    36   with assms show ?thesis unfolding `A = B` by (simp cong: cong)
```
```    37 next
```
```    38   assume "\<not> finite A"
```
```    39   then show ?thesis unfolding `A = B` by simp
```
```    40 qed
```
```    41
```
```    42 lemma If_cases:
```
```    43   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
```
```    44   assumes fA: "finite A"
```
```    45   shows "F (\<lambda>x. if P x then h x else g x) A =
```
```    46          F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
```
```    47 proof-
```
```    48   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
```
```    49           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
```
```    50     by blast+
```
```    51   from fA
```
```    52   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
```
```    53   let ?g = "\<lambda>x. if P x then h x else g x"
```
```    54   from union_disjoint[OF f a(2), of ?g] a(1)
```
```    55   show ?thesis
```
```    56     by (subst (1 2) F_cong) simp_all
```
```    57 qed
```
```    58
```
```    59 end
```
```    60
```
```    61 text {* for ad-hoc proofs for @{const fold_image} *}
```
```    62
```
```    63 lemma (in comm_monoid_add) comm_monoid_mult:
```
```    64   "class.comm_monoid_mult (op +) 0"
```
```    65 proof qed (auto intro: add_assoc add_commute)
```
```    66
```
```    67 notation times (infixl "*" 70)
```
```    68 notation Groups.one ("1")
```
```    69
```
```    70
```
```    71 subsection {* Generalized summation over a set *}
```
```    72
```
```    73 definition (in comm_monoid_add) setsum :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
```
```    74   "setsum f A = (if finite A then fold_image (op +) f 0 A else 0)"
```
```    75
```
```    76 sublocale comm_monoid_add < setsum!: comm_monoid_big "op +" 0 setsum proof
```
```    77 qed (fact setsum_def)
```
```    78
```
```    79 abbreviation
```
```    80   Setsum  ("\<Sum>_"  999) where
```
```    81   "\<Sum>A == setsum (%x. x) A"
```
```    82
```
```    83 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```    84 written @{text"\<Sum>x\<in>A. e"}. *}
```
```    85
```
```    86 syntax
```
```    87   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```    88 syntax (xsymbols)
```
```    89   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```    90 syntax (HTML output)
```
```    91   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```    92
```
```    93 translations -- {* Beware of argument permutation! *}
```
```    94   "SUM i:A. b" == "CONST setsum (%i. b) A"
```
```    95   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
```
```    96
```
```    97 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```    98  @{text"\<Sum>x|P. e"}. *}
```
```    99
```
```   100 syntax
```
```   101   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   102 syntax (xsymbols)
```
```   103   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   104 syntax (HTML output)
```
```   105   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   106
```
```   107 translations
```
```   108   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   109   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   110
```
```   111 print_translation {*
```
```   112 let
```
```   113   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```   114         if x <> y then raise Match
```
```   115         else
```
```   116           let
```
```   117             val x' = Syntax_Trans.mark_bound x;
```
```   118             val t' = subst_bound (x', t);
```
```   119             val P' = subst_bound (x', P);
```
```   120           in Syntax.const @{syntax_const "_qsetsum"} \$ Syntax_Trans.mark_bound x \$ P' \$ t' end
```
```   121     | setsum_tr' _ = raise Match;
```
```   122 in [(@{const_syntax setsum}, setsum_tr')] end
```
```   123 *}
```
```   124
```
```   125 lemma setsum_empty:
```
```   126   "setsum f {} = 0"
```
```   127   by (fact setsum.empty)
```
```   128
```
```   129 lemma setsum_insert:
```
```   130   "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   131   by (fact setsum.insert)
```
```   132
```
```   133 lemma setsum_infinite:
```
```   134   "~ finite A ==> setsum f A = 0"
```
```   135   by (fact setsum.infinite)
```
```   136
```
```   137 lemma (in comm_monoid_add) setsum_reindex:
```
```   138   assumes "inj_on f B" shows "setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   139 proof -
```
```   140   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   141   from assms show ?thesis by (auto simp add: setsum_def fold_image_reindex dest!:finite_imageD)
```
```   142 qed
```
```   143
```
```   144 lemma (in comm_monoid_add) setsum_reindex_id:
```
```   145   "inj_on f B ==> setsum f B = setsum id (f ` B)"
```
```   146   by (simp add: setsum_reindex)
```
```   147
```
```   148 lemma (in comm_monoid_add) setsum_reindex_nonzero:
```
```   149   assumes fS: "finite S"
```
```   150   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"
```
```   151   shows "setsum h (f ` S) = setsum (h o f) S"
```
```   152 using nz
```
```   153 proof(induct rule: finite_induct[OF fS])
```
```   154   case 1 thus ?case by simp
```
```   155 next
```
```   156   case (2 x F)
```
```   157   {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
```
```   158     then obtain y where y: "y \<in> F" "f x = f y" by auto
```
```   159     from "2.hyps" y have xy: "x \<noteq> y" by auto
```
```   160
```
```   161     from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
```
```   162     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
```
```   163     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   164       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   165       using h0
```
```   166       apply simp
```
```   167       apply (rule "2.hyps"(3))
```
```   168       apply (rule_tac y="y" in  "2.prems")
```
```   169       apply simp_all
```
```   170       done
```
```   171     finally have ?case .}
```
```   172   moreover
```
```   173   {assume fxF: "f x \<notin> f ` F"
```
```   174     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)"
```
```   175       using fxF "2.hyps" by simp
```
```   176     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   177       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   178       apply simp
```
```   179       apply (rule cong [OF refl [of "op + (h (f x))"]])
```
```   180       apply (rule "2.hyps"(3))
```
```   181       apply (rule_tac y="y" in  "2.prems")
```
```   182       apply simp_all
```
```   183       done
```
```   184     finally have ?case .}
```
```   185   ultimately show ?case by blast
```
```   186 qed
```
```   187
```
```   188 lemma (in comm_monoid_add) setsum_cong:
```
```   189   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   190   by (cases "finite A") (auto intro: setsum.cong)
```
```   191
```
```   192 lemma (in comm_monoid_add) strong_setsum_cong [cong]:
```
```   193   "A = B ==> (!!x. x:B =simp=> f x = g x)
```
```   194    ==> setsum (%x. f x) A = setsum (%x. g x) B"
```
```   195   by (rule setsum_cong) (simp_all add: simp_implies_def)
```
```   196
```
```   197 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"
```
```   198   by (auto intro: setsum_cong)
```
```   199
```
```   200 lemma (in comm_monoid_add) setsum_reindex_cong:
```
```   201    "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
```
```   202     ==> setsum h B = setsum g A"
```
```   203   by (simp add: setsum_reindex)
```
```   204
```
```   205 lemma (in comm_monoid_add) setsum_0[simp]: "setsum (%i. 0) A = 0"
```
```   206   by (cases "finite A") (erule finite_induct, auto)
```
```   207
```
```   208 lemma (in comm_monoid_add) setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
```
```   209   by (simp add:setsum_cong)
```
```   210
```
```   211 lemma (in comm_monoid_add) setsum_Un_Int: "finite A ==> finite B ==>
```
```   212   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   213   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   214   by (fact setsum.union_inter)
```
```   215
```
```   216 lemma (in comm_monoid_add) setsum_Un_disjoint: "finite A ==> finite B
```
```   217   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   218   by (fact setsum.union_disjoint)
```
```   219
```
```   220 lemma setsum_mono_zero_left:
```
```   221   assumes fT: "finite T" and ST: "S \<subseteq> T"
```
```   222   and z: "\<forall>i \<in> T - S. f i = 0"
```
```   223   shows "setsum f S = setsum f T"
```
```   224 proof-
```
```   225   have eq: "T = S \<union> (T - S)" using ST by blast
```
```   226   have d: "S \<inter> (T - S) = {}" using ST by blast
```
```   227   from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
```
```   228   show ?thesis
```
```   229   by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
```
```   230 qed
```
```   231
```
```   232 lemma setsum_mono_zero_right:
```
```   233   "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. f i = 0 \<Longrightarrow> setsum f T = setsum f S"
```
```   234 by(blast intro!: setsum_mono_zero_left[symmetric])
```
```   235
```
```   236 lemma setsum_mono_zero_cong_left:
```
```   237   assumes fT: "finite T" and ST: "S \<subseteq> T"
```
```   238   and z: "\<forall>i \<in> T - S. g i = 0"
```
```   239   and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
```
```   240   shows "setsum f S = setsum g T"
```
```   241 proof-
```
```   242   have eq: "T = S \<union> (T - S)" using ST by blast
```
```   243   have d: "S \<inter> (T - S) = {}" using ST by blast
```
```   244   from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
```
```   245   show ?thesis
```
```   246     using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
```
```   247 qed
```
```   248
```
```   249 lemma setsum_mono_zero_cong_right:
```
```   250   assumes fT: "finite T" and ST: "S \<subseteq> T"
```
```   251   and z: "\<forall>i \<in> T - S. f i = 0"
```
```   252   and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
```
```   253   shows "setsum f T = setsum g S"
```
```   254 using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto
```
```   255
```
```   256 lemma setsum_delta:
```
```   257   assumes fS: "finite S"
```
```   258   shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
```
```   259 proof-
```
```   260   let ?f = "(\<lambda>k. if k=a then b k else 0)"
```
```   261   {assume a: "a \<notin> S"
```
```   262     hence "\<forall> k\<in> S. ?f k = 0" by simp
```
```   263     hence ?thesis  using a by simp}
```
```   264   moreover
```
```   265   {assume a: "a \<in> S"
```
```   266     let ?A = "S - {a}"
```
```   267     let ?B = "{a}"
```
```   268     have eq: "S = ?A \<union> ?B" using a by blast
```
```   269     have dj: "?A \<inter> ?B = {}" by simp
```
```   270     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   271     have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
```
```   272       using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```   273       by simp
```
```   274     then have ?thesis  using a by simp}
```
```   275   ultimately show ?thesis by blast
```
```   276 qed
```
```   277 lemma setsum_delta':
```
```   278   assumes fS: "finite S" shows
```
```   279   "setsum (\<lambda>k. if a = k then b k else 0) S =
```
```   280      (if a\<in> S then b a else 0)"
```
```   281   using setsum_delta[OF fS, of a b, symmetric]
```
```   282   by (auto intro: setsum_cong)
```
```   283
```
```   284 lemma setsum_restrict_set:
```
```   285   assumes fA: "finite A"
```
```   286   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
```
```   287 proof-
```
```   288   from fA have fab: "finite (A \<inter> B)" by auto
```
```   289   have aba: "A \<inter> B \<subseteq> A" by blast
```
```   290   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
```
```   291   from setsum_mono_zero_left[OF fA aba, of ?g]
```
```   292   show ?thesis by simp
```
```   293 qed
```
```   294
```
```   295 lemma setsum_cases:
```
```   296   assumes fA: "finite A"
```
```   297   shows "setsum (\<lambda>x. if P x then f x else g x) A =
```
```   298          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
```
```   299   using setsum.If_cases[OF fA] .
```
```   300
```
```   301 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   302   the lhs need not be, since UNION I A could still be finite.*)
```
```   303 lemma (in comm_monoid_add) setsum_UN_disjoint:
```
```   304   assumes "finite I" and "ALL i:I. finite (A i)"
```
```   305     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
```
```   306   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   307 proof -
```
```   308   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   309   from assms show ?thesis by (simp add: setsum_def fold_image_UN_disjoint)
```
```   310 qed
```
```   311
```
```   312 text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
```
```   313 directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
```
```   314 lemma setsum_Union_disjoint:
```
```   315   "[| (ALL A:C. finite A);
```
```   316       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```   317    ==> setsum f (Union C) = setsum (setsum f) C"
```
```   318 apply (cases "finite C")
```
```   319  prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
```
```   320   apply (frule setsum_UN_disjoint [of C id f])
```
```   321  apply (unfold Union_def id_def, assumption+)
```
```   322 done
```
```   323
```
```   324 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```   325   the rhs need not be, since SIGMA A B could still be finite.*)
```
```   326 lemma (in comm_monoid_add) setsum_Sigma:
```
```   327   assumes "finite A" and  "ALL x:A. finite (B x)"
```
```   328   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)"
```
```   329 proof -
```
```   330   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   331   from assms show ?thesis by (simp add: setsum_def fold_image_Sigma split_def)
```
```   332 qed
```
```   333
```
```   334 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```   335 lemma setsum_cartesian_product:
```
```   336    "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
```
```   337 apply (cases "finite A")
```
```   338  apply (cases "finite B")
```
```   339   apply (simp add: setsum_Sigma)
```
```   340  apply (cases "A={}", simp)
```
```   341  apply (simp)
```
```   342 apply (auto simp add: setsum_def
```
```   343             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   344 done
```
```   345
```
```   346 lemma (in comm_monoid_add) setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```   347   by (cases "finite A") (simp_all add: setsum.distrib)
```
```   348
```
```   349
```
```   350 subsubsection {* Properties in more restricted classes of structures *}
```
```   351
```
```   352 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   353 apply (case_tac "finite A")
```
```   354  prefer 2 apply (simp add: setsum_def)
```
```   355 apply (erule rev_mp)
```
```   356 apply (erule finite_induct, auto)
```
```   357 done
```
```   358
```
```   359 lemma setsum_eq_0_iff [simp]:
```
```   360     "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   361 by (induct set: finite) auto
```
```   362
```
```   363 lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
```
```   364   (setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
```
```   365 apply(erule finite_induct)
```
```   366 apply (auto simp add:add_is_1)
```
```   367 done
```
```   368
```
```   369 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   370
```
```   371 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   372   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   373   -- {* For the natural numbers, we have subtraction. *}
```
```   374 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   375
```
```   376 lemma setsum_Un: "finite A ==> finite B ==>
```
```   377   (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   378    setsum f A + setsum f B - setsum f (A Int B)"
```
```   379 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   380
```
```   381 lemma (in comm_monoid_add) setsum_eq_general_reverses:
```
```   382   assumes fS: "finite S" and fT: "finite T"
```
```   383   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```   384   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
```
```   385   shows "setsum f S = setsum g T"
```
```   386 proof -
```
```   387   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   388   show ?thesis
```
```   389   apply (simp add: setsum_def fS fT)
```
```   390   apply (rule fold_image_eq_general_inverses)
```
```   391   apply (rule fS)
```
```   392   apply (erule kh)
```
```   393   apply (erule hk)
```
```   394   done
```
```   395 qed
```
```   396
```
```   397 lemma (in comm_monoid_add) setsum_Un_zero:
```
```   398   assumes fS: "finite S" and fT: "finite T"
```
```   399   and I0: "\<forall>x \<in> S\<inter>T. f x = 0"
```
```   400   shows "setsum f (S \<union> T) = setsum f S  + setsum f T"
```
```   401 proof -
```
```   402   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   403   show ?thesis
```
```   404   using fS fT
```
```   405   apply (simp add: setsum_def)
```
```   406   apply (rule fold_image_Un_one)
```
```   407   using I0 by auto
```
```   408 qed
```
```   409
```
```   410 lemma setsum_UNION_zero:
```
```   411   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
```
```   412   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"
```
```   413   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
```
```   414   using fSS f0
```
```   415 proof(induct rule: finite_induct[OF fS])
```
```   416   case 1 thus ?case by simp
```
```   417 next
```
```   418   case (2 T F)
```
```   419   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F"
```
```   420     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
```
```   421   from fTF have fUF: "finite (\<Union>F)" by auto
```
```   422   from "2.prems" TF fTF
```
```   423   show ?case
```
```   424     by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f])
```
```   425 qed
```
```   426
```
```   427 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   428   (if a:A then setsum f A - f a else setsum f A)"
```
```   429 apply (case_tac "finite A")
```
```   430  prefer 2 apply (simp add: setsum_def)
```
```   431 apply (erule finite_induct)
```
```   432  apply (auto simp add: insert_Diff_if)
```
```   433 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   434 done
```
```   435
```
```   436 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   437   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   438   (if a:A then setsum f A - f a else setsum f A)"
```
```   439 by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   440
```
```   441 lemma setsum_diff1'[rule_format]:
```
```   442   "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
```
```   443 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))"])
```
```   444 apply (auto simp add: insert_Diff_if add_ac)
```
```   445 done
```
```   446
```
```   447 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
```
```   448   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
```
```   449 unfolding setsum_diff1'[OF assms] by auto
```
```   450
```
```   451 (* By Jeremy Siek: *)
```
```   452
```
```   453 lemma setsum_diff_nat:
```
```   454 assumes "finite B" and "B \<subseteq> A"
```
```   455 shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```   456 using assms
```
```   457 proof induct
```
```   458   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```   459 next
```
```   460   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```   461     and xFinA: "insert x F \<subseteq> A"
```
```   462     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```   463   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```   464   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```   465     by (simp add: setsum_diff1_nat)
```
```   466   from xFinA have "F \<subseteq> A" by simp
```
```   467   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```   468   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```   469     by simp
```
```   470   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```   471   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```   472     by simp
```
```   473   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```   474   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```   475     by simp
```
```   476   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```   477 qed
```
```   478
```
```   479 lemma setsum_diff:
```
```   480   assumes le: "finite A" "B \<subseteq> A"
```
```   481   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```   482 proof -
```
```   483   from le have finiteB: "finite B" using finite_subset by auto
```
```   484   show ?thesis using finiteB le
```
```   485   proof induct
```
```   486     case empty
```
```   487     thus ?case by auto
```
```   488   next
```
```   489     case (insert x F)
```
```   490     thus ?case using le finiteB
```
```   491       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```   492   qed
```
```   493 qed
```
```   494
```
```   495 lemma setsum_mono:
```
```   496   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
```
```   497   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   498 proof (cases "finite K")
```
```   499   case True
```
```   500   thus ?thesis using le
```
```   501   proof induct
```
```   502     case empty
```
```   503     thus ?case by simp
```
```   504   next
```
```   505     case insert
```
```   506     thus ?case using add_mono by fastsimp
```
```   507   qed
```
```   508 next
```
```   509   case False
```
```   510   thus ?thesis
```
```   511     by (simp add: setsum_def)
```
```   512 qed
```
```   513
```
```   514 lemma setsum_strict_mono:
```
```   515   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```   516   assumes "finite A"  "A \<noteq> {}"
```
```   517     and "!!x. x:A \<Longrightarrow> f x < g x"
```
```   518   shows "setsum f A < setsum g A"
```
```   519   using assms
```
```   520 proof (induct rule: finite_ne_induct)
```
```   521   case singleton thus ?case by simp
```
```   522 next
```
```   523   case insert thus ?case by (auto simp: add_strict_mono)
```
```   524 qed
```
```   525
```
```   526 lemma setsum_negf:
```
```   527   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```   528 proof (cases "finite A")
```
```   529   case True thus ?thesis by (induct set: finite) auto
```
```   530 next
```
```   531   case False thus ?thesis by (simp add: setsum_def)
```
```   532 qed
```
```   533
```
```   534 lemma setsum_subtractf:
```
```   535   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```   536     setsum f A - setsum g A"
```
```   537 proof (cases "finite A")
```
```   538   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
```
```   539 next
```
```   540   case False thus ?thesis by (simp add: setsum_def)
```
```   541 qed
```
```   542
```
```   543 lemma setsum_nonneg:
```
```   544   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```   545   shows "0 \<le> setsum f A"
```
```   546 proof (cases "finite A")
```
```   547   case True thus ?thesis using nn
```
```   548   proof induct
```
```   549     case empty then show ?case by simp
```
```   550   next
```
```   551     case (insert x F)
```
```   552     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   553     with insert show ?case by simp
```
```   554   qed
```
```   555 next
```
```   556   case False thus ?thesis by (simp add: setsum_def)
```
```   557 qed
```
```   558
```
```   559 lemma setsum_nonpos:
```
```   560   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
```
```   561   shows "setsum f A \<le> 0"
```
```   562 proof (cases "finite A")
```
```   563   case True thus ?thesis using np
```
```   564   proof induct
```
```   565     case empty then show ?case by simp
```
```   566   next
```
```   567     case (insert x F)
```
```   568     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   569     with insert show ?case by simp
```
```   570   qed
```
```   571 next
```
```   572   case False thus ?thesis by (simp add: setsum_def)
```
```   573 qed
```
```   574
```
```   575 lemma setsum_nonneg_leq_bound:
```
```   576   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   577   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   578   shows "f i \<le> B"
```
```   579 proof -
```
```   580   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
```
```   581     using assms by (auto intro!: setsum_nonneg)
```
```   582   moreover
```
```   583   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
```
```   584     using assms by (simp add: setsum_diff1)
```
```   585   ultimately show ?thesis by auto
```
```   586 qed
```
```   587
```
```   588 lemma setsum_nonneg_0:
```
```   589   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   590   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
```
```   591   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
```
```   592   shows "f i = 0"
```
```   593   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
```
```   594
```
```   595 lemma setsum_mono2:
```
```   596 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
```
```   597 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   598 shows "setsum f A \<le> setsum f B"
```
```   599 proof -
```
```   600   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   601     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   602   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```   603     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```   604   also have "A \<union> (B-A) = B" using sub by blast
```
```   605   finally show ?thesis .
```
```   606 qed
```
```   607
```
```   608 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```   609     ALL x: B - A.
```
```   610       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
```
```   611         setsum f A <= setsum f B"
```
```   612   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```   613   apply (erule ssubst)
```
```   614   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```   615   apply simp
```
```   616   apply (rule add_left_mono)
```
```   617   apply (erule setsum_nonneg)
```
```   618   apply (subst setsum_Un_disjoint [THEN sym])
```
```   619   apply (erule finite_subset, assumption)
```
```   620   apply (rule finite_subset)
```
```   621   prefer 2
```
```   622   apply assumption
```
```   623   apply (auto simp add: sup_absorb2)
```
```   624 done
```
```   625
```
```   626 lemma setsum_right_distrib:
```
```   627   fixes f :: "'a => ('b::semiring_0)"
```
```   628   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```   629 proof (cases "finite A")
```
```   630   case True
```
```   631   thus ?thesis
```
```   632   proof induct
```
```   633     case empty thus ?case by simp
```
```   634   next
```
```   635     case (insert x A) thus ?case by (simp add: right_distrib)
```
```   636   qed
```
```   637 next
```
```   638   case False thus ?thesis by (simp add: setsum_def)
```
```   639 qed
```
```   640
```
```   641 lemma setsum_left_distrib:
```
```   642   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```   643 proof (cases "finite A")
```
```   644   case True
```
```   645   then show ?thesis
```
```   646   proof induct
```
```   647     case empty thus ?case by simp
```
```   648   next
```
```   649     case (insert x A) thus ?case by (simp add: left_distrib)
```
```   650   qed
```
```   651 next
```
```   652   case False thus ?thesis by (simp add: setsum_def)
```
```   653 qed
```
```   654
```
```   655 lemma setsum_divide_distrib:
```
```   656   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```   657 proof (cases "finite A")
```
```   658   case True
```
```   659   then show ?thesis
```
```   660   proof induct
```
```   661     case empty thus ?case by simp
```
```   662   next
```
```   663     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```   664   qed
```
```   665 next
```
```   666   case False thus ?thesis by (simp add: setsum_def)
```
```   667 qed
```
```   668
```
```   669 lemma setsum_abs[iff]:
```
```   670   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   671   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```   672 proof (cases "finite A")
```
```   673   case True
```
```   674   thus ?thesis
```
```   675   proof induct
```
```   676     case empty thus ?case by simp
```
```   677   next
```
```   678     case (insert x A)
```
```   679     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   680   qed
```
```   681 next
```
```   682   case False thus ?thesis by (simp add: setsum_def)
```
```   683 qed
```
```   684
```
```   685 lemma setsum_abs_ge_zero[iff]:
```
```   686   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   687   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```   688 proof (cases "finite A")
```
```   689   case True
```
```   690   thus ?thesis
```
```   691   proof induct
```
```   692     case empty thus ?case by simp
```
```   693   next
```
```   694     case (insert x A) thus ?case by auto
```
```   695   qed
```
```   696 next
```
```   697   case False thus ?thesis by (simp add: setsum_def)
```
```   698 qed
```
```   699
```
```   700 lemma abs_setsum_abs[simp]:
```
```   701   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   702   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```   703 proof (cases "finite A")
```
```   704   case True
```
```   705   thus ?thesis
```
```   706   proof induct
```
```   707     case empty thus ?case by simp
```
```   708   next
```
```   709     case (insert a A)
```
```   710     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
```
```   711     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```   712     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```   713       by (simp del: abs_of_nonneg)
```
```   714     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```   715     finally show ?case .
```
```   716   qed
```
```   717 next
```
```   718   case False thus ?thesis by (simp add: setsum_def)
```
```   719 qed
```
```   720
```
```   721 lemma setsum_Plus:
```
```   722   fixes A :: "'a set" and B :: "'b set"
```
```   723   assumes fin: "finite A" "finite B"
```
```   724   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
```
```   725 proof -
```
```   726   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   727   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
```
```   728     by auto
```
```   729   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
```
```   730   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
```
```   731   ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)
```
```   732 qed
```
```   733
```
```   734
```
```   735 text {* Commuting outer and inner summation *}
```
```   736
```
```   737 lemma setsum_commute:
```
```   738   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
```
```   739 proof (simp add: setsum_cartesian_product)
```
```   740   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
```
```   741     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
```
```   742     (is "?s = _")
```
```   743     apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
```
```   744     apply (simp add: split_def)
```
```   745     done
```
```   746   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
```
```   747     (is "_ = ?t")
```
```   748     apply (simp add: swap_product)
```
```   749     done
```
```   750   finally show "?s = ?t" .
```
```   751 qed
```
```   752
```
```   753 lemma setsum_product:
```
```   754   fixes f :: "'a => ('b::semiring_0)"
```
```   755   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   756   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
```
```   757
```
```   758 lemma setsum_mult_setsum_if_inj:
```
```   759 fixes f :: "'a => ('b::semiring_0)"
```
```   760 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
```
```   761   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
```
```   762 by(auto simp: setsum_product setsum_cartesian_product
```
```   763         intro!:  setsum_reindex_cong[symmetric])
```
```   764
```
```   765 lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
```
```   766 apply (cases "finite A")
```
```   767 apply (erule finite_induct)
```
```   768 apply (auto simp add: algebra_simps)
```
```   769 done
```
```   770
```
```   771 lemma setsum_bounded:
```
```   772   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
```
```   773   shows "setsum f A \<le> of_nat(card A) * K"
```
```   774 proof (cases "finite A")
```
```   775   case True
```
```   776   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```   777 next
```
```   778   case False thus ?thesis by (simp add: setsum_def)
```
```   779 qed
```
```   780
```
```   781
```
```   782 subsubsection {* Cardinality as special case of @{const setsum} *}
```
```   783
```
```   784 lemma card_eq_setsum:
```
```   785   "card A = setsum (\<lambda>x. 1) A"
```
```   786   by (simp only: card_def setsum_def)
```
```   787
```
```   788 lemma card_UN_disjoint:
```
```   789   "finite I ==> (ALL i:I. finite (A i)) ==>
```
```   790    (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {})
```
```   791    ==> card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```   792 apply (simp add: card_eq_setsum del: setsum_constant)
```
```   793 apply (subgoal_tac
```
```   794          "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
```
```   795 apply (simp add: setsum_UN_disjoint del: setsum_constant)
```
```   796 apply simp
```
```   797 done
```
```   798
```
```   799 lemma card_Union_disjoint:
```
```   800   "finite C ==> (ALL A:C. finite A) ==>
```
```   801    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
```
```   802    ==> card (Union C) = setsum card C"
```
```   803 apply (frule card_UN_disjoint [of C id])
```
```   804 apply (unfold Union_def id_def, assumption+)
```
```   805 done
```
```   806
```
```   807 text{*The image of a finite set can be expressed using @{term fold_image}.*}
```
```   808 lemma image_eq_fold_image:
```
```   809   "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A"
```
```   810 proof (induct rule: finite_induct)
```
```   811   case empty then show ?case by simp
```
```   812 next
```
```   813   interpret ab_semigroup_mult "op Un"
```
```   814     proof qed auto
```
```   815   case insert
```
```   816   then show ?case by simp
```
```   817 qed
```
```   818
```
```   819 subsubsection {* Cardinality of products *}
```
```   820
```
```   821 lemma card_SigmaI [simp]:
```
```   822   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```   823   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```   824 by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
```
```   825
```
```   826 (*
```
```   827 lemma SigmaI_insert: "y \<notin> A ==>
```
```   828   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```   829   by auto
```
```   830 *)
```
```   831
```
```   832 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```   833   by (cases "finite A \<and> finite B")
```
```   834     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   835
```
```   836 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```   837 by (simp add: card_cartesian_product)
```
```   838
```
```   839
```
```   840 subsection {* Generalized product over a set *}
```
```   841
```
```   842 definition (in comm_monoid_mult) setprod :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
```
```   843   "setprod f A = (if finite A then fold_image (op *) f 1 A else 1)"
```
```   844
```
```   845 sublocale comm_monoid_mult < setprod!: comm_monoid_big "op *" 1 setprod proof
```
```   846 qed (fact setprod_def)
```
```   847
```
```   848 abbreviation
```
```   849   Setprod  ("\<Prod>_"  999) where
```
```   850   "\<Prod>A == setprod (%x. x) A"
```
```   851
```
```   852 syntax
```
```   853   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```   854 syntax (xsymbols)
```
```   855   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```   856 syntax (HTML output)
```
```   857   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```   858
```
```   859 translations -- {* Beware of argument permutation! *}
```
```   860   "PROD i:A. b" == "CONST setprod (%i. b) A"
```
```   861   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
```
```   862
```
```   863 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```   864  @{text"\<Prod>x|P. e"}. *}
```
```   865
```
```   866 syntax
```
```   867   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```   868 syntax (xsymbols)
```
```   869   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```   870 syntax (HTML output)
```
```   871   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```   872
```
```   873 translations
```
```   874   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```   875   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```   876
```
```   877 lemma setprod_empty: "setprod f {} = 1"
```
```   878   by (fact setprod.empty)
```
```   879
```
```   880 lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
```
```   881     setprod f (insert a A) = f a * setprod f A"
```
```   882   by (fact setprod.insert)
```
```   883
```
```   884 lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
```
```   885   by (fact setprod.infinite)
```
```   886
```
```   887 lemma setprod_reindex:
```
```   888    "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```   889 by(auto simp: setprod_def fold_image_reindex dest!:finite_imageD)
```
```   890
```
```   891 lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```   892 by (auto simp add: setprod_reindex)
```
```   893
```
```   894 lemma setprod_cong:
```
```   895   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```   896 by(fastsimp simp: setprod_def intro: fold_image_cong)
```
```   897
```
```   898 lemma strong_setprod_cong[cong]:
```
```   899   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
```
```   900 by(fastsimp simp: simp_implies_def setprod_def intro: fold_image_cong)
```
```   901
```
```   902 lemma setprod_reindex_cong: "inj_on f A ==>
```
```   903     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```   904 by (frule setprod_reindex, simp)
```
```   905
```
```   906 lemma strong_setprod_reindex_cong: assumes i: "inj_on f A"
```
```   907   and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
```
```   908   shows "setprod h B = setprod g A"
```
```   909 proof-
```
```   910     have "setprod h B = setprod (h o f) A"
```
```   911       by (simp add: B setprod_reindex[OF i, of h])
```
```   912     then show ?thesis apply simp
```
```   913       apply (rule setprod_cong)
```
```   914       apply simp
```
```   915       by (simp add: eq)
```
```   916 qed
```
```   917
```
```   918 lemma setprod_Un_one:
```
```   919   assumes fS: "finite S" and fT: "finite T"
```
```   920   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
```
```   921   shows "setprod f (S \<union> T) = setprod f S  * setprod f T"
```
```   922   using fS fT
```
```   923   apply (simp add: setprod_def)
```
```   924   apply (rule fold_image_Un_one)
```
```   925   using I0 by auto
```
```   926
```
```   927
```
```   928 lemma setprod_1: "setprod (%i. 1) A = 1"
```
```   929 apply (case_tac "finite A")
```
```   930 apply (erule finite_induct, auto simp add: mult_ac)
```
```   931 done
```
```   932
```
```   933 lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
```
```   934 apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
```
```   935 apply (erule ssubst, rule setprod_1)
```
```   936 apply (rule setprod_cong, auto)
```
```   937 done
```
```   938
```
```   939 lemma setprod_Un_Int: "finite A ==> finite B
```
```   940     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```   941 by(simp add: setprod_def fold_image_Un_Int[symmetric])
```
```   942
```
```   943 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```   944   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```   945 by (subst setprod_Un_Int [symmetric], auto)
```
```   946
```
```   947 lemma setprod_mono_one_left:
```
```   948   assumes fT: "finite T" and ST: "S \<subseteq> T"
```
```   949   and z: "\<forall>i \<in> T - S. f i = 1"
```
```   950   shows "setprod f S = setprod f T"
```
```   951 proof-
```
```   952   have eq: "T = S \<union> (T - S)" using ST by blast
```
```   953   have d: "S \<inter> (T - S) = {}" using ST by blast
```
```   954   from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
```
```   955   show ?thesis
```
```   956   by (simp add: setprod_Un_disjoint[OF f d, unfolded eq[symmetric]] setprod_1'[OF z])
```
```   957 qed
```
```   958
```
```   959 lemmas setprod_mono_one_right = setprod_mono_one_left [THEN sym]
```
```   960
```
```   961 lemma setprod_delta:
```
```   962   assumes fS: "finite S"
```
```   963   shows "setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   964 proof-
```
```   965   let ?f = "(\<lambda>k. if k=a then b k else 1)"
```
```   966   {assume a: "a \<notin> S"
```
```   967     hence "\<forall> k\<in> S. ?f k = 1" by simp
```
```   968     hence ?thesis  using a by (simp add: setprod_1) }
```
```   969   moreover
```
```   970   {assume a: "a \<in> S"
```
```   971     let ?A = "S - {a}"
```
```   972     let ?B = "{a}"
```
```   973     have eq: "S = ?A \<union> ?B" using a by blast
```
```   974     have dj: "?A \<inter> ?B = {}" by simp
```
```   975     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   976     have fA1: "setprod ?f ?A = 1" apply (rule setprod_1') by auto
```
```   977     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```   978       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```   979       by simp
```
```   980     then have ?thesis using a by (simp add: fA1 cong: setprod_cong cong del: if_weak_cong)}
```
```   981   ultimately show ?thesis by blast
```
```   982 qed
```
```   983
```
```   984 lemma setprod_delta':
```
```   985   assumes fS: "finite S" shows
```
```   986   "setprod (\<lambda>k. if a = k then b k else 1) S =
```
```   987      (if a\<in> S then b a else 1)"
```
```   988   using setprod_delta[OF fS, of a b, symmetric]
```
```   989   by (auto intro: setprod_cong)
```
```   990
```
```   991
```
```   992 lemma setprod_UN_disjoint:
```
```   993     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```   994         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```   995       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```   996   by (simp add: setprod_def fold_image_UN_disjoint)
```
```   997
```
```   998 lemma setprod_Union_disjoint:
```
```   999   "[| (ALL A:C. finite A);
```
```  1000       (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
```
```  1001    ==> setprod f (Union C) = setprod (setprod f) C"
```
```  1002 apply (cases "finite C")
```
```  1003  prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
```
```  1004   apply (frule setprod_UN_disjoint [of C id f])
```
```  1005  apply (unfold Union_def id_def, assumption+)
```
```  1006 done
```
```  1007
```
```  1008 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1009     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
```
```  1010     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```  1011 by(simp add:setprod_def fold_image_Sigma split_def)
```
```  1012
```
```  1013 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1014 lemma setprod_cartesian_product:
```
```  1015      "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
```
```  1016 apply (cases "finite A")
```
```  1017  apply (cases "finite B")
```
```  1018   apply (simp add: setprod_Sigma)
```
```  1019  apply (cases "A={}", simp)
```
```  1020  apply (simp add: setprod_1)
```
```  1021 apply (auto simp add: setprod_def
```
```  1022             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1023 done
```
```  1024
```
```  1025 lemma setprod_timesf:
```
```  1026      "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
```
```  1027 by(simp add:setprod_def fold_image_distrib)
```
```  1028
```
```  1029
```
```  1030 subsubsection {* Properties in more restricted classes of structures *}
```
```  1031
```
```  1032 lemma setprod_eq_1_iff [simp]:
```
```  1033   "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
```
```  1034 by (induct set: finite) auto
```
```  1035
```
```  1036 lemma setprod_zero:
```
```  1037      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
```
```  1038 apply (induct set: finite, force, clarsimp)
```
```  1039 apply (erule disjE, auto)
```
```  1040 done
```
```  1041
```
```  1042 lemma setprod_nonneg [rule_format]:
```
```  1043    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1044 by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
```
```  1045
```
```  1046 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
```
```  1047   --> 0 < setprod f A"
```
```  1048 by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
```
```  1049
```
```  1050 lemma setprod_zero_iff[simp]: "finite A ==>
```
```  1051   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
```
```  1052   (EX x: A. f x = 0)"
```
```  1053 by (erule finite_induct, auto simp:no_zero_divisors)
```
```  1054
```
```  1055 lemma setprod_pos_nat:
```
```  1056   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
```
```  1057 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1058
```
```  1059 lemma setprod_pos_nat_iff[simp]:
```
```  1060   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
```
```  1061 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1062
```
```  1063 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1064   (setprod f (A Un B) :: 'a ::{field})
```
```  1065    = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1066 by (subst setprod_Un_Int [symmetric], auto)
```
```  1067
```
```  1068 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1069   (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1070   (if a:A then setprod f A / f a else setprod f A)"
```
```  1071   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```  1072
```
```  1073 lemma setprod_inversef:
```
```  1074   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1075   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1076 by (erule finite_induct) auto
```
```  1077
```
```  1078 lemma setprod_dividef:
```
```  1079   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1080   shows "finite A
```
```  1081     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1082 apply (subgoal_tac
```
```  1083          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1084 apply (erule ssubst)
```
```  1085 apply (subst divide_inverse)
```
```  1086 apply (subst setprod_timesf)
```
```  1087 apply (subst setprod_inversef, assumption+, rule refl)
```
```  1088 apply (rule setprod_cong, rule refl)
```
```  1089 apply (subst divide_inverse, auto)
```
```  1090 done
```
```  1091
```
```  1092 lemma setprod_dvd_setprod [rule_format]:
```
```  1093     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
```
```  1094   apply (cases "finite A")
```
```  1095   apply (induct set: finite)
```
```  1096   apply (auto simp add: dvd_def)
```
```  1097   apply (rule_tac x = "k * ka" in exI)
```
```  1098   apply (simp add: algebra_simps)
```
```  1099 done
```
```  1100
```
```  1101 lemma setprod_dvd_setprod_subset:
```
```  1102   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1103   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
```
```  1104   apply (unfold dvd_def, blast)
```
```  1105   apply (subst setprod_Un_disjoint [symmetric])
```
```  1106   apply (auto elim: finite_subset intro: setprod_cong)
```
```  1107 done
```
```  1108
```
```  1109 lemma setprod_dvd_setprod_subset2:
```
```  1110   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow>
```
```  1111       setprod f A dvd setprod g B"
```
```  1112   apply (rule dvd_trans)
```
```  1113   apply (rule setprod_dvd_setprod, erule (1) bspec)
```
```  1114   apply (erule (1) setprod_dvd_setprod_subset)
```
```  1115 done
```
```  1116
```
```  1117 lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow>
```
```  1118     (f i ::'a::comm_semiring_1) dvd setprod f A"
```
```  1119 by (induct set: finite) (auto intro: dvd_mult)
```
```  1120
```
```  1121 lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow>
```
```  1122     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
```
```  1123   apply (cases "finite A")
```
```  1124   apply (induct set: finite)
```
```  1125   apply auto
```
```  1126 done
```
```  1127
```
```  1128 lemma setprod_mono:
```
```  1129   fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
```
```  1130   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
```
```  1131   shows "setprod f A \<le> setprod g A"
```
```  1132 proof (cases "finite A")
```
```  1133   case True
```
```  1134   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
```
```  1135   proof (induct A rule: finite_subset_induct)
```
```  1136     case (insert a F)
```
```  1137     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
```
```  1138       unfolding setprod_insert[OF insert(1,3)]
```
```  1139       using assms[rule_format,OF insert(2)] insert
```
```  1140       by (auto intro: mult_mono mult_nonneg_nonneg)
```
```  1141   qed auto
```
```  1142   thus ?thesis by simp
```
```  1143 qed auto
```
```  1144
```
```  1145 lemma abs_setprod:
```
```  1146   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
```
```  1147   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
```
```  1148 proof (cases "finite A")
```
```  1149   case True thus ?thesis
```
```  1150     by induct (auto simp add: field_simps abs_mult)
```
```  1151 qed auto
```
```  1152
```
```  1153 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
```
```  1154 apply (erule finite_induct)
```
```  1155 apply auto
```
```  1156 done
```
```  1157
```
```  1158 lemma setprod_gen_delta:
```
```  1159   assumes fS: "finite S"
```
```  1160   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)"
```
```  1161 proof-
```
```  1162   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```  1163   {assume a: "a \<notin> S"
```
```  1164     hence "\<forall> k\<in> S. ?f k = c" by simp
```
```  1165     hence ?thesis  using a setprod_constant[OF fS, of c] by (simp add: setprod_1 cong add: setprod_cong) }
```
```  1166   moreover
```
```  1167   {assume a: "a \<in> S"
```
```  1168     let ?A = "S - {a}"
```
```  1169     let ?B = "{a}"
```
```  1170     have eq: "S = ?A \<union> ?B" using a by blast
```
```  1171     have dj: "?A \<inter> ?B = {}" by simp
```
```  1172     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```  1173     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```  1174       apply (rule setprod_cong) by auto
```
```  1175     have cA: "card ?A = card S - 1" using fS a by auto
```
```  1176     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
```
```  1177     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```  1178       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```  1179       by simp
```
```  1180     then have ?thesis using a cA
```
```  1181       by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
```
```  1182   ultimately show ?thesis by blast
```
```  1183 qed
```
```  1184
```
```  1185
```
```  1186 subsection {* Versions of @{const inf} and @{const sup} on non-empty sets *}
```
```  1187
```
```  1188 no_notation times (infixl "*" 70)
```
```  1189 no_notation Groups.one ("1")
```
```  1190
```
```  1191 locale semilattice_big = semilattice +
```
```  1192   fixes F :: "'a set \<Rightarrow> 'a"
```
```  1193   assumes F_eq: "finite A \<Longrightarrow> F A = fold1 (op *) A"
```
```  1194
```
```  1195 sublocale semilattice_big < folding_one_idem proof
```
```  1196 qed (simp_all add: F_eq)
```
```  1197
```
```  1198 notation times (infixl "*" 70)
```
```  1199 notation Groups.one ("1")
```
```  1200
```
```  1201 context lattice
```
```  1202 begin
```
```  1203
```
```  1204 definition Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_"  900) where
```
```  1205   "Inf_fin = fold1 inf"
```
```  1206
```
```  1207 definition Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_"  900) where
```
```  1208   "Sup_fin = fold1 sup"
```
```  1209
```
```  1210 end
```
```  1211
```
```  1212 sublocale lattice < Inf_fin!: semilattice_big inf Inf_fin proof
```
```  1213 qed (simp add: Inf_fin_def)
```
```  1214
```
```  1215 sublocale lattice < Sup_fin!: semilattice_big sup Sup_fin proof
```
```  1216 qed (simp add: Sup_fin_def)
```
```  1217
```
```  1218 context semilattice_inf
```
```  1219 begin
```
```  1220
```
```  1221 lemma ab_semigroup_idem_mult_inf:
```
```  1222   "class.ab_semigroup_idem_mult inf"
```
```  1223 proof qed (rule inf_assoc inf_commute inf_idem)+
```
```  1224
```
```  1225 lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> fold inf b (insert a A) = inf a (fold inf b A)"
```
```  1226 by(rule comp_fun_idem.fold_insert_idem[OF ab_semigroup_idem_mult.comp_fun_idem[OF ab_semigroup_idem_mult_inf]])
```
```  1227
```
```  1228 lemma inf_le_fold_inf: "finite A \<Longrightarrow> ALL a:A. b \<le> a \<Longrightarrow> inf b c \<le> fold inf c A"
```
```  1229 by (induct pred: finite) (auto intro: le_infI1)
```
```  1230
```
```  1231 lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> fold inf b A \<le> inf a b"
```
```  1232 proof(induct arbitrary: a pred:finite)
```
```  1233   case empty thus ?case by simp
```
```  1234 next
```
```  1235   case (insert x A)
```
```  1236   show ?case
```
```  1237   proof cases
```
```  1238     assume "A = {}" thus ?thesis using insert by simp
```
```  1239   next
```
```  1240     assume "A \<noteq> {}" thus ?thesis using insert by (auto intro: le_infI2)
```
```  1241   qed
```
```  1242 qed
```
```  1243
```
```  1244 lemma below_fold1_iff:
```
```  1245   assumes "finite A" "A \<noteq> {}"
```
```  1246   shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  1247 proof -
```
```  1248   interpret ab_semigroup_idem_mult inf
```
```  1249     by (rule ab_semigroup_idem_mult_inf)
```
```  1250   show ?thesis using assms by (induct rule: finite_ne_induct) simp_all
```
```  1251 qed
```
```  1252
```
```  1253 lemma fold1_belowI:
```
```  1254   assumes "finite A"
```
```  1255     and "a \<in> A"
```
```  1256   shows "fold1 inf A \<le> a"
```
```  1257 proof -
```
```  1258   from assms have "A \<noteq> {}" by auto
```
```  1259   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```  1260   proof (induct rule: finite_ne_induct)
```
```  1261     case singleton thus ?case by simp
```
```  1262   next
```
```  1263     interpret ab_semigroup_idem_mult inf
```
```  1264       by (rule ab_semigroup_idem_mult_inf)
```
```  1265     case (insert x F)
```
```  1266     from insert(5) have "a = x \<or> a \<in> F" by simp
```
```  1267     thus ?case
```
```  1268     proof
```
```  1269       assume "a = x" thus ?thesis using insert
```
```  1270         by (simp add: mult_ac)
```
```  1271     next
```
```  1272       assume "a \<in> F"
```
```  1273       hence bel: "fold1 inf F \<le> a" by (rule insert)
```
```  1274       have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
```
```  1275         using insert by (simp add: mult_ac)
```
```  1276       also have "inf (fold1 inf F) a = fold1 inf F"
```
```  1277         using bel by (auto intro: antisym)
```
```  1278       also have "inf x \<dots> = fold1 inf (insert x F)"
```
```  1279         using insert by (simp add: mult_ac)
```
```  1280       finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
```
```  1281       moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
```
```  1282       ultimately show ?thesis by simp
```
```  1283     qed
```
```  1284   qed
```
```  1285 qed
```
```  1286
```
```  1287 end
```
```  1288
```
```  1289 context semilattice_sup
```
```  1290 begin
```
```  1291
```
```  1292 lemma ab_semigroup_idem_mult_sup: "class.ab_semigroup_idem_mult sup"
```
```  1293 by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
```
```  1294
```
```  1295 lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> fold sup b (insert a A) = sup a (fold sup b A)"
```
```  1296 by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)
```
```  1297
```
```  1298 lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> fold sup c A \<le> sup b c"
```
```  1299 by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)
```
```  1300
```
```  1301 lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> fold sup b A"
```
```  1302 by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)
```
```  1303
```
```  1304 end
```
```  1305
```
```  1306 context lattice
```
```  1307 begin
```
```  1308
```
```  1309 lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
```
```  1310 apply(unfold Sup_fin_def Inf_fin_def)
```
```  1311 apply(subgoal_tac "EX a. a:A")
```
```  1312 prefer 2 apply blast
```
```  1313 apply(erule exE)
```
```  1314 apply(rule order_trans)
```
```  1315 apply(erule (1) fold1_belowI)
```
```  1316 apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice])
```
```  1317 done
```
```  1318
```
```  1319 lemma sup_Inf_absorb [simp]:
```
```  1320   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
```
```  1321 apply(subst sup_commute)
```
```  1322 apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)
```
```  1323 done
```
```  1324
```
```  1325 lemma inf_Sup_absorb [simp]:
```
```  1326   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
```
```  1327 by (simp add: Sup_fin_def inf_absorb1
```
```  1328   semilattice_inf.fold1_belowI [OF dual_semilattice])
```
```  1329
```
```  1330 end
```
```  1331
```
```  1332 context distrib_lattice
```
```  1333 begin
```
```  1334
```
```  1335 lemma sup_Inf1_distrib:
```
```  1336   assumes "finite A"
```
```  1337     and "A \<noteq> {}"
```
```  1338   shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
```
```  1339 proof -
```
```  1340   interpret ab_semigroup_idem_mult inf
```
```  1341     by (rule ab_semigroup_idem_mult_inf)
```
```  1342   from assms show ?thesis
```
```  1343     by (simp add: Inf_fin_def image_def
```
```  1344       hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])
```
```  1345         (rule arg_cong [where f="fold1 inf"], blast)
```
```  1346 qed
```
```  1347
```
```  1348 lemma sup_Inf2_distrib:
```
```  1349   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1350   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}"
```
```  1351 using A proof (induct rule: finite_ne_induct)
```
```  1352   case singleton thus ?case
```
```  1353     by (simp add: sup_Inf1_distrib [OF B])
```
```  1354 next
```
```  1355   interpret ab_semigroup_idem_mult inf
```
```  1356     by (rule ab_semigroup_idem_mult_inf)
```
```  1357   case (insert x A)
```
```  1358   have finB: "finite {sup x b |b. b \<in> B}"
```
```  1359     by(rule finite_surj[where f = "sup x", OF B(1)], auto)
```
```  1360   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1361   proof -
```
```  1362     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
```
```  1363       by blast
```
```  1364     thus ?thesis by(simp add: insert(1) B(1))
```
```  1365   qed
```
```  1366   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1367   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)"
```
```  1368     using insert by simp
```
```  1369   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)
```
```  1370   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})"
```
```  1371     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  1372   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})"
```
```  1373     (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
```
```  1374     using B insert
```
```  1375     by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
```
```  1376   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1377     by blast
```
```  1378   finally show ?case .
```
```  1379 qed
```
```  1380
```
```  1381 lemma inf_Sup1_distrib:
```
```  1382   assumes "finite A" and "A \<noteq> {}"
```
```  1383   shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
```
```  1384 proof -
```
```  1385   interpret ab_semigroup_idem_mult sup
```
```  1386     by (rule ab_semigroup_idem_mult_sup)
```
```  1387   from assms show ?thesis
```
```  1388     by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])
```
```  1389       (rule arg_cong [where f="fold1 sup"], blast)
```
```  1390 qed
```
```  1391
```
```  1392 lemma inf_Sup2_distrib:
```
```  1393   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1394   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}"
```
```  1395 using A proof (induct rule: finite_ne_induct)
```
```  1396   case singleton thus ?case
```
```  1397     by(simp add: inf_Sup1_distrib [OF B] fold1_singleton_def [OF Sup_fin_def])
```
```  1398 next
```
```  1399   case (insert x A)
```
```  1400   have finB: "finite {inf x b |b. b \<in> B}"
```
```  1401     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
```
```  1402   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1403   proof -
```
```  1404     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
```
```  1405       by blast
```
```  1406     thus ?thesis by(simp add: insert(1) B(1))
```
```  1407   qed
```
```  1408   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1409   interpret ab_semigroup_idem_mult sup
```
```  1410     by (rule ab_semigroup_idem_mult_sup)
```
```  1411   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)"
```
```  1412     using insert by simp
```
```  1413   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)
```
```  1414   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})"
```
```  1415     using insert by(simp add:inf_Sup1_distrib[OF B])
```
```  1416   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})"
```
```  1417     (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
```
```  1418     using B insert
```
```  1419     by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
```
```  1420   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1421     by blast
```
```  1422   finally show ?case .
```
```  1423 qed
```
```  1424
```
```  1425 end
```
```  1426
```
```  1427 context complete_lattice
```
```  1428 begin
```
```  1429
```
```  1430 lemma Inf_fin_Inf:
```
```  1431   assumes "finite A" and "A \<noteq> {}"
```
```  1432   shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
```
```  1433 proof -
```
```  1434   interpret ab_semigroup_idem_mult inf
```
```  1435     by (rule ab_semigroup_idem_mult_inf)
```
```  1436   from `A \<noteq> {}` obtain b B where "A = insert b B" by auto
```
```  1437   moreover with `finite A` have "finite B" by simp
```
```  1438   ultimately show ?thesis
```
```  1439   by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])
```
```  1440     (simp add: Inf_fold_inf)
```
```  1441 qed
```
```  1442
```
```  1443 lemma Sup_fin_Sup:
```
```  1444   assumes "finite A" and "A \<noteq> {}"
```
```  1445   shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
```
```  1446 proof -
```
```  1447   interpret ab_semigroup_idem_mult sup
```
```  1448     by (rule ab_semigroup_idem_mult_sup)
```
```  1449   from `A \<noteq> {}` obtain b B where "A = insert b B" by auto
```
```  1450   moreover with `finite A` have "finite B" by simp
```
```  1451   ultimately show ?thesis
```
```  1452   by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric])
```
```  1453     (simp add: Sup_fold_sup)
```
```  1454 qed
```
```  1455
```
```  1456 end
```
```  1457
```
```  1458
```
```  1459 subsection {* Versions of @{const min} and @{const max} on non-empty sets *}
```
```  1460
```
```  1461 definition (in linorder) Min :: "'a set \<Rightarrow> 'a" where
```
```  1462   "Min = fold1 min"
```
```  1463
```
```  1464 definition (in linorder) Max :: "'a set \<Rightarrow> 'a" where
```
```  1465   "Max = fold1 max"
```
```  1466
```
```  1467 sublocale linorder < Min!: semilattice_big min Min proof
```
```  1468 qed (simp add: Min_def)
```
```  1469
```
```  1470 sublocale linorder < Max!: semilattice_big max Max proof
```
```  1471 qed (simp add: Max_def)
```
```  1472
```
```  1473 context linorder
```
```  1474 begin
```
```  1475
```
```  1476 lemmas Min_singleton = Min.singleton
```
```  1477 lemmas Max_singleton = Max.singleton
```
```  1478
```
```  1479 lemma Min_insert:
```
```  1480   assumes "finite A" and "A \<noteq> {}"
```
```  1481   shows "Min (insert x A) = min x (Min A)"
```
```  1482   using assms by simp
```
```  1483
```
```  1484 lemma Max_insert:
```
```  1485   assumes "finite A" and "A \<noteq> {}"
```
```  1486   shows "Max (insert x A) = max x (Max A)"
```
```  1487   using assms by simp
```
```  1488
```
```  1489 lemma Min_Un:
```
```  1490   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
```
```  1491   shows "Min (A \<union> B) = min (Min A) (Min B)"
```
```  1492   using assms by (rule Min.union_idem)
```
```  1493
```
```  1494 lemma Max_Un:
```
```  1495   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
```
```  1496   shows "Max (A \<union> B) = max (Max A) (Max B)"
```
```  1497   using assms by (rule Max.union_idem)
```
```  1498
```
```  1499 lemma hom_Min_commute:
```
```  1500   assumes "\<And>x y. h (min x y) = min (h x) (h y)"
```
```  1501     and "finite N" and "N \<noteq> {}"
```
```  1502   shows "h (Min N) = Min (h ` N)"
```
```  1503   using assms by (rule Min.hom_commute)
```
```  1504
```
```  1505 lemma hom_Max_commute:
```
```  1506   assumes "\<And>x y. h (max x y) = max (h x) (h y)"
```
```  1507     and "finite N" and "N \<noteq> {}"
```
```  1508   shows "h (Max N) = Max (h ` N)"
```
```  1509   using assms by (rule Max.hom_commute)
```
```  1510
```
```  1511 lemma ab_semigroup_idem_mult_min:
```
```  1512   "class.ab_semigroup_idem_mult min"
```
```  1513   proof qed (auto simp add: min_def)
```
```  1514
```
```  1515 lemma ab_semigroup_idem_mult_max:
```
```  1516   "class.ab_semigroup_idem_mult max"
```
```  1517   proof qed (auto simp add: max_def)
```
```  1518
```
```  1519 lemma max_lattice:
```
```  1520   "class.semilattice_inf max (op \<ge>) (op >)"
```
```  1521   by (fact min_max.dual_semilattice)
```
```  1522
```
```  1523 lemma dual_max:
```
```  1524   "ord.max (op \<ge>) = min"
```
```  1525   by (auto simp add: ord.max_def_raw min_def fun_eq_iff)
```
```  1526
```
```  1527 lemma dual_min:
```
```  1528   "ord.min (op \<ge>) = max"
```
```  1529   by (auto simp add: ord.min_def_raw max_def fun_eq_iff)
```
```  1530
```
```  1531 lemma strict_below_fold1_iff:
```
```  1532   assumes "finite A" and "A \<noteq> {}"
```
```  1533   shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  1534 proof -
```
```  1535   interpret ab_semigroup_idem_mult min
```
```  1536     by (rule ab_semigroup_idem_mult_min)
```
```  1537   from assms show ?thesis
```
```  1538   by (induct rule: finite_ne_induct)
```
```  1539     (simp_all add: fold1_insert)
```
```  1540 qed
```
```  1541
```
```  1542 lemma fold1_below_iff:
```
```  1543   assumes "finite A" and "A \<noteq> {}"
```
```  1544   shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  1545 proof -
```
```  1546   interpret ab_semigroup_idem_mult min
```
```  1547     by (rule ab_semigroup_idem_mult_min)
```
```  1548   from assms show ?thesis
```
```  1549   by (induct rule: finite_ne_induct)
```
```  1550     (simp_all add: fold1_insert min_le_iff_disj)
```
```  1551 qed
```
```  1552
```
```  1553 lemma fold1_strict_below_iff:
```
```  1554   assumes "finite A" and "A \<noteq> {}"
```
```  1555   shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  1556 proof -
```
```  1557   interpret ab_semigroup_idem_mult min
```
```  1558     by (rule ab_semigroup_idem_mult_min)
```
```  1559   from assms show ?thesis
```
```  1560   by (induct rule: finite_ne_induct)
```
```  1561     (simp_all add: fold1_insert min_less_iff_disj)
```
```  1562 qed
```
```  1563
```
```  1564 lemma fold1_antimono:
```
```  1565   assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
```
```  1566   shows "fold1 min B \<le> fold1 min A"
```
```  1567 proof cases
```
```  1568   assume "A = B" thus ?thesis by simp
```
```  1569 next
```
```  1570   interpret ab_semigroup_idem_mult min
```
```  1571     by (rule ab_semigroup_idem_mult_min)
```
```  1572   assume neq: "A \<noteq> B"
```
```  1573   have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
```
```  1574   have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
```
```  1575   also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
```
```  1576   proof -
```
```  1577     have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
```
```  1578     moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`])
```
```  1579     moreover have "(B-A) \<noteq> {}" using assms neq by blast
```
```  1580     moreover have "A Int (B-A) = {}" using assms by blast
```
```  1581     ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
```
```  1582   qed
```
```  1583   also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
```
```  1584   finally show ?thesis .
```
```  1585 qed
```
```  1586
```
```  1587 lemma Min_in [simp]:
```
```  1588   assumes "finite A" and "A \<noteq> {}"
```
```  1589   shows "Min A \<in> A"
```
```  1590 proof -
```
```  1591   interpret ab_semigroup_idem_mult min
```
```  1592     by (rule ab_semigroup_idem_mult_min)
```
```  1593   from assms fold1_in show ?thesis by (fastsimp simp: Min_def min_def)
```
```  1594 qed
```
```  1595
```
```  1596 lemma Max_in [simp]:
```
```  1597   assumes "finite A" and "A \<noteq> {}"
```
```  1598   shows "Max A \<in> A"
```
```  1599 proof -
```
```  1600   interpret ab_semigroup_idem_mult max
```
```  1601     by (rule ab_semigroup_idem_mult_max)
```
```  1602   from assms fold1_in [of A] show ?thesis by (fastsimp simp: Max_def max_def)
```
```  1603 qed
```
```  1604
```
```  1605 lemma Min_le [simp]:
```
```  1606   assumes "finite A" and "x \<in> A"
```
```  1607   shows "Min A \<le> x"
```
```  1608   using assms by (simp add: Min_def min_max.fold1_belowI)
```
```  1609
```
```  1610 lemma Max_ge [simp]:
```
```  1611   assumes "finite A" and "x \<in> A"
```
```  1612   shows "x \<le> Max A"
```
```  1613 proof -
```
```  1614   interpret semilattice_inf max "op \<ge>" "op >"
```
```  1615     by (rule max_lattice)
```
```  1616   from assms show ?thesis by (simp add: Max_def fold1_belowI)
```
```  1617 qed
```
```  1618
```
```  1619 lemma Min_ge_iff [simp, no_atp]:
```
```  1620   assumes "finite A" and "A \<noteq> {}"
```
```  1621   shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  1622   using assms by (simp add: Min_def min_max.below_fold1_iff)
```
```  1623
```
```  1624 lemma Max_le_iff [simp, no_atp]:
```
```  1625   assumes "finite A" and "A \<noteq> {}"
```
```  1626   shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
```
```  1627 proof -
```
```  1628   interpret semilattice_inf max "op \<ge>" "op >"
```
```  1629     by (rule max_lattice)
```
```  1630   from assms show ?thesis by (simp add: Max_def below_fold1_iff)
```
```  1631 qed
```
```  1632
```
```  1633 lemma Min_gr_iff [simp, no_atp]:
```
```  1634   assumes "finite A" and "A \<noteq> {}"
```
```  1635   shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  1636   using assms by (simp add: Min_def strict_below_fold1_iff)
```
```  1637
```
```  1638 lemma Max_less_iff [simp, no_atp]:
```
```  1639   assumes "finite A" and "A \<noteq> {}"
```
```  1640   shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
```
```  1641 proof -
```
```  1642   interpret dual: linorder "op \<ge>" "op >"
```
```  1643     by (rule dual_linorder)
```
```  1644   from assms show ?thesis
```
```  1645     by (simp add: Max_def dual.strict_below_fold1_iff [folded dual.dual_max])
```
```  1646 qed
```
```  1647
```
```  1648 lemma Min_le_iff [no_atp]:
```
```  1649   assumes "finite A" and "A \<noteq> {}"
```
```  1650   shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  1651   using assms by (simp add: Min_def fold1_below_iff)
```
```  1652
```
```  1653 lemma Max_ge_iff [no_atp]:
```
```  1654   assumes "finite A" and "A \<noteq> {}"
```
```  1655   shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
```
```  1656 proof -
```
```  1657   interpret dual: linorder "op \<ge>" "op >"
```
```  1658     by (rule dual_linorder)
```
```  1659   from assms show ?thesis
```
```  1660     by (simp add: Max_def dual.fold1_below_iff [folded dual.dual_max])
```
```  1661 qed
```
```  1662
```
```  1663 lemma Min_less_iff [no_atp]:
```
```  1664   assumes "finite A" and "A \<noteq> {}"
```
```  1665   shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  1666   using assms by (simp add: Min_def fold1_strict_below_iff)
```
```  1667
```
```  1668 lemma Max_gr_iff [no_atp]:
```
```  1669   assumes "finite A" and "A \<noteq> {}"
```
```  1670   shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
```
```  1671 proof -
```
```  1672   interpret dual: linorder "op \<ge>" "op >"
```
```  1673     by (rule dual_linorder)
```
```  1674   from assms show ?thesis
```
```  1675     by (simp add: Max_def dual.fold1_strict_below_iff [folded dual.dual_max])
```
```  1676 qed
```
```  1677
```
```  1678 lemma Min_eqI:
```
```  1679   assumes "finite A"
```
```  1680   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
```
```  1681     and "x \<in> A"
```
```  1682   shows "Min A = x"
```
```  1683 proof (rule antisym)
```
```  1684   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1685   with assms show "Min A \<ge> x" by simp
```
```  1686 next
```
```  1687   from assms show "x \<ge> Min A" by simp
```
```  1688 qed
```
```  1689
```
```  1690 lemma Max_eqI:
```
```  1691   assumes "finite A"
```
```  1692   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
```
```  1693     and "x \<in> A"
```
```  1694   shows "Max A = x"
```
```  1695 proof (rule antisym)
```
```  1696   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1697   with assms show "Max A \<le> x" by simp
```
```  1698 next
```
```  1699   from assms show "x \<le> Max A" by simp
```
```  1700 qed
```
```  1701
```
```  1702 lemma Min_antimono:
```
```  1703   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  1704   shows "Min N \<le> Min M"
```
```  1705   using assms by (simp add: Min_def fold1_antimono)
```
```  1706
```
```  1707 lemma Max_mono:
```
```  1708   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  1709   shows "Max M \<le> Max N"
```
```  1710 proof -
```
```  1711   interpret dual: linorder "op \<ge>" "op >"
```
```  1712     by (rule dual_linorder)
```
```  1713   from assms show ?thesis
```
```  1714     by (simp add: Max_def dual.fold1_antimono [folded dual.dual_max])
```
```  1715 qed
```
```  1716
```
```  1717 lemma finite_linorder_max_induct[consumes 1, case_names empty insert]:
```
```  1718  assumes fin: "finite A"
```
```  1719  and   empty: "P {}"
```
```  1720  and  insert: "(!!b A. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))"
```
```  1721  shows "P A"
```
```  1722 using fin empty insert
```
```  1723 proof (induct rule: finite_psubset_induct)
```
```  1724   case (psubset A)
```
```  1725   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
```
```  1726   have fin: "finite A" by fact
```
```  1727   have empty: "P {}" by fact
```
```  1728   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
```
```  1729   show "P A"
```
```  1730   proof (cases "A = {}")
```
```  1731     assume "A = {}"
```
```  1732     then show "P A" using `P {}` by simp
```
```  1733   next
```
```  1734     let ?B = "A - {Max A}"
```
```  1735     let ?A = "insert (Max A) ?B"
```
```  1736     have "finite ?B" using `finite A` by simp
```
```  1737     assume "A \<noteq> {}"
```
```  1738     with `finite A` have "Max A : A" by auto
```
```  1739     then have A: "?A = A" using insert_Diff_single insert_absorb by auto
```
```  1740     then have "P ?B" using `P {}` step IH[of ?B] by blast
```
```  1741     moreover
```
```  1742     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastsimp
```
```  1743     ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastsimp
```
```  1744   qed
```
```  1745 qed
```
```  1746
```
```  1747 lemma finite_linorder_min_induct[consumes 1, case_names empty insert]:
```
```  1748  "\<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"
```
```  1749 by(rule linorder.finite_linorder_max_induct[OF dual_linorder])
```
```  1750
```
```  1751 end
```
```  1752
```
```  1753 context linordered_ab_semigroup_add
```
```  1754 begin
```
```  1755
```
```  1756 lemma add_Min_commute:
```
```  1757   fixes k
```
```  1758   assumes "finite N" and "N \<noteq> {}"
```
```  1759   shows "k + Min N = Min {k + m | m. m \<in> N}"
```
```  1760 proof -
```
```  1761   have "\<And>x y. k + min x y = min (k + x) (k + y)"
```
```  1762     by (simp add: min_def not_le)
```
```  1763       (blast intro: antisym less_imp_le add_left_mono)
```
```  1764   with assms show ?thesis
```
```  1765     using hom_Min_commute [of "plus k" N]
```
```  1766     by simp (blast intro: arg_cong [where f = Min])
```
```  1767 qed
```
```  1768
```
```  1769 lemma add_Max_commute:
```
```  1770   fixes k
```
```  1771   assumes "finite N" and "N \<noteq> {}"
```
```  1772   shows "k + Max N = Max {k + m | m. m \<in> N}"
```
```  1773 proof -
```
```  1774   have "\<And>x y. k + max x y = max (k + x) (k + y)"
```
```  1775     by (simp add: max_def not_le)
```
```  1776       (blast intro: antisym less_imp_le add_left_mono)
```
```  1777   with assms show ?thesis
```
```  1778     using hom_Max_commute [of "plus k" N]
```
```  1779     by simp (blast intro: arg_cong [where f = Max])
```
```  1780 qed
```
```  1781
```
```  1782 end
```
```  1783
```
```  1784 context linordered_ab_group_add
```
```  1785 begin
```
```  1786
```
```  1787 lemma minus_Max_eq_Min [simp]:
```
```  1788   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Max S) = Min (uminus ` S)"
```
```  1789   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
```
```  1790
```
```  1791 lemma minus_Min_eq_Max [simp]:
```
```  1792   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Min S) = Max (uminus ` S)"
```
```  1793   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
```
```  1794
```
```  1795 end
```
```  1796
```
```  1797 end
```