src/HOL/Big_Operators.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 49962 a8cc904a6820 child 51112 da97167e03f7 permissions -rw-r--r--
introduce order topology
```     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 strong_F_cong [cong]:
```
```    43   "\<lbrakk> A = B; !!x. x:B =simp=> g x = h x \<rbrakk>
```
```    44    \<Longrightarrow> F (%x. g x) A = F (%x. h x) B"
```
```    45 by (rule F_cong) (simp_all add: simp_implies_def)
```
```    46
```
```    47 lemma F_neutral[simp]: "F (%i. 1) A = 1"
```
```    48 by (cases "finite A") (simp_all add: neutral)
```
```    49
```
```    50 lemma F_neutral': "ALL a:A. g a = 1 \<Longrightarrow> F g A = 1"
```
```    51 by simp
```
```    52
```
```    53 lemma F_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow> F g A = F g (A - B) * F g B"
```
```    54 by (metis Diff_partition union_disjoint Diff_disjoint finite_Un inf_commute sup_commute)
```
```    55
```
```    56 lemma F_mono_neutral_cong_left:
```
```    57   assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
```
```    58   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
```
```    59 proof-
```
```    60   have eq: "T = S \<union> (T - S)" using `S \<subseteq> T` by blast
```
```    61   have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast
```
```    62   from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
```
```    63     by (auto intro: finite_subset)
```
```    64   show ?thesis using assms(4)
```
```    65     by (simp add: union_disjoint[OF f d, unfolded eq[symmetric]] F_neutral'[OF assms(3)])
```
```    66 qed
```
```    67
```
```    68 lemma F_mono_neutral_cong_right:
```
```    69   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk>
```
```    70    \<Longrightarrow> F g T = F h S"
```
```    71 by(auto intro!: F_mono_neutral_cong_left[symmetric])
```
```    72
```
```    73 lemma F_mono_neutral_left:
```
```    74   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
```
```    75 by(blast intro: F_mono_neutral_cong_left)
```
```    76
```
```    77 lemma F_mono_neutral_right:
```
```    78   "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
```
```    79 by(blast intro!: F_mono_neutral_left[symmetric])
```
```    80
```
```    81 lemma F_delta:
```
```    82   assumes fS: "finite S"
```
```    83   shows "F (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```    84 proof-
```
```    85   let ?f = "(\<lambda>k. if k=a then b k else 1)"
```
```    86   { assume a: "a \<notin> S"
```
```    87     hence "\<forall>k\<in>S. ?f k = 1" by simp
```
```    88     hence ?thesis  using a by simp }
```
```    89   moreover
```
```    90   { assume a: "a \<in> S"
```
```    91     let ?A = "S - {a}"
```
```    92     let ?B = "{a}"
```
```    93     have eq: "S = ?A \<union> ?B" using a by blast
```
```    94     have dj: "?A \<inter> ?B = {}" by simp
```
```    95     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```    96     have "F ?f S = F ?f ?A * F ?f ?B"
```
```    97       using union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```    98       by simp
```
```    99     then have ?thesis  using a by simp }
```
```   100   ultimately show ?thesis by blast
```
```   101 qed
```
```   102
```
```   103 lemma F_delta':
```
```   104   assumes fS: "finite S" shows
```
```   105   "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   106 using F_delta[OF fS, of a b, symmetric] by (auto intro: F_cong)
```
```   107
```
```   108 lemma F_fun_f: "F (%x. g x * h x) A = (F g A * F h A)"
```
```   109 by (cases "finite A") (simp_all add: distrib)
```
```   110
```
```   111
```
```   112 text {* for ad-hoc proofs for @{const fold_image} *}
```
```   113 lemma comm_monoid_mult:  "class.comm_monoid_mult (op *) 1"
```
```   114 proof qed (auto intro: assoc commute)
```
```   115
```
```   116 lemma F_Un_neutral:
```
```   117   assumes fS: "finite S" and fT: "finite T"
```
```   118   and I1: "\<forall>x \<in> S\<inter>T. g x = 1"
```
```   119   shows "F g (S \<union> T) = F g S  * F g T"
```
```   120 proof -
```
```   121   interpret comm_monoid_mult "op *" 1 by (fact comm_monoid_mult)
```
```   122   show ?thesis
```
```   123   using fS fT
```
```   124   apply (simp add: F_eq)
```
```   125   apply (rule fold_image_Un_one)
```
```   126   using I1 by auto
```
```   127 qed
```
```   128
```
```   129 lemma If_cases:
```
```   130   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
```
```   131   assumes fA: "finite A"
```
```   132   shows "F (\<lambda>x. if P x then h x else g x) A =
```
```   133          F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
```
```   134 proof-
```
```   135   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
```
```   136           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
```
```   137     by blast+
```
```   138   from fA
```
```   139   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
```
```   140   let ?g = "\<lambda>x. if P x then h x else g x"
```
```   141   from union_disjoint[OF f a(2), of ?g] a(1)
```
```   142   show ?thesis
```
```   143     by (subst (1 2) F_cong) simp_all
```
```   144 qed
```
```   145
```
```   146 end
```
```   147
```
```   148 text {* for ad-hoc proofs for @{const fold_image} *}
```
```   149
```
```   150 lemma (in comm_monoid_add) comm_monoid_mult:
```
```   151   "class.comm_monoid_mult (op +) 0"
```
```   152 proof qed (auto intro: add_assoc add_commute)
```
```   153
```
```   154 notation times (infixl "*" 70)
```
```   155 notation Groups.one ("1")
```
```   156
```
```   157
```
```   158 subsection {* Generalized summation over a set *}
```
```   159
```
```   160 definition (in comm_monoid_add) setsum :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
```
```   161   "setsum f A = (if finite A then fold_image (op +) f 0 A else 0)"
```
```   162
```
```   163 sublocale comm_monoid_add < setsum!: comm_monoid_big "op +" 0 setsum proof
```
```   164 qed (fact setsum_def)
```
```   165
```
```   166 abbreviation
```
```   167   Setsum  ("\<Sum>_" [1000] 999) where
```
```   168   "\<Sum>A == setsum (%x. x) A"
```
```   169
```
```   170 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
```
```   171 written @{text"\<Sum>x\<in>A. e"}. *}
```
```   172
```
```   173 syntax
```
```   174   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)
```
```   175 syntax (xsymbols)
```
```   176   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   177 syntax (HTML output)
```
```   178   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
```
```   179
```
```   180 translations -- {* Beware of argument permutation! *}
```
```   181   "SUM i:A. b" == "CONST setsum (%i. b) A"
```
```   182   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
```
```   183
```
```   184 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
```
```   185  @{text"\<Sum>x|P. e"}. *}
```
```   186
```
```   187 syntax
```
```   188   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
```
```   189 syntax (xsymbols)
```
```   190   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   191 syntax (HTML output)
```
```   192   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
```
```   193
```
```   194 translations
```
```   195   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   196   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
```
```   197
```
```   198 print_translation {*
```
```   199 let
```
```   200   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
```
```   201         if x <> y then raise Match
```
```   202         else
```
```   203           let
```
```   204             val x' = Syntax_Trans.mark_bound_body (x, Tx);
```
```   205             val t' = subst_bound (x', t);
```
```   206             val P' = subst_bound (x', P);
```
```   207           in
```
```   208             Syntax.const @{syntax_const "_qsetsum"} \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
```
```   209           end
```
```   210     | setsum_tr' _ = raise Match;
```
```   211 in [(@{const_syntax setsum}, setsum_tr')] end
```
```   212 *}
```
```   213
```
```   214 lemma setsum_empty:
```
```   215   "setsum f {} = 0"
```
```   216   by (fact setsum.empty)
```
```   217
```
```   218 lemma setsum_insert:
```
```   219   "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
```
```   220   by (fact setsum.insert)
```
```   221
```
```   222 lemma setsum_infinite:
```
```   223   "~ finite A ==> setsum f A = 0"
```
```   224   by (fact setsum.infinite)
```
```   225
```
```   226 lemma (in comm_monoid_add) setsum_reindex:
```
```   227   assumes "inj_on f B" shows "setsum h (f ` B) = setsum (h \<circ> f) B"
```
```   228 proof -
```
```   229   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   230   from assms show ?thesis by (auto simp add: setsum_def fold_image_reindex o_def dest!:finite_imageD)
```
```   231 qed
```
```   232
```
```   233 lemma setsum_reindex_id:
```
```   234   "inj_on f B ==> setsum f B = setsum id (f ` B)"
```
```   235 by (simp add: setsum_reindex)
```
```   236
```
```   237 lemma setsum_reindex_nonzero:
```
```   238   assumes fS: "finite S"
```
```   239   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"
```
```   240   shows "setsum h (f ` S) = setsum (h o f) S"
```
```   241 using nz
```
```   242 proof(induct rule: finite_induct[OF fS])
```
```   243   case 1 thus ?case by simp
```
```   244 next
```
```   245   case (2 x F)
```
```   246   { assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
```
```   247     then obtain y where y: "y \<in> F" "f x = f y" by auto
```
```   248     from "2.hyps" y have xy: "x \<noteq> y" by auto
```
```   249
```
```   250     from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
```
```   251     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
```
```   252     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   253       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   254       using h0
```
```   255       apply (simp cong del:setsum.strong_F_cong)
```
```   256       apply (rule "2.hyps"(3))
```
```   257       apply (rule_tac y="y" in  "2.prems")
```
```   258       apply simp_all
```
```   259       done
```
```   260     finally have ?case . }
```
```   261   moreover
```
```   262   { assume fxF: "f x \<notin> f ` F"
```
```   263     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)"
```
```   264       using fxF "2.hyps" by simp
```
```   265     also have "\<dots> = setsum (h o f) (insert x F)"
```
```   266       unfolding setsum.insert[OF `finite F` `x\<notin>F`]
```
```   267       apply (simp cong del:setsum.strong_F_cong)
```
```   268       apply (rule cong [OF refl [of "op + (h (f x))"]])
```
```   269       apply (rule "2.hyps"(3))
```
```   270       apply (rule_tac y="y" in  "2.prems")
```
```   271       apply simp_all
```
```   272       done
```
```   273     finally have ?case . }
```
```   274   ultimately show ?case by blast
```
```   275 qed
```
```   276
```
```   277 lemma setsum_cong:
```
```   278   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
```
```   279 by (fact setsum.F_cong)
```
```   280
```
```   281 lemma strong_setsum_cong:
```
```   282   "A = B ==> (!!x. x:B =simp=> f x = g x)
```
```   283    ==> setsum (%x. f x) A = setsum (%x. g x) B"
```
```   284 by (fact setsum.strong_F_cong)
```
```   285
```
```   286 lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"
```
```   287 by (auto intro: setsum_cong)
```
```   288
```
```   289 lemma setsum_reindex_cong:
```
```   290    "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
```
```   291     ==> setsum h B = setsum g A"
```
```   292 by (simp add: setsum_reindex)
```
```   293
```
```   294 lemmas setsum_0 = setsum.F_neutral
```
```   295 lemmas setsum_0' = setsum.F_neutral'
```
```   296
```
```   297 lemma setsum_Un_Int: "finite A ==> finite B ==>
```
```   298   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
```
```   299   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
```
```   300 by (fact setsum.union_inter)
```
```   301
```
```   302 lemma setsum_Un_disjoint: "finite A ==> finite B
```
```   303   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
```
```   304 by (fact setsum.union_disjoint)
```
```   305
```
```   306 lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
```
```   307     setsum f A = setsum f (A - B) + setsum f B"
```
```   308 by(fact setsum.F_subset_diff)
```
```   309
```
```   310 lemma setsum_mono_zero_left:
```
```   311   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
```
```   312 by(fact setsum.F_mono_neutral_left)
```
```   313
```
```   314 lemmas setsum_mono_zero_right = setsum.F_mono_neutral_right
```
```   315
```
```   316 lemma setsum_mono_zero_cong_left:
```
```   317   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 0; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
```
```   318   \<Longrightarrow> setsum f S = setsum g T"
```
```   319 by(fact setsum.F_mono_neutral_cong_left)
```
```   320
```
```   321 lemmas setsum_mono_zero_cong_right = setsum.F_mono_neutral_cong_right
```
```   322
```
```   323 lemma setsum_delta: "finite S \<Longrightarrow>
```
```   324   setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
```
```   325 by(fact setsum.F_delta)
```
```   326
```
```   327 lemma setsum_delta': "finite S \<Longrightarrow>
```
```   328   setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
```
```   329 by(fact setsum.F_delta')
```
```   330
```
```   331 lemma setsum_restrict_set:
```
```   332   assumes fA: "finite A"
```
```   333   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
```
```   334 proof-
```
```   335   from fA have fab: "finite (A \<inter> B)" by auto
```
```   336   have aba: "A \<inter> B \<subseteq> A" by blast
```
```   337   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
```
```   338   from setsum_mono_zero_left[OF fA aba, of ?g]
```
```   339   show ?thesis by simp
```
```   340 qed
```
```   341
```
```   342 lemma setsum_cases:
```
```   343   assumes fA: "finite A"
```
```   344   shows "setsum (\<lambda>x. if P x then f x else g x) A =
```
```   345          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
```
```   346   using setsum.If_cases[OF fA] .
```
```   347
```
```   348 (*But we can't get rid of finite I. If infinite, although the rhs is 0,
```
```   349   the lhs need not be, since UNION I A could still be finite.*)
```
```   350 lemma (in comm_monoid_add) setsum_UN_disjoint:
```
```   351   assumes "finite I" and "ALL i:I. finite (A i)"
```
```   352     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
```
```   353   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
```
```   354 proof -
```
```   355   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   356   from assms show ?thesis by (simp add: setsum_def fold_image_UN_disjoint)
```
```   357 qed
```
```   358
```
```   359 text{*No need to assume that @{term C} is finite.  If infinite, the rhs is
```
```   360 directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
```
```   361 lemma setsum_Union_disjoint:
```
```   362   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```   363   shows "setsum f (Union C) = setsum (setsum f) C"
```
```   364 proof cases
```
```   365   assume "finite C"
```
```   366   from setsum_UN_disjoint[OF this assms]
```
```   367   show ?thesis
```
```   368     by (simp add: SUP_def)
```
```   369 qed (force dest: finite_UnionD simp add: setsum_def)
```
```   370
```
```   371 (*But we can't get rid of finite A. If infinite, although the lhs is 0,
```
```   372   the rhs need not be, since SIGMA A B could still be finite.*)
```
```   373 lemma (in comm_monoid_add) setsum_Sigma:
```
```   374   assumes "finite A" and  "ALL x:A. finite (B x)"
```
```   375   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)"
```
```   376 proof -
```
```   377   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   378   from assms show ?thesis by (simp add: setsum_def fold_image_Sigma split_def)
```
```   379 qed
```
```   380
```
```   381 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```   382 lemma setsum_cartesian_product:
```
```   383    "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
```
```   384 apply (cases "finite A")
```
```   385  apply (cases "finite B")
```
```   386   apply (simp add: setsum_Sigma)
```
```   387  apply (cases "A={}", simp)
```
```   388  apply (simp)
```
```   389 apply (auto simp add: setsum_def
```
```   390             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   391 done
```
```   392
```
```   393 lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
```
```   394 by (fact setsum.F_fun_f)
```
```   395
```
```   396 lemma setsum_Un_zero:
```
```   397   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
```
```   398   setsum f (S \<union> T) = setsum f S + setsum f T"
```
```   399 by(fact setsum.F_Un_neutral)
```
```   400
```
```   401 lemma setsum_UNION_zero:
```
```   402   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
```
```   403   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"
```
```   404   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
```
```   405   using fSS f0
```
```   406 proof(induct rule: finite_induct[OF fS])
```
```   407   case 1 thus ?case by simp
```
```   408 next
```
```   409   case (2 T F)
```
```   410   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F"
```
```   411     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto
```
```   412   from fTF have fUF: "finite (\<Union>F)" by auto
```
```   413   from "2.prems" TF fTF
```
```   414   show ?case
```
```   415     by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f])
```
```   416 qed
```
```   417
```
```   418
```
```   419 subsubsection {* Properties in more restricted classes of structures *}
```
```   420
```
```   421 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
```
```   422 apply (case_tac "finite A")
```
```   423  prefer 2 apply (simp add: setsum_def)
```
```   424 apply (erule rev_mp)
```
```   425 apply (erule finite_induct, auto)
```
```   426 done
```
```   427
```
```   428 lemma setsum_eq_0_iff [simp]:
```
```   429     "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
```
```   430 by (induct set: finite) auto
```
```   431
```
```   432 lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
```
```   433   (setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
```
```   434 apply(erule finite_induct)
```
```   435 apply (auto simp add:add_is_1)
```
```   436 done
```
```   437
```
```   438 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
```
```   439
```
```   440 lemma setsum_Un_nat: "finite A ==> finite B ==>
```
```   441   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
```
```   442   -- {* For the natural numbers, we have subtraction. *}
```
```   443 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   444
```
```   445 lemma setsum_Un: "finite A ==> finite B ==>
```
```   446   (setsum f (A Un B) :: 'a :: ab_group_add) =
```
```   447    setsum f A + setsum f B - setsum f (A Int B)"
```
```   448 by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
```
```   449
```
```   450 lemma setsum_Un2:
```
```   451   assumes "finite (A \<union> B)"
```
```   452   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
```
```   453 proof -
```
```   454   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```   455     by auto
```
```   456   with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
```
```   457 qed
```
```   458
```
```   459 lemma (in comm_monoid_add) setsum_eq_general_reverses:
```
```   460   assumes fS: "finite S" and fT: "finite T"
```
```   461   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
```
```   462   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
```
```   463   shows "setsum f S = setsum g T"
```
```   464 proof -
```
```   465   interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
```
```   466   show ?thesis
```
```   467   apply (simp add: setsum_def fS fT)
```
```   468   apply (rule fold_image_eq_general_inverses)
```
```   469   apply (rule fS)
```
```   470   apply (erule kh)
```
```   471   apply (erule hk)
```
```   472   done
```
```   473 qed
```
```   474
```
```   475 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
```
```   476   (if a:A then setsum f A - f a else setsum f A)"
```
```   477 apply (case_tac "finite A")
```
```   478  prefer 2 apply (simp add: setsum_def)
```
```   479 apply (erule finite_induct)
```
```   480  apply (auto simp add: insert_Diff_if)
```
```   481 apply (drule_tac a = a in mk_disjoint_insert, auto)
```
```   482 done
```
```   483
```
```   484 lemma setsum_diff1: "finite A \<Longrightarrow>
```
```   485   (setsum f (A - {a}) :: ('a::ab_group_add)) =
```
```   486   (if a:A then setsum f A - f a else setsum f A)"
```
```   487 by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```   488
```
```   489 lemma setsum_diff1'[rule_format]:
```
```   490   "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
```
```   491 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))"])
```
```   492 apply (auto simp add: insert_Diff_if add_ac)
```
```   493 done
```
```   494
```
```   495 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
```
```   496   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
```
```   497 unfolding setsum_diff1'[OF assms] by auto
```
```   498
```
```   499 (* By Jeremy Siek: *)
```
```   500
```
```   501 lemma setsum_diff_nat:
```
```   502 assumes "finite B" and "B \<subseteq> A"
```
```   503 shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
```
```   504 using assms
```
```   505 proof induct
```
```   506   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
```
```   507 next
```
```   508   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
```
```   509     and xFinA: "insert x F \<subseteq> A"
```
```   510     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
```
```   511   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
```
```   512   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
```
```   513     by (simp add: setsum_diff1_nat)
```
```   514   from xFinA have "F \<subseteq> A" by simp
```
```   515   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
```
```   516   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
```
```   517     by simp
```
```   518   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
```
```   519   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
```
```   520     by simp
```
```   521   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
```
```   522   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
```
```   523     by simp
```
```   524   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
```
```   525 qed
```
```   526
```
```   527 lemma setsum_diff:
```
```   528   assumes le: "finite A" "B \<subseteq> A"
```
```   529   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
```
```   530 proof -
```
```   531   from le have finiteB: "finite B" using finite_subset by auto
```
```   532   show ?thesis using finiteB le
```
```   533   proof induct
```
```   534     case empty
```
```   535     thus ?case by auto
```
```   536   next
```
```   537     case (insert x F)
```
```   538     thus ?case using le finiteB
```
```   539       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
```
```   540   qed
```
```   541 qed
```
```   542
```
```   543 lemma setsum_mono:
```
```   544   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
```
```   545   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
```
```   546 proof (cases "finite K")
```
```   547   case True
```
```   548   thus ?thesis using le
```
```   549   proof induct
```
```   550     case empty
```
```   551     thus ?case by simp
```
```   552   next
```
```   553     case insert
```
```   554     thus ?case using add_mono by fastforce
```
```   555   qed
```
```   556 next
```
```   557   case False
```
```   558   thus ?thesis
```
```   559     by (simp add: setsum_def)
```
```   560 qed
```
```   561
```
```   562 lemma setsum_strict_mono:
```
```   563   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
```
```   564   assumes "finite A"  "A \<noteq> {}"
```
```   565     and "!!x. x:A \<Longrightarrow> f x < g x"
```
```   566   shows "setsum f A < setsum g A"
```
```   567   using assms
```
```   568 proof (induct rule: finite_ne_induct)
```
```   569   case singleton thus ?case by simp
```
```   570 next
```
```   571   case insert thus ?case by (auto simp: add_strict_mono)
```
```   572 qed
```
```   573
```
```   574 lemma setsum_strict_mono_ex1:
```
```   575 fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
```
```   576 assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
```
```   577 shows "setsum f A < setsum g A"
```
```   578 proof-
```
```   579   from assms(3) obtain a where a: "a:A" "f a < g a" by blast
```
```   580   have "setsum f A = setsum f ((A-{a}) \<union> {a})"
```
```   581     by(simp add:insert_absorb[OF `a:A`])
```
```   582   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
```
```   583     using `finite A` by(subst setsum_Un_disjoint) auto
```
```   584   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
```
```   585     by(rule setsum_mono)(simp add: assms(2))
```
```   586   also have "setsum f {a} < setsum g {a}" using a by simp
```
```   587   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
```
```   588     using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
```
```   589   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
```
```   590   finally show ?thesis by (metis add_right_mono add_strict_left_mono)
```
```   591 qed
```
```   592
```
```   593 lemma setsum_negf:
```
```   594   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```   595 proof (cases "finite A")
```
```   596   case True thus ?thesis by (induct set: finite) auto
```
```   597 next
```
```   598   case False thus ?thesis by (simp add: setsum_def)
```
```   599 qed
```
```   600
```
```   601 lemma setsum_subtractf:
```
```   602   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```   603     setsum f A - setsum g A"
```
```   604 proof (cases "finite A")
```
```   605   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
```
```   606 next
```
```   607   case False thus ?thesis by (simp add: setsum_def)
```
```   608 qed
```
```   609
```
```   610 lemma setsum_nonneg:
```
```   611   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```   612   shows "0 \<le> setsum f A"
```
```   613 proof (cases "finite A")
```
```   614   case True thus ?thesis using nn
```
```   615   proof induct
```
```   616     case empty then show ?case by simp
```
```   617   next
```
```   618     case (insert x F)
```
```   619     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   620     with insert show ?case by simp
```
```   621   qed
```
```   622 next
```
```   623   case False thus ?thesis by (simp add: setsum_def)
```
```   624 qed
```
```   625
```
```   626 lemma setsum_nonpos:
```
```   627   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
```
```   628   shows "setsum f A \<le> 0"
```
```   629 proof (cases "finite A")
```
```   630   case True thus ?thesis using np
```
```   631   proof induct
```
```   632     case empty then show ?case by simp
```
```   633   next
```
```   634     case (insert x F)
```
```   635     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   636     with insert show ?case by simp
```
```   637   qed
```
```   638 next
```
```   639   case False thus ?thesis by (simp add: setsum_def)
```
```   640 qed
```
```   641
```
```   642 lemma setsum_nonneg_leq_bound:
```
```   643   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   644   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   645   shows "f i \<le> B"
```
```   646 proof -
```
```   647   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
```
```   648     using assms by (auto intro!: setsum_nonneg)
```
```   649   moreover
```
```   650   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
```
```   651     using assms by (simp add: setsum_diff1)
```
```   652   ultimately show ?thesis by auto
```
```   653 qed
```
```   654
```
```   655 lemma setsum_nonneg_0:
```
```   656   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   657   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
```
```   658   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
```
```   659   shows "f i = 0"
```
```   660   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
```
```   661
```
```   662 lemma setsum_mono2:
```
```   663 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
```
```   664 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   665 shows "setsum f A \<le> setsum f B"
```
```   666 proof -
```
```   667   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   668     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   669   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```   670     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```   671   also have "A \<union> (B-A) = B" using sub by blast
```
```   672   finally show ?thesis .
```
```   673 qed
```
```   674
```
```   675 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```   676     ALL x: B - A.
```
```   677       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
```
```   678         setsum f A <= setsum f B"
```
```   679   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```   680   apply (erule ssubst)
```
```   681   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```   682   apply simp
```
```   683   apply (rule add_left_mono)
```
```   684   apply (erule setsum_nonneg)
```
```   685   apply (subst setsum_Un_disjoint [THEN sym])
```
```   686   apply (erule finite_subset, assumption)
```
```   687   apply (rule finite_subset)
```
```   688   prefer 2
```
```   689   apply assumption
```
```   690   apply (auto simp add: sup_absorb2)
```
```   691 done
```
```   692
```
```   693 lemma setsum_right_distrib:
```
```   694   fixes f :: "'a => ('b::semiring_0)"
```
```   695   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```   696 proof (cases "finite A")
```
```   697   case True
```
```   698   thus ?thesis
```
```   699   proof induct
```
```   700     case empty thus ?case by simp
```
```   701   next
```
```   702     case (insert x A) thus ?case by (simp add: distrib_left)
```
```   703   qed
```
```   704 next
```
```   705   case False thus ?thesis by (simp add: setsum_def)
```
```   706 qed
```
```   707
```
```   708 lemma setsum_left_distrib:
```
```   709   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```   710 proof (cases "finite A")
```
```   711   case True
```
```   712   then show ?thesis
```
```   713   proof induct
```
```   714     case empty thus ?case by simp
```
```   715   next
```
```   716     case (insert x A) thus ?case by (simp add: distrib_right)
```
```   717   qed
```
```   718 next
```
```   719   case False thus ?thesis by (simp add: setsum_def)
```
```   720 qed
```
```   721
```
```   722 lemma setsum_divide_distrib:
```
```   723   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```   724 proof (cases "finite A")
```
```   725   case True
```
```   726   then show ?thesis
```
```   727   proof induct
```
```   728     case empty thus ?case by simp
```
```   729   next
```
```   730     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```   731   qed
```
```   732 next
```
```   733   case False thus ?thesis by (simp add: setsum_def)
```
```   734 qed
```
```   735
```
```   736 lemma setsum_abs[iff]:
```
```   737   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   738   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```   739 proof (cases "finite A")
```
```   740   case True
```
```   741   thus ?thesis
```
```   742   proof induct
```
```   743     case empty thus ?case by simp
```
```   744   next
```
```   745     case (insert x A)
```
```   746     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   747   qed
```
```   748 next
```
```   749   case False thus ?thesis by (simp add: setsum_def)
```
```   750 qed
```
```   751
```
```   752 lemma setsum_abs_ge_zero[iff]:
```
```   753   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   754   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```   755 proof (cases "finite A")
```
```   756   case True
```
```   757   thus ?thesis
```
```   758   proof induct
```
```   759     case empty thus ?case by simp
```
```   760   next
```
```   761     case (insert x A) thus ?case by auto
```
```   762   qed
```
```   763 next
```
```   764   case False thus ?thesis by (simp add: setsum_def)
```
```   765 qed
```
```   766
```
```   767 lemma abs_setsum_abs[simp]:
```
```   768   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   769   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```   770 proof (cases "finite A")
```
```   771   case True
```
```   772   thus ?thesis
```
```   773   proof induct
```
```   774     case empty thus ?case by simp
```
```   775   next
```
```   776     case (insert a A)
```
```   777     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
```
```   778     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```   779     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```   780       by (simp del: abs_of_nonneg)
```
```   781     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```   782     finally show ?case .
```
```   783   qed
```
```   784 next
```
```   785   case False thus ?thesis by (simp add: setsum_def)
```
```   786 qed
```
```   787
```
```   788 lemma setsum_Plus:
```
```   789   fixes A :: "'a set" and B :: "'b set"
```
```   790   assumes fin: "finite A" "finite B"
```
```   791   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
```
```   792 proof -
```
```   793   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   794   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
```
```   795     by auto
```
```   796   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
```
```   797   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
```
```   798   ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)
```
```   799 qed
```
```   800
```
```   801
```
```   802 text {* Commuting outer and inner summation *}
```
```   803
```
```   804 lemma setsum_commute:
```
```   805   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
```
```   806 proof (simp add: setsum_cartesian_product)
```
```   807   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
```
```   808     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
```
```   809     (is "?s = _")
```
```   810     apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
```
```   811     apply (simp add: split_def)
```
```   812     done
```
```   813   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
```
```   814     (is "_ = ?t")
```
```   815     apply (simp add: swap_product)
```
```   816     done
```
```   817   finally show "?s = ?t" .
```
```   818 qed
```
```   819
```
```   820 lemma setsum_product:
```
```   821   fixes f :: "'a => ('b::semiring_0)"
```
```   822   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   823   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
```
```   824
```
```   825 lemma setsum_mult_setsum_if_inj:
```
```   826 fixes f :: "'a => ('b::semiring_0)"
```
```   827 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
```
```   828   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
```
```   829 by(auto simp: setsum_product setsum_cartesian_product
```
```   830         intro!:  setsum_reindex_cong[symmetric])
```
```   831
```
```   832 lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
```
```   833 apply (cases "finite A")
```
```   834 apply (erule finite_induct)
```
```   835 apply (auto simp add: algebra_simps)
```
```   836 done
```
```   837
```
```   838 lemma setsum_bounded:
```
```   839   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
```
```   840   shows "setsum f A \<le> of_nat(card A) * K"
```
```   841 proof (cases "finite A")
```
```   842   case True
```
```   843   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```   844 next
```
```   845   case False thus ?thesis by (simp add: setsum_def)
```
```   846 qed
```
```   847
```
```   848
```
```   849 subsubsection {* Cardinality as special case of @{const setsum} *}
```
```   850
```
```   851 lemma card_eq_setsum:
```
```   852   "card A = setsum (\<lambda>x. 1) A"
```
```   853   by (simp only: card_def setsum_def)
```
```   854
```
```   855 lemma card_UN_disjoint:
```
```   856   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   857     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   858   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```   859 proof -
```
```   860   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
```
```   861   with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
```
```   862 qed
```
```   863
```
```   864 lemma card_Union_disjoint:
```
```   865   "finite C ==> (ALL A:C. finite A) ==>
```
```   866    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
```
```   867    ==> card (Union C) = setsum card C"
```
```   868 apply (frule card_UN_disjoint [of C id])
```
```   869 apply (simp_all add: SUP_def id_def)
```
```   870 done
```
```   871
```
```   872 text{*The image of a finite set can be expressed using @{term fold_image}.*}
```
```   873 lemma image_eq_fold_image:
```
```   874   "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A"
```
```   875 proof (induct rule: finite_induct)
```
```   876   case empty then show ?case by simp
```
```   877 next
```
```   878   interpret ab_semigroup_mult "op Un"
```
```   879     proof qed auto
```
```   880   case insert
```
```   881   then show ?case by simp
```
```   882 qed
```
```   883
```
```   884 subsubsection {* Cardinality of products *}
```
```   885
```
```   886 lemma card_SigmaI [simp]:
```
```   887   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```   888   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```   889 by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
```
```   890
```
```   891 (*
```
```   892 lemma SigmaI_insert: "y \<notin> A ==>
```
```   893   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```   894   by auto
```
```   895 *)
```
```   896
```
```   897 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```   898   by (cases "finite A \<and> finite B")
```
```   899     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   900
```
```   901 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```   902 by (simp add: card_cartesian_product)
```
```   903
```
```   904
```
```   905 subsection {* Generalized product over a set *}
```
```   906
```
```   907 definition (in comm_monoid_mult) setprod :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
```
```   908   "setprod f A = (if finite A then fold_image (op *) f 1 A else 1)"
```
```   909
```
```   910 sublocale comm_monoid_mult < setprod!: comm_monoid_big "op *" 1 setprod proof
```
```   911 qed (fact setprod_def)
```
```   912
```
```   913 abbreviation
```
```   914   Setprod  ("\<Prod>_" [1000] 999) where
```
```   915   "\<Prod>A == setprod (%x. x) A"
```
```   916
```
```   917 syntax
```
```   918   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```   919 syntax (xsymbols)
```
```   920   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```   921 syntax (HTML output)
```
```   922   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```   923
```
```   924 translations -- {* Beware of argument permutation! *}
```
```   925   "PROD i:A. b" == "CONST setprod (%i. b) A"
```
```   926   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
```
```   927
```
```   928 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```   929  @{text"\<Prod>x|P. e"}. *}
```
```   930
```
```   931 syntax
```
```   932   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```   933 syntax (xsymbols)
```
```   934   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```   935 syntax (HTML output)
```
```   936   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```   937
```
```   938 translations
```
```   939   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```   940   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```   941
```
```   942 lemma setprod_empty: "setprod f {} = 1"
```
```   943   by (fact setprod.empty)
```
```   944
```
```   945 lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
```
```   946     setprod f (insert a A) = f a * setprod f A"
```
```   947   by (fact setprod.insert)
```
```   948
```
```   949 lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
```
```   950   by (fact setprod.infinite)
```
```   951
```
```   952 lemma setprod_reindex:
```
```   953    "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```   954 by(auto simp: setprod_def fold_image_reindex o_def dest!:finite_imageD)
```
```   955
```
```   956 lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```   957 by (auto simp add: setprod_reindex)
```
```   958
```
```   959 lemma setprod_cong:
```
```   960   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```   961 by(fact setprod.F_cong)
```
```   962
```
```   963 lemma strong_setprod_cong:
```
```   964   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
```
```   965 by(fact setprod.strong_F_cong)
```
```   966
```
```   967 lemma setprod_reindex_cong: "inj_on f A ==>
```
```   968     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```   969 by (frule setprod_reindex, simp)
```
```   970
```
```   971 lemma strong_setprod_reindex_cong: assumes i: "inj_on f A"
```
```   972   and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
```
```   973   shows "setprod h B = setprod g A"
```
```   974 proof-
```
```   975     have "setprod h B = setprod (h o f) A"
```
```   976       by (simp add: B setprod_reindex[OF i, of h])
```
```   977     then show ?thesis apply simp
```
```   978       apply (rule setprod_cong)
```
```   979       apply simp
```
```   980       by (simp add: eq)
```
```   981 qed
```
```   982
```
```   983 lemma setprod_Un_one: "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
```
```   984   \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"
```
```   985 by(fact setprod.F_Un_neutral)
```
```   986
```
```   987 lemmas setprod_1 = setprod.F_neutral
```
```   988 lemmas setprod_1' = setprod.F_neutral'
```
```   989
```
```   990
```
```   991 lemma setprod_Un_Int: "finite A ==> finite B
```
```   992     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```   993 by (fact setprod.union_inter)
```
```   994
```
```   995 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```   996   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```   997 by (fact setprod.union_disjoint)
```
```   998
```
```   999 lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
```
```  1000     setprod f A = setprod f (A - B) * setprod f B"
```
```  1001 by(fact setprod.F_subset_diff)
```
```  1002
```
```  1003 lemma setprod_mono_one_left:
```
```  1004   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
```
```  1005 by(fact setprod.F_mono_neutral_left)
```
```  1006
```
```  1007 lemmas setprod_mono_one_right = setprod.F_mono_neutral_right
```
```  1008
```
```  1009 lemma setprod_mono_one_cong_left:
```
```  1010   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
```
```  1011   \<Longrightarrow> setprod f S = setprod g T"
```
```  1012 by(fact setprod.F_mono_neutral_cong_left)
```
```  1013
```
```  1014 lemmas setprod_mono_one_cong_right = setprod.F_mono_neutral_cong_right
```
```  1015
```
```  1016 lemma setprod_delta: "finite S \<Longrightarrow>
```
```  1017   setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```  1018 by(fact setprod.F_delta)
```
```  1019
```
```  1020 lemma setprod_delta': "finite S \<Longrightarrow>
```
```  1021   setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
```
```  1022 by(fact setprod.F_delta')
```
```  1023
```
```  1024 lemma setprod_UN_disjoint:
```
```  1025     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1026         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1027       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```  1028   by (simp add: setprod_def fold_image_UN_disjoint)
```
```  1029
```
```  1030 lemma setprod_Union_disjoint:
```
```  1031   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```  1032   shows "setprod f (Union C) = setprod (setprod f) C"
```
```  1033 proof cases
```
```  1034   assume "finite C"
```
```  1035   from setprod_UN_disjoint[OF this assms]
```
```  1036   show ?thesis
```
```  1037     by (simp add: SUP_def)
```
```  1038 qed (force dest: finite_UnionD simp add: setprod_def)
```
```  1039
```
```  1040 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1041     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
```
```  1042     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```  1043 by(simp add:setprod_def fold_image_Sigma split_def)
```
```  1044
```
```  1045 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1046 lemma setprod_cartesian_product:
```
```  1047      "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
```
```  1048 apply (cases "finite A")
```
```  1049  apply (cases "finite B")
```
```  1050   apply (simp add: setprod_Sigma)
```
```  1051  apply (cases "A={}", simp)
```
```  1052  apply (simp)
```
```  1053 apply (auto simp add: setprod_def
```
```  1054             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1055 done
```
```  1056
```
```  1057 lemma setprod_timesf: "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
```
```  1058 by (fact setprod.F_fun_f)
```
```  1059
```
```  1060
```
```  1061 subsubsection {* Properties in more restricted classes of structures *}
```
```  1062
```
```  1063 lemma setprod_eq_1_iff [simp]:
```
```  1064   "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
```
```  1065 by (induct set: finite) auto
```
```  1066
```
```  1067 lemma setprod_zero:
```
```  1068      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
```
```  1069 apply (induct set: finite, force, clarsimp)
```
```  1070 apply (erule disjE, auto)
```
```  1071 done
```
```  1072
```
```  1073 lemma setprod_nonneg [rule_format]:
```
```  1074    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1075 by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
```
```  1076
```
```  1077 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
```
```  1078   --> 0 < setprod f A"
```
```  1079 by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
```
```  1080
```
```  1081 lemma setprod_zero_iff[simp]: "finite A ==>
```
```  1082   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
```
```  1083   (EX x: A. f x = 0)"
```
```  1084 by (erule finite_induct, auto simp:no_zero_divisors)
```
```  1085
```
```  1086 lemma setprod_pos_nat:
```
```  1087   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
```
```  1088 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1089
```
```  1090 lemma setprod_pos_nat_iff[simp]:
```
```  1091   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
```
```  1092 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1093
```
```  1094 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1095   (setprod f (A Un B) :: 'a ::{field})
```
```  1096    = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1097 by (subst setprod_Un_Int [symmetric], auto)
```
```  1098
```
```  1099 lemma setprod_Un2:
```
```  1100   assumes "finite (A \<union> B)"
```
```  1101   shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
```
```  1102 proof -
```
```  1103   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
```
```  1104     by auto
```
```  1105   with assms show ?thesis by simp (subst setprod_Un_disjoint, auto)+
```
```  1106 qed
```
```  1107
```
```  1108 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1109   (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1110   (if a:A then setprod f A / f a else setprod f A)"
```
```  1111   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```  1112
```
```  1113 lemma setprod_inversef:
```
```  1114   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1115   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1116 by (erule finite_induct) auto
```
```  1117
```
```  1118 lemma setprod_dividef:
```
```  1119   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1120   shows "finite A
```
```  1121     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1122 apply (subgoal_tac
```
```  1123          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1124 apply (erule ssubst)
```
```  1125 apply (subst divide_inverse)
```
```  1126 apply (subst setprod_timesf)
```
```  1127 apply (subst setprod_inversef, assumption+, rule refl)
```
```  1128 apply (rule setprod_cong, rule refl)
```
```  1129 apply (subst divide_inverse, auto)
```
```  1130 done
```
```  1131
```
```  1132 lemma setprod_dvd_setprod [rule_format]:
```
```  1133     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
```
```  1134   apply (cases "finite A")
```
```  1135   apply (induct set: finite)
```
```  1136   apply (auto simp add: dvd_def)
```
```  1137   apply (rule_tac x = "k * ka" in exI)
```
```  1138   apply (simp add: algebra_simps)
```
```  1139 done
```
```  1140
```
```  1141 lemma setprod_dvd_setprod_subset:
```
```  1142   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1143   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
```
```  1144   apply (unfold dvd_def, blast)
```
```  1145   apply (subst setprod_Un_disjoint [symmetric])
```
```  1146   apply (auto elim: finite_subset intro: setprod_cong)
```
```  1147 done
```
```  1148
```
```  1149 lemma setprod_dvd_setprod_subset2:
```
```  1150   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow>
```
```  1151       setprod f A dvd setprod g B"
```
```  1152   apply (rule dvd_trans)
```
```  1153   apply (rule setprod_dvd_setprod, erule (1) bspec)
```
```  1154   apply (erule (1) setprod_dvd_setprod_subset)
```
```  1155 done
```
```  1156
```
```  1157 lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow>
```
```  1158     (f i ::'a::comm_semiring_1) dvd setprod f A"
```
```  1159 by (induct set: finite) (auto intro: dvd_mult)
```
```  1160
```
```  1161 lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow>
```
```  1162     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
```
```  1163   apply (cases "finite A")
```
```  1164   apply (induct set: finite)
```
```  1165   apply auto
```
```  1166 done
```
```  1167
```
```  1168 lemma setprod_mono:
```
```  1169   fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
```
```  1170   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
```
```  1171   shows "setprod f A \<le> setprod g A"
```
```  1172 proof (cases "finite A")
```
```  1173   case True
```
```  1174   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
```
```  1175   proof (induct A rule: finite_subset_induct)
```
```  1176     case (insert a F)
```
```  1177     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
```
```  1178       unfolding setprod_insert[OF insert(1,3)]
```
```  1179       using assms[rule_format,OF insert(2)] insert
```
```  1180       by (auto intro: mult_mono mult_nonneg_nonneg)
```
```  1181   qed auto
```
```  1182   thus ?thesis by simp
```
```  1183 qed auto
```
```  1184
```
```  1185 lemma abs_setprod:
```
```  1186   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
```
```  1187   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
```
```  1188 proof (cases "finite A")
```
```  1189   case True thus ?thesis
```
```  1190     by induct (auto simp add: field_simps abs_mult)
```
```  1191 qed auto
```
```  1192
```
```  1193 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
```
```  1194 apply (erule finite_induct)
```
```  1195 apply auto
```
```  1196 done
```
```  1197
```
```  1198 lemma setprod_gen_delta:
```
```  1199   assumes fS: "finite S"
```
```  1200   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)"
```
```  1201 proof-
```
```  1202   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```  1203   {assume a: "a \<notin> S"
```
```  1204     hence "\<forall> k\<in> S. ?f k = c" by simp
```
```  1205     hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
```
```  1206   moreover
```
```  1207   {assume a: "a \<in> S"
```
```  1208     let ?A = "S - {a}"
```
```  1209     let ?B = "{a}"
```
```  1210     have eq: "S = ?A \<union> ?B" using a by blast
```
```  1211     have dj: "?A \<inter> ?B = {}" by simp
```
```  1212     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```  1213     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```  1214       apply (rule setprod_cong) by auto
```
```  1215     have cA: "card ?A = card S - 1" using fS a by auto
```
```  1216     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
```
```  1217     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```  1218       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```  1219       by simp
```
```  1220     then have ?thesis using a cA
```
```  1221       by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
```
```  1222   ultimately show ?thesis by blast
```
```  1223 qed
```
```  1224
```
```  1225
```
```  1226 subsection {* Versions of @{const inf} and @{const sup} on non-empty sets *}
```
```  1227
```
```  1228 no_notation times (infixl "*" 70)
```
```  1229 no_notation Groups.one ("1")
```
```  1230
```
```  1231 locale semilattice_big = semilattice +
```
```  1232   fixes F :: "'a set \<Rightarrow> 'a"
```
```  1233   assumes F_eq: "finite A \<Longrightarrow> F A = fold1 (op *) A"
```
```  1234
```
```  1235 sublocale semilattice_big < folding_one_idem proof
```
```  1236 qed (simp_all add: F_eq)
```
```  1237
```
```  1238 notation times (infixl "*" 70)
```
```  1239 notation Groups.one ("1")
```
```  1240
```
```  1241 context lattice
```
```  1242 begin
```
```  1243
```
```  1244 definition Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900) where
```
```  1245   "Inf_fin = fold1 inf"
```
```  1246
```
```  1247 definition Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900) where
```
```  1248   "Sup_fin = fold1 sup"
```
```  1249
```
```  1250 end
```
```  1251
```
```  1252 sublocale lattice < Inf_fin!: semilattice_big inf Inf_fin proof
```
```  1253 qed (simp add: Inf_fin_def)
```
```  1254
```
```  1255 sublocale lattice < Sup_fin!: semilattice_big sup Sup_fin proof
```
```  1256 qed (simp add: Sup_fin_def)
```
```  1257
```
```  1258 context semilattice_inf
```
```  1259 begin
```
```  1260
```
```  1261 lemma ab_semigroup_idem_mult_inf:
```
```  1262   "class.ab_semigroup_idem_mult inf"
```
```  1263 proof qed (rule inf_assoc inf_commute inf_idem)+
```
```  1264
```
```  1265 lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> Finite_Set.fold inf b (insert a A) = inf a (Finite_Set.fold inf b A)"
```
```  1266 by(rule comp_fun_idem.fold_insert_idem[OF ab_semigroup_idem_mult.comp_fun_idem[OF ab_semigroup_idem_mult_inf]])
```
```  1267
```
```  1268 lemma inf_le_fold_inf: "finite A \<Longrightarrow> ALL a:A. b \<le> a \<Longrightarrow> inf b c \<le> Finite_Set.fold inf c A"
```
```  1269 by (induct pred: finite) (auto intro: le_infI1)
```
```  1270
```
```  1271 lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> Finite_Set.fold inf b A \<le> inf a b"
```
```  1272 proof(induct arbitrary: a pred:finite)
```
```  1273   case empty thus ?case by simp
```
```  1274 next
```
```  1275   case (insert x A)
```
```  1276   show ?case
```
```  1277   proof cases
```
```  1278     assume "A = {}" thus ?thesis using insert by simp
```
```  1279   next
```
```  1280     assume "A \<noteq> {}" thus ?thesis using insert by (auto intro: le_infI2)
```
```  1281   qed
```
```  1282 qed
```
```  1283
```
```  1284 lemma below_fold1_iff:
```
```  1285   assumes "finite A" "A \<noteq> {}"
```
```  1286   shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  1287 proof -
```
```  1288   interpret ab_semigroup_idem_mult inf
```
```  1289     by (rule ab_semigroup_idem_mult_inf)
```
```  1290   show ?thesis using assms by (induct rule: finite_ne_induct) simp_all
```
```  1291 qed
```
```  1292
```
```  1293 lemma fold1_belowI:
```
```  1294   assumes "finite A"
```
```  1295     and "a \<in> A"
```
```  1296   shows "fold1 inf A \<le> a"
```
```  1297 proof -
```
```  1298   from assms have "A \<noteq> {}" by auto
```
```  1299   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```  1300   proof (induct rule: finite_ne_induct)
```
```  1301     case singleton thus ?case by simp
```
```  1302   next
```
```  1303     interpret ab_semigroup_idem_mult inf
```
```  1304       by (rule ab_semigroup_idem_mult_inf)
```
```  1305     case (insert x F)
```
```  1306     from insert(5) have "a = x \<or> a \<in> F" by simp
```
```  1307     thus ?case
```
```  1308     proof
```
```  1309       assume "a = x" thus ?thesis using insert
```
```  1310         by (simp add: mult_ac)
```
```  1311     next
```
```  1312       assume "a \<in> F"
```
```  1313       hence bel: "fold1 inf F \<le> a" by (rule insert)
```
```  1314       have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
```
```  1315         using insert by (simp add: mult_ac)
```
```  1316       also have "inf (fold1 inf F) a = fold1 inf F"
```
```  1317         using bel by (auto intro: antisym)
```
```  1318       also have "inf x \<dots> = fold1 inf (insert x F)"
```
```  1319         using insert by (simp add: mult_ac)
```
```  1320       finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
```
```  1321       moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
```
```  1322       ultimately show ?thesis by simp
```
```  1323     qed
```
```  1324   qed
```
```  1325 qed
```
```  1326
```
```  1327 end
```
```  1328
```
```  1329 context semilattice_sup
```
```  1330 begin
```
```  1331
```
```  1332 lemma ab_semigroup_idem_mult_sup: "class.ab_semigroup_idem_mult sup"
```
```  1333 by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
```
```  1334
```
```  1335 lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> Finite_Set.fold sup b (insert a A) = sup a (Finite_Set.fold sup b A)"
```
```  1336 by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)
```
```  1337
```
```  1338 lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> Finite_Set.fold sup c A \<le> sup b c"
```
```  1339 by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)
```
```  1340
```
```  1341 lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> Finite_Set.fold sup b A"
```
```  1342 by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)
```
```  1343
```
```  1344 end
```
```  1345
```
```  1346 context lattice
```
```  1347 begin
```
```  1348
```
```  1349 lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
```
```  1350 apply(unfold Sup_fin_def Inf_fin_def)
```
```  1351 apply(subgoal_tac "EX a. a:A")
```
```  1352 prefer 2 apply blast
```
```  1353 apply(erule exE)
```
```  1354 apply(rule order_trans)
```
```  1355 apply(erule (1) fold1_belowI)
```
```  1356 apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice])
```
```  1357 done
```
```  1358
```
```  1359 lemma sup_Inf_absorb [simp]:
```
```  1360   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
```
```  1361 apply(subst sup_commute)
```
```  1362 apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)
```
```  1363 done
```
```  1364
```
```  1365 lemma inf_Sup_absorb [simp]:
```
```  1366   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
```
```  1367 by (simp add: Sup_fin_def inf_absorb1
```
```  1368   semilattice_inf.fold1_belowI [OF dual_semilattice])
```
```  1369
```
```  1370 end
```
```  1371
```
```  1372 context distrib_lattice
```
```  1373 begin
```
```  1374
```
```  1375 lemma sup_Inf1_distrib:
```
```  1376   assumes "finite A"
```
```  1377     and "A \<noteq> {}"
```
```  1378   shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
```
```  1379 proof -
```
```  1380   interpret ab_semigroup_idem_mult inf
```
```  1381     by (rule ab_semigroup_idem_mult_inf)
```
```  1382   from assms show ?thesis
```
```  1383     by (simp add: Inf_fin_def image_def
```
```  1384       hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])
```
```  1385         (rule arg_cong [where f="fold1 inf"], blast)
```
```  1386 qed
```
```  1387
```
```  1388 lemma sup_Inf2_distrib:
```
```  1389   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1390   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}"
```
```  1391 using A proof (induct rule: finite_ne_induct)
```
```  1392   case singleton thus ?case
```
```  1393     by (simp add: sup_Inf1_distrib [OF B])
```
```  1394 next
```
```  1395   interpret ab_semigroup_idem_mult inf
```
```  1396     by (rule ab_semigroup_idem_mult_inf)
```
```  1397   case (insert x A)
```
```  1398   have finB: "finite {sup x b |b. b \<in> B}"
```
```  1399     by(rule finite_surj[where f = "sup x", OF B(1)], auto)
```
```  1400   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1401   proof -
```
```  1402     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
```
```  1403       by blast
```
```  1404     thus ?thesis by(simp add: insert(1) B(1))
```
```  1405   qed
```
```  1406   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1407   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)"
```
```  1408     using insert by simp
```
```  1409   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)
```
```  1410   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})"
```
```  1411     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  1412   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})"
```
```  1413     (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
```
```  1414     using B insert
```
```  1415     by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
```
```  1416   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1417     by blast
```
```  1418   finally show ?case .
```
```  1419 qed
```
```  1420
```
```  1421 lemma inf_Sup1_distrib:
```
```  1422   assumes "finite A" and "A \<noteq> {}"
```
```  1423   shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
```
```  1424 proof -
```
```  1425   interpret ab_semigroup_idem_mult sup
```
```  1426     by (rule ab_semigroup_idem_mult_sup)
```
```  1427   from assms show ?thesis
```
```  1428     by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])
```
```  1429       (rule arg_cong [where f="fold1 sup"], blast)
```
```  1430 qed
```
```  1431
```
```  1432 lemma inf_Sup2_distrib:
```
```  1433   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1434   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}"
```
```  1435 using A proof (induct rule: finite_ne_induct)
```
```  1436   case singleton thus ?case
```
```  1437     by(simp add: inf_Sup1_distrib [OF B])
```
```  1438 next
```
```  1439   case (insert x A)
```
```  1440   have finB: "finite {inf x b |b. b \<in> B}"
```
```  1441     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
```
```  1442   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1443   proof -
```
```  1444     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
```
```  1445       by blast
```
```  1446     thus ?thesis by(simp add: insert(1) B(1))
```
```  1447   qed
```
```  1448   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1449   interpret ab_semigroup_idem_mult sup
```
```  1450     by (rule ab_semigroup_idem_mult_sup)
```
```  1451   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)"
```
```  1452     using insert by simp
```
```  1453   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)
```
```  1454   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})"
```
```  1455     using insert by(simp add:inf_Sup1_distrib[OF B])
```
```  1456   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})"
```
```  1457     (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
```
```  1458     using B insert
```
```  1459     by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
```
```  1460   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1461     by blast
```
```  1462   finally show ?case .
```
```  1463 qed
```
```  1464
```
```  1465 end
```
```  1466
```
```  1467 context complete_lattice
```
```  1468 begin
```
```  1469
```
```  1470 lemma Inf_fin_Inf:
```
```  1471   assumes "finite A" and "A \<noteq> {}"
```
```  1472   shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
```
```  1473 proof -
```
```  1474   interpret ab_semigroup_idem_mult inf
```
```  1475     by (rule ab_semigroup_idem_mult_inf)
```
```  1476   from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
```
```  1477   moreover with `finite A` have "finite B" by simp
```
```  1478   ultimately show ?thesis
```
```  1479     by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])
```
```  1480 qed
```
```  1481
```
```  1482 lemma Sup_fin_Sup:
```
```  1483   assumes "finite A" and "A \<noteq> {}"
```
```  1484   shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
```
```  1485 proof -
```
```  1486   interpret ab_semigroup_idem_mult sup
```
```  1487     by (rule ab_semigroup_idem_mult_sup)
```
```  1488   from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
```
```  1489   moreover with `finite A` have "finite B" by simp
```
```  1490   ultimately show ?thesis
```
```  1491   by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric])
```
```  1492 qed
```
```  1493
```
```  1494 end
```
```  1495
```
```  1496
```
```  1497 subsection {* Versions of @{const min} and @{const max} on non-empty sets *}
```
```  1498
```
```  1499 definition (in linorder) Min :: "'a set \<Rightarrow> 'a" where
```
```  1500   "Min = fold1 min"
```
```  1501
```
```  1502 definition (in linorder) Max :: "'a set \<Rightarrow> 'a" where
```
```  1503   "Max = fold1 max"
```
```  1504
```
```  1505 sublocale linorder < Min!: semilattice_big min Min proof
```
```  1506 qed (simp add: Min_def)
```
```  1507
```
```  1508 sublocale linorder < Max!: semilattice_big max Max proof
```
```  1509 qed (simp add: Max_def)
```
```  1510
```
```  1511 context linorder
```
```  1512 begin
```
```  1513
```
```  1514 lemmas Min_singleton = Min.singleton
```
```  1515 lemmas Max_singleton = Max.singleton
```
```  1516
```
```  1517 lemma Min_insert:
```
```  1518   assumes "finite A" and "A \<noteq> {}"
```
```  1519   shows "Min (insert x A) = min x (Min A)"
```
```  1520   using assms by simp
```
```  1521
```
```  1522 lemma Max_insert:
```
```  1523   assumes "finite A" and "A \<noteq> {}"
```
```  1524   shows "Max (insert x A) = max x (Max A)"
```
```  1525   using assms by simp
```
```  1526
```
```  1527 lemma Min_Un:
```
```  1528   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
```
```  1529   shows "Min (A \<union> B) = min (Min A) (Min B)"
```
```  1530   using assms by (rule Min.union_idem)
```
```  1531
```
```  1532 lemma Max_Un:
```
```  1533   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
```
```  1534   shows "Max (A \<union> B) = max (Max A) (Max B)"
```
```  1535   using assms by (rule Max.union_idem)
```
```  1536
```
```  1537 lemma hom_Min_commute:
```
```  1538   assumes "\<And>x y. h (min x y) = min (h x) (h y)"
```
```  1539     and "finite N" and "N \<noteq> {}"
```
```  1540   shows "h (Min N) = Min (h ` N)"
```
```  1541   using assms by (rule Min.hom_commute)
```
```  1542
```
```  1543 lemma hom_Max_commute:
```
```  1544   assumes "\<And>x y. h (max x y) = max (h x) (h y)"
```
```  1545     and "finite N" and "N \<noteq> {}"
```
```  1546   shows "h (Max N) = Max (h ` N)"
```
```  1547   using assms by (rule Max.hom_commute)
```
```  1548
```
```  1549 lemma ab_semigroup_idem_mult_min:
```
```  1550   "class.ab_semigroup_idem_mult min"
```
```  1551   proof qed (auto simp add: min_def)
```
```  1552
```
```  1553 lemma ab_semigroup_idem_mult_max:
```
```  1554   "class.ab_semigroup_idem_mult max"
```
```  1555   proof qed (auto simp add: max_def)
```
```  1556
```
```  1557 lemma max_lattice:
```
```  1558   "class.semilattice_inf max (op \<ge>) (op >)"
```
```  1559   by (fact min_max.dual_semilattice)
```
```  1560
```
```  1561 lemma dual_max:
```
```  1562   "ord.max (op \<ge>) = min"
```
```  1563   by (auto simp add: ord.max_def min_def fun_eq_iff)
```
```  1564
```
```  1565 lemma dual_min:
```
```  1566   "ord.min (op \<ge>) = max"
```
```  1567   by (auto simp add: ord.min_def max_def fun_eq_iff)
```
```  1568
```
```  1569 lemma strict_below_fold1_iff:
```
```  1570   assumes "finite A" and "A \<noteq> {}"
```
```  1571   shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  1572 proof -
```
```  1573   interpret ab_semigroup_idem_mult min
```
```  1574     by (rule ab_semigroup_idem_mult_min)
```
```  1575   from assms show ?thesis
```
```  1576   by (induct rule: finite_ne_induct)
```
```  1577     (simp_all add: fold1_insert)
```
```  1578 qed
```
```  1579
```
```  1580 lemma fold1_below_iff:
```
```  1581   assumes "finite A" and "A \<noteq> {}"
```
```  1582   shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  1583 proof -
```
```  1584   interpret ab_semigroup_idem_mult min
```
```  1585     by (rule ab_semigroup_idem_mult_min)
```
```  1586   from assms show ?thesis
```
```  1587   by (induct rule: finite_ne_induct)
```
```  1588     (simp_all add: fold1_insert min_le_iff_disj)
```
```  1589 qed
```
```  1590
```
```  1591 lemma fold1_strict_below_iff:
```
```  1592   assumes "finite A" and "A \<noteq> {}"
```
```  1593   shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  1594 proof -
```
```  1595   interpret ab_semigroup_idem_mult min
```
```  1596     by (rule ab_semigroup_idem_mult_min)
```
```  1597   from assms show ?thesis
```
```  1598   by (induct rule: finite_ne_induct)
```
```  1599     (simp_all add: fold1_insert min_less_iff_disj)
```
```  1600 qed
```
```  1601
```
```  1602 lemma fold1_antimono:
```
```  1603   assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
```
```  1604   shows "fold1 min B \<le> fold1 min A"
```
```  1605 proof cases
```
```  1606   assume "A = B" thus ?thesis by simp
```
```  1607 next
```
```  1608   interpret ab_semigroup_idem_mult min
```
```  1609     by (rule ab_semigroup_idem_mult_min)
```
```  1610   assume neq: "A \<noteq> B"
```
```  1611   have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
```
```  1612   have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
```
```  1613   also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
```
```  1614   proof -
```
```  1615     have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
```
```  1616     moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`])
```
```  1617     moreover have "(B-A) \<noteq> {}" using assms neq by blast
```
```  1618     moreover have "A Int (B-A) = {}" using assms by blast
```
```  1619     ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
```
```  1620   qed
```
```  1621   also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
```
```  1622   finally show ?thesis .
```
```  1623 qed
```
```  1624
```
```  1625 lemma Min_in [simp]:
```
```  1626   assumes "finite A" and "A \<noteq> {}"
```
```  1627   shows "Min A \<in> A"
```
```  1628 proof -
```
```  1629   interpret ab_semigroup_idem_mult min
```
```  1630     by (rule ab_semigroup_idem_mult_min)
```
```  1631   from assms fold1_in show ?thesis by (fastforce simp: Min_def min_def)
```
```  1632 qed
```
```  1633
```
```  1634 lemma Max_in [simp]:
```
```  1635   assumes "finite A" and "A \<noteq> {}"
```
```  1636   shows "Max A \<in> A"
```
```  1637 proof -
```
```  1638   interpret ab_semigroup_idem_mult max
```
```  1639     by (rule ab_semigroup_idem_mult_max)
```
```  1640   from assms fold1_in [of A] show ?thesis by (fastforce simp: Max_def max_def)
```
```  1641 qed
```
```  1642
```
```  1643 lemma Min_le [simp]:
```
```  1644   assumes "finite A" and "x \<in> A"
```
```  1645   shows "Min A \<le> x"
```
```  1646   using assms by (simp add: Min_def min_max.fold1_belowI)
```
```  1647
```
```  1648 lemma Max_ge [simp]:
```
```  1649   assumes "finite A" and "x \<in> A"
```
```  1650   shows "x \<le> Max A"
```
```  1651   by (simp add: Max_def semilattice_inf.fold1_belowI [OF max_lattice] assms)
```
```  1652
```
```  1653 lemma Min_ge_iff [simp, no_atp]:
```
```  1654   assumes "finite A" and "A \<noteq> {}"
```
```  1655   shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  1656   using assms by (simp add: Min_def min_max.below_fold1_iff)
```
```  1657
```
```  1658 lemma Max_le_iff [simp, no_atp]:
```
```  1659   assumes "finite A" and "A \<noteq> {}"
```
```  1660   shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
```
```  1661   by (simp add: Max_def semilattice_inf.below_fold1_iff [OF max_lattice] assms)
```
```  1662
```
```  1663 lemma Min_gr_iff [simp, no_atp]:
```
```  1664   assumes "finite A" and "A \<noteq> {}"
```
```  1665   shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  1666   using assms by (simp add: Min_def strict_below_fold1_iff)
```
```  1667
```
```  1668 lemma Max_less_iff [simp, no_atp]:
```
```  1669   assumes "finite A" and "A \<noteq> {}"
```
```  1670   shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
```
```  1671   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
```
```  1672     linorder.strict_below_fold1_iff [OF dual_linorder] assms)
```
```  1673
```
```  1674 lemma Min_le_iff [no_atp]:
```
```  1675   assumes "finite A" and "A \<noteq> {}"
```
```  1676   shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  1677   using assms by (simp add: Min_def fold1_below_iff)
```
```  1678
```
```  1679 lemma Max_ge_iff [no_atp]:
```
```  1680   assumes "finite A" and "A \<noteq> {}"
```
```  1681   shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
```
```  1682   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
```
```  1683     linorder.fold1_below_iff [OF dual_linorder] assms)
```
```  1684
```
```  1685 lemma Min_less_iff [no_atp]:
```
```  1686   assumes "finite A" and "A \<noteq> {}"
```
```  1687   shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  1688   using assms by (simp add: Min_def fold1_strict_below_iff)
```
```  1689
```
```  1690 lemma Max_gr_iff [no_atp]:
```
```  1691   assumes "finite A" and "A \<noteq> {}"
```
```  1692   shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
```
```  1693   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
```
```  1694     linorder.fold1_strict_below_iff [OF dual_linorder] assms)
```
```  1695
```
```  1696 lemma Min_eqI:
```
```  1697   assumes "finite A"
```
```  1698   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
```
```  1699     and "x \<in> A"
```
```  1700   shows "Min A = x"
```
```  1701 proof (rule antisym)
```
```  1702   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1703   with assms show "Min A \<ge> x" by simp
```
```  1704 next
```
```  1705   from assms show "x \<ge> Min A" by simp
```
```  1706 qed
```
```  1707
```
```  1708 lemma Max_eqI:
```
```  1709   assumes "finite A"
```
```  1710   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
```
```  1711     and "x \<in> A"
```
```  1712   shows "Max A = x"
```
```  1713 proof (rule antisym)
```
```  1714   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1715   with assms show "Max A \<le> x" by simp
```
```  1716 next
```
```  1717   from assms show "x \<le> Max A" by simp
```
```  1718 qed
```
```  1719
```
```  1720 lemma Min_antimono:
```
```  1721   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  1722   shows "Min N \<le> Min M"
```
```  1723   using assms by (simp add: Min_def fold1_antimono)
```
```  1724
```
```  1725 lemma Max_mono:
```
```  1726   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  1727   shows "Max M \<le> Max N"
```
```  1728   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
```
```  1729     linorder.fold1_antimono [OF dual_linorder] assms)
```
```  1730
```
```  1731 lemma finite_linorder_max_induct[consumes 1, case_names empty insert]:
```
```  1732  assumes fin: "finite A"
```
```  1733  and   empty: "P {}"
```
```  1734  and  insert: "(!!b A. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))"
```
```  1735  shows "P A"
```
```  1736 using fin empty insert
```
```  1737 proof (induct rule: finite_psubset_induct)
```
```  1738   case (psubset A)
```
```  1739   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
```
```  1740   have fin: "finite A" by fact
```
```  1741   have empty: "P {}" by fact
```
```  1742   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
```
```  1743   show "P A"
```
```  1744   proof (cases "A = {}")
```
```  1745     assume "A = {}"
```
```  1746     then show "P A" using `P {}` by simp
```
```  1747   next
```
```  1748     let ?B = "A - {Max A}"
```
```  1749     let ?A = "insert (Max A) ?B"
```
```  1750     have "finite ?B" using `finite A` by simp
```
```  1751     assume "A \<noteq> {}"
```
```  1752     with `finite A` have "Max A : A" by auto
```
```  1753     then have A: "?A = A" using insert_Diff_single insert_absorb by auto
```
```  1754     then have "P ?B" using `P {}` step IH[of ?B] by blast
```
```  1755     moreover
```
```  1756     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
```
```  1757     ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastforce
```
```  1758   qed
```
```  1759 qed
```
```  1760
```
```  1761 lemma finite_linorder_min_induct[consumes 1, case_names empty insert]:
```
```  1762  "\<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"
```
```  1763 by(rule linorder.finite_linorder_max_induct[OF dual_linorder])
```
```  1764
```
```  1765 end
```
```  1766
```
```  1767 context linordered_ab_semigroup_add
```
```  1768 begin
```
```  1769
```
```  1770 lemma add_Min_commute:
```
```  1771   fixes k
```
```  1772   assumes "finite N" and "N \<noteq> {}"
```
```  1773   shows "k + Min N = Min {k + m | m. m \<in> N}"
```
```  1774 proof -
```
```  1775   have "\<And>x y. k + min x y = min (k + x) (k + y)"
```
```  1776     by (simp add: min_def not_le)
```
```  1777       (blast intro: antisym less_imp_le add_left_mono)
```
```  1778   with assms show ?thesis
```
```  1779     using hom_Min_commute [of "plus k" N]
```
```  1780     by simp (blast intro: arg_cong [where f = Min])
```
```  1781 qed
```
```  1782
```
```  1783 lemma add_Max_commute:
```
```  1784   fixes k
```
```  1785   assumes "finite N" and "N \<noteq> {}"
```
```  1786   shows "k + Max N = Max {k + m | m. m \<in> N}"
```
```  1787 proof -
```
```  1788   have "\<And>x y. k + max x y = max (k + x) (k + y)"
```
```  1789     by (simp add: max_def not_le)
```
```  1790       (blast intro: antisym less_imp_le add_left_mono)
```
```  1791   with assms show ?thesis
```
```  1792     using hom_Max_commute [of "plus k" N]
```
```  1793     by simp (blast intro: arg_cong [where f = Max])
```
```  1794 qed
```
```  1795
```
```  1796 end
```
```  1797
```
```  1798 context linordered_ab_group_add
```
```  1799 begin
```
```  1800
```
```  1801 lemma minus_Max_eq_Min [simp]:
```
```  1802   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Max S) = Min (uminus ` S)"
```
```  1803   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
```
```  1804
```
```  1805 lemma minus_Min_eq_Max [simp]:
```
```  1806   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Min S) = Max (uminus ` S)"
```
```  1807   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
```
```  1808
```
```  1809 end
```
```  1810
```
```  1811 end
```