src/HOL/Big_Operators.thy
 author huffman Fri Mar 30 12:32:35 2012 +0200 (2012-03-30) changeset 47220 52426c62b5d0 parent 46904 f30e941b4512 child 48819 6cf7a9d8bbaf permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
```     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   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```   316   shows "setsum f (Union C) = setsum (setsum f) C"
```
```   317 proof cases
```
```   318   assume "finite C"
```
```   319   from setsum_UN_disjoint[OF this assms]
```
```   320   show ?thesis
```
```   321     by (simp add: SUP_def)
```
```   322 qed (force dest: finite_UnionD simp add: setsum_def)
```
```   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 fastforce
```
```   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_strict_mono_ex1:
```
```   527 fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
```
```   528 assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
```
```   529 shows "setsum f A < setsum g A"
```
```   530 proof-
```
```   531   from assms(3) obtain a where a: "a:A" "f a < g a" by blast
```
```   532   have "setsum f A = setsum f ((A-{a}) \<union> {a})"
```
```   533     by(simp add:insert_absorb[OF `a:A`])
```
```   534   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
```
```   535     using `finite A` by(subst setsum_Un_disjoint) auto
```
```   536   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
```
```   537     by(rule setsum_mono)(simp add: assms(2))
```
```   538   also have "setsum f {a} < setsum g {a}" using a by simp
```
```   539   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
```
```   540     using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto
```
```   541   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])
```
```   542   finally show ?thesis by (metis add_right_mono add_strict_left_mono)
```
```   543 qed
```
```   544
```
```   545 lemma setsum_negf:
```
```   546   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
```
```   547 proof (cases "finite A")
```
```   548   case True thus ?thesis by (induct set: finite) auto
```
```   549 next
```
```   550   case False thus ?thesis by (simp add: setsum_def)
```
```   551 qed
```
```   552
```
```   553 lemma setsum_subtractf:
```
```   554   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
```
```   555     setsum f A - setsum g A"
```
```   556 proof (cases "finite A")
```
```   557   case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
```
```   558 next
```
```   559   case False thus ?thesis by (simp add: setsum_def)
```
```   560 qed
```
```   561
```
```   562 lemma setsum_nonneg:
```
```   563   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
```
```   564   shows "0 \<le> setsum f A"
```
```   565 proof (cases "finite A")
```
```   566   case True thus ?thesis using nn
```
```   567   proof induct
```
```   568     case empty then show ?case by simp
```
```   569   next
```
```   570     case (insert x F)
```
```   571     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
```
```   572     with insert show ?case by simp
```
```   573   qed
```
```   574 next
```
```   575   case False thus ?thesis by (simp add: setsum_def)
```
```   576 qed
```
```   577
```
```   578 lemma setsum_nonpos:
```
```   579   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
```
```   580   shows "setsum f A \<le> 0"
```
```   581 proof (cases "finite A")
```
```   582   case True thus ?thesis using np
```
```   583   proof induct
```
```   584     case empty then show ?case by simp
```
```   585   next
```
```   586     case (insert x F)
```
```   587     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
```
```   588     with insert show ?case by simp
```
```   589   qed
```
```   590 next
```
```   591   case False thus ?thesis by (simp add: setsum_def)
```
```   592 qed
```
```   593
```
```   594 lemma setsum_nonneg_leq_bound:
```
```   595   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   596   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
```
```   597   shows "f i \<le> B"
```
```   598 proof -
```
```   599   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
```
```   600     using assms by (auto intro!: setsum_nonneg)
```
```   601   moreover
```
```   602   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
```
```   603     using assms by (simp add: setsum_diff1)
```
```   604   ultimately show ?thesis by auto
```
```   605 qed
```
```   606
```
```   607 lemma setsum_nonneg_0:
```
```   608   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
```
```   609   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
```
```   610   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
```
```   611   shows "f i = 0"
```
```   612   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
```
```   613
```
```   614 lemma setsum_mono2:
```
```   615 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
```
```   616 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
```
```   617 shows "setsum f A \<le> setsum f B"
```
```   618 proof -
```
```   619   have "setsum f A \<le> setsum f A + setsum f (B-A)"
```
```   620     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
```
```   621   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
```
```   622     by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
```
```   623   also have "A \<union> (B-A) = B" using sub by blast
```
```   624   finally show ?thesis .
```
```   625 qed
```
```   626
```
```   627 lemma setsum_mono3: "finite B ==> A <= B ==>
```
```   628     ALL x: B - A.
```
```   629       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>
```
```   630         setsum f A <= setsum f B"
```
```   631   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
```
```   632   apply (erule ssubst)
```
```   633   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
```
```   634   apply simp
```
```   635   apply (rule add_left_mono)
```
```   636   apply (erule setsum_nonneg)
```
```   637   apply (subst setsum_Un_disjoint [THEN sym])
```
```   638   apply (erule finite_subset, assumption)
```
```   639   apply (rule finite_subset)
```
```   640   prefer 2
```
```   641   apply assumption
```
```   642   apply (auto simp add: sup_absorb2)
```
```   643 done
```
```   644
```
```   645 lemma setsum_right_distrib:
```
```   646   fixes f :: "'a => ('b::semiring_0)"
```
```   647   shows "r * setsum f A = setsum (%n. r * f n) A"
```
```   648 proof (cases "finite A")
```
```   649   case True
```
```   650   thus ?thesis
```
```   651   proof induct
```
```   652     case empty thus ?case by simp
```
```   653   next
```
```   654     case (insert x A) thus ?case by (simp add: right_distrib)
```
```   655   qed
```
```   656 next
```
```   657   case False thus ?thesis by (simp add: setsum_def)
```
```   658 qed
```
```   659
```
```   660 lemma setsum_left_distrib:
```
```   661   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
```
```   662 proof (cases "finite A")
```
```   663   case True
```
```   664   then show ?thesis
```
```   665   proof induct
```
```   666     case empty thus ?case by simp
```
```   667   next
```
```   668     case (insert x A) thus ?case by (simp add: left_distrib)
```
```   669   qed
```
```   670 next
```
```   671   case False thus ?thesis by (simp add: setsum_def)
```
```   672 qed
```
```   673
```
```   674 lemma setsum_divide_distrib:
```
```   675   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
```
```   676 proof (cases "finite A")
```
```   677   case True
```
```   678   then show ?thesis
```
```   679   proof induct
```
```   680     case empty thus ?case by simp
```
```   681   next
```
```   682     case (insert x A) thus ?case by (simp add: add_divide_distrib)
```
```   683   qed
```
```   684 next
```
```   685   case False thus ?thesis by (simp add: setsum_def)
```
```   686 qed
```
```   687
```
```   688 lemma setsum_abs[iff]:
```
```   689   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   690   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
```
```   691 proof (cases "finite A")
```
```   692   case True
```
```   693   thus ?thesis
```
```   694   proof induct
```
```   695     case empty thus ?case by simp
```
```   696   next
```
```   697     case (insert x A)
```
```   698     thus ?case by (auto intro: abs_triangle_ineq order_trans)
```
```   699   qed
```
```   700 next
```
```   701   case False thus ?thesis by (simp add: setsum_def)
```
```   702 qed
```
```   703
```
```   704 lemma setsum_abs_ge_zero[iff]:
```
```   705   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   706   shows "0 \<le> setsum (%i. abs(f i)) A"
```
```   707 proof (cases "finite A")
```
```   708   case True
```
```   709   thus ?thesis
```
```   710   proof induct
```
```   711     case empty thus ?case by simp
```
```   712   next
```
```   713     case (insert x A) thus ?case by auto
```
```   714   qed
```
```   715 next
```
```   716   case False thus ?thesis by (simp add: setsum_def)
```
```   717 qed
```
```   718
```
```   719 lemma abs_setsum_abs[simp]:
```
```   720   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
```
```   721   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
```
```   722 proof (cases "finite A")
```
```   723   case True
```
```   724   thus ?thesis
```
```   725   proof induct
```
```   726     case empty thus ?case by simp
```
```   727   next
```
```   728     case (insert a A)
```
```   729     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
```
```   730     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
```
```   731     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
```
```   732       by (simp del: abs_of_nonneg)
```
```   733     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
```
```   734     finally show ?case .
```
```   735   qed
```
```   736 next
```
```   737   case False thus ?thesis by (simp add: setsum_def)
```
```   738 qed
```
```   739
```
```   740 lemma setsum_Plus:
```
```   741   fixes A :: "'a set" and B :: "'b set"
```
```   742   assumes fin: "finite A" "finite B"
```
```   743   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
```
```   744 proof -
```
```   745   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
```
```   746   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
```
```   747     by auto
```
```   748   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
```
```   749   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
```
```   750   ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)
```
```   751 qed
```
```   752
```
```   753
```
```   754 text {* Commuting outer and inner summation *}
```
```   755
```
```   756 lemma setsum_commute:
```
```   757   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
```
```   758 proof (simp add: setsum_cartesian_product)
```
```   759   have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
```
```   760     (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
```
```   761     (is "?s = _")
```
```   762     apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
```
```   763     apply (simp add: split_def)
```
```   764     done
```
```   765   also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
```
```   766     (is "_ = ?t")
```
```   767     apply (simp add: swap_product)
```
```   768     done
```
```   769   finally show "?s = ?t" .
```
```   770 qed
```
```   771
```
```   772 lemma setsum_product:
```
```   773   fixes f :: "'a => ('b::semiring_0)"
```
```   774   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
```
```   775   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
```
```   776
```
```   777 lemma setsum_mult_setsum_if_inj:
```
```   778 fixes f :: "'a => ('b::semiring_0)"
```
```   779 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
```
```   780   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
```
```   781 by(auto simp: setsum_product setsum_cartesian_product
```
```   782         intro!:  setsum_reindex_cong[symmetric])
```
```   783
```
```   784 lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
```
```   785 apply (cases "finite A")
```
```   786 apply (erule finite_induct)
```
```   787 apply (auto simp add: algebra_simps)
```
```   788 done
```
```   789
```
```   790 lemma setsum_bounded:
```
```   791   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
```
```   792   shows "setsum f A \<le> of_nat(card A) * K"
```
```   793 proof (cases "finite A")
```
```   794   case True
```
```   795   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
```
```   796 next
```
```   797   case False thus ?thesis by (simp add: setsum_def)
```
```   798 qed
```
```   799
```
```   800
```
```   801 subsubsection {* Cardinality as special case of @{const setsum} *}
```
```   802
```
```   803 lemma card_eq_setsum:
```
```   804   "card A = setsum (\<lambda>x. 1) A"
```
```   805   by (simp only: card_def setsum_def)
```
```   806
```
```   807 lemma card_UN_disjoint:
```
```   808   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
```
```   809     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
```
```   810   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
```
```   811 proof -
```
```   812   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
```
```   813   with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant)
```
```   814 qed
```
```   815
```
```   816 lemma card_Union_disjoint:
```
```   817   "finite C ==> (ALL A:C. finite A) ==>
```
```   818    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
```
```   819    ==> card (Union C) = setsum card C"
```
```   820 apply (frule card_UN_disjoint [of C id])
```
```   821 apply (simp_all add: SUP_def id_def)
```
```   822 done
```
```   823
```
```   824 text{*The image of a finite set can be expressed using @{term fold_image}.*}
```
```   825 lemma image_eq_fold_image:
```
```   826   "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A"
```
```   827 proof (induct rule: finite_induct)
```
```   828   case empty then show ?case by simp
```
```   829 next
```
```   830   interpret ab_semigroup_mult "op Un"
```
```   831     proof qed auto
```
```   832   case insert
```
```   833   then show ?case by simp
```
```   834 qed
```
```   835
```
```   836 subsubsection {* Cardinality of products *}
```
```   837
```
```   838 lemma card_SigmaI [simp]:
```
```   839   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
```
```   840   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
```
```   841 by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant)
```
```   842
```
```   843 (*
```
```   844 lemma SigmaI_insert: "y \<notin> A ==>
```
```   845   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
```
```   846   by auto
```
```   847 *)
```
```   848
```
```   849 lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
```
```   850   by (cases "finite A \<and> finite B")
```
```   851     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```   852
```
```   853 lemma card_cartesian_product_singleton:  "card({x} <*> A) = card(A)"
```
```   854 by (simp add: card_cartesian_product)
```
```   855
```
```   856
```
```   857 subsection {* Generalized product over a set *}
```
```   858
```
```   859 definition (in comm_monoid_mult) setprod :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
```
```   860   "setprod f A = (if finite A then fold_image (op *) f 1 A else 1)"
```
```   861
```
```   862 sublocale comm_monoid_mult < setprod!: comm_monoid_big "op *" 1 setprod proof
```
```   863 qed (fact setprod_def)
```
```   864
```
```   865 abbreviation
```
```   866   Setprod  ("\<Prod>_"  999) where
```
```   867   "\<Prod>A == setprod (%x. x) A"
```
```   868
```
```   869 syntax
```
```   870   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)
```
```   871 syntax (xsymbols)
```
```   872   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```   873 syntax (HTML output)
```
```   874   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
```
```   875
```
```   876 translations -- {* Beware of argument permutation! *}
```
```   877   "PROD i:A. b" == "CONST setprod (%i. b) A"
```
```   878   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
```
```   879
```
```   880 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
```
```   881  @{text"\<Prod>x|P. e"}. *}
```
```   882
```
```   883 syntax
```
```   884   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
```
```   885 syntax (xsymbols)
```
```   886   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```   887 syntax (HTML output)
```
```   888   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
```
```   889
```
```   890 translations
```
```   891   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```   892   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
```
```   893
```
```   894 lemma setprod_empty: "setprod f {} = 1"
```
```   895   by (fact setprod.empty)
```
```   896
```
```   897 lemma setprod_insert: "[| finite A; a \<notin> A |] ==>
```
```   898     setprod f (insert a A) = f a * setprod f A"
```
```   899   by (fact setprod.insert)
```
```   900
```
```   901 lemma setprod_infinite: "~ finite A ==> setprod f A = 1"
```
```   902   by (fact setprod.infinite)
```
```   903
```
```   904 lemma setprod_reindex:
```
```   905    "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
```
```   906 by(auto simp: setprod_def fold_image_reindex dest!:finite_imageD)
```
```   907
```
```   908 lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
```
```   909 by (auto simp add: setprod_reindex)
```
```   910
```
```   911 lemma setprod_cong:
```
```   912   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
```
```   913 by(fastforce simp: setprod_def intro: fold_image_cong)
```
```   914
```
```   915 lemma strong_setprod_cong[cong]:
```
```   916   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
```
```   917 by(fastforce simp: simp_implies_def setprod_def intro: fold_image_cong)
```
```   918
```
```   919 lemma setprod_reindex_cong: "inj_on f A ==>
```
```   920     B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
```
```   921 by (frule setprod_reindex, simp)
```
```   922
```
```   923 lemma strong_setprod_reindex_cong: assumes i: "inj_on f A"
```
```   924   and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
```
```   925   shows "setprod h B = setprod g A"
```
```   926 proof-
```
```   927     have "setprod h B = setprod (h o f) A"
```
```   928       by (simp add: B setprod_reindex[OF i, of h])
```
```   929     then show ?thesis apply simp
```
```   930       apply (rule setprod_cong)
```
```   931       apply simp
```
```   932       by (simp add: eq)
```
```   933 qed
```
```   934
```
```   935 lemma setprod_Un_one:
```
```   936   assumes fS: "finite S" and fT: "finite T"
```
```   937   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
```
```   938   shows "setprod f (S \<union> T) = setprod f S  * setprod f T"
```
```   939   using fS fT
```
```   940   apply (simp add: setprod_def)
```
```   941   apply (rule fold_image_Un_one)
```
```   942   using I0 by auto
```
```   943
```
```   944
```
```   945 lemma setprod_1: "setprod (%i. 1) A = 1"
```
```   946 apply (case_tac "finite A")
```
```   947 apply (erule finite_induct, auto simp add: mult_ac)
```
```   948 done
```
```   949
```
```   950 lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
```
```   951 apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
```
```   952 apply (erule ssubst, rule setprod_1)
```
```   953 apply (rule setprod_cong, auto)
```
```   954 done
```
```   955
```
```   956 lemma setprod_Un_Int: "finite A ==> finite B
```
```   957     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
```
```   958 by(simp add: setprod_def fold_image_Un_Int[symmetric])
```
```   959
```
```   960 lemma setprod_Un_disjoint: "finite A ==> finite B
```
```   961   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
```
```   962 by (subst setprod_Un_Int [symmetric], auto)
```
```   963
```
```   964 lemma setprod_mono_one_left:
```
```   965   assumes fT: "finite T" and ST: "S \<subseteq> T"
```
```   966   and z: "\<forall>i \<in> T - S. f i = 1"
```
```   967   shows "setprod f S = setprod f T"
```
```   968 proof-
```
```   969   have eq: "T = S \<union> (T - S)" using ST by blast
```
```   970   have d: "S \<inter> (T - S) = {}" using ST by blast
```
```   971   from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
```
```   972   show ?thesis
```
```   973   by (simp add: setprod_Un_disjoint[OF f d, unfolded eq[symmetric]] setprod_1'[OF z])
```
```   974 qed
```
```   975
```
```   976 lemmas setprod_mono_one_right = setprod_mono_one_left [THEN sym]
```
```   977
```
```   978 lemma setprod_delta:
```
```   979   assumes fS: "finite S"
```
```   980   shows "setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
```
```   981 proof-
```
```   982   let ?f = "(\<lambda>k. if k=a then b k else 1)"
```
```   983   {assume a: "a \<notin> S"
```
```   984     hence "\<forall> k\<in> S. ?f k = 1" by simp
```
```   985     hence ?thesis  using a by (simp add: setprod_1) }
```
```   986   moreover
```
```   987   {assume a: "a \<in> S"
```
```   988     let ?A = "S - {a}"
```
```   989     let ?B = "{a}"
```
```   990     have eq: "S = ?A \<union> ?B" using a by blast
```
```   991     have dj: "?A \<inter> ?B = {}" by simp
```
```   992     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   993     have fA1: "setprod ?f ?A = 1" apply (rule setprod_1') by auto
```
```   994     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```   995       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```   996       by simp
```
```   997     then have ?thesis using a by (simp add: fA1 cong: setprod_cong cong del: if_weak_cong)}
```
```   998   ultimately show ?thesis by blast
```
```   999 qed
```
```  1000
```
```  1001 lemma setprod_delta':
```
```  1002   assumes fS: "finite S" shows
```
```  1003   "setprod (\<lambda>k. if a = k then b k else 1) S =
```
```  1004      (if a\<in> S then b a else 1)"
```
```  1005   using setprod_delta[OF fS, of a b, symmetric]
```
```  1006   by (auto intro: setprod_cong)
```
```  1007
```
```  1008
```
```  1009 lemma setprod_UN_disjoint:
```
```  1010     "finite I ==> (ALL i:I. finite (A i)) ==>
```
```  1011         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
```
```  1012       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
```
```  1013   by (simp add: setprod_def fold_image_UN_disjoint)
```
```  1014
```
```  1015 lemma setprod_Union_disjoint:
```
```  1016   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```
```  1017   shows "setprod f (Union C) = setprod (setprod f) C"
```
```  1018 proof cases
```
```  1019   assume "finite C"
```
```  1020   from setprod_UN_disjoint[OF this assms]
```
```  1021   show ?thesis
```
```  1022     by (simp add: SUP_def)
```
```  1023 qed (force dest: finite_UnionD simp add: setprod_def)
```
```  1024
```
```  1025 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
```
```  1026     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
```
```  1027     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
```
```  1028 by(simp add:setprod_def fold_image_Sigma split_def)
```
```  1029
```
```  1030 text{*Here we can eliminate the finiteness assumptions, by cases.*}
```
```  1031 lemma setprod_cartesian_product:
```
```  1032      "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
```
```  1033 apply (cases "finite A")
```
```  1034  apply (cases "finite B")
```
```  1035   apply (simp add: setprod_Sigma)
```
```  1036  apply (cases "A={}", simp)
```
```  1037  apply (simp add: setprod_1)
```
```  1038 apply (auto simp add: setprod_def
```
```  1039             dest: finite_cartesian_productD1 finite_cartesian_productD2)
```
```  1040 done
```
```  1041
```
```  1042 lemma setprod_timesf:
```
```  1043      "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
```
```  1044 by(simp add:setprod_def fold_image_distrib)
```
```  1045
```
```  1046
```
```  1047 subsubsection {* Properties in more restricted classes of structures *}
```
```  1048
```
```  1049 lemma setprod_eq_1_iff [simp]:
```
```  1050   "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
```
```  1051 by (induct set: finite) auto
```
```  1052
```
```  1053 lemma setprod_zero:
```
```  1054      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
```
```  1055 apply (induct set: finite, force, clarsimp)
```
```  1056 apply (erule disjE, auto)
```
```  1057 done
```
```  1058
```
```  1059 lemma setprod_nonneg [rule_format]:
```
```  1060    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
```
```  1061 by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
```
```  1062
```
```  1063 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)
```
```  1064   --> 0 < setprod f A"
```
```  1065 by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
```
```  1066
```
```  1067 lemma setprod_zero_iff[simp]: "finite A ==>
```
```  1068   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
```
```  1069   (EX x: A. f x = 0)"
```
```  1070 by (erule finite_induct, auto simp:no_zero_divisors)
```
```  1071
```
```  1072 lemma setprod_pos_nat:
```
```  1073   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
```
```  1074 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1075
```
```  1076 lemma setprod_pos_nat_iff[simp]:
```
```  1077   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
```
```  1078 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
```
```  1079
```
```  1080 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
```
```  1081   (setprod f (A Un B) :: 'a ::{field})
```
```  1082    = setprod f A * setprod f B / setprod f (A Int B)"
```
```  1083 by (subst setprod_Un_Int [symmetric], auto)
```
```  1084
```
```  1085 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
```
```  1086   (setprod f (A - {a}) :: 'a :: {field}) =
```
```  1087   (if a:A then setprod f A / f a else setprod f A)"
```
```  1088   by (erule finite_induct) (auto simp add: insert_Diff_if)
```
```  1089
```
```  1090 lemma setprod_inversef:
```
```  1091   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1092   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
```
```  1093 by (erule finite_induct) auto
```
```  1094
```
```  1095 lemma setprod_dividef:
```
```  1096   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"
```
```  1097   shows "finite A
```
```  1098     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
```
```  1099 apply (subgoal_tac
```
```  1100          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
```
```  1101 apply (erule ssubst)
```
```  1102 apply (subst divide_inverse)
```
```  1103 apply (subst setprod_timesf)
```
```  1104 apply (subst setprod_inversef, assumption+, rule refl)
```
```  1105 apply (rule setprod_cong, rule refl)
```
```  1106 apply (subst divide_inverse, auto)
```
```  1107 done
```
```  1108
```
```  1109 lemma setprod_dvd_setprod [rule_format]:
```
```  1110     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
```
```  1111   apply (cases "finite A")
```
```  1112   apply (induct set: finite)
```
```  1113   apply (auto simp add: dvd_def)
```
```  1114   apply (rule_tac x = "k * ka" in exI)
```
```  1115   apply (simp add: algebra_simps)
```
```  1116 done
```
```  1117
```
```  1118 lemma setprod_dvd_setprod_subset:
```
```  1119   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
```
```  1120   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
```
```  1121   apply (unfold dvd_def, blast)
```
```  1122   apply (subst setprod_Un_disjoint [symmetric])
```
```  1123   apply (auto elim: finite_subset intro: setprod_cong)
```
```  1124 done
```
```  1125
```
```  1126 lemma setprod_dvd_setprod_subset2:
```
```  1127   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow>
```
```  1128       setprod f A dvd setprod g B"
```
```  1129   apply (rule dvd_trans)
```
```  1130   apply (rule setprod_dvd_setprod, erule (1) bspec)
```
```  1131   apply (erule (1) setprod_dvd_setprod_subset)
```
```  1132 done
```
```  1133
```
```  1134 lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow>
```
```  1135     (f i ::'a::comm_semiring_1) dvd setprod f A"
```
```  1136 by (induct set: finite) (auto intro: dvd_mult)
```
```  1137
```
```  1138 lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow>
```
```  1139     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
```
```  1140   apply (cases "finite A")
```
```  1141   apply (induct set: finite)
```
```  1142   apply auto
```
```  1143 done
```
```  1144
```
```  1145 lemma setprod_mono:
```
```  1146   fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"
```
```  1147   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
```
```  1148   shows "setprod f A \<le> setprod g A"
```
```  1149 proof (cases "finite A")
```
```  1150   case True
```
```  1151   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]
```
```  1152   proof (induct A rule: finite_subset_induct)
```
```  1153     case (insert a F)
```
```  1154     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"
```
```  1155       unfolding setprod_insert[OF insert(1,3)]
```
```  1156       using assms[rule_format,OF insert(2)] insert
```
```  1157       by (auto intro: mult_mono mult_nonneg_nonneg)
```
```  1158   qed auto
```
```  1159   thus ?thesis by simp
```
```  1160 qed auto
```
```  1161
```
```  1162 lemma abs_setprod:
```
```  1163   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"
```
```  1164   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"
```
```  1165 proof (cases "finite A")
```
```  1166   case True thus ?thesis
```
```  1167     by induct (auto simp add: field_simps abs_mult)
```
```  1168 qed auto
```
```  1169
```
```  1170 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
```
```  1171 apply (erule finite_induct)
```
```  1172 apply auto
```
```  1173 done
```
```  1174
```
```  1175 lemma setprod_gen_delta:
```
```  1176   assumes fS: "finite S"
```
```  1177   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)"
```
```  1178 proof-
```
```  1179   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```  1180   {assume a: "a \<notin> S"
```
```  1181     hence "\<forall> k\<in> S. ?f k = c" by simp
```
```  1182     hence ?thesis  using a setprod_constant[OF fS, of c] by (simp add: setprod_1 cong add: setprod_cong) }
```
```  1183   moreover
```
```  1184   {assume a: "a \<in> S"
```
```  1185     let ?A = "S - {a}"
```
```  1186     let ?B = "{a}"
```
```  1187     have eq: "S = ?A \<union> ?B" using a by blast
```
```  1188     have dj: "?A \<inter> ?B = {}" by simp
```
```  1189     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```  1190     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```  1191       apply (rule setprod_cong) by auto
```
```  1192     have cA: "card ?A = card S - 1" using fS a by auto
```
```  1193     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
```
```  1194     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```  1195       using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```  1196       by simp
```
```  1197     then have ?thesis using a cA
```
```  1198       by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
```
```  1199   ultimately show ?thesis by blast
```
```  1200 qed
```
```  1201
```
```  1202
```
```  1203 subsection {* Versions of @{const inf} and @{const sup} on non-empty sets *}
```
```  1204
```
```  1205 no_notation times (infixl "*" 70)
```
```  1206 no_notation Groups.one ("1")
```
```  1207
```
```  1208 locale semilattice_big = semilattice +
```
```  1209   fixes F :: "'a set \<Rightarrow> 'a"
```
```  1210   assumes F_eq: "finite A \<Longrightarrow> F A = fold1 (op *) A"
```
```  1211
```
```  1212 sublocale semilattice_big < folding_one_idem proof
```
```  1213 qed (simp_all add: F_eq)
```
```  1214
```
```  1215 notation times (infixl "*" 70)
```
```  1216 notation Groups.one ("1")
```
```  1217
```
```  1218 context lattice
```
```  1219 begin
```
```  1220
```
```  1221 definition Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_"  900) where
```
```  1222   "Inf_fin = fold1 inf"
```
```  1223
```
```  1224 definition Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_"  900) where
```
```  1225   "Sup_fin = fold1 sup"
```
```  1226
```
```  1227 end
```
```  1228
```
```  1229 sublocale lattice < Inf_fin!: semilattice_big inf Inf_fin proof
```
```  1230 qed (simp add: Inf_fin_def)
```
```  1231
```
```  1232 sublocale lattice < Sup_fin!: semilattice_big sup Sup_fin proof
```
```  1233 qed (simp add: Sup_fin_def)
```
```  1234
```
```  1235 context semilattice_inf
```
```  1236 begin
```
```  1237
```
```  1238 lemma ab_semigroup_idem_mult_inf:
```
```  1239   "class.ab_semigroup_idem_mult inf"
```
```  1240 proof qed (rule inf_assoc inf_commute inf_idem)+
```
```  1241
```
```  1242 lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> Finite_Set.fold inf b (insert a A) = inf a (Finite_Set.fold inf b A)"
```
```  1243 by(rule comp_fun_idem.fold_insert_idem[OF ab_semigroup_idem_mult.comp_fun_idem[OF ab_semigroup_idem_mult_inf]])
```
```  1244
```
```  1245 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"
```
```  1246 by (induct pred: finite) (auto intro: le_infI1)
```
```  1247
```
```  1248 lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> Finite_Set.fold inf b A \<le> inf a b"
```
```  1249 proof(induct arbitrary: a pred:finite)
```
```  1250   case empty thus ?case by simp
```
```  1251 next
```
```  1252   case (insert x A)
```
```  1253   show ?case
```
```  1254   proof cases
```
```  1255     assume "A = {}" thus ?thesis using insert by simp
```
```  1256   next
```
```  1257     assume "A \<noteq> {}" thus ?thesis using insert by (auto intro: le_infI2)
```
```  1258   qed
```
```  1259 qed
```
```  1260
```
```  1261 lemma below_fold1_iff:
```
```  1262   assumes "finite A" "A \<noteq> {}"
```
```  1263   shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  1264 proof -
```
```  1265   interpret ab_semigroup_idem_mult inf
```
```  1266     by (rule ab_semigroup_idem_mult_inf)
```
```  1267   show ?thesis using assms by (induct rule: finite_ne_induct) simp_all
```
```  1268 qed
```
```  1269
```
```  1270 lemma fold1_belowI:
```
```  1271   assumes "finite A"
```
```  1272     and "a \<in> A"
```
```  1273   shows "fold1 inf A \<le> a"
```
```  1274 proof -
```
```  1275   from assms have "A \<noteq> {}" by auto
```
```  1276   from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
```
```  1277   proof (induct rule: finite_ne_induct)
```
```  1278     case singleton thus ?case by simp
```
```  1279   next
```
```  1280     interpret ab_semigroup_idem_mult inf
```
```  1281       by (rule ab_semigroup_idem_mult_inf)
```
```  1282     case (insert x F)
```
```  1283     from insert(5) have "a = x \<or> a \<in> F" by simp
```
```  1284     thus ?case
```
```  1285     proof
```
```  1286       assume "a = x" thus ?thesis using insert
```
```  1287         by (simp add: mult_ac)
```
```  1288     next
```
```  1289       assume "a \<in> F"
```
```  1290       hence bel: "fold1 inf F \<le> a" by (rule insert)
```
```  1291       have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
```
```  1292         using insert by (simp add: mult_ac)
```
```  1293       also have "inf (fold1 inf F) a = fold1 inf F"
```
```  1294         using bel by (auto intro: antisym)
```
```  1295       also have "inf x \<dots> = fold1 inf (insert x F)"
```
```  1296         using insert by (simp add: mult_ac)
```
```  1297       finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
```
```  1298       moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
```
```  1299       ultimately show ?thesis by simp
```
```  1300     qed
```
```  1301   qed
```
```  1302 qed
```
```  1303
```
```  1304 end
```
```  1305
```
```  1306 context semilattice_sup
```
```  1307 begin
```
```  1308
```
```  1309 lemma ab_semigroup_idem_mult_sup: "class.ab_semigroup_idem_mult sup"
```
```  1310 by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
```
```  1311
```
```  1312 lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> Finite_Set.fold sup b (insert a A) = sup a (Finite_Set.fold sup b A)"
```
```  1313 by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)
```
```  1314
```
```  1315 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"
```
```  1316 by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)
```
```  1317
```
```  1318 lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> Finite_Set.fold sup b A"
```
```  1319 by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)
```
```  1320
```
```  1321 end
```
```  1322
```
```  1323 context lattice
```
```  1324 begin
```
```  1325
```
```  1326 lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
```
```  1327 apply(unfold Sup_fin_def Inf_fin_def)
```
```  1328 apply(subgoal_tac "EX a. a:A")
```
```  1329 prefer 2 apply blast
```
```  1330 apply(erule exE)
```
```  1331 apply(rule order_trans)
```
```  1332 apply(erule (1) fold1_belowI)
```
```  1333 apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice])
```
```  1334 done
```
```  1335
```
```  1336 lemma sup_Inf_absorb [simp]:
```
```  1337   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
```
```  1338 apply(subst sup_commute)
```
```  1339 apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)
```
```  1340 done
```
```  1341
```
```  1342 lemma inf_Sup_absorb [simp]:
```
```  1343   "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
```
```  1344 by (simp add: Sup_fin_def inf_absorb1
```
```  1345   semilattice_inf.fold1_belowI [OF dual_semilattice])
```
```  1346
```
```  1347 end
```
```  1348
```
```  1349 context distrib_lattice
```
```  1350 begin
```
```  1351
```
```  1352 lemma sup_Inf1_distrib:
```
```  1353   assumes "finite A"
```
```  1354     and "A \<noteq> {}"
```
```  1355   shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
```
```  1356 proof -
```
```  1357   interpret ab_semigroup_idem_mult inf
```
```  1358     by (rule ab_semigroup_idem_mult_inf)
```
```  1359   from assms show ?thesis
```
```  1360     by (simp add: Inf_fin_def image_def
```
```  1361       hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])
```
```  1362         (rule arg_cong [where f="fold1 inf"], blast)
```
```  1363 qed
```
```  1364
```
```  1365 lemma sup_Inf2_distrib:
```
```  1366   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1367   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}"
```
```  1368 using A proof (induct rule: finite_ne_induct)
```
```  1369   case singleton thus ?case
```
```  1370     by (simp add: sup_Inf1_distrib [OF B])
```
```  1371 next
```
```  1372   interpret ab_semigroup_idem_mult inf
```
```  1373     by (rule ab_semigroup_idem_mult_inf)
```
```  1374   case (insert x A)
```
```  1375   have finB: "finite {sup x b |b. b \<in> B}"
```
```  1376     by(rule finite_surj[where f = "sup x", OF B(1)], auto)
```
```  1377   have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1378   proof -
```
```  1379     have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
```
```  1380       by blast
```
```  1381     thus ?thesis by(simp add: insert(1) B(1))
```
```  1382   qed
```
```  1383   have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1384   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)"
```
```  1385     using insert by simp
```
```  1386   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)
```
```  1387   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})"
```
```  1388     using insert by(simp add:sup_Inf1_distrib[OF B])
```
```  1389   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})"
```
```  1390     (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
```
```  1391     using B insert
```
```  1392     by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
```
```  1393   also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1394     by blast
```
```  1395   finally show ?case .
```
```  1396 qed
```
```  1397
```
```  1398 lemma inf_Sup1_distrib:
```
```  1399   assumes "finite A" and "A \<noteq> {}"
```
```  1400   shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
```
```  1401 proof -
```
```  1402   interpret ab_semigroup_idem_mult sup
```
```  1403     by (rule ab_semigroup_idem_mult_sup)
```
```  1404   from assms show ?thesis
```
```  1405     by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])
```
```  1406       (rule arg_cong [where f="fold1 sup"], blast)
```
```  1407 qed
```
```  1408
```
```  1409 lemma inf_Sup2_distrib:
```
```  1410   assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
```
```  1411   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}"
```
```  1412 using A proof (induct rule: finite_ne_induct)
```
```  1413   case singleton thus ?case
```
```  1414     by(simp add: inf_Sup1_distrib [OF B])
```
```  1415 next
```
```  1416   case (insert x A)
```
```  1417   have finB: "finite {inf x b |b. b \<in> B}"
```
```  1418     by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
```
```  1419   have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
```
```  1420   proof -
```
```  1421     have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
```
```  1422       by blast
```
```  1423     thus ?thesis by(simp add: insert(1) B(1))
```
```  1424   qed
```
```  1425   have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
```
```  1426   interpret ab_semigroup_idem_mult sup
```
```  1427     by (rule ab_semigroup_idem_mult_sup)
```
```  1428   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)"
```
```  1429     using insert by simp
```
```  1430   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)
```
```  1431   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})"
```
```  1432     using insert by(simp add:inf_Sup1_distrib[OF B])
```
```  1433   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})"
```
```  1434     (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
```
```  1435     using B insert
```
```  1436     by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
```
```  1437   also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
```
```  1438     by blast
```
```  1439   finally show ?case .
```
```  1440 qed
```
```  1441
```
```  1442 end
```
```  1443
```
```  1444 context complete_lattice
```
```  1445 begin
```
```  1446
```
```  1447 lemma Inf_fin_Inf:
```
```  1448   assumes "finite A" and "A \<noteq> {}"
```
```  1449   shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
```
```  1450 proof -
```
```  1451   interpret ab_semigroup_idem_mult inf
```
```  1452     by (rule ab_semigroup_idem_mult_inf)
```
```  1453   from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
```
```  1454   moreover with `finite A` have "finite B" by simp
```
```  1455   ultimately show ?thesis
```
```  1456     by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])
```
```  1457 qed
```
```  1458
```
```  1459 lemma Sup_fin_Sup:
```
```  1460   assumes "finite A" and "A \<noteq> {}"
```
```  1461   shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
```
```  1462 proof -
```
```  1463   interpret ab_semigroup_idem_mult sup
```
```  1464     by (rule ab_semigroup_idem_mult_sup)
```
```  1465   from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
```
```  1466   moreover with `finite A` have "finite B" by simp
```
```  1467   ultimately show ?thesis
```
```  1468   by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric])
```
```  1469 qed
```
```  1470
```
```  1471 end
```
```  1472
```
```  1473
```
```  1474 subsection {* Versions of @{const min} and @{const max} on non-empty sets *}
```
```  1475
```
```  1476 definition (in linorder) Min :: "'a set \<Rightarrow> 'a" where
```
```  1477   "Min = fold1 min"
```
```  1478
```
```  1479 definition (in linorder) Max :: "'a set \<Rightarrow> 'a" where
```
```  1480   "Max = fold1 max"
```
```  1481
```
```  1482 sublocale linorder < Min!: semilattice_big min Min proof
```
```  1483 qed (simp add: Min_def)
```
```  1484
```
```  1485 sublocale linorder < Max!: semilattice_big max Max proof
```
```  1486 qed (simp add: Max_def)
```
```  1487
```
```  1488 context linorder
```
```  1489 begin
```
```  1490
```
```  1491 lemmas Min_singleton = Min.singleton
```
```  1492 lemmas Max_singleton = Max.singleton
```
```  1493
```
```  1494 lemma Min_insert:
```
```  1495   assumes "finite A" and "A \<noteq> {}"
```
```  1496   shows "Min (insert x A) = min x (Min A)"
```
```  1497   using assms by simp
```
```  1498
```
```  1499 lemma Max_insert:
```
```  1500   assumes "finite A" and "A \<noteq> {}"
```
```  1501   shows "Max (insert x A) = max x (Max A)"
```
```  1502   using assms by simp
```
```  1503
```
```  1504 lemma Min_Un:
```
```  1505   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
```
```  1506   shows "Min (A \<union> B) = min (Min A) (Min B)"
```
```  1507   using assms by (rule Min.union_idem)
```
```  1508
```
```  1509 lemma Max_Un:
```
```  1510   assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
```
```  1511   shows "Max (A \<union> B) = max (Max A) (Max B)"
```
```  1512   using assms by (rule Max.union_idem)
```
```  1513
```
```  1514 lemma hom_Min_commute:
```
```  1515   assumes "\<And>x y. h (min x y) = min (h x) (h y)"
```
```  1516     and "finite N" and "N \<noteq> {}"
```
```  1517   shows "h (Min N) = Min (h ` N)"
```
```  1518   using assms by (rule Min.hom_commute)
```
```  1519
```
```  1520 lemma hom_Max_commute:
```
```  1521   assumes "\<And>x y. h (max x y) = max (h x) (h y)"
```
```  1522     and "finite N" and "N \<noteq> {}"
```
```  1523   shows "h (Max N) = Max (h ` N)"
```
```  1524   using assms by (rule Max.hom_commute)
```
```  1525
```
```  1526 lemma ab_semigroup_idem_mult_min:
```
```  1527   "class.ab_semigroup_idem_mult min"
```
```  1528   proof qed (auto simp add: min_def)
```
```  1529
```
```  1530 lemma ab_semigroup_idem_mult_max:
```
```  1531   "class.ab_semigroup_idem_mult max"
```
```  1532   proof qed (auto simp add: max_def)
```
```  1533
```
```  1534 lemma max_lattice:
```
```  1535   "class.semilattice_inf max (op \<ge>) (op >)"
```
```  1536   by (fact min_max.dual_semilattice)
```
```  1537
```
```  1538 lemma dual_max:
```
```  1539   "ord.max (op \<ge>) = min"
```
```  1540   by (auto simp add: ord.max_def min_def fun_eq_iff)
```
```  1541
```
```  1542 lemma dual_min:
```
```  1543   "ord.min (op \<ge>) = max"
```
```  1544   by (auto simp add: ord.min_def max_def fun_eq_iff)
```
```  1545
```
```  1546 lemma strict_below_fold1_iff:
```
```  1547   assumes "finite A" and "A \<noteq> {}"
```
```  1548   shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  1549 proof -
```
```  1550   interpret ab_semigroup_idem_mult min
```
```  1551     by (rule ab_semigroup_idem_mult_min)
```
```  1552   from assms show ?thesis
```
```  1553   by (induct rule: finite_ne_induct)
```
```  1554     (simp_all add: fold1_insert)
```
```  1555 qed
```
```  1556
```
```  1557 lemma fold1_below_iff:
```
```  1558   assumes "finite A" and "A \<noteq> {}"
```
```  1559   shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  1560 proof -
```
```  1561   interpret ab_semigroup_idem_mult min
```
```  1562     by (rule ab_semigroup_idem_mult_min)
```
```  1563   from assms show ?thesis
```
```  1564   by (induct rule: finite_ne_induct)
```
```  1565     (simp_all add: fold1_insert min_le_iff_disj)
```
```  1566 qed
```
```  1567
```
```  1568 lemma fold1_strict_below_iff:
```
```  1569   assumes "finite A" and "A \<noteq> {}"
```
```  1570   shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  1571 proof -
```
```  1572   interpret ab_semigroup_idem_mult min
```
```  1573     by (rule ab_semigroup_idem_mult_min)
```
```  1574   from assms show ?thesis
```
```  1575   by (induct rule: finite_ne_induct)
```
```  1576     (simp_all add: fold1_insert min_less_iff_disj)
```
```  1577 qed
```
```  1578
```
```  1579 lemma fold1_antimono:
```
```  1580   assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
```
```  1581   shows "fold1 min B \<le> fold1 min A"
```
```  1582 proof cases
```
```  1583   assume "A = B" thus ?thesis by simp
```
```  1584 next
```
```  1585   interpret ab_semigroup_idem_mult min
```
```  1586     by (rule ab_semigroup_idem_mult_min)
```
```  1587   assume neq: "A \<noteq> B"
```
```  1588   have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
```
```  1589   have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
```
```  1590   also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
```
```  1591   proof -
```
```  1592     have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
```
```  1593     moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`])
```
```  1594     moreover have "(B-A) \<noteq> {}" using assms neq by blast
```
```  1595     moreover have "A Int (B-A) = {}" using assms by blast
```
```  1596     ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
```
```  1597   qed
```
```  1598   also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
```
```  1599   finally show ?thesis .
```
```  1600 qed
```
```  1601
```
```  1602 lemma Min_in [simp]:
```
```  1603   assumes "finite A" and "A \<noteq> {}"
```
```  1604   shows "Min A \<in> A"
```
```  1605 proof -
```
```  1606   interpret ab_semigroup_idem_mult min
```
```  1607     by (rule ab_semigroup_idem_mult_min)
```
```  1608   from assms fold1_in show ?thesis by (fastforce simp: Min_def min_def)
```
```  1609 qed
```
```  1610
```
```  1611 lemma Max_in [simp]:
```
```  1612   assumes "finite A" and "A \<noteq> {}"
```
```  1613   shows "Max A \<in> A"
```
```  1614 proof -
```
```  1615   interpret ab_semigroup_idem_mult max
```
```  1616     by (rule ab_semigroup_idem_mult_max)
```
```  1617   from assms fold1_in [of A] show ?thesis by (fastforce simp: Max_def max_def)
```
```  1618 qed
```
```  1619
```
```  1620 lemma Min_le [simp]:
```
```  1621   assumes "finite A" and "x \<in> A"
```
```  1622   shows "Min A \<le> x"
```
```  1623   using assms by (simp add: Min_def min_max.fold1_belowI)
```
```  1624
```
```  1625 lemma Max_ge [simp]:
```
```  1626   assumes "finite A" and "x \<in> A"
```
```  1627   shows "x \<le> Max A"
```
```  1628   by (simp add: Max_def semilattice_inf.fold1_belowI [OF max_lattice] assms)
```
```  1629
```
```  1630 lemma Min_ge_iff [simp, no_atp]:
```
```  1631   assumes "finite A" and "A \<noteq> {}"
```
```  1632   shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
```
```  1633   using assms by (simp add: Min_def min_max.below_fold1_iff)
```
```  1634
```
```  1635 lemma Max_le_iff [simp, no_atp]:
```
```  1636   assumes "finite A" and "A \<noteq> {}"
```
```  1637   shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
```
```  1638   by (simp add: Max_def semilattice_inf.below_fold1_iff [OF max_lattice] assms)
```
```  1639
```
```  1640 lemma Min_gr_iff [simp, no_atp]:
```
```  1641   assumes "finite A" and "A \<noteq> {}"
```
```  1642   shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
```
```  1643   using assms by (simp add: Min_def strict_below_fold1_iff)
```
```  1644
```
```  1645 lemma Max_less_iff [simp, no_atp]:
```
```  1646   assumes "finite A" and "A \<noteq> {}"
```
```  1647   shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
```
```  1648   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
```
```  1649     linorder.strict_below_fold1_iff [OF dual_linorder] assms)
```
```  1650
```
```  1651 lemma Min_le_iff [no_atp]:
```
```  1652   assumes "finite A" and "A \<noteq> {}"
```
```  1653   shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
```
```  1654   using assms by (simp add: Min_def fold1_below_iff)
```
```  1655
```
```  1656 lemma Max_ge_iff [no_atp]:
```
```  1657   assumes "finite A" and "A \<noteq> {}"
```
```  1658   shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
```
```  1659   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
```
```  1660     linorder.fold1_below_iff [OF dual_linorder] assms)
```
```  1661
```
```  1662 lemma Min_less_iff [no_atp]:
```
```  1663   assumes "finite A" and "A \<noteq> {}"
```
```  1664   shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
```
```  1665   using assms by (simp add: Min_def fold1_strict_below_iff)
```
```  1666
```
```  1667 lemma Max_gr_iff [no_atp]:
```
```  1668   assumes "finite A" and "A \<noteq> {}"
```
```  1669   shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
```
```  1670   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
```
```  1671     linorder.fold1_strict_below_iff [OF dual_linorder] assms)
```
```  1672
```
```  1673 lemma Min_eqI:
```
```  1674   assumes "finite A"
```
```  1675   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
```
```  1676     and "x \<in> A"
```
```  1677   shows "Min A = x"
```
```  1678 proof (rule antisym)
```
```  1679   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1680   with assms show "Min A \<ge> x" by simp
```
```  1681 next
```
```  1682   from assms show "x \<ge> Min A" by simp
```
```  1683 qed
```
```  1684
```
```  1685 lemma Max_eqI:
```
```  1686   assumes "finite A"
```
```  1687   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
```
```  1688     and "x \<in> A"
```
```  1689   shows "Max A = x"
```
```  1690 proof (rule antisym)
```
```  1691   from `x \<in> A` have "A \<noteq> {}" by auto
```
```  1692   with assms show "Max A \<le> x" by simp
```
```  1693 next
```
```  1694   from assms show "x \<le> Max A" by simp
```
```  1695 qed
```
```  1696
```
```  1697 lemma Min_antimono:
```
```  1698   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  1699   shows "Min N \<le> Min M"
```
```  1700   using assms by (simp add: Min_def fold1_antimono)
```
```  1701
```
```  1702 lemma Max_mono:
```
```  1703   assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
```
```  1704   shows "Max M \<le> Max N"
```
```  1705   by (simp add: Max_def linorder.dual_max [OF dual_linorder]
```
```  1706     linorder.fold1_antimono [OF dual_linorder] assms)
```
```  1707
```
```  1708 lemma finite_linorder_max_induct[consumes 1, case_names empty insert]:
```
```  1709  assumes fin: "finite A"
```
```  1710  and   empty: "P {}"
```
```  1711  and  insert: "(!!b A. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))"
```
```  1712  shows "P A"
```
```  1713 using fin empty insert
```
```  1714 proof (induct rule: finite_psubset_induct)
```
```  1715   case (psubset A)
```
```  1716   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
```
```  1717   have fin: "finite A" by fact
```
```  1718   have empty: "P {}" by fact
```
```  1719   have step: "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" by fact
```
```  1720   show "P A"
```
```  1721   proof (cases "A = {}")
```
```  1722     assume "A = {}"
```
```  1723     then show "P A" using `P {}` by simp
```
```  1724   next
```
```  1725     let ?B = "A - {Max A}"
```
```  1726     let ?A = "insert (Max A) ?B"
```
```  1727     have "finite ?B" using `finite A` by simp
```
```  1728     assume "A \<noteq> {}"
```
```  1729     with `finite A` have "Max A : A" by auto
```
```  1730     then have A: "?A = A" using insert_Diff_single insert_absorb by auto
```
```  1731     then have "P ?B" using `P {}` step IH[of ?B] by blast
```
```  1732     moreover
```
```  1733     have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
```
```  1734     ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastforce
```
```  1735   qed
```
```  1736 qed
```
```  1737
```
```  1738 lemma finite_linorder_min_induct[consumes 1, case_names empty insert]:
```
```  1739  "\<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"
```
```  1740 by(rule linorder.finite_linorder_max_induct[OF dual_linorder])
```
```  1741
```
```  1742 end
```
```  1743
```
```  1744 context linordered_ab_semigroup_add
```
```  1745 begin
```
```  1746
```
```  1747 lemma add_Min_commute:
```
```  1748   fixes k
```
```  1749   assumes "finite N" and "N \<noteq> {}"
```
```  1750   shows "k + Min N = Min {k + m | m. m \<in> N}"
```
```  1751 proof -
```
```  1752   have "\<And>x y. k + min x y = min (k + x) (k + y)"
```
```  1753     by (simp add: min_def not_le)
```
```  1754       (blast intro: antisym less_imp_le add_left_mono)
```
```  1755   with assms show ?thesis
```
```  1756     using hom_Min_commute [of "plus k" N]
```
```  1757     by simp (blast intro: arg_cong [where f = Min])
```
```  1758 qed
```
```  1759
```
```  1760 lemma add_Max_commute:
```
```  1761   fixes k
```
```  1762   assumes "finite N" and "N \<noteq> {}"
```
```  1763   shows "k + Max N = Max {k + m | m. m \<in> N}"
```
```  1764 proof -
```
```  1765   have "\<And>x y. k + max x y = max (k + x) (k + y)"
```
```  1766     by (simp add: max_def not_le)
```
```  1767       (blast intro: antisym less_imp_le add_left_mono)
```
```  1768   with assms show ?thesis
```
```  1769     using hom_Max_commute [of "plus k" N]
```
```  1770     by simp (blast intro: arg_cong [where f = Max])
```
```  1771 qed
```
```  1772
```
```  1773 end
```
```  1774
```
```  1775 context linordered_ab_group_add
```
```  1776 begin
```
```  1777
```
```  1778 lemma minus_Max_eq_Min [simp]:
```
```  1779   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Max S) = Min (uminus ` S)"
```
```  1780   by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
```
```  1781
```
```  1782 lemma minus_Min_eq_Max [simp]:
```
```  1783   "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Min S) = Max (uminus ` S)"
```
```  1784   by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
```
```  1785
```
```  1786 end
```
```  1787
```
```  1788 end
```