src/HOL/Big_Operators.thy
 author haftmann Wed Mar 10 16:53:27 2010 +0100 (2010-03-10) changeset 35719 99b6152aedf5 parent 35577 src/HOL/Finite_Set.thy@43b93e294522 child 35722 69419a09a7ff permissions -rw-r--r--
split off theory Big_Operators from theory Finite_Set
 haftmann@35719 ` 1` ```(* Title: HOL/Big_Operators.thy ``` wenzelm@12396 ` 2` ``` Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel ``` avigad@16775 ` 3` ``` with contributions by Jeremy Avigad ``` wenzelm@12396 ` 4` ```*) ``` wenzelm@12396 ` 5` haftmann@35719 ` 6` ```header {* Big operators and finite (non-empty) sets *} ``` haftmann@26041 ` 7` haftmann@35719 ` 8` ```theory Big_Operators ``` haftmann@35719 ` 9` ```imports Finite_Set ``` haftmann@26041 ` 10` ```begin ``` haftmann@26041 ` 11` nipkow@15402 ` 12` ```subsection {* Generalized summation over a set *} ``` nipkow@15402 ` 13` haftmann@35267 ` 14` ```interpretation comm_monoid_add: comm_monoid_mult "op +" "0::'a::comm_monoid_add" ``` haftmann@28823 ` 15` ``` proof qed (auto intro: add_assoc add_commute) ``` haftmann@26041 ` 16` nipkow@28853 ` 17` ```definition setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add" ``` nipkow@28853 ` 18` ```where "setsum f A == if finite A then fold_image (op +) f 0 A else 0" ``` nipkow@15402 ` 19` wenzelm@19535 ` 20` ```abbreviation ``` wenzelm@21404 ` 21` ``` Setsum ("\_" [1000] 999) where ``` wenzelm@19535 ` 22` ``` "\A == setsum (%x. x) A" ``` wenzelm@19535 ` 23` nipkow@15402 ` 24` ```text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is ``` nipkow@15402 ` 25` ```written @{text"\x\A. e"}. *} ``` nipkow@15402 ` 26` nipkow@15402 ` 27` ```syntax ``` paulson@17189 ` 28` ``` "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10) ``` nipkow@15402 ` 29` ```syntax (xsymbols) ``` paulson@17189 ` 30` ``` "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\_\_. _)" [0, 51, 10] 10) ``` nipkow@15402 ` 31` ```syntax (HTML output) ``` paulson@17189 ` 32` ``` "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\_\_. _)" [0, 51, 10] 10) ``` nipkow@15402 ` 33` nipkow@15402 ` 34` ```translations -- {* Beware of argument permutation! *} ``` nipkow@28853 ` 35` ``` "SUM i:A. b" == "CONST setsum (%i. b) A" ``` nipkow@28853 ` 36` ``` "\i\A. b" == "CONST setsum (%i. b) A" ``` nipkow@15402 ` 37` nipkow@15402 ` 38` ```text{* Instead of @{term"\x\{x. P}. e"} we introduce the shorter ``` nipkow@15402 ` 39` ``` @{text"\x|P. e"}. *} ``` nipkow@15402 ` 40` nipkow@15402 ` 41` ```syntax ``` paulson@17189 ` 42` ``` "_qsetsum" :: "pttrn \ bool \ 'a \ 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10) ``` nipkow@15402 ` 43` ```syntax (xsymbols) ``` paulson@17189 ` 44` ``` "_qsetsum" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) ``` nipkow@15402 ` 45` ```syntax (HTML output) ``` paulson@17189 ` 46` ``` "_qsetsum" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) ``` nipkow@15402 ` 47` nipkow@15402 ` 48` ```translations ``` nipkow@28853 ` 49` ``` "SUM x|P. t" => "CONST setsum (%x. t) {x. P}" ``` nipkow@28853 ` 50` ``` "\x|P. t" => "CONST setsum (%x. t) {x. P}" ``` nipkow@15402 ` 51` nipkow@15402 ` 52` ```print_translation {* ``` nipkow@15402 ` 53` ```let ``` wenzelm@35115 ` 54` ``` fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] = ``` wenzelm@35115 ` 55` ``` if x <> y then raise Match ``` wenzelm@35115 ` 56` ``` else ``` wenzelm@35115 ` 57` ``` let ``` wenzelm@35115 ` 58` ``` val x' = Syntax.mark_bound x; ``` wenzelm@35115 ` 59` ``` val t' = subst_bound (x', t); ``` wenzelm@35115 ` 60` ``` val P' = subst_bound (x', P); ``` wenzelm@35115 ` 61` ``` in Syntax.const @{syntax_const "_qsetsum"} \$ Syntax.mark_bound x \$ P' \$ t' end ``` wenzelm@35115 ` 62` ``` | setsum_tr' _ = raise Match; ``` wenzelm@35115 ` 63` ```in [(@{const_syntax setsum}, setsum_tr')] end ``` nipkow@15402 ` 64` ```*} ``` nipkow@15402 ` 65` wenzelm@19535 ` 66` nipkow@15402 ` 67` ```lemma setsum_empty [simp]: "setsum f {} = 0" ``` nipkow@28853 ` 68` ```by (simp add: setsum_def) ``` nipkow@15402 ` 69` nipkow@15402 ` 70` ```lemma setsum_insert [simp]: ``` nipkow@28853 ` 71` ``` "finite F ==> a \ F ==> setsum f (insert a F) = f a + setsum f F" ``` nipkow@28853 ` 72` ```by (simp add: setsum_def) ``` nipkow@15402 ` 73` paulson@15409 ` 74` ```lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0" ``` nipkow@28853 ` 75` ```by (simp add: setsum_def) ``` paulson@15409 ` 76` nipkow@15402 ` 77` ```lemma setsum_reindex: ``` nipkow@15402 ` 78` ``` "inj_on f B ==> setsum h (f ` B) = setsum (h \ f) B" ``` nipkow@28853 ` 79` ```by(auto simp add: setsum_def comm_monoid_add.fold_image_reindex dest!:finite_imageD) ``` nipkow@15402 ` 80` nipkow@15402 ` 81` ```lemma setsum_reindex_id: ``` nipkow@15402 ` 82` ``` "inj_on f B ==> setsum f B = setsum id (f ` B)" ``` nipkow@15402 ` 83` ```by (auto simp add: setsum_reindex) ``` nipkow@15402 ` 84` chaieb@29674 ` 85` ```lemma setsum_reindex_nonzero: ``` chaieb@29674 ` 86` ``` assumes fS: "finite S" ``` chaieb@29674 ` 87` ``` and nz: "\ x y. x \ S \ y \ S \ x \ y \ f x = f y \ h (f x) = 0" ``` chaieb@29674 ` 88` ``` shows "setsum h (f ` S) = setsum (h o f) S" ``` chaieb@29674 ` 89` ```using nz ``` chaieb@29674 ` 90` ```proof(induct rule: finite_induct[OF fS]) ``` chaieb@29674 ` 91` ``` case 1 thus ?case by simp ``` chaieb@29674 ` 92` ```next ``` chaieb@29674 ` 93` ``` case (2 x F) ``` chaieb@29674 ` 94` ``` {assume fxF: "f x \ f ` F" hence "\y \ F . f y = f x" by auto ``` chaieb@29674 ` 95` ``` then obtain y where y: "y \ F" "f x = f y" by auto ``` chaieb@29674 ` 96` ``` from "2.hyps" y have xy: "x \ y" by auto ``` chaieb@29674 ` 97` ``` ``` chaieb@29674 ` 98` ``` from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp ``` chaieb@29674 ` 99` ``` have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto ``` chaieb@29674 ` 100` ``` also have "\ = setsum (h o f) (insert x F)" ``` chaieb@29674 ` 101` ``` unfolding setsum_insert[OF `finite F` `x\F`] ``` chaieb@29674 ` 102` ``` using h0 ``` chaieb@29674 ` 103` ``` apply simp ``` chaieb@29674 ` 104` ``` apply (rule "2.hyps"(3)) ``` chaieb@29674 ` 105` ``` apply (rule_tac y="y" in "2.prems") ``` chaieb@29674 ` 106` ``` apply simp_all ``` chaieb@29674 ` 107` ``` done ``` chaieb@29674 ` 108` ``` finally have ?case .} ``` chaieb@29674 ` 109` ``` moreover ``` chaieb@29674 ` 110` ``` {assume fxF: "f x \ f ` F" ``` chaieb@29674 ` 111` ``` have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" ``` chaieb@29674 ` 112` ``` using fxF "2.hyps" by simp ``` chaieb@29674 ` 113` ``` also have "\ = setsum (h o f) (insert x F)" ``` chaieb@29674 ` 114` ``` unfolding setsum_insert[OF `finite F` `x\F`] ``` chaieb@29674 ` 115` ``` apply simp ``` chaieb@29674 ` 116` ``` apply (rule cong[OF refl[of "op + (h (f x))"]]) ``` chaieb@29674 ` 117` ``` apply (rule "2.hyps"(3)) ``` chaieb@29674 ` 118` ``` apply (rule_tac y="y" in "2.prems") ``` chaieb@29674 ` 119` ``` apply simp_all ``` chaieb@29674 ` 120` ``` done ``` chaieb@29674 ` 121` ``` finally have ?case .} ``` chaieb@29674 ` 122` ``` ultimately show ?case by blast ``` chaieb@29674 ` 123` ```qed ``` chaieb@29674 ` 124` nipkow@15402 ` 125` ```lemma setsum_cong: ``` nipkow@15402 ` 126` ``` "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" ``` nipkow@28853 ` 127` ```by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_image_cong) ``` nipkow@15402 ` 128` nipkow@16733 ` 129` ```lemma strong_setsum_cong[cong]: ``` nipkow@16733 ` 130` ``` "A = B ==> (!!x. x:B =simp=> f x = g x) ``` nipkow@16733 ` 131` ``` ==> setsum (%x. f x) A = setsum (%x. g x) B" ``` nipkow@28853 ` 132` ```by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_image_cong) ``` berghofe@16632 ` 133` haftmann@33960 ` 134` ```lemma setsum_cong2: "\\x. x \ A \ f x = g x\ \ setsum f A = setsum g A" ``` haftmann@33960 ` 135` ```by (rule setsum_cong[OF refl], auto) ``` nipkow@15554 ` 136` nipkow@15402 ` 137` ```lemma setsum_reindex_cong: ``` nipkow@28853 ` 138` ``` "[|inj_on f A; B = f ` A; !!a. a:A \ g a = h (f a)|] ``` nipkow@28853 ` 139` ``` ==> setsum h B = setsum g A" ``` nipkow@28853 ` 140` ```by (simp add: setsum_reindex cong: setsum_cong) ``` nipkow@15402 ` 141` chaieb@29674 ` 142` nipkow@15542 ` 143` ```lemma setsum_0[simp]: "setsum (%i. 0) A = 0" ``` nipkow@15402 ` 144` ```apply (clarsimp simp: setsum_def) ``` ballarin@15765 ` 145` ```apply (erule finite_induct, auto) ``` nipkow@15402 ` 146` ```done ``` nipkow@15402 ` 147` nipkow@15543 ` 148` ```lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0" ``` nipkow@15543 ` 149` ```by(simp add:setsum_cong) ``` nipkow@15402 ` 150` nipkow@15402 ` 151` ```lemma setsum_Un_Int: "finite A ==> finite B ==> ``` nipkow@15402 ` 152` ``` setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" ``` nipkow@15402 ` 153` ``` -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} ``` nipkow@28853 ` 154` ```by(simp add: setsum_def comm_monoid_add.fold_image_Un_Int [symmetric]) ``` nipkow@15402 ` 155` nipkow@15402 ` 156` ```lemma setsum_Un_disjoint: "finite A ==> finite B ``` nipkow@15402 ` 157` ``` ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" ``` nipkow@15402 ` 158` ```by (subst setsum_Un_Int [symmetric], auto) ``` nipkow@15402 ` 159` chaieb@29674 ` 160` ```lemma setsum_mono_zero_left: ``` chaieb@29674 ` 161` ``` assumes fT: "finite T" and ST: "S \ T" ``` chaieb@29674 ` 162` ``` and z: "\i \ T - S. f i = 0" ``` chaieb@29674 ` 163` ``` shows "setsum f S = setsum f T" ``` chaieb@29674 ` 164` ```proof- ``` chaieb@29674 ` 165` ``` have eq: "T = S \ (T - S)" using ST by blast ``` chaieb@29674 ` 166` ``` have d: "S \ (T - S) = {}" using ST by blast ``` chaieb@29674 ` 167` ``` from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) ``` chaieb@29674 ` 168` ``` show ?thesis ``` chaieb@29674 ` 169` ``` by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) ``` chaieb@29674 ` 170` ```qed ``` chaieb@29674 ` 171` chaieb@29674 ` 172` ```lemma setsum_mono_zero_right: ``` nipkow@30837 ` 173` ``` "finite T \ S \ T \ \i \ T - S. f i = 0 \ setsum f T = setsum f S" ``` nipkow@30837 ` 174` ```by(blast intro!: setsum_mono_zero_left[symmetric]) ``` chaieb@29674 ` 175` chaieb@29674 ` 176` ```lemma setsum_mono_zero_cong_left: ``` chaieb@29674 ` 177` ``` assumes fT: "finite T" and ST: "S \ T" ``` chaieb@29674 ` 178` ``` and z: "\i \ T - S. g i = 0" ``` chaieb@29674 ` 179` ``` and fg: "\x. x \ S \ f x = g x" ``` chaieb@29674 ` 180` ``` shows "setsum f S = setsum g T" ``` chaieb@29674 ` 181` ```proof- ``` chaieb@29674 ` 182` ``` have eq: "T = S \ (T - S)" using ST by blast ``` chaieb@29674 ` 183` ``` have d: "S \ (T - S) = {}" using ST by blast ``` chaieb@29674 ` 184` ``` from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) ``` chaieb@29674 ` 185` ``` show ?thesis ``` chaieb@29674 ` 186` ``` using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) ``` chaieb@29674 ` 187` ```qed ``` chaieb@29674 ` 188` chaieb@29674 ` 189` ```lemma setsum_mono_zero_cong_right: ``` chaieb@29674 ` 190` ``` assumes fT: "finite T" and ST: "S \ T" ``` chaieb@29674 ` 191` ``` and z: "\i \ T - S. f i = 0" ``` chaieb@29674 ` 192` ``` and fg: "\x. x \ S \ f x = g x" ``` chaieb@29674 ` 193` ``` shows "setsum f T = setsum g S" ``` chaieb@29674 ` 194` ```using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto ``` chaieb@29674 ` 195` chaieb@29674 ` 196` ```lemma setsum_delta: ``` chaieb@29674 ` 197` ``` assumes fS: "finite S" ``` chaieb@29674 ` 198` ``` shows "setsum (\k. if k=a then b k else 0) S = (if a \ S then b a else 0)" ``` chaieb@29674 ` 199` ```proof- ``` chaieb@29674 ` 200` ``` let ?f = "(\k. if k=a then b k else 0)" ``` chaieb@29674 ` 201` ``` {assume a: "a \ S" ``` chaieb@29674 ` 202` ``` hence "\ k\ S. ?f k = 0" by simp ``` chaieb@29674 ` 203` ``` hence ?thesis using a by simp} ``` chaieb@29674 ` 204` ``` moreover ``` chaieb@29674 ` 205` ``` {assume a: "a \ S" ``` chaieb@29674 ` 206` ``` let ?A = "S - {a}" ``` chaieb@29674 ` 207` ``` let ?B = "{a}" ``` chaieb@29674 ` 208` ``` have eq: "S = ?A \ ?B" using a by blast ``` chaieb@29674 ` 209` ``` have dj: "?A \ ?B = {}" by simp ``` chaieb@29674 ` 210` ``` from fS have fAB: "finite ?A" "finite ?B" by auto ``` chaieb@29674 ` 211` ``` have "setsum ?f S = setsum ?f ?A + setsum ?f ?B" ``` chaieb@29674 ` 212` ``` using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] ``` chaieb@29674 ` 213` ``` by simp ``` chaieb@29674 ` 214` ``` then have ?thesis using a by simp} ``` chaieb@29674 ` 215` ``` ultimately show ?thesis by blast ``` chaieb@29674 ` 216` ```qed ``` chaieb@29674 ` 217` ```lemma setsum_delta': ``` chaieb@29674 ` 218` ``` assumes fS: "finite S" shows ``` chaieb@29674 ` 219` ``` "setsum (\k. if a = k then b k else 0) S = ``` chaieb@29674 ` 220` ``` (if a\ S then b a else 0)" ``` chaieb@29674 ` 221` ``` using setsum_delta[OF fS, of a b, symmetric] ``` chaieb@29674 ` 222` ``` by (auto intro: setsum_cong) ``` chaieb@29674 ` 223` chaieb@30260 ` 224` ```lemma setsum_restrict_set: ``` chaieb@30260 ` 225` ``` assumes fA: "finite A" ``` chaieb@30260 ` 226` ``` shows "setsum f (A \ B) = setsum (\x. if x \ B then f x else 0) A" ``` chaieb@30260 ` 227` ```proof- ``` chaieb@30260 ` 228` ``` from fA have fab: "finite (A \ B)" by auto ``` chaieb@30260 ` 229` ``` have aba: "A \ B \ A" by blast ``` chaieb@30260 ` 230` ``` let ?g = "\x. if x \ A\B then f x else 0" ``` chaieb@30260 ` 231` ``` from setsum_mono_zero_left[OF fA aba, of ?g] ``` chaieb@30260 ` 232` ``` show ?thesis by simp ``` chaieb@30260 ` 233` ```qed ``` chaieb@30260 ` 234` chaieb@30260 ` 235` ```lemma setsum_cases: ``` chaieb@30260 ` 236` ``` assumes fA: "finite A" ``` hoelzl@35577 ` 237` ``` shows "setsum (\x. if P x then f x else g x) A = ``` hoelzl@35577 ` 238` ``` setsum f (A \ {x. P x}) + setsum g (A \ - {x. P x})" ``` chaieb@30260 ` 239` ```proof- ``` hoelzl@35577 ` 240` ``` have a: "A = A \ {x. P x} \ A \ -{x. P x}" ``` hoelzl@35577 ` 241` ``` "(A \ {x. P x}) \ (A \ -{x. P x}) = {}" ``` chaieb@30260 ` 242` ``` by blast+ ``` chaieb@30260 ` 243` ``` from fA ``` hoelzl@35577 ` 244` ``` have f: "finite (A \ {x. P x})" "finite (A \ -{x. P x})" by auto ``` hoelzl@35577 ` 245` ``` let ?g = "\x. if P x then f x else g x" ``` chaieb@30260 ` 246` ``` from setsum_Un_disjoint[OF f a(2), of ?g] a(1) ``` chaieb@30260 ` 247` ``` show ?thesis by simp ``` chaieb@30260 ` 248` ```qed ``` chaieb@30260 ` 249` chaieb@29674 ` 250` paulson@15409 ` 251` ```(*But we can't get rid of finite I. If infinite, although the rhs is 0, ``` paulson@15409 ` 252` ``` the lhs need not be, since UNION I A could still be finite.*) ``` nipkow@15402 ` 253` ```lemma setsum_UN_disjoint: ``` nipkow@15402 ` 254` ``` "finite I ==> (ALL i:I. finite (A i)) ==> ``` nipkow@15402 ` 255` ``` (ALL i:I. ALL j:I. i \ j --> A i Int A j = {}) ==> ``` nipkow@15402 ` 256` ``` setsum f (UNION I A) = (\i\I. setsum f (A i))" ``` nipkow@28853 ` 257` ```by(simp add: setsum_def comm_monoid_add.fold_image_UN_disjoint cong: setsum_cong) ``` nipkow@15402 ` 258` paulson@15409 ` 259` ```text{*No need to assume that @{term C} is finite. If infinite, the rhs is ``` paulson@15409 ` 260` ```directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*} ``` nipkow@15402 ` 261` ```lemma setsum_Union_disjoint: ``` paulson@15409 ` 262` ``` "[| (ALL A:C. finite A); ``` paulson@15409 ` 263` ``` (ALL A:C. ALL B:C. A \ B --> A Int B = {}) |] ``` paulson@15409 ` 264` ``` ==> setsum f (Union C) = setsum (setsum f) C" ``` paulson@15409 ` 265` ```apply (cases "finite C") ``` paulson@15409 ` 266` ``` prefer 2 apply (force dest: finite_UnionD simp add: setsum_def) ``` nipkow@15402 ` 267` ``` apply (frule setsum_UN_disjoint [of C id f]) ``` paulson@15409 ` 268` ``` apply (unfold Union_def id_def, assumption+) ``` paulson@15409 ` 269` ```done ``` nipkow@15402 ` 270` paulson@15409 ` 271` ```(*But we can't get rid of finite A. If infinite, although the lhs is 0, ``` paulson@15409 ` 272` ``` the rhs need not be, since SIGMA A B could still be finite.*) ``` nipkow@15402 ` 273` ```lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==> ``` paulson@17189 ` 274` ``` (\x\A. (\y\B x. f x y)) = (\(x,y)\(SIGMA x:A. B x). f x y)" ``` nipkow@28853 ` 275` ```by(simp add:setsum_def comm_monoid_add.fold_image_Sigma split_def cong:setsum_cong) ``` nipkow@15402 ` 276` paulson@15409 ` 277` ```text{*Here we can eliminate the finiteness assumptions, by cases.*} ``` paulson@15409 ` 278` ```lemma setsum_cartesian_product: ``` paulson@17189 ` 279` ``` "(\x\A. (\y\B. f x y)) = (\(x,y) \ A <*> B. f x y)" ``` paulson@15409 ` 280` ```apply (cases "finite A") ``` paulson@15409 ` 281` ``` apply (cases "finite B") ``` paulson@15409 ` 282` ``` apply (simp add: setsum_Sigma) ``` paulson@15409 ` 283` ``` apply (cases "A={}", simp) ``` nipkow@15543 ` 284` ``` apply (simp) ``` paulson@15409 ` 285` ```apply (auto simp add: setsum_def ``` paulson@15409 ` 286` ``` dest: finite_cartesian_productD1 finite_cartesian_productD2) ``` paulson@15409 ` 287` ```done ``` nipkow@15402 ` 288` nipkow@15402 ` 289` ```lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)" ``` nipkow@28853 ` 290` ```by(simp add:setsum_def comm_monoid_add.fold_image_distrib) ``` nipkow@15402 ` 291` nipkow@15402 ` 292` nipkow@15402 ` 293` ```subsubsection {* Properties in more restricted classes of structures *} ``` nipkow@15402 ` 294` nipkow@15402 ` 295` ```lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" ``` nipkow@28853 ` 296` ```apply (case_tac "finite A") ``` nipkow@28853 ` 297` ``` prefer 2 apply (simp add: setsum_def) ``` nipkow@28853 ` 298` ```apply (erule rev_mp) ``` nipkow@28853 ` 299` ```apply (erule finite_induct, auto) ``` nipkow@28853 ` 300` ```done ``` nipkow@15402 ` 301` nipkow@15402 ` 302` ```lemma setsum_eq_0_iff [simp]: ``` nipkow@15402 ` 303` ``` "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" ``` nipkow@28853 ` 304` ```by (induct set: finite) auto ``` nipkow@15402 ` 305` nipkow@30859 ` 306` ```lemma setsum_eq_Suc0_iff: "finite A \ ``` nipkow@30859 ` 307` ``` (setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\b \ f b = 0))" ``` nipkow@30859 ` 308` ```apply(erule finite_induct) ``` nipkow@30859 ` 309` ```apply (auto simp add:add_is_1) ``` nipkow@30859 ` 310` ```done ``` nipkow@30859 ` 311` nipkow@30859 ` 312` ```lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]] ``` nipkow@30859 ` 313` nipkow@15402 ` 314` ```lemma setsum_Un_nat: "finite A ==> finite B ==> ``` nipkow@28853 ` 315` ``` (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" ``` nipkow@15402 ` 316` ``` -- {* For the natural numbers, we have subtraction. *} ``` nipkow@29667 ` 317` ```by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps) ``` nipkow@15402 ` 318` nipkow@15402 ` 319` ```lemma setsum_Un: "finite A ==> finite B ==> ``` nipkow@28853 ` 320` ``` (setsum f (A Un B) :: 'a :: ab_group_add) = ``` nipkow@28853 ` 321` ``` setsum f A + setsum f B - setsum f (A Int B)" ``` nipkow@29667 ` 322` ```by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps) ``` nipkow@15402 ` 323` chaieb@30260 ` 324` ```lemma (in comm_monoid_mult) fold_image_1: "finite S \ (\x\S. f x = 1) \ fold_image op * f 1 S = 1" ``` chaieb@30260 ` 325` ``` apply (induct set: finite) ``` huffman@35216 ` 326` ``` apply simp by auto ``` chaieb@30260 ` 327` chaieb@30260 ` 328` ```lemma (in comm_monoid_mult) fold_image_Un_one: ``` chaieb@30260 ` 329` ``` assumes fS: "finite S" and fT: "finite T" ``` chaieb@30260 ` 330` ``` and I0: "\x \ S\T. f x = 1" ``` chaieb@30260 ` 331` ``` shows "fold_image (op *) f 1 (S \ T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T" ``` chaieb@30260 ` 332` ```proof- ``` chaieb@30260 ` 333` ``` have "fold_image op * f 1 (S \ T) = 1" ``` chaieb@30260 ` 334` ``` apply (rule fold_image_1) ``` chaieb@30260 ` 335` ``` using fS fT I0 by auto ``` chaieb@30260 ` 336` ``` with fold_image_Un_Int[OF fS fT] show ?thesis by simp ``` chaieb@30260 ` 337` ```qed ``` chaieb@30260 ` 338` chaieb@30260 ` 339` ```lemma setsum_eq_general_reverses: ``` chaieb@30260 ` 340` ``` assumes fS: "finite S" and fT: "finite T" ``` chaieb@30260 ` 341` ``` and kh: "\y. y \ T \ k y \ S \ h (k y) = y" ``` chaieb@30260 ` 342` ``` and hk: "\x. x \ S \ h x \ T \ k (h x) = x \ g (h x) = f x" ``` chaieb@30260 ` 343` ``` shows "setsum f S = setsum g T" ``` chaieb@30260 ` 344` ``` apply (simp add: setsum_def fS fT) ``` chaieb@30260 ` 345` ``` apply (rule comm_monoid_add.fold_image_eq_general_inverses[OF fS]) ``` chaieb@30260 ` 346` ``` apply (erule kh) ``` chaieb@30260 ` 347` ``` apply (erule hk) ``` chaieb@30260 ` 348` ``` done ``` chaieb@30260 ` 349` chaieb@30260 ` 350` chaieb@30260 ` 351` chaieb@30260 ` 352` ```lemma setsum_Un_zero: ``` chaieb@30260 ` 353` ``` assumes fS: "finite S" and fT: "finite T" ``` chaieb@30260 ` 354` ``` and I0: "\x \ S\T. f x = 0" ``` chaieb@30260 ` 355` ``` shows "setsum f (S \ T) = setsum f S + setsum f T" ``` chaieb@30260 ` 356` ``` using fS fT ``` chaieb@30260 ` 357` ``` apply (simp add: setsum_def) ``` chaieb@30260 ` 358` ``` apply (rule comm_monoid_add.fold_image_Un_one) ``` chaieb@30260 ` 359` ``` using I0 by auto ``` chaieb@30260 ` 360` chaieb@30260 ` 361` chaieb@30260 ` 362` ```lemma setsum_UNION_zero: ``` chaieb@30260 ` 363` ``` assumes fS: "finite S" and fSS: "\T \ S. finite T" ``` chaieb@30260 ` 364` ``` and f0: "\T1 T2 x. T1\S \ T2\S \ T1 \ T2 \ x \ T1 \ x \ T2 \ f x = 0" ``` chaieb@30260 ` 365` ``` shows "setsum f (\S) = setsum (\T. setsum f T) S" ``` chaieb@30260 ` 366` ``` using fSS f0 ``` chaieb@30260 ` 367` ```proof(induct rule: finite_induct[OF fS]) ``` chaieb@30260 ` 368` ``` case 1 thus ?case by simp ``` chaieb@30260 ` 369` ```next ``` chaieb@30260 ` 370` ``` case (2 T F) ``` chaieb@30260 ` 371` ``` then have fTF: "finite T" "\T\F. finite T" "finite F" and TF: "T \ F" ``` huffman@35216 ` 372` ``` and H: "setsum f (\ F) = setsum (setsum f) F" by auto ``` huffman@35216 ` 373` ``` from fTF have fUF: "finite (\F)" by auto ``` chaieb@30260 ` 374` ``` from "2.prems" TF fTF ``` chaieb@30260 ` 375` ``` show ?case ``` chaieb@30260 ` 376` ``` by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f]) ``` chaieb@30260 ` 377` ```qed ``` chaieb@30260 ` 378` chaieb@30260 ` 379` nipkow@15402 ` 380` ```lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) = ``` nipkow@28853 ` 381` ``` (if a:A then setsum f A - f a else setsum f A)" ``` nipkow@28853 ` 382` ```apply (case_tac "finite A") ``` nipkow@28853 ` 383` ``` prefer 2 apply (simp add: setsum_def) ``` nipkow@28853 ` 384` ```apply (erule finite_induct) ``` nipkow@28853 ` 385` ``` apply (auto simp add: insert_Diff_if) ``` nipkow@28853 ` 386` ```apply (drule_tac a = a in mk_disjoint_insert, auto) ``` nipkow@28853 ` 387` ```done ``` nipkow@15402 ` 388` nipkow@15402 ` 389` ```lemma setsum_diff1: "finite A \ ``` nipkow@15402 ` 390` ``` (setsum f (A - {a}) :: ('a::ab_group_add)) = ``` nipkow@15402 ` 391` ``` (if a:A then setsum f A - f a else setsum f A)" ``` nipkow@28853 ` 392` ```by (erule finite_induct) (auto simp add: insert_Diff_if) ``` nipkow@28853 ` 393` nipkow@28853 ` 394` ```lemma setsum_diff1'[rule_format]: ``` nipkow@28853 ` 395` ``` "finite A \ a \ A \ (\ x \ A. f x) = f a + (\ x \ (A - {a}). f x)" ``` nipkow@28853 ` 396` ```apply (erule finite_induct[where F=A and P="% A. (a \ A \ (\ x \ A. f x) = f a + (\ x \ (A - {a}). f x))"]) ``` nipkow@28853 ` 397` ```apply (auto simp add: insert_Diff_if add_ac) ``` nipkow@28853 ` 398` ```done ``` obua@15552 ` 399` nipkow@31438 ` 400` ```lemma setsum_diff1_ring: assumes "finite A" "a \ A" ``` nipkow@31438 ` 401` ``` shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)" ``` nipkow@31438 ` 402` ```unfolding setsum_diff1'[OF assms] by auto ``` nipkow@31438 ` 403` nipkow@15402 ` 404` ```(* By Jeremy Siek: *) ``` nipkow@15402 ` 405` nipkow@15402 ` 406` ```lemma setsum_diff_nat: ``` nipkow@28853 ` 407` ```assumes "finite B" and "B \ A" ``` nipkow@28853 ` 408` ```shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)" ``` nipkow@28853 ` 409` ```using assms ``` wenzelm@19535 ` 410` ```proof induct ``` nipkow@15402 ` 411` ``` show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp ``` nipkow@15402 ` 412` ```next ``` nipkow@15402 ` 413` ``` fix F x assume finF: "finite F" and xnotinF: "x \ F" ``` nipkow@15402 ` 414` ``` and xFinA: "insert x F \ A" ``` nipkow@15402 ` 415` ``` and IH: "F \ A \ setsum f (A - F) = setsum f A - setsum f F" ``` nipkow@15402 ` 416` ``` from xnotinF xFinA have xinAF: "x \ (A - F)" by simp ``` nipkow@15402 ` 417` ``` from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x" ``` nipkow@15402 ` 418` ``` by (simp add: setsum_diff1_nat) ``` nipkow@15402 ` 419` ``` from xFinA have "F \ A" by simp ``` nipkow@15402 ` 420` ``` with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp ``` nipkow@15402 ` 421` ``` with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x" ``` nipkow@15402 ` 422` ``` by simp ``` nipkow@15402 ` 423` ``` from xnotinF have "A - insert x F = (A - F) - {x}" by auto ``` nipkow@15402 ` 424` ``` with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x" ``` nipkow@15402 ` 425` ``` by simp ``` nipkow@15402 ` 426` ``` from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp ``` nipkow@15402 ` 427` ``` with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" ``` nipkow@15402 ` 428` ``` by simp ``` nipkow@15402 ` 429` ``` thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp ``` nipkow@15402 ` 430` ```qed ``` nipkow@15402 ` 431` nipkow@15402 ` 432` ```lemma setsum_diff: ``` nipkow@15402 ` 433` ``` assumes le: "finite A" "B \ A" ``` nipkow@15402 ` 434` ``` shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))" ``` nipkow@15402 ` 435` ```proof - ``` nipkow@15402 ` 436` ``` from le have finiteB: "finite B" using finite_subset by auto ``` nipkow@15402 ` 437` ``` show ?thesis using finiteB le ``` wenzelm@21575 ` 438` ``` proof induct ``` wenzelm@19535 ` 439` ``` case empty ``` wenzelm@19535 ` 440` ``` thus ?case by auto ``` wenzelm@19535 ` 441` ``` next ``` wenzelm@19535 ` 442` ``` case (insert x F) ``` wenzelm@19535 ` 443` ``` thus ?case using le finiteB ``` wenzelm@19535 ` 444` ``` by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb) ``` nipkow@15402 ` 445` ``` qed ``` wenzelm@19535 ` 446` ```qed ``` nipkow@15402 ` 447` nipkow@15402 ` 448` ```lemma setsum_mono: ``` haftmann@35028 ` 449` ``` assumes le: "\i. i\K \ f (i::'a) \ ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))" ``` nipkow@15402 ` 450` ``` shows "(\i\K. f i) \ (\i\K. g i)" ``` nipkow@15402 ` 451` ```proof (cases "finite K") ``` nipkow@15402 ` 452` ``` case True ``` nipkow@15402 ` 453` ``` thus ?thesis using le ``` wenzelm@19535 ` 454` ``` proof induct ``` nipkow@15402 ` 455` ``` case empty ``` nipkow@15402 ` 456` ``` thus ?case by simp ``` nipkow@15402 ` 457` ``` next ``` nipkow@15402 ` 458` ``` case insert ``` wenzelm@19535 ` 459` ``` thus ?case using add_mono by fastsimp ``` nipkow@15402 ` 460` ``` qed ``` nipkow@15402 ` 461` ```next ``` nipkow@15402 ` 462` ``` case False ``` nipkow@15402 ` 463` ``` thus ?thesis ``` nipkow@15402 ` 464` ``` by (simp add: setsum_def) ``` nipkow@15402 ` 465` ```qed ``` nipkow@15402 ` 466` nipkow@15554 ` 467` ```lemma setsum_strict_mono: ``` haftmann@35028 ` 468` ``` fixes f :: "'a \ 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}" ``` wenzelm@19535 ` 469` ``` assumes "finite A" "A \ {}" ``` wenzelm@19535 ` 470` ``` and "!!x. x:A \ f x < g x" ``` wenzelm@19535 ` 471` ``` shows "setsum f A < setsum g A" ``` wenzelm@19535 ` 472` ``` using prems ``` nipkow@15554 ` 473` ```proof (induct rule: finite_ne_induct) ``` nipkow@15554 ` 474` ``` case singleton thus ?case by simp ``` nipkow@15554 ` 475` ```next ``` nipkow@15554 ` 476` ``` case insert thus ?case by (auto simp: add_strict_mono) ``` nipkow@15554 ` 477` ```qed ``` nipkow@15554 ` 478` nipkow@15535 ` 479` ```lemma setsum_negf: ``` wenzelm@19535 ` 480` ``` "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A" ``` nipkow@15535 ` 481` ```proof (cases "finite A") ``` berghofe@22262 ` 482` ``` case True thus ?thesis by (induct set: finite) auto ``` nipkow@15535 ` 483` ```next ``` nipkow@15535 ` 484` ``` case False thus ?thesis by (simp add: setsum_def) ``` nipkow@15535 ` 485` ```qed ``` nipkow@15402 ` 486` nipkow@15535 ` 487` ```lemma setsum_subtractf: ``` wenzelm@19535 ` 488` ``` "setsum (%x. ((f x)::'a::ab_group_add) - g x) A = ``` wenzelm@19535 ` 489` ``` setsum f A - setsum g A" ``` nipkow@15535 ` 490` ```proof (cases "finite A") ``` nipkow@15535 ` 491` ``` case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf) ``` nipkow@15535 ` 492` ```next ``` nipkow@15535 ` 493` ``` case False thus ?thesis by (simp add: setsum_def) ``` nipkow@15535 ` 494` ```qed ``` nipkow@15402 ` 495` nipkow@15535 ` 496` ```lemma setsum_nonneg: ``` haftmann@35028 ` 497` ``` assumes nn: "\x\A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \ f x" ``` wenzelm@19535 ` 498` ``` shows "0 \ setsum f A" ``` nipkow@15535 ` 499` ```proof (cases "finite A") ``` nipkow@15535 ` 500` ``` case True thus ?thesis using nn ``` wenzelm@21575 ` 501` ``` proof induct ``` wenzelm@19535 ` 502` ``` case empty then show ?case by simp ``` wenzelm@19535 ` 503` ``` next ``` wenzelm@19535 ` 504` ``` case (insert x F) ``` wenzelm@19535 ` 505` ``` then have "0 + 0 \ f x + setsum f F" by (blast intro: add_mono) ``` wenzelm@19535 ` 506` ``` with insert show ?case by simp ``` wenzelm@19535 ` 507` ``` qed ``` nipkow@15535 ` 508` ```next ``` nipkow@15535 ` 509` ``` case False thus ?thesis by (simp add: setsum_def) ``` nipkow@15535 ` 510` ```qed ``` nipkow@15402 ` 511` nipkow@15535 ` 512` ```lemma setsum_nonpos: ``` haftmann@35028 ` 513` ``` assumes np: "\x\A. f x \ (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})" ``` wenzelm@19535 ` 514` ``` shows "setsum f A \ 0" ``` nipkow@15535 ` 515` ```proof (cases "finite A") ``` nipkow@15535 ` 516` ``` case True thus ?thesis using np ``` wenzelm@21575 ` 517` ``` proof induct ``` wenzelm@19535 ` 518` ``` case empty then show ?case by simp ``` wenzelm@19535 ` 519` ``` next ``` wenzelm@19535 ` 520` ``` case (insert x F) ``` wenzelm@19535 ` 521` ``` then have "f x + setsum f F \ 0 + 0" by (blast intro: add_mono) ``` wenzelm@19535 ` 522` ``` with insert show ?case by simp ``` wenzelm@19535 ` 523` ``` qed ``` nipkow@15535 ` 524` ```next ``` nipkow@15535 ` 525` ``` case False thus ?thesis by (simp add: setsum_def) ``` nipkow@15535 ` 526` ```qed ``` nipkow@15402 ` 527` nipkow@15539 ` 528` ```lemma setsum_mono2: ``` haftmann@35028 ` 529` ```fixes f :: "'a \ 'b :: {ordered_ab_semigroup_add_imp_le,comm_monoid_add}" ``` nipkow@15539 ` 530` ```assumes fin: "finite B" and sub: "A \ B" and nn: "\b. b \ B-A \ 0 \ f b" ``` nipkow@15539 ` 531` ```shows "setsum f A \ setsum f B" ``` nipkow@15539 ` 532` ```proof - ``` nipkow@15539 ` 533` ``` have "setsum f A \ setsum f A + setsum f (B-A)" ``` nipkow@15539 ` 534` ``` by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def) ``` nipkow@15539 ` 535` ``` also have "\ = setsum f (A \ (B-A))" using fin finite_subset[OF sub fin] ``` nipkow@15539 ` 536` ``` by (simp add:setsum_Un_disjoint del:Un_Diff_cancel) ``` nipkow@15539 ` 537` ``` also have "A \ (B-A) = B" using sub by blast ``` nipkow@15539 ` 538` ``` finally show ?thesis . ``` nipkow@15539 ` 539` ```qed ``` nipkow@15542 ` 540` avigad@16775 ` 541` ```lemma setsum_mono3: "finite B ==> A <= B ==> ``` avigad@16775 ` 542` ``` ALL x: B - A. ``` haftmann@35028 ` 543` ``` 0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==> ``` avigad@16775 ` 544` ``` setsum f A <= setsum f B" ``` avigad@16775 ` 545` ``` apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)") ``` avigad@16775 ` 546` ``` apply (erule ssubst) ``` avigad@16775 ` 547` ``` apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)") ``` avigad@16775 ` 548` ``` apply simp ``` avigad@16775 ` 549` ``` apply (rule add_left_mono) ``` avigad@16775 ` 550` ``` apply (erule setsum_nonneg) ``` avigad@16775 ` 551` ``` apply (subst setsum_Un_disjoint [THEN sym]) ``` avigad@16775 ` 552` ``` apply (erule finite_subset, assumption) ``` avigad@16775 ` 553` ``` apply (rule finite_subset) ``` avigad@16775 ` 554` ``` prefer 2 ``` avigad@16775 ` 555` ``` apply assumption ``` haftmann@32698 ` 556` ``` apply (auto simp add: sup_absorb2) ``` avigad@16775 ` 557` ```done ``` avigad@16775 ` 558` ballarin@19279 ` 559` ```lemma setsum_right_distrib: ``` huffman@22934 ` 560` ``` fixes f :: "'a => ('b::semiring_0)" ``` nipkow@15402 ` 561` ``` shows "r * setsum f A = setsum (%n. r * f n) A" ``` nipkow@15402 ` 562` ```proof (cases "finite A") ``` nipkow@15402 ` 563` ``` case True ``` nipkow@15402 ` 564` ``` thus ?thesis ``` wenzelm@21575 ` 565` ``` proof induct ``` nipkow@15402 ` 566` ``` case empty thus ?case by simp ``` nipkow@15402 ` 567` ``` next ``` nipkow@15402 ` 568` ``` case (insert x A) thus ?case by (simp add: right_distrib) ``` nipkow@15402 ` 569` ``` qed ``` nipkow@15402 ` 570` ```next ``` nipkow@15402 ` 571` ``` case False thus ?thesis by (simp add: setsum_def) ``` nipkow@15402 ` 572` ```qed ``` nipkow@15402 ` 573` ballarin@17149 ` 574` ```lemma setsum_left_distrib: ``` huffman@22934 ` 575` ``` "setsum f A * (r::'a::semiring_0) = (\n\A. f n * r)" ``` ballarin@17149 ` 576` ```proof (cases "finite A") ``` ballarin@17149 ` 577` ``` case True ``` ballarin@17149 ` 578` ``` then show ?thesis ``` ballarin@17149 ` 579` ``` proof induct ``` ballarin@17149 ` 580` ``` case empty thus ?case by simp ``` ballarin@17149 ` 581` ``` next ``` ballarin@17149 ` 582` ``` case (insert x A) thus ?case by (simp add: left_distrib) ``` ballarin@17149 ` 583` ``` qed ``` ballarin@17149 ` 584` ```next ``` ballarin@17149 ` 585` ``` case False thus ?thesis by (simp add: setsum_def) ``` ballarin@17149 ` 586` ```qed ``` ballarin@17149 ` 587` ballarin@17149 ` 588` ```lemma setsum_divide_distrib: ``` ballarin@17149 ` 589` ``` "setsum f A / (r::'a::field) = (\n\A. f n / r)" ``` ballarin@17149 ` 590` ```proof (cases "finite A") ``` ballarin@17149 ` 591` ``` case True ``` ballarin@17149 ` 592` ``` then show ?thesis ``` ballarin@17149 ` 593` ``` proof induct ``` ballarin@17149 ` 594` ``` case empty thus ?case by simp ``` ballarin@17149 ` 595` ``` next ``` ballarin@17149 ` 596` ``` case (insert x A) thus ?case by (simp add: add_divide_distrib) ``` ballarin@17149 ` 597` ``` qed ``` ballarin@17149 ` 598` ```next ``` ballarin@17149 ` 599` ``` case False thus ?thesis by (simp add: setsum_def) ``` ballarin@17149 ` 600` ```qed ``` ballarin@17149 ` 601` nipkow@15535 ` 602` ```lemma setsum_abs[iff]: ``` haftmann@35028 ` 603` ``` fixes f :: "'a => ('b::ordered_ab_group_add_abs)" ``` nipkow@15402 ` 604` ``` shows "abs (setsum f A) \ setsum (%i. abs(f i)) A" ``` nipkow@15535 ` 605` ```proof (cases "finite A") ``` nipkow@15535 ` 606` ``` case True ``` nipkow@15535 ` 607` ``` thus ?thesis ``` wenzelm@21575 ` 608` ``` proof induct ``` nipkow@15535 ` 609` ``` case empty thus ?case by simp ``` nipkow@15535 ` 610` ``` next ``` nipkow@15535 ` 611` ``` case (insert x A) ``` nipkow@15535 ` 612` ``` thus ?case by (auto intro: abs_triangle_ineq order_trans) ``` nipkow@15535 ` 613` ``` qed ``` nipkow@15402 ` 614` ```next ``` nipkow@15535 ` 615` ``` case False thus ?thesis by (simp add: setsum_def) ``` nipkow@15402 ` 616` ```qed ``` nipkow@15402 ` 617` nipkow@15535 ` 618` ```lemma setsum_abs_ge_zero[iff]: ``` haftmann@35028 ` 619` ``` fixes f :: "'a => ('b::ordered_ab_group_add_abs)" ``` nipkow@15402 ` 620` ``` shows "0 \ setsum (%i. abs(f i)) A" ``` nipkow@15535 ` 621` ```proof (cases "finite A") ``` nipkow@15535 ` 622` ``` case True ``` nipkow@15535 ` 623` ``` thus ?thesis ``` wenzelm@21575 ` 624` ``` proof induct ``` nipkow@15535 ` 625` ``` case empty thus ?case by simp ``` nipkow@15535 ` 626` ``` next ``` nipkow@21733 ` 627` ``` case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg) ``` nipkow@15535 ` 628` ``` qed ``` nipkow@15402 ` 629` ```next ``` nipkow@15535 ` 630` ``` case False thus ?thesis by (simp add: setsum_def) ``` nipkow@15402 ` 631` ```qed ``` nipkow@15402 ` 632` nipkow@15539 ` 633` ```lemma abs_setsum_abs[simp]: ``` haftmann@35028 ` 634` ``` fixes f :: "'a => ('b::ordered_ab_group_add_abs)" ``` nipkow@15539 ` 635` ``` shows "abs (\a\A. abs(f a)) = (\a\A. abs(f a))" ``` nipkow@15539 ` 636` ```proof (cases "finite A") ``` nipkow@15539 ` 637` ``` case True ``` nipkow@15539 ` 638` ``` thus ?thesis ``` wenzelm@21575 ` 639` ``` proof induct ``` nipkow@15539 ` 640` ``` case empty thus ?case by simp ``` nipkow@15539 ` 641` ``` next ``` nipkow@15539 ` 642` ``` case (insert a A) ``` nipkow@15539 ` 643` ``` hence "\\a\insert a A. \f a\\ = \\f a\ + (\a\A. \f a\)\" by simp ``` nipkow@15539 ` 644` ``` also have "\ = \\f a\ + \\a\A. \f a\\\" using insert by simp ``` avigad@16775 ` 645` ``` also have "\ = \f a\ + \\a\A. \f a\\" ``` avigad@16775 ` 646` ``` by (simp del: abs_of_nonneg) ``` nipkow@15539 ` 647` ``` also have "\ = (\a\insert a A. \f a\)" using insert by simp ``` nipkow@15539 ` 648` ``` finally show ?case . ``` nipkow@15539 ` 649` ``` qed ``` nipkow@15539 ` 650` ```next ``` nipkow@15539 ` 651` ``` case False thus ?thesis by (simp add: setsum_def) ``` nipkow@15539 ` 652` ```qed ``` nipkow@15539 ` 653` nipkow@15402 ` 654` nipkow@31080 ` 655` ```lemma setsum_Plus: ``` nipkow@31080 ` 656` ``` fixes A :: "'a set" and B :: "'b set" ``` nipkow@31080 ` 657` ``` assumes fin: "finite A" "finite B" ``` nipkow@31080 ` 658` ``` shows "setsum f (A <+> B) = setsum (f \ Inl) A + setsum (f \ Inr) B" ``` nipkow@31080 ` 659` ```proof - ``` nipkow@31080 ` 660` ``` have "A <+> B = Inl ` A \ Inr ` B" by auto ``` nipkow@31080 ` 661` ``` moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)" ``` nipkow@31080 ` 662` ``` by(auto intro: finite_imageI) ``` nipkow@31080 ` 663` ``` moreover have "Inl ` A \ Inr ` B = ({} :: ('a + 'b) set)" by auto ``` nipkow@31080 ` 664` ``` moreover have "inj_on (Inl :: 'a \ 'a + 'b) A" "inj_on (Inr :: 'b \ 'a + 'b) B" by(auto intro: inj_onI) ``` nipkow@31080 ` 665` ``` ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex) ``` nipkow@31080 ` 666` ```qed ``` nipkow@31080 ` 667` nipkow@31080 ` 668` ballarin@17149 ` 669` ```text {* Commuting outer and inner summation *} ``` ballarin@17149 ` 670` ballarin@17149 ` 671` ```lemma swap_inj_on: ``` ballarin@17149 ` 672` ``` "inj_on (%(i, j). (j, i)) (A \ B)" ``` ballarin@17149 ` 673` ``` by (unfold inj_on_def) fast ``` ballarin@17149 ` 674` ballarin@17149 ` 675` ```lemma swap_product: ``` ballarin@17149 ` 676` ``` "(%(i, j). (j, i)) ` (A \ B) = B \ A" ``` ballarin@17149 ` 677` ``` by (simp add: split_def image_def) blast ``` ballarin@17149 ` 678` ballarin@17149 ` 679` ```lemma setsum_commute: ``` ballarin@17149 ` 680` ``` "(\i\A. \j\B. f i j) = (\j\B. \i\A. f i j)" ``` ballarin@17149 ` 681` ```proof (simp add: setsum_cartesian_product) ``` paulson@17189 ` 682` ``` have "(\(x,y) \ A <*> B. f x y) = ``` paulson@17189 ` 683` ``` (\(y,x) \ (%(i, j). (j, i)) ` (A \ B). f x y)" ``` ballarin@17149 ` 684` ``` (is "?s = _") ``` ballarin@17149 ` 685` ``` apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on) ``` ballarin@17149 ` 686` ``` apply (simp add: split_def) ``` ballarin@17149 ` 687` ``` done ``` paulson@17189 ` 688` ``` also have "... = (\(y,x)\B \ A. f x y)" ``` ballarin@17149 ` 689` ``` (is "_ = ?t") ``` ballarin@17149 ` 690` ``` apply (simp add: swap_product) ``` ballarin@17149 ` 691` ``` done ``` ballarin@17149 ` 692` ``` finally show "?s = ?t" . ``` ballarin@17149 ` 693` ```qed ``` ballarin@17149 ` 694` ballarin@19279 ` 695` ```lemma setsum_product: ``` huffman@22934 ` 696` ``` fixes f :: "'a => ('b::semiring_0)" ``` ballarin@19279 ` 697` ``` shows "setsum f A * setsum g B = (\i\A. \j\B. f i * g j)" ``` ballarin@19279 ` 698` ``` by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute) ``` ballarin@19279 ` 699` nipkow@34223 ` 700` ```lemma setsum_mult_setsum_if_inj: ``` nipkow@34223 ` 701` ```fixes f :: "'a => ('b::semiring_0)" ``` nipkow@34223 ` 702` ```shows "inj_on (%(a,b). f a * g b) (A \ B) ==> ``` nipkow@34223 ` 703` ``` setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}" ``` nipkow@34223 ` 704` ```by(auto simp: setsum_product setsum_cartesian_product ``` nipkow@34223 ` 705` ``` intro!: setsum_reindex_cong[symmetric]) ``` nipkow@34223 ` 706` ballarin@17149 ` 707` nipkow@15402 ` 708` ```subsection {* Generalized product over a set *} ``` nipkow@15402 ` 709` nipkow@28853 ` 710` ```definition setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult" ``` nipkow@28853 ` 711` ```where "setprod f A == if finite A then fold_image (op *) f 1 A else 1" ``` nipkow@15402 ` 712` wenzelm@19535 ` 713` ```abbreviation ``` wenzelm@21404 ` 714` ``` Setprod ("\_" [1000] 999) where ``` wenzelm@19535 ` 715` ``` "\A == setprod (%x. x) A" ``` wenzelm@19535 ` 716` nipkow@15402 ` 717` ```syntax ``` paulson@17189 ` 718` ``` "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3PROD _:_. _)" [0, 51, 10] 10) ``` nipkow@15402 ` 719` ```syntax (xsymbols) ``` paulson@17189 ` 720` ``` "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\_\_. _)" [0, 51, 10] 10) ``` nipkow@15402 ` 721` ```syntax (HTML output) ``` paulson@17189 ` 722` ``` "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\_\_. _)" [0, 51, 10] 10) ``` nipkow@16550 ` 723` nipkow@16550 ` 724` ```translations -- {* Beware of argument permutation! *} ``` nipkow@28853 ` 725` ``` "PROD i:A. b" == "CONST setprod (%i. b) A" ``` nipkow@28853 ` 726` ``` "\i\A. b" == "CONST setprod (%i. b) A" ``` nipkow@16550 ` 727` nipkow@16550 ` 728` ```text{* Instead of @{term"\x\{x. P}. e"} we introduce the shorter ``` nipkow@16550 ` 729` ``` @{text"\x|P. e"}. *} ``` nipkow@16550 ` 730` nipkow@16550 ` 731` ```syntax ``` paulson@17189 ` 732` ``` "_qsetprod" :: "pttrn \ bool \ 'a \ 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10) ``` nipkow@16550 ` 733` ```syntax (xsymbols) ``` paulson@17189 ` 734` ``` "_qsetprod" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) ``` nipkow@16550 ` 735` ```syntax (HTML output) ``` paulson@17189 ` 736` ``` "_qsetprod" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) ``` nipkow@16550 ` 737` nipkow@15402 ` 738` ```translations ``` nipkow@28853 ` 739` ``` "PROD x|P. t" => "CONST setprod (%x. t) {x. P}" ``` nipkow@28853 ` 740` ``` "\x|P. t" => "CONST setprod (%x. t) {x. P}" ``` nipkow@16550 ` 741` nipkow@15402 ` 742` nipkow@15402 ` 743` ```lemma setprod_empty [simp]: "setprod f {} = 1" ``` nipkow@28853 ` 744` ```by (auto simp add: setprod_def) ``` nipkow@15402 ` 745` nipkow@15402 ` 746` ```lemma setprod_insert [simp]: "[| finite A; a \ A |] ==> ``` nipkow@15402 ` 747` ``` setprod f (insert a A) = f a * setprod f A" ``` nipkow@28853 ` 748` ```by (simp add: setprod_def) ``` nipkow@15402 ` 749` paulson@15409 ` 750` ```lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1" ``` nipkow@28853 ` 751` ```by (simp add: setprod_def) ``` paulson@15409 ` 752` nipkow@15402 ` 753` ```lemma setprod_reindex: ``` nipkow@28853 ` 754` ``` "inj_on f B ==> setprod h (f ` B) = setprod (h \ f) B" ``` nipkow@28853 ` 755` ```by(auto simp: setprod_def fold_image_reindex dest!:finite_imageD) ``` nipkow@15402 ` 756` nipkow@15402 ` 757` ```lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)" ``` nipkow@15402 ` 758` ```by (auto simp add: setprod_reindex) ``` nipkow@15402 ` 759` nipkow@15402 ` 760` ```lemma setprod_cong: ``` nipkow@15402 ` 761` ``` "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B" ``` nipkow@28853 ` 762` ```by(fastsimp simp: setprod_def intro: fold_image_cong) ``` nipkow@15402 ` 763` nipkow@30837 ` 764` ```lemma strong_setprod_cong[cong]: ``` berghofe@16632 ` 765` ``` "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B" ``` nipkow@28853 ` 766` ```by(fastsimp simp: simp_implies_def setprod_def intro: fold_image_cong) ``` berghofe@16632 ` 767` nipkow@15402 ` 768` ```lemma setprod_reindex_cong: "inj_on f A ==> ``` nipkow@15402 ` 769` ``` B = f ` A ==> g = h \ f ==> setprod h B = setprod g A" ``` nipkow@28853 ` 770` ```by (frule setprod_reindex, simp) ``` nipkow@15402 ` 771` chaieb@29674 ` 772` ```lemma strong_setprod_reindex_cong: assumes i: "inj_on f A" ``` chaieb@29674 ` 773` ``` and B: "B = f ` A" and eq: "\x. x \ A \ g x = (h \ f) x" ``` chaieb@29674 ` 774` ``` shows "setprod h B = setprod g A" ``` chaieb@29674 ` 775` ```proof- ``` chaieb@29674 ` 776` ``` have "setprod h B = setprod (h o f) A" ``` chaieb@29674 ` 777` ``` by (simp add: B setprod_reindex[OF i, of h]) ``` chaieb@29674 ` 778` ``` then show ?thesis apply simp ``` chaieb@29674 ` 779` ``` apply (rule setprod_cong) ``` chaieb@29674 ` 780` ``` apply simp ``` nipkow@30837 ` 781` ``` by (simp add: eq) ``` chaieb@29674 ` 782` ```qed ``` chaieb@29674 ` 783` chaieb@30260 ` 784` ```lemma setprod_Un_one: ``` chaieb@30260 ` 785` ``` assumes fS: "finite S" and fT: "finite T" ``` chaieb@30260 ` 786` ``` and I0: "\x \ S\T. f x = 1" ``` chaieb@30260 ` 787` ``` shows "setprod f (S \ T) = setprod f S * setprod f T" ``` chaieb@30260 ` 788` ``` using fS fT ``` chaieb@30260 ` 789` ``` apply (simp add: setprod_def) ``` chaieb@30260 ` 790` ``` apply (rule fold_image_Un_one) ``` chaieb@30260 ` 791` ``` using I0 by auto ``` chaieb@30260 ` 792` nipkow@15402 ` 793` nipkow@15402 ` 794` ```lemma setprod_1: "setprod (%i. 1) A = 1" ``` nipkow@28853 ` 795` ```apply (case_tac "finite A") ``` nipkow@28853 ` 796` ```apply (erule finite_induct, auto simp add: mult_ac) ``` nipkow@28853 ` 797` ```done ``` nipkow@15402 ` 798` nipkow@15402 ` 799` ```lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1" ``` nipkow@28853 ` 800` ```apply (subgoal_tac "setprod f F = setprod (%x. 1) F") ``` nipkow@28853 ` 801` ```apply (erule ssubst, rule setprod_1) ``` nipkow@28853 ` 802` ```apply (rule setprod_cong, auto) ``` nipkow@28853 ` 803` ```done ``` nipkow@15402 ` 804` nipkow@15402 ` 805` ```lemma setprod_Un_Int: "finite A ==> finite B ``` nipkow@15402 ` 806` ``` ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B" ``` nipkow@28853 ` 807` ```by(simp add: setprod_def fold_image_Un_Int[symmetric]) ``` nipkow@15402 ` 808` nipkow@15402 ` 809` ```lemma setprod_Un_disjoint: "finite A ==> finite B ``` nipkow@15402 ` 810` ``` ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B" ``` nipkow@15402 ` 811` ```by (subst setprod_Un_Int [symmetric], auto) ``` nipkow@15402 ` 812` nipkow@30837 ` 813` ```lemma setprod_mono_one_left: ``` nipkow@30837 ` 814` ``` assumes fT: "finite T" and ST: "S \ T" ``` nipkow@30837 ` 815` ``` and z: "\i \ T - S. f i = 1" ``` nipkow@30837 ` 816` ``` shows "setprod f S = setprod f T" ``` nipkow@30837 ` 817` ```proof- ``` nipkow@30837 ` 818` ``` have eq: "T = S \ (T - S)" using ST by blast ``` nipkow@30837 ` 819` ``` have d: "S \ (T - S) = {}" using ST by blast ``` nipkow@30837 ` 820` ``` from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) ``` nipkow@30837 ` 821` ``` show ?thesis ``` nipkow@30837 ` 822` ``` by (simp add: setprod_Un_disjoint[OF f d, unfolded eq[symmetric]] setprod_1'[OF z]) ``` nipkow@30837 ` 823` ```qed ``` nipkow@30837 ` 824` nipkow@30837 ` 825` ```lemmas setprod_mono_one_right = setprod_mono_one_left [THEN sym] ``` nipkow@30837 ` 826` chaieb@29674 ` 827` ```lemma setprod_delta: ``` chaieb@29674 ` 828` ``` assumes fS: "finite S" ``` chaieb@29674 ` 829` ``` shows "setprod (\k. if k=a then b k else 1) S = (if a \ S then b a else 1)" ``` chaieb@29674 ` 830` ```proof- ``` chaieb@29674 ` 831` ``` let ?f = "(\k. if k=a then b k else 1)" ``` chaieb@29674 ` 832` ``` {assume a: "a \ S" ``` chaieb@29674 ` 833` ``` hence "\ k\ S. ?f k = 1" by simp ``` chaieb@29674 ` 834` ``` hence ?thesis using a by (simp add: setprod_1 cong add: setprod_cong) } ``` chaieb@29674 ` 835` ``` moreover ``` chaieb@29674 ` 836` ``` {assume a: "a \ S" ``` chaieb@29674 ` 837` ``` let ?A = "S - {a}" ``` chaieb@29674 ` 838` ``` let ?B = "{a}" ``` chaieb@29674 ` 839` ``` have eq: "S = ?A \ ?B" using a by blast ``` chaieb@29674 ` 840` ``` have dj: "?A \ ?B = {}" by simp ``` chaieb@29674 ` 841` ``` from fS have fAB: "finite ?A" "finite ?B" by auto ``` chaieb@29674 ` 842` ``` have fA1: "setprod ?f ?A = 1" apply (rule setprod_1') by auto ``` chaieb@29674 ` 843` ``` have "setprod ?f ?A * setprod ?f ?B = setprod ?f S" ``` chaieb@29674 ` 844` ``` using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] ``` chaieb@29674 ` 845` ``` by simp ``` chaieb@29674 ` 846` ``` then have ?thesis using a by (simp add: fA1 cong add: setprod_cong cong del: if_weak_cong)} ``` chaieb@29674 ` 847` ``` ultimately show ?thesis by blast ``` chaieb@29674 ` 848` ```qed ``` chaieb@29674 ` 849` chaieb@29674 ` 850` ```lemma setprod_delta': ``` chaieb@29674 ` 851` ``` assumes fS: "finite S" shows ``` chaieb@29674 ` 852` ``` "setprod (\k. if a = k then b k else 1) S = ``` chaieb@29674 ` 853` ``` (if a\ S then b a else 1)" ``` chaieb@29674 ` 854` ``` using setprod_delta[OF fS, of a b, symmetric] ``` chaieb@29674 ` 855` ``` by (auto intro: setprod_cong) ``` chaieb@29674 ` 856` chaieb@29674 ` 857` nipkow@15402 ` 858` ```lemma setprod_UN_disjoint: ``` nipkow@15402 ` 859` ``` "finite I ==> (ALL i:I. finite (A i)) ==> ``` nipkow@15402 ` 860` ``` (ALL i:I. ALL j:I. i \ j --> A i Int A j = {}) ==> ``` nipkow@15402 ` 861` ``` setprod f (UNION I A) = setprod (%i. setprod f (A i)) I" ``` nipkow@28853 ` 862` ```by(simp add: setprod_def fold_image_UN_disjoint cong: setprod_cong) ``` nipkow@15402 ` 863` nipkow@15402 ` 864` ```lemma setprod_Union_disjoint: ``` paulson@15409 ` 865` ``` "[| (ALL A:C. finite A); ``` paulson@15409 ` 866` ``` (ALL A:C. ALL B:C. A \ B --> A Int B = {}) |] ``` paulson@15409 ` 867` ``` ==> setprod f (Union C) = setprod (setprod f) C" ``` paulson@15409 ` 868` ```apply (cases "finite C") ``` paulson@15409 ` 869` ``` prefer 2 apply (force dest: finite_UnionD simp add: setprod_def) ``` nipkow@15402 ` 870` ``` apply (frule setprod_UN_disjoint [of C id f]) ``` paulson@15409 ` 871` ``` apply (unfold Union_def id_def, assumption+) ``` paulson@15409 ` 872` ```done ``` nipkow@15402 ` 873` nipkow@15402 ` 874` ```lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==> ``` nipkow@16550 ` 875` ``` (\x\A. (\y\ B x. f x y)) = ``` paulson@17189 ` 876` ``` (\(x,y)\(SIGMA x:A. B x). f x y)" ``` nipkow@28853 ` 877` ```by(simp add:setprod_def fold_image_Sigma split_def cong:setprod_cong) ``` nipkow@15402 ` 878` paulson@15409 ` 879` ```text{*Here we can eliminate the finiteness assumptions, by cases.*} ``` paulson@15409 ` 880` ```lemma setprod_cartesian_product: ``` paulson@17189 ` 881` ``` "(\x\A. (\y\ B. f x y)) = (\(x,y)\(A <*> B). f x y)" ``` paulson@15409 ` 882` ```apply (cases "finite A") ``` paulson@15409 ` 883` ``` apply (cases "finite B") ``` paulson@15409 ` 884` ``` apply (simp add: setprod_Sigma) ``` paulson@15409 ` 885` ``` apply (cases "A={}", simp) ``` paulson@15409 ` 886` ``` apply (simp add: setprod_1) ``` paulson@15409 ` 887` ```apply (auto simp add: setprod_def ``` paulson@15409 ` 888` ``` dest: finite_cartesian_productD1 finite_cartesian_productD2) ``` paulson@15409 ` 889` ```done ``` nipkow@15402 ` 890` nipkow@15402 ` 891` ```lemma setprod_timesf: ``` paulson@15409 ` 892` ``` "setprod (%x. f x * g x) A = (setprod f A * setprod g A)" ``` nipkow@28853 ` 893` ```by(simp add:setprod_def fold_image_distrib) ``` nipkow@15402 ` 894` nipkow@15402 ` 895` nipkow@15402 ` 896` ```subsubsection {* Properties in more restricted classes of structures *} ``` nipkow@15402 ` 897` nipkow@15402 ` 898` ```lemma setprod_eq_1_iff [simp]: ``` nipkow@28853 ` 899` ``` "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))" ``` nipkow@28853 ` 900` ```by (induct set: finite) auto ``` nipkow@15402 ` 901` nipkow@15402 ` 902` ```lemma setprod_zero: ``` huffman@23277 ` 903` ``` "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0" ``` nipkow@28853 ` 904` ```apply (induct set: finite, force, clarsimp) ``` nipkow@28853 ` 905` ```apply (erule disjE, auto) ``` nipkow@28853 ` 906` ```done ``` nipkow@15402 ` 907` nipkow@15402 ` 908` ```lemma setprod_nonneg [rule_format]: ``` haftmann@35028 ` 909` ``` "(ALL x: A. (0::'a::linordered_semidom) \ f x) --> 0 \ setprod f A" ``` huffman@30841 ` 910` ```by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg) ``` huffman@30841 ` 911` haftmann@35028 ` 912` ```lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x) ``` nipkow@28853 ` 913` ``` --> 0 < setprod f A" ``` huffman@30841 ` 914` ```by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos) ``` nipkow@15402 ` 915` nipkow@30843 ` 916` ```lemma setprod_zero_iff[simp]: "finite A ==> ``` nipkow@30843 ` 917` ``` (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) = ``` nipkow@30843 ` 918` ``` (EX x: A. f x = 0)" ``` nipkow@30843 ` 919` ```by (erule finite_induct, auto simp:no_zero_divisors) ``` nipkow@30843 ` 920` nipkow@30843 ` 921` ```lemma setprod_pos_nat: ``` nipkow@30843 ` 922` ``` "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0" ``` nipkow@30843 ` 923` ```using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric]) ``` nipkow@15402 ` 924` nipkow@30863 ` 925` ```lemma setprod_pos_nat_iff[simp]: ``` nipkow@30863 ` 926` ``` "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))" ``` nipkow@30863 ` 927` ```using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric]) ``` nipkow@30863 ` 928` nipkow@15402 ` 929` ```lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \ 0) ==> ``` nipkow@28853 ` 930` ``` (setprod f (A Un B) :: 'a ::{field}) ``` nipkow@28853 ` 931` ``` = setprod f A * setprod f B / setprod f (A Int B)" ``` nipkow@30843 ` 932` ```by (subst setprod_Un_Int [symmetric], auto) ``` nipkow@15402 ` 933` nipkow@15402 ` 934` ```lemma setprod_diff1: "finite A ==> f a \ 0 ==> ``` nipkow@28853 ` 935` ``` (setprod f (A - {a}) :: 'a :: {field}) = ``` nipkow@28853 ` 936` ``` (if a:A then setprod f A / f a else setprod f A)" ``` nipkow@23413 ` 937` ```by (erule finite_induct) (auto simp add: insert_Diff_if) ``` nipkow@15402 ` 938` paulson@31906 ` 939` ```lemma setprod_inversef: ``` paulson@31906 ` 940` ``` fixes f :: "'b \ 'a::{field,division_by_zero}" ``` paulson@31906 ` 941` ``` shows "finite A ==> setprod (inverse \ f) A = inverse (setprod f A)" ``` nipkow@28853 ` 942` ```by (erule finite_induct) auto ``` nipkow@15402 ` 943` nipkow@15402 ` 944` ```lemma setprod_dividef: ``` paulson@31906 ` 945` ``` fixes f :: "'b \ 'a::{field,division_by_zero}" ``` wenzelm@31916 ` 946` ``` shows "finite A ``` nipkow@28853 ` 947` ``` ==> setprod (%x. f x / g x) A = setprod f A / setprod g A" ``` nipkow@28853 ` 948` ```apply (subgoal_tac ``` nipkow@15402 ` 949` ``` "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \ g) x) A") ``` nipkow@28853 ` 950` ```apply (erule ssubst) ``` nipkow@28853 ` 951` ```apply (subst divide_inverse) ``` nipkow@28853 ` 952` ```apply (subst setprod_timesf) ``` nipkow@28853 ` 953` ```apply (subst setprod_inversef, assumption+, rule refl) ``` nipkow@28853 ` 954` ```apply (rule setprod_cong, rule refl) ``` nipkow@28853 ` 955` ```apply (subst divide_inverse, auto) ``` nipkow@28853 ` 956` ```done ``` nipkow@28853 ` 957` nipkow@29925 ` 958` ```lemma setprod_dvd_setprod [rule_format]: ``` nipkow@29925 ` 959` ``` "(ALL x : A. f x dvd g x) \ setprod f A dvd setprod g A" ``` nipkow@29925 ` 960` ``` apply (cases "finite A") ``` nipkow@29925 ` 961` ``` apply (induct set: finite) ``` nipkow@29925 ` 962` ``` apply (auto simp add: dvd_def) ``` nipkow@29925 ` 963` ``` apply (rule_tac x = "k * ka" in exI) ``` nipkow@29925 ` 964` ``` apply (simp add: algebra_simps) ``` nipkow@29925 ` 965` ```done ``` nipkow@29925 ` 966` nipkow@29925 ` 967` ```lemma setprod_dvd_setprod_subset: ``` nipkow@29925 ` 968` ``` "finite B \ A <= B \ setprod f A dvd setprod f B" ``` nipkow@29925 ` 969` ``` apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)") ``` nipkow@29925 ` 970` ``` apply (unfold dvd_def, blast) ``` nipkow@29925 ` 971` ``` apply (subst setprod_Un_disjoint [symmetric]) ``` nipkow@29925 ` 972` ``` apply (auto elim: finite_subset intro: setprod_cong) ``` nipkow@29925 ` 973` ```done ``` nipkow@29925 ` 974` nipkow@29925 ` 975` ```lemma setprod_dvd_setprod_subset2: ``` nipkow@29925 ` 976` ``` "finite B \ A <= B \ ALL x : A. (f x::'a::comm_semiring_1) dvd g x \ ``` nipkow@29925 ` 977` ``` setprod f A dvd setprod g B" ``` nipkow@29925 ` 978` ``` apply (rule dvd_trans) ``` nipkow@29925 ` 979` ``` apply (rule setprod_dvd_setprod, erule (1) bspec) ``` nipkow@29925 ` 980` ``` apply (erule (1) setprod_dvd_setprod_subset) ``` nipkow@29925 ` 981` ```done ``` nipkow@29925 ` 982` nipkow@29925 ` 983` ```lemma dvd_setprod: "finite A \ i:A \ ``` nipkow@29925 ` 984` ``` (f i ::'a::comm_semiring_1) dvd setprod f A" ``` nipkow@29925 ` 985` ```by (induct set: finite) (auto intro: dvd_mult) ``` nipkow@29925 ` 986` nipkow@29925 ` 987` ```lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \ ``` nipkow@29925 ` 988` ``` (d::'a::comm_semiring_1) dvd (SUM x : A. f x)" ``` nipkow@29925 ` 989` ``` apply (cases "finite A") ``` nipkow@29925 ` 990` ``` apply (induct set: finite) ``` nipkow@29925 ` 991` ``` apply auto ``` nipkow@29925 ` 992` ```done ``` nipkow@29925 ` 993` hoelzl@35171 ` 994` ```lemma setprod_mono: ``` hoelzl@35171 ` 995` ``` fixes f :: "'a \ 'b\linordered_semidom" ``` hoelzl@35171 ` 996` ``` assumes "\i\A. 0 \ f i \ f i \ g i" ``` hoelzl@35171 ` 997` ``` shows "setprod f A \ setprod g A" ``` hoelzl@35171 ` 998` ```proof (cases "finite A") ``` hoelzl@35171 ` 999` ``` case True ``` hoelzl@35171 ` 1000` ``` hence ?thesis "setprod f A \ 0" using subset_refl[of A] ``` hoelzl@35171 ` 1001` ``` proof (induct A rule: finite_subset_induct) ``` hoelzl@35171 ` 1002` ``` case (insert a F) ``` hoelzl@35171 ` 1003` ``` thus "setprod f (insert a F) \ setprod g (insert a F)" "0 \ setprod f (insert a F)" ``` hoelzl@35171 ` 1004` ``` unfolding setprod_insert[OF insert(1,3)] ``` hoelzl@35171 ` 1005` ``` using assms[rule_format,OF insert(2)] insert ``` hoelzl@35171 ` 1006` ``` by (auto intro: mult_mono mult_nonneg_nonneg) ``` hoelzl@35171 ` 1007` ``` qed auto ``` hoelzl@35171 ` 1008` ``` thus ?thesis by simp ``` hoelzl@35171 ` 1009` ```qed auto ``` hoelzl@35171 ` 1010` hoelzl@35171 ` 1011` ```lemma abs_setprod: ``` hoelzl@35171 ` 1012` ``` fixes f :: "'a \ 'b\{linordered_field,abs}" ``` hoelzl@35171 ` 1013` ``` shows "abs (setprod f A) = setprod (\x. abs (f x)) A" ``` hoelzl@35171 ` 1014` ```proof (cases "finite A") ``` hoelzl@35171 ` 1015` ``` case True thus ?thesis ``` huffman@35216 ` 1016` ``` by induct (auto simp add: field_simps abs_mult) ``` hoelzl@35171 ` 1017` ```qed auto ``` hoelzl@35171 ` 1018` nipkow@15402 ` 1019` wenzelm@12396 ` 1020` ```subsection {* Finite cardinality *} ``` wenzelm@12396 ` 1021` nipkow@15402 ` 1022` ```text {* This definition, although traditional, is ugly to work with: ``` nipkow@15402 ` 1023` ```@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}. ``` nipkow@15402 ` 1024` ```But now that we have @{text setsum} things are easy: ``` wenzelm@12396 ` 1025` ```*} ``` wenzelm@12396 ` 1026` haftmann@31380 ` 1027` ```definition card :: "'a set \ nat" where ``` haftmann@31380 ` 1028` ``` "card A = setsum (\x. 1) A" ``` haftmann@31380 ` 1029` haftmann@31380 ` 1030` ```lemmas card_eq_setsum = card_def ``` wenzelm@12396 ` 1031` wenzelm@12396 ` 1032` ```lemma card_empty [simp]: "card {} = 0" ``` haftmann@31380 ` 1033` ``` by (simp add: card_def) ``` wenzelm@12396 ` 1034` wenzelm@12396 ` 1035` ```lemma card_insert_disjoint [simp]: ``` wenzelm@12396 ` 1036` ``` "finite A ==> x \ A ==> card (insert x A) = Suc(card A)" ``` haftmann@31380 ` 1037` ``` by (simp add: card_def) ``` nipkow@15402 ` 1038` nipkow@15402 ` 1039` ```lemma card_insert_if: ``` nipkow@28853 ` 1040` ``` "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))" ``` haftmann@31380 ` 1041` ``` by (simp add: insert_absorb) ``` haftmann@31380 ` 1042` haftmann@31380 ` 1043` ```lemma card_infinite [simp]: "~ finite A ==> card A = 0" ``` haftmann@31380 ` 1044` ``` by (simp add: card_def) ``` haftmann@31380 ` 1045` haftmann@31380 ` 1046` ```lemma card_ge_0_finite: ``` haftmann@31380 ` 1047` ``` "card A > 0 \ finite A" ``` haftmann@31380 ` 1048` ``` by (rule ccontr) simp ``` wenzelm@12396 ` 1049` paulson@24286 ` 1050` ```lemma card_0_eq [simp,noatp]: "finite A ==> (card A = 0) = (A = {})" ``` haftmann@31380 ` 1051` ``` apply auto ``` haftmann@31380 ` 1052` ``` apply (drule_tac a = x in mk_disjoint_insert, clarify, auto) ``` haftmann@31380 ` 1053` ``` done ``` haftmann@31380 ` 1054` haftmann@31380 ` 1055` ```lemma finite_UNIV_card_ge_0: ``` haftmann@31380 ` 1056` ``` "finite (UNIV :: 'a set) \ card (UNIV :: 'a set) > 0" ``` haftmann@31380 ` 1057` ``` by (rule ccontr) simp ``` wenzelm@12396 ` 1058` paulson@15409 ` 1059` ```lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)" ``` haftmann@31380 ` 1060` ``` by auto ``` nipkow@24853 ` 1061` paulson@34106 ` 1062` ```lemma card_gt_0_iff: "(0 < card A) = (A \ {} & finite A)" ``` paulson@34106 ` 1063` ``` by (simp add: neq0_conv [symmetric] card_eq_0_iff) ``` paulson@34106 ` 1064` wenzelm@12396 ` 1065` ```lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A" ``` nipkow@14302 ` 1066` ```apply(rule_tac t = A in insert_Diff [THEN subst], assumption) ``` nipkow@14302 ` 1067` ```apply(simp del:insert_Diff_single) ``` nipkow@14302 ` 1068` ```done ``` wenzelm@12396 ` 1069` wenzelm@12396 ` 1070` ```lemma card_Diff_singleton: ``` nipkow@24853 ` 1071` ``` "finite A ==> x: A ==> card (A - {x}) = card A - 1" ``` nipkow@24853 ` 1072` ```by (simp add: card_Suc_Diff1 [symmetric]) ``` wenzelm@12396 ` 1073` wenzelm@12396 ` 1074` ```lemma card_Diff_singleton_if: ``` nipkow@24853 ` 1075` ``` "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)" ``` nipkow@24853 ` 1076` ```by (simp add: card_Diff_singleton) ``` nipkow@24853 ` 1077` nipkow@24853 ` 1078` ```lemma card_Diff_insert[simp]: ``` nipkow@24853 ` 1079` ```assumes "finite A" and "a:A" and "a ~: B" ``` nipkow@24853 ` 1080` ```shows "card(A - insert a B) = card(A - B) - 1" ``` nipkow@24853 ` 1081` ```proof - ``` nipkow@24853 ` 1082` ``` have "A - insert a B = (A - B) - {a}" using assms by blast ``` nipkow@24853 ` 1083` ``` then show ?thesis using assms by(simp add:card_Diff_singleton) ``` nipkow@24853 ` 1084` ```qed ``` wenzelm@12396 ` 1085` wenzelm@12396 ` 1086` ```lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))" ``` nipkow@24853 ` 1087` ```by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert) ``` wenzelm@12396 ` 1088` wenzelm@12396 ` 1089` ```lemma card_insert_le: "finite A ==> card A <= card (insert x A)" ``` nipkow@24853 ` 1090` ```by (simp add: card_insert_if) ``` wenzelm@12396 ` 1091` nipkow@15402 ` 1092` ```lemma card_mono: "\ finite B; A \ B \ \ card A \ card B" ``` nipkow@15539 ` 1093` ```by (simp add: card_def setsum_mono2) ``` nipkow@15402 ` 1094` wenzelm@12396 ` 1095` ```lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)" ``` nipkow@28853 ` 1096` ```apply (induct set: finite, simp, clarify) ``` nipkow@28853 ` 1097` ```apply (subgoal_tac "finite A & A - {x} <= F") ``` nipkow@28853 ` 1098` ``` prefer 2 apply (blast intro: finite_subset, atomize) ``` nipkow@28853 ` 1099` ```apply (drule_tac x = "A - {x}" in spec) ``` nipkow@28853 ` 1100` ```apply (simp add: card_Diff_singleton_if split add: split_if_asm) ``` nipkow@28853 ` 1101` ```apply (case_tac "card A", auto) ``` nipkow@28853 ` 1102` ```done ``` wenzelm@12396 ` 1103` wenzelm@12396 ` 1104` ```lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B" ``` berghofe@26792 ` 1105` ```apply (simp add: psubset_eq linorder_not_le [symmetric]) ``` nipkow@24853 ` 1106` ```apply (blast dest: card_seteq) ``` nipkow@24853 ` 1107` ```done ``` wenzelm@12396 ` 1108` wenzelm@12396 ` 1109` ```lemma card_Un_Int: "finite A ==> finite B ``` wenzelm@12396 ` 1110` ``` ==> card A + card B = card (A Un B) + card (A Int B)" ``` nipkow@15402 ` 1111` ```by(simp add:card_def setsum_Un_Int) ``` wenzelm@12396 ` 1112` wenzelm@12396 ` 1113` ```lemma card_Un_disjoint: "finite A ==> finite B ``` wenzelm@12396 ` 1114` ``` ==> A Int B = {} ==> card (A Un B) = card A + card B" ``` nipkow@24853 ` 1115` ```by (simp add: card_Un_Int) ``` wenzelm@12396 ` 1116` wenzelm@12396 ` 1117` ```lemma card_Diff_subset: ``` nipkow@15402 ` 1118` ``` "finite B ==> B <= A ==> card (A - B) = card A - card B" ``` nipkow@15402 ` 1119` ```by(simp add:card_def setsum_diff_nat) ``` wenzelm@12396 ` 1120` paulson@34106 ` 1121` ```lemma card_Diff_subset_Int: ``` paulson@34106 ` 1122` ``` assumes AB: "finite (A \ B)" shows "card (A - B) = card A - card (A \ B)" ``` paulson@34106 ` 1123` ```proof - ``` paulson@34106 ` 1124` ``` have "A - B = A - A \ B" by auto ``` paulson@34106 ` 1125` ``` thus ?thesis ``` paulson@34106 ` 1126` ``` by (simp add: card_Diff_subset AB) ``` paulson@34106 ` 1127` ```qed ``` paulson@34106 ` 1128` wenzelm@12396 ` 1129` ```lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A" ``` nipkow@28853 ` 1130` ```apply (rule Suc_less_SucD) ``` nipkow@28853 ` 1131` ```apply (simp add: card_Suc_Diff1 del:card_Diff_insert) ``` nipkow@28853 ` 1132` ```done ``` wenzelm@12396 ` 1133` wenzelm@12396 ` 1134` ```lemma card_Diff2_less: ``` nipkow@28853 ` 1135` ``` "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A" ``` nipkow@28853 ` 1136` ```apply (case_tac "x = y") ``` nipkow@28853 ` 1137` ``` apply (simp add: card_Diff1_less del:card_Diff_insert) ``` nipkow@28853 ` 1138` ```apply (rule less_trans) ``` nipkow@28853 ` 1139` ``` prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert) ``` nipkow@28853 ` 1140` ```done ``` wenzelm@12396 ` 1141` wenzelm@12396 ` 1142` ```lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A" ``` nipkow@28853 ` 1143` ```apply (case_tac "x : A") ``` nipkow@28853 ` 1144` ``` apply (simp_all add: card_Diff1_less less_imp_le) ``` nipkow@28853 ` 1145` ```done ``` wenzelm@12396 ` 1146` wenzelm@12396 ` 1147` ```lemma card_psubset: "finite B ==> A \ B ==> card A < card B ==> A < B" ``` paulson@14208 ` 1148` ```by (erule psubsetI, blast) ``` wenzelm@12396 ` 1149` paulson@14889 ` 1150` ```lemma insert_partition: ``` nipkow@15402 ` 1151` ``` "\ x \ F; \c1 \ insert x F. \c2 \ insert x F. c1 \ c2 \ c1 \ c2 = {} \ ``` nipkow@15402 ` 1152` ``` \ x \ \ F = {}" ``` paulson@14889 ` 1153` ```by auto ``` paulson@14889 ` 1154` nipkow@32006 ` 1155` ```lemma finite_psubset_induct[consumes 1, case_names psubset]: ``` nipkow@32006 ` 1156` ``` assumes "finite A" and "!!A. finite A \ (!!B. finite B \ B \ A \ P(B)) \ P(A)" shows "P A" ``` nipkow@32006 ` 1157` ```using assms(1) ``` nipkow@32006 ` 1158` ```proof (induct A rule: measure_induct_rule[where f=card]) ``` nipkow@32006 ` 1159` ``` case (less A) ``` nipkow@32006 ` 1160` ``` show ?case ``` nipkow@32006 ` 1161` ``` proof(rule assms(2)[OF less(2)]) ``` nipkow@32006 ` 1162` ``` fix B assume "finite B" "B \ A" ``` nipkow@32006 ` 1163` ``` show "P B" by(rule less(1)[OF psubset_card_mono[OF less(2) `B \ A`] `finite B`]) ``` nipkow@32006 ` 1164` ``` qed ``` nipkow@32006 ` 1165` ```qed ``` nipkow@32006 ` 1166` paulson@19793 ` 1167` ```text{* main cardinality theorem *} ``` paulson@14889 ` 1168` ```lemma card_partition [rule_format]: ``` nipkow@28853 ` 1169` ``` "finite C ==> ``` nipkow@28853 ` 1170` ``` finite (\ C) --> ``` nipkow@28853 ` 1171` ``` (\c\C. card c = k) --> ``` nipkow@28853 ` 1172` ``` (\c1 \ C. \c2 \ C. c1 \ c2 --> c1 \ c2 = {}) --> ``` nipkow@28853 ` 1173` ``` k * card(C) = card (\ C)" ``` paulson@14889 ` 1174` ```apply (erule finite_induct, simp) ``` huffman@35216 ` 1175` ```apply (simp add: card_Un_disjoint insert_partition ``` paulson@14889 ` 1176` ``` finite_subset [of _ "\ (insert x F)"]) ``` paulson@14889 ` 1177` ```done ``` paulson@14889 ` 1178` haftmann@31380 ` 1179` ```lemma card_eq_UNIV_imp_eq_UNIV: ``` haftmann@31380 ` 1180` ``` assumes fin: "finite (UNIV :: 'a set)" ``` haftmann@31380 ` 1181` ``` and card: "card A = card (UNIV :: 'a set)" ``` haftmann@31380 ` 1182` ``` shows "A = (UNIV :: 'a set)" ``` haftmann@31380 ` 1183` ```proof ``` haftmann@31380 ` 1184` ``` show "A \ UNIV" by simp ``` haftmann@31380 ` 1185` ``` show "UNIV \ A" ``` haftmann@31380 ` 1186` ``` proof ``` haftmann@31380 ` 1187` ``` fix x ``` haftmann@31380 ` 1188` ``` show "x \ A" ``` haftmann@31380 ` 1189` ``` proof (rule ccontr) ``` haftmann@31380 ` 1190` ``` assume "x \ A" ``` haftmann@31380 ` 1191` ``` then have "A \ UNIV" by auto ``` haftmann@31380 ` 1192` ``` with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono) ``` haftmann@31380 ` 1193` ``` with card show False by simp ``` haftmann@31380 ` 1194` ``` qed ``` haftmann@31380 ` 1195` ``` qed ``` haftmann@31380 ` 1196` ```qed ``` wenzelm@12396 ` 1197` paulson@19793 ` 1198` ```text{*The form of a finite set of given cardinality*} ``` paulson@19793 ` 1199` paulson@19793 ` 1200` ```lemma card_eq_SucD: ``` nipkow@24853 ` 1201` ```assumes "card A = Suc k" ``` nipkow@24853 ` 1202` ```shows "\b B. A = insert b B & b \ B & card B = k & (k=0 \ B={})" ``` paulson@19793 ` 1203` ```proof - ``` nipkow@24853 ` 1204` ``` have fin: "finite A" using assms by (auto intro: ccontr) ``` nipkow@24853 ` 1205` ``` moreover have "card A \ 0" using assms by auto ``` nipkow@24853 ` 1206` ``` ultimately obtain b where b: "b \ A" by auto ``` paulson@19793 ` 1207` ``` show ?thesis ``` paulson@19793 ` 1208` ``` proof (intro exI conjI) ``` paulson@19793 ` 1209` ``` show "A = insert b (A-{b})" using b by blast ``` paulson@19793 ` 1210` ``` show "b \ A - {b}" by blast ``` nipkow@24853 ` 1211` ``` show "card (A - {b}) = k" and "k = 0 \ A - {b} = {}" ``` nipkow@24853 ` 1212` ``` using assms b fin by(fastsimp dest:mk_disjoint_insert)+ ``` paulson@19793 ` 1213` ``` qed ``` paulson@19793 ` 1214` ```qed ``` paulson@19793 ` 1215` paulson@19793 ` 1216` ```lemma card_Suc_eq: ``` nipkow@24853 ` 1217` ``` "(card A = Suc k) = ``` nipkow@24853 ` 1218` ``` (\b B. A = insert b B & b \ B & card B = k & (k=0 \ B={}))" ``` nipkow@24853 ` 1219` ```apply(rule iffI) ``` nipkow@24853 ` 1220` ``` apply(erule card_eq_SucD) ``` nipkow@24853 ` 1221` ```apply(auto) ``` nipkow@24853 ` 1222` ```apply(subst card_insert) ``` nipkow@24853 ` 1223` ``` apply(auto intro:ccontr) ``` nipkow@24853 ` 1224` ```done ``` paulson@19793 ` 1225` haftmann@31380 ` 1226` ```lemma finite_fun_UNIVD2: ``` haftmann@31380 ` 1227` ``` assumes fin: "finite (UNIV :: ('a \ 'b) set)" ``` haftmann@31380 ` 1228` ``` shows "finite (UNIV :: 'b set)" ``` haftmann@31380 ` 1229` ```proof - ``` haftmann@31380 ` 1230` ``` from fin have "finite (range (\f :: 'a \ 'b. f arbitrary))" ``` haftmann@31380 ` 1231` ``` by(rule finite_imageI) ``` haftmann@31380 ` 1232` ``` moreover have "UNIV = range (\f :: 'a \ 'b. f arbitrary)" ``` haftmann@31380 ` 1233` ``` by(rule UNIV_eq_I) auto ``` haftmann@31380 ` 1234` ``` ultimately show "finite (UNIV :: 'b set)" by simp ``` haftmann@31380 ` 1235` ```qed ``` haftmann@31380 ` 1236` nipkow@15539 ` 1237` ```lemma setsum_constant [simp]: "(\x \ A. y) = of_nat(card A) * y" ``` nipkow@15539 ` 1238` ```apply (cases "finite A") ``` nipkow@15539 ` 1239` ```apply (erule finite_induct) ``` nipkow@29667 ` 1240` ```apply (auto simp add: algebra_simps) ``` paulson@15409 ` 1241` ```done ``` nipkow@15402 ` 1242` haftmann@31017 ` 1243` ```lemma setprod_constant: "finite A ==> (\x\ A. (y::'a::{comm_monoid_mult})) = y^(card A)" ``` nipkow@28853 ` 1244` ```apply (erule finite_induct) ``` huffman@35216 ` 1245` ```apply auto ``` nipkow@28853 ` 1246` ```done ``` nipkow@15402 ` 1247` chaieb@29674 ` 1248` ```lemma setprod_gen_delta: ``` chaieb@29674 ` 1249` ``` assumes fS: "finite S" ``` haftmann@31017 ` 1250` ``` shows "setprod (\k. if k=a then b k else c) S = (if a \ S then (b a ::'a::{comm_monoid_mult}) * c^ (card S - 1) else c^ card S)" ``` chaieb@29674 ` 1251` ```proof- ``` chaieb@29674 ` 1252` ``` let ?f = "(\k. if k=a then b k else c)" ``` chaieb@29674 ` 1253` ``` {assume a: "a \ S" ``` chaieb@29674 ` 1254` ``` hence "\ k\ S. ?f k = c" by simp ``` chaieb@29674 ` 1255` ``` hence ?thesis using a setprod_constant[OF fS, of c] by (simp add: setprod_1 cong add: setprod_cong) } ``` chaieb@29674 ` 1256` ``` moreover ``` chaieb@29674 ` 1257` ``` {assume a: "a \ S" ``` chaieb@29674 ` 1258` ``` let ?A = "S - {a}" ``` chaieb@29674 ` 1259` ``` let ?B = "{a}" ``` chaieb@29674 ` 1260` ``` have eq: "S = ?A \ ?B" using a by blast ``` chaieb@29674 ` 1261` ``` have dj: "?A \ ?B = {}" by simp ``` chaieb@29674 ` 1262` ``` from fS have fAB: "finite ?A" "finite ?B" by auto ``` chaieb@29674 ` 1263` ``` have fA0:"setprod ?f ?A = setprod (\i. c) ?A" ``` chaieb@29674 ` 1264` ``` apply (rule setprod_cong) by auto ``` chaieb@29674 ` 1265` ``` have cA: "card ?A = card S - 1" using fS a by auto ``` chaieb@29674 ` 1266` ``` have fA1: "setprod ?f ?A = c ^ card ?A" unfolding fA0 apply (rule setprod_constant) using fS by auto ``` chaieb@29674 ` 1267` ``` have "setprod ?f ?A * setprod ?f ?B = setprod ?f S" ``` chaieb@29674 ` 1268` ``` using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] ``` chaieb@29674 ` 1269` ``` by simp ``` chaieb@29674 ` 1270` ``` then have ?thesis using a cA ``` chaieb@29674 ` 1271` ``` by (simp add: fA1 ring_simps cong add: setprod_cong cong del: if_weak_cong)} ``` chaieb@29674 ` 1272` ``` ultimately show ?thesis by blast ``` chaieb@29674 ` 1273` ```qed ``` chaieb@29674 ` 1274` chaieb@29674 ` 1275` nipkow@15542 ` 1276` ```lemma setsum_bounded: ``` haftmann@35028 ` 1277` ``` assumes le: "\i. i\A \ f i \ (K::'a::{semiring_1, ordered_ab_semigroup_add})" ``` nipkow@15542 ` 1278` ``` shows "setsum f A \ of_nat(card A) * K" ``` nipkow@15542 ` 1279` ```proof (cases "finite A") ``` nipkow@15542 ` 1280` ``` case True ``` nipkow@15542 ` 1281` ``` thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp ``` nipkow@15542 ` 1282` ```next ``` nipkow@15542 ` 1283` ``` case False thus ?thesis by (simp add: setsum_def) ``` nipkow@15542 ` 1284` ```qed ``` nipkow@15542 ` 1285` nipkow@15402 ` 1286` nipkow@31080 ` 1287` ```lemma card_UNIV_unit: "card (UNIV :: unit set) = 1" ``` nipkow@31080 ` 1288` ``` unfolding UNIV_unit by simp ``` nipkow@31080 ` 1289` nipkow@31080 ` 1290` nipkow@15402 ` 1291` ```subsubsection {* Cardinality of unions *} ``` nipkow@15402 ` 1292` nipkow@15402 ` 1293` ```lemma card_UN_disjoint: ``` nipkow@28853 ` 1294` ``` "finite I ==> (ALL i:I. finite (A i)) ==> ``` nipkow@28853 ` 1295` ``` (ALL i:I. ALL j:I. i \ j --> A i Int A j = {}) ``` nipkow@28853 ` 1296` ``` ==> card (UNION I A) = (\i\I. card(A i))" ``` nipkow@28853 ` 1297` ```apply (simp add: card_def del: setsum_constant) ``` nipkow@28853 ` 1298` ```apply (subgoal_tac ``` nipkow@28853 ` 1299` ``` "setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I") ``` nipkow@28853 ` 1300` ```apply (simp add: setsum_UN_disjoint del: setsum_constant) ``` nipkow@28853 ` 1301` ```apply (simp cong: setsum_cong) ``` nipkow@28853 ` 1302` ```done ``` nipkow@15402 ` 1303` nipkow@15402 ` 1304` ```lemma card_Union_disjoint: ``` nipkow@15402 ` 1305` ``` "finite C ==> (ALL A:C. finite A) ==> ``` nipkow@28853 ` 1306` ``` (ALL A:C. ALL B:C. A \ B --> A Int B = {}) ``` nipkow@28853 ` 1307` ``` ==> card (Union C) = setsum card C" ``` nipkow@28853 ` 1308` ```apply (frule card_UN_disjoint [of C id]) ``` nipkow@28853 ` 1309` ```apply (unfold Union_def id_def, assumption+) ``` nipkow@28853 ` 1310` ```done ``` nipkow@28853 ` 1311` nipkow@15402 ` 1312` wenzelm@12396 ` 1313` ```subsubsection {* Cardinality of image *} ``` wenzelm@12396 ` 1314` nipkow@28853 ` 1315` ```text{*The image of a finite set can be expressed using @{term fold_image}.*} ``` nipkow@28853 ` 1316` ```lemma image_eq_fold_image: ``` nipkow@28853 ` 1317` ``` "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A" ``` haftmann@26041 ` 1318` ```proof (induct rule: finite_induct) ``` haftmann@26041 ` 1319` ``` case empty then show ?case by simp ``` haftmann@26041 ` 1320` ```next ``` haftmann@29509 ` 1321` ``` interpret ab_semigroup_mult "op Un" ``` haftmann@28823 ` 1322` ``` proof qed auto ``` haftmann@26041 ` 1323` ``` case insert ``` haftmann@26041 ` 1324` ``` then show ?case by simp ``` haftmann@26041 ` 1325` ```qed ``` paulson@15447 ` 1326` wenzelm@12396 ` 1327` ```lemma card_image_le: "finite A ==> card (f ` A) <= card A" ``` nipkow@28853 ` 1328` ```apply (induct set: finite) ``` nipkow@28853 ` 1329` ``` apply simp ``` huffman@35216 ` 1330` ```apply (simp add: le_SucI card_insert_if) ``` nipkow@28853 ` 1331` ```done ``` wenzelm@12396 ` 1332` nipkow@15402 ` 1333` ```lemma card_image: "inj_on f A ==> card (f ` A) = card A" ``` nipkow@15539 ` 1334` ```by(simp add:card_def setsum_reindex o_def del:setsum_constant) ``` wenzelm@12396 ` 1335` nipkow@31451 ` 1336` ```lemma bij_betw_same_card: "bij_betw f A B \ card A = card B" ``` nipkow@31451 ` 1337` ```by(auto simp: card_image bij_betw_def) ``` nipkow@31451 ` 1338` wenzelm@12396 ` 1339` ```lemma endo_inj_surj: "finite A ==> f ` A \ A ==> inj_on f A ==> f ` A = A" ``` nipkow@25162 ` 1340` ```by (simp add: card_seteq card_image) ``` wenzelm@12396 ` 1341` nipkow@15111 ` 1342` ```lemma eq_card_imp_inj_on: ``` nipkow@15111 ` 1343` ``` "[| finite A; card(f ` A) = card A |] ==> inj_on f A" ``` wenzelm@21575 ` 1344` ```apply (induct rule:finite_induct) ``` wenzelm@21575 ` 1345` ```apply simp ``` nipkow@15111 ` 1346` ```apply(frule card_image_le[where f = f]) ``` nipkow@15111 ` 1347` ```apply(simp add:card_insert_if split:if_splits) ``` nipkow@15111 ` 1348` ```done ``` nipkow@15111 ` 1349` nipkow@15111 ` 1350` ```lemma inj_on_iff_eq_card: ``` nipkow@15111 ` 1351` ``` "finite A ==> inj_on f A = (card(f ` A) = card A)" ``` nipkow@15111 ` 1352` ```by(blast intro: card_image eq_card_imp_inj_on) ``` nipkow@15111 ` 1353` wenzelm@12396 ` 1354` nipkow@15402 ` 1355` ```lemma card_inj_on_le: ``` nipkow@28853 ` 1356` ``` "[|inj_on f A; f ` A \ B; finite B |] ==> card A \ card B" ``` nipkow@15402 ` 1357` ```apply (subgoal_tac "finite A") ``` nipkow@15402 ` 1358` ``` apply (force intro: card_mono simp add: card_image [symmetric]) ``` nipkow@15402 ` 1359` ```apply (blast intro: finite_imageD dest: finite_subset) ``` nipkow@15402 ` 1360` ```done ``` nipkow@15402 ` 1361` nipkow@15402 ` 1362` ```lemma card_bij_eq: ``` nipkow@28853 ` 1363` ``` "[|inj_on f A; f ` A \ B; inj_on g B; g ` B \ A; ``` nipkow@28853 ` 1364` ``` finite A; finite B |] ==> card A = card B" ``` nipkow@33657 ` 1365` ```by (auto intro: le_antisym card_inj_on_le) ``` nipkow@15402 ` 1366` nipkow@15402 ` 1367` nipkow@15402 ` 1368` ```subsubsection {* Cardinality of products *} ``` nipkow@15402 ` 1369` nipkow@15402 ` 1370` ```(* ``` nipkow@15402 ` 1371` ```lemma SigmaI_insert: "y \ A ==> ``` nipkow@15402 ` 1372` ``` (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \ (SIGMA x: A. B x))" ``` nipkow@15402 ` 1373` ``` by auto ``` nipkow@15402 ` 1374` ```*) ``` nipkow@15402 ` 1375` nipkow@15402 ` 1376` ```lemma card_SigmaI [simp]: ``` nipkow@15402 ` 1377` ``` "\ finite A; ALL a:A. finite (B a) \ ``` nipkow@15402 ` 1378` ``` \ card (SIGMA x: A. B x) = (\a\A. card (B a))" ``` nipkow@15539 ` 1379` ```by(simp add:card_def setsum_Sigma del:setsum_constant) ``` nipkow@15402 ` 1380` paulson@15409 ` 1381` ```lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)" ``` paulson@15409 ` 1382` ```apply (cases "finite A") ``` paulson@15409 ` 1383` ```apply (cases "finite B") ``` paulson@15409 ` 1384` ```apply (auto simp add: card_eq_0_iff ``` nipkow@15539 ` 1385` ``` dest: finite_cartesian_productD1 finite_cartesian_productD2) ``` paulson@15409 ` 1386` ```done ``` nipkow@15402 ` 1387` nipkow@15402 ` 1388` ```lemma card_cartesian_product_singleton: "card({x} <*> A) = card(A)" ``` nipkow@15539 ` 1389` ```by (simp add: card_cartesian_product) ``` paulson@15409 ` 1390` nipkow@15402 ` 1391` huffman@29025 ` 1392` ```subsubsection {* Cardinality of sums *} ``` huffman@29025 ` 1393` huffman@29025 ` 1394` ```lemma card_Plus: ``` huffman@29025 ` 1395` ``` assumes "finite A" and "finite B" ``` huffman@29025 ` 1396` ``` shows "card (A <+> B) = card A + card B" ``` huffman@29025 ` 1397` ```proof - ``` huffman@29025 ` 1398` ``` have "Inl`A \ Inr`B = {}" by fast ``` huffman@29025 ` 1399` ``` with assms show ?thesis ``` huffman@29025 ` 1400` ``` unfolding Plus_def ``` huffman@29025 ` 1401` ``` by (simp add: card_Un_disjoint card_image) ``` huffman@29025 ` 1402` ```qed ``` huffman@29025 ` 1403` nipkow@31080 ` 1404` ```lemma card_Plus_conv_if: ``` nipkow@31080 ` 1405` ``` "card (A <+> B) = (if finite A \ finite B then card(A) + card(B) else 0)" ``` nipkow@31080 ` 1406` ```by(auto simp: card_def setsum_Plus simp del: setsum_constant) ``` nipkow@31080 ` 1407` nipkow@15402 ` 1408` wenzelm@12396 ` 1409` ```subsubsection {* Cardinality of the Powerset *} ``` wenzelm@12396 ` 1410` wenzelm@12396 ` 1411` ```lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *) ``` nipkow@28853 ` 1412` ```apply (induct set: finite) ``` nipkow@28853 ` 1413` ``` apply (simp_all add: Pow_insert) ``` nipkow@28853 ` 1414` ```apply (subst card_Un_disjoint, blast) ``` nipkow@28853 ` 1415` ``` apply (blast intro: finite_imageI, blast) ``` nipkow@28853 ` 1416` ```apply (subgoal_tac "inj_on (insert x) (Pow F)") ``` nipkow@28853 ` 1417` ``` apply (simp add: card_image Pow_insert) ``` nipkow@28853 ` 1418` ```apply (unfold inj_on_def) ``` nipkow@28853 ` 1419` ```apply (blast elim!: equalityE) ``` nipkow@28853 ` 1420` ```done ``` wenzelm@12396 ` 1421` haftmann@24342 ` 1422` ```text {* Relates to equivalence classes. Based on a theorem of F. Kammüller. *} ``` wenzelm@12396 ` 1423` wenzelm@12396 ` 1424` ```lemma dvd_partition: ``` nipkow@15392 ` 1425` ``` "finite (Union C) ==> ``` wenzelm@12396 ` 1426` ``` ALL c : C. k dvd card c ==> ``` paulson@14430 ` 1427` ``` (ALL c1: C. ALL c2: C. c1 \ c2 --> c1 Int c2 = {}) ==> ``` wenzelm@12396 ` 1428` ``` k dvd card (Union C)" ``` nipkow@15392 ` 1429` ```apply(frule finite_UnionD) ``` nipkow@15392 ` 1430` ```apply(rotate_tac -1) ``` nipkow@28853 ` 1431` ```apply (induct set: finite, simp_all, clarify) ``` nipkow@28853 ` 1432` ```apply (subst card_Un_disjoint) ``` huffman@35216 ` 1433` ``` apply (auto simp add: disjoint_eq_subset_Compl) ``` nipkow@28853 ` 1434` ```done ``` wenzelm@12396 ` 1435` wenzelm@12396 ` 1436` nipkow@25162 ` 1437` ```subsubsection {* Relating injectivity and surjectivity *} ``` nipkow@25162 ` 1438` nipkow@25162 ` 1439` ```lemma finite_surj_inj: "finite(A) \ A <= f`A \ inj_on f A" ``` nipkow@25162 ` 1440` ```apply(rule eq_card_imp_inj_on, assumption) ``` nipkow@25162 ` 1441` ```apply(frule finite_imageI) ``` nipkow@25162 ` 1442` ```apply(drule (1) card_seteq) ``` nipkow@28853 ` 1443` ``` apply(erule card_image_le) ``` nipkow@25162 ` 1444` ```apply simp ``` nipkow@25162 ` 1445` ```done ``` nipkow@25162 ` 1446` nipkow@25162 ` 1447` ```lemma finite_UNIV_surj_inj: fixes f :: "'a \ 'a" ``` nipkow@25162 ` 1448` ```shows "finite(UNIV:: 'a set) \ surj f \ inj f" ``` nipkow@25162 ` 1449` ```by (blast intro: finite_surj_inj subset_UNIV dest:surj_range) ``` nipkow@25162 ` 1450` nipkow@25162 ` 1451` ```lemma finite_UNIV_inj_surj: fixes f :: "'a \ 'a" ``` nipkow@25162 ` 1452` ```shows "finite(UNIV:: 'a set) \ inj f \ surj f" ``` nipkow@25162 ` 1453` ```by(fastsimp simp:surj_def dest!: endo_inj_surj) ``` nipkow@25162 ` 1454` nipkow@31992 ` 1455` ```corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)" ``` nipkow@25162 ` 1456` ```proof ``` nipkow@25162 ` 1457` ``` assume "finite(UNIV::nat set)" ``` nipkow@25162 ` 1458` ``` with finite_UNIV_inj_surj[of Suc] ``` nipkow@25162 ` 1459` ``` show False by simp (blast dest: Suc_neq_Zero surjD) ``` nipkow@25162 ` 1460` ```qed ``` nipkow@25162 ` 1461` nipkow@31992 ` 1462` ```(* Often leads to bogus ATP proofs because of reduced type information, hence noatp *) ``` nipkow@31992 ` 1463` ```lemma infinite_UNIV_char_0[noatp]: ``` nipkow@29879 ` 1464` ``` "\ finite (UNIV::'a::semiring_char_0 set)" ``` nipkow@29879 ` 1465` ```proof ``` nipkow@29879 ` 1466` ``` assume "finite (UNIV::'a set)" ``` nipkow@29879 ` 1467` ``` with subset_UNIV have "finite (range of_nat::'a set)" ``` nipkow@29879 ` 1468` ``` by (rule finite_subset) ``` nipkow@29879 ` 1469` ``` moreover have "inj (of_nat::nat \ 'a)" ``` nipkow@29879 ` 1470` ``` by (simp add: inj_on_def) ``` nipkow@29879 ` 1471` ``` ultimately have "finite (UNIV::nat set)" ``` nipkow@29879 ` 1472` ``` by (rule finite_imageD) ``` nipkow@29879 ` 1473` ``` then show "False" ``` huffman@35216 ` 1474` ``` by simp ``` nipkow@29879 ` 1475` ```qed ``` nipkow@25162 ` 1476` haftmann@22917 ` 1477` ```subsubsection {* Fold1 in lattices with @{const inf} and @{const sup} *} ``` haftmann@22917 ` 1478` haftmann@22917 ` 1479` ```text{* ``` haftmann@22917 ` 1480` ``` As an application of @{text fold1} we define infimum ``` haftmann@22917 ` 1481` ``` and supremum in (not necessarily complete!) lattices ``` haftmann@22917 ` 1482` ``` over (non-empty) sets by means of @{text fold1}. ``` haftmann@22917 ` 1483` ```*} ``` haftmann@22917 ` 1484` haftmann@35028 ` 1485` ```context semilattice_inf ``` haftmann@26041 ` 1486` ```begin ``` haftmann@26041 ` 1487` haftmann@26041 ` 1488` ```lemma below_fold1_iff: ``` haftmann@26041 ` 1489` ``` assumes "finite A" "A \ {}" ``` haftmann@26041 ` 1490` ``` shows "x \ fold1 inf A \ (\a\A. x \ a)" ``` haftmann@26041 ` 1491` ```proof - ``` haftmann@29509 ` 1492` ``` interpret ab_semigroup_idem_mult inf ``` haftmann@26041 ` 1493` ``` by (rule ab_semigroup_idem_mult_inf) ``` haftmann@26041 ` 1494` ``` show ?thesis using assms by (induct rule: finite_ne_induct) simp_all ``` haftmann@26041 ` 1495` ```qed ``` haftmann@26041 ` 1496` haftmann@26041 ` 1497` ```lemma fold1_belowI: ``` haftmann@26757 ` 1498` ``` assumes "finite A" ``` haftmann@26041 ` 1499` ``` and "a \ A" ``` haftmann@26041 ` 1500` ``` shows "fold1 inf A \ a" ``` haftmann@26757 ` 1501` ```proof - ``` haftmann@26757 ` 1502` ``` from assms have "A \ {}" by auto ``` haftmann@26757 ` 1503` ``` from `finite A` `A \ {}` `a \ A` show ?thesis ``` haftmann@26757 ` 1504` ``` proof (induct rule: finite_ne_induct) ``` haftmann@26757 ` 1505` ``` case singleton thus ?case by simp ``` haftmann@26041 ` 1506` ``` next ``` haftmann@29509 ` 1507` ``` interpret ab_semigroup_idem_mult inf ``` haftmann@26757 ` 1508` ``` by (rule ab_semigroup_idem_mult_inf) ``` haftmann@26757 ` 1509` ``` case (insert x F) ``` haftmann@26757 ` 1510` ``` from insert(5) have "a = x \ a \ F" by simp ``` haftmann@26757 ` 1511` ``` thus ?case ``` haftmann@26757 ` 1512` ``` proof ``` haftmann@26757 ` 1513` ``` assume "a = x" thus ?thesis using insert ``` nipkow@29667 ` 1514` ``` by (simp add: mult_ac) ``` haftmann@26757 ` 1515` ``` next ``` haftmann@26757 ` 1516` ``` assume "a \ F" ``` haftmann@26757 ` 1517` ``` hence bel: "fold1 inf F \ a" by (rule insert) ``` haftmann@26757 ` 1518` ``` have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)" ``` nipkow@29667 ` 1519` ``` using insert by (simp add: mult_ac) ``` haftmann@26757 ` 1520` ``` also have "inf (fold1 inf F) a = fold1 inf F" ``` haftmann@26757 ` 1521` ``` using bel by (auto intro: antisym) ``` haftmann@26757 ` 1522` ``` also have "inf x \ = fold1 inf (insert x F)" ``` nipkow@29667 ` 1523` ``` using insert by (simp add: mult_ac) ``` haftmann@26757 ` 1524` ``` finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" . ``` haftmann@26757 ` 1525` ``` moreover have "inf (fold1 inf (insert x F)) a \ a" by simp ``` haftmann@26757 ` 1526` ``` ultimately show ?thesis by simp ``` haftmann@26757 ` 1527` ``` qed ``` haftmann@26041 ` 1528` ``` qed ``` haftmann@26041 ` 1529` ```qed ``` haftmann@26041 ` 1530` haftmann@26041 ` 1531` ```end ``` haftmann@26041 ` 1532` haftmann@24342 ` 1533` ```context lattice ``` haftmann@22917 ` 1534` ```begin ``` haftmann@22917 ` 1535` haftmann@22917 ` 1536` ```definition ``` wenzelm@31916 ` 1537` ``` Inf_fin :: "'a set \ 'a" ("\\<^bsub>fin\<^esub>_" [900] 900) ``` haftmann@22917 ` 1538` ```where ``` haftmann@25062 ` 1539` ``` "Inf_fin = fold1 inf" ``` haftmann@22917 ` 1540` haftmann@22917 ` 1541` ```definition ``` wenzelm@31916 ` 1542` ``` Sup_fin :: "'a set \ 'a" ("\\<^bsub>fin\<^esub>_" [900] 900) ``` haftmann@22917 ` 1543` ```where ``` haftmann@25062 ` 1544` ``` "Sup_fin = fold1 sup" ``` haftmann@25062 ` 1545` wenzelm@31916 ` 1546` ```lemma Inf_le_Sup [simp]: "\ finite A; A \ {} \ \ \\<^bsub>fin\<^esub>A \ \\<^bsub>fin\<^esub>A" ``` haftmann@24342 ` 1547` ```apply(unfold Sup_fin_def Inf_fin_def) ``` nipkow@15500 ` 1548` ```apply(subgoal_tac "EX a. a:A") ``` nipkow@15500 ` 1549` ```prefer 2 apply blast ``` nipkow@15500 ` 1550` ```apply(erule exE) ``` haftmann@22388 ` 1551` ```apply(rule order_trans) ``` haftmann@26757 ` 1552` ```apply(erule (1) fold1_belowI) ``` haftmann@35028 ` 1553` ```apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice]) ``` nipkow@15500 ` 1554` ```done ``` nipkow@15500 ` 1555` haftmann@24342 ` 1556` ```lemma sup_Inf_absorb [simp]: ``` wenzelm@31916 ` 1557` ``` "finite A \ a \ A \ sup a (\\<^bsub>fin\<^esub>A) = a" ``` nipkow@15512 ` 1558` ```apply(subst sup_commute) ``` haftmann@26041 ` 1559` ```apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI) ``` nipkow@15504 ` 1560` ```done ``` nipkow@15504 ` 1561` haftmann@24342 ` 1562` ```lemma inf_Sup_absorb [simp]: ``` wenzelm@31916 ` 1563` ``` "finite A \ a \ A \ inf a (\\<^bsub>fin\<^esub>A) = a" ``` haftmann@26041 ` 1564` ```by (simp add: Sup_fin_def inf_absorb1 ``` haftmann@35028 ` 1565` ``` semilattice_inf.fold1_belowI [OF dual_semilattice]) ``` haftmann@24342 ` 1566` haftmann@24342 ` 1567` ```end ``` haftmann@24342 ` 1568` haftmann@24342 ` 1569` ```context distrib_lattice ``` haftmann@24342 ` 1570` ```begin ``` haftmann@24342 ` 1571` haftmann@24342 ` 1572` ```lemma sup_Inf1_distrib: ``` haftmann@26041 ` 1573` ``` assumes "finite A" ``` haftmann@26041 ` 1574` ``` and "A \ {}" ``` wenzelm@31916 ` 1575` ``` shows "sup x (\\<^bsub>fin\<^esub>A) = \\<^bsub>fin\<^esub>{sup x a|a. a \ A}" ``` haftmann@26041 ` 1576` ```proof - ``` haftmann@29509 ` 1577` ``` interpret ab_semigroup_idem_mult inf ``` haftmann@26041 ` 1578` ``` by (rule ab_semigroup_idem_mult_inf) ``` haftmann@26041 ` 1579` ``` from assms show ?thesis ``` haftmann@26041 ` 1580` ``` by (simp add: Inf_fin_def image_def ``` haftmann@26041 ` 1581` ``` hom_fold1_commute [where h="sup x", OF sup_inf_distrib1]) ``` berghofe@26792 ` 1582` ``` (rule arg_cong [where f="fold1 inf"], blast) ``` haftmann@26041 ` 1583` ```qed ``` nipkow@18423 ` 1584` haftmann@24342 ` 1585` ```lemma sup_Inf2_distrib: ``` haftmann@24342 ` 1586` ``` assumes A: "finite A" "A \ {}" and B: "finite B" "B \ {}" ``` wenzelm@31916 ` 1587` ``` shows "sup (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B) = \\<^bsub>fin\<^esub>{sup a b|a b. a \ A \ b \ B}" ``` haftmann@24342 ` 1588` ```using A proof (induct rule: finite_ne_induct) ``` nipkow@15500 ` 1589` ``` case singleton thus ?case ``` haftmann@24342 ` 1590` ``` by (simp add: sup_Inf1_distrib [OF B] fold1_singleton_def [OF Inf_fin_def]) ``` nipkow@15500 ` 1591` ```next ``` haftmann@29509 ` 1592` ``` interpret ab_semigroup_idem_mult inf ``` haftmann@26041 ` 1593` ``` by (rule ab_semigroup_idem_mult_inf) ``` nipkow@15500 ` 1594` ``` case (insert x A) ``` haftmann@25062 ` 1595` ``` have finB: "finite {sup x b |b. b \ B}" ``` haftmann@25062 ` 1596` ``` by(rule finite_surj[where f = "sup x", OF B(1)], auto) ``` haftmann@25062 ` 1597` ``` have finAB: "finite {sup a b |a b. a \ A \ b \ B}" ``` nipkow@15500 ` 1598` ``` proof - ``` haftmann@25062 ` 1599` ``` have "{sup a b |a b. a \ A \ b \ B} = (UN a:A. UN b:B. {sup a b})" ``` nipkow@15500 ` 1600` ``` by blast ``` berghofe@15517 ` 1601` ``` thus ?thesis by(simp add: insert(1) B(1)) ``` nipkow@15500 ` 1602` ``` qed ``` haftmann@25062 ` 1603` ``` have ne: "{sup a b |a b. a \ A \ b \ B} \ {}" using insert B by blast ``` wenzelm@31916 ` 1604` ``` have "sup (\\<^bsub>fin\<^esub>(insert x A)) (\\<^bsub>fin\<^esub>B) = sup (inf x (\\<^bsub>fin\<^esub>A)) (\\<^bsub>fin\<^esub>B)" ``` haftmann@26041 ` 1605` ``` using insert by (simp add: fold1_insert_idem_def [OF Inf_fin_def]) ``` wenzelm@31916 ` 1606` ``` also have "\ = inf (sup x (\\<^bsub>fin\<^esub>B)) (sup (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2) ``` wenzelm@31916 ` 1607` ``` also have "\ = inf (\\<^bsub>fin\<^esub>{sup x b|b. b \ B}) (\\<^bsub>fin\<^esub>{sup a b|a b. a \ A \ b \ B})" ``` nipkow@15500 ` 1608` ``` using insert by(simp add:sup_Inf1_distrib[OF B]) ``` wenzelm@31916 ` 1609` ``` also have "\ = \\<^bsub>fin\<^esub>({sup x b |b. b \ B} \ {sup a b |a b. a \ A \ b \ B})" ``` wenzelm@31916 ` 1610` ``` (is "_ = \\<^bsub>fin\<^esub>?M") ``` nipkow@15500 ` 1611` ``` using B insert ``` haftmann@26041 ` 1612` ``` by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne]) ``` haftmann@25062 ` 1613` ``` also have "?M = {sup a b |a b. a \ insert x A \ b \ B}" ``` nipkow@15500 ` 1614` ``` by blast ``` nipkow@15500 ` 1615` ``` finally show ?case . ``` nipkow@15500 ` 1616` ```qed ``` nipkow@15500 ` 1617` haftmann@24342 ` 1618` ```lemma inf_Sup1_distrib: ``` haftmann@26041 ` 1619` ``` assumes "finite A" and "A \ {}" ``` wenzelm@31916 ` 1620` ``` shows "inf x (\\<^bsub>fin\<^esub>A) = \\<^bsub>fin\<^esub>{inf x a|a. a \ A}" ``` haftmann@26041 ` 1621` ```proof - ``` haftmann@29509 ` 1622` ``` interpret ab_semigroup_idem_mult sup ``` haftmann@26041 ` 1623` ``` by (rule ab_semigroup_idem_mult_sup) ``` haftmann@26041 ` 1624` ``` from assms show ?thesis ``` haftmann@26041 ` 1625` ``` by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1]) ``` berghofe@26792 ` 1626` ``` (rule arg_cong [where f="fold1 sup"], blast) ``` haftmann@26041 ` 1627` ```qed ``` nipkow@18423 ` 1628` haftmann@24342 ` 1629` ```lemma inf_Sup2_distrib: ``` haftmann@24342 ` 1630` ``` assumes A: "finite A" "A \ {}" and B: "finite B" "B \ {}" ``` wenzelm@31916 ` 1631` ``` shows "inf (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B) = \\<^bsub>fin\<^esub>{inf a b|a b. a \ A \ b \ B}" ``` haftmann@24342 ` 1632` ```using A proof (induct rule: finite_ne_induct) ``` nipkow@18423 ` 1633` ``` case singleton thus ?case ``` haftmann@24342 ` 1634` ``` by(simp add: inf_Sup1_distrib [OF B] fold1_singleton_def [OF Sup_fin_def]) ``` nipkow@18423 ` 1635` ```next ``` nipkow@18423 ` 1636` ``` case (insert x A) ``` haftmann@25062 ` 1637` ``` have finB: "finite {inf x b |b. b \ B}" ``` haftmann@25062 ` 1638` ``` by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto) ``` haftmann@25062 ` 1639` ``` have finAB: "finite {inf a b |a b. a \ A \ b \ B}" ``` nipkow@18423 ` 1640` ``` proof - ``` haftmann@25062 ` 1641` ``` have "{inf a b |a b. a \ A \ b \ B} = (UN a:A. UN b:B. {inf a b})" ``` nipkow@18423 ` 1642` ``` by blast ``` nipkow@18423 ` 1643` ``` thus ?thesis by(simp add: insert(1) B(1)) ``` nipkow@18423 ` 1644` ``` qed ``` haftmann@25062 ` 1645` ``` have ne: "{inf a b |a b. a \ A \ b \ B} \ {}" using insert B by blast ``` haftmann@29509 ` 1646` ``` interpret ab_semigroup_idem_mult sup ``` haftmann@26041 ` 1647` ``` by (rule ab_semigroup_idem_mult_sup) ``` wenzelm@31916 ` 1648` ``` have "inf (\\<^bsub>fin\<^esub>(insert x A)) (\\<^bsub>fin\<^esub>B) = inf (sup x (\\<^bsub>fin\<^esub>A)) (\\<^bsub>fin\<^esub>B)" ``` haftmann@26041 ` 1649` ``` using insert by (simp add: fold1_insert_idem_def [OF Sup_fin_def]) ``` wenzelm@31916 ` 1650` ``` also have "\ = sup (inf x (\\<^bsub>fin\<^esub>B)) (inf (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2) ``` wenzelm@31916 ` 1651` ``` also have "\ = sup (\\<^bsub>fin\<^esub>{inf x b|b. b \ B}) (\\<^bsub>fin\<^esub>{inf a b|a b. a \ A \ b \ B})" ``` nipkow@18423 ` 1652` ``` using insert by(simp add:inf_Sup1_distrib[OF B]) ``` wenzelm@31916 ` 1653` ``` also have "\ = \\<^bsub>fin\<^esub>({inf x b |b. b \ B} \ {inf a b |a b. a \ A \ b \ B})" ``` wenzelm@31916 ` 1654` ``` (is "_ = \\<^bsub>fin\<^esub>?M") ``` nipkow@18423 ` 1655` ``` using B insert ``` haftmann@26041 ` 1656` ``` by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne]) ``` haftmann@25062 ` 1657` ``` also have "?M = {inf a b |a b. a \ insert x A \ b \ B}" ``` nipkow@18423 ` 1658` ``` by blast ``` nipkow@18423 ` 1659` ``` finally show ?case . ``` nipkow@18423 ` 1660` ```qed ``` nipkow@18423 ` 1661` haftmann@24342 ` 1662` ```end ``` haftmann@24342 ` 1663` haftmann@35719 ` 1664` ```context complete_lattice ``` haftmann@35719 ` 1665` ```begin ``` haftmann@35719 ` 1666` haftmann@35719 ` 1667` ```lemma Inf_fin_Inf: ``` haftmann@35719 ` 1668` ``` assumes "finite A" and "A \ {}" ``` haftmann@35719 ` 1669` ``` shows "\\<^bsub>fin\<^esub>A = Inf A" ``` haftmann@35719 ` 1670` ```proof - ``` haftmann@35719 ` 1671` ``` interpret ab_semigroup_idem_mult inf ``` haftmann@35719 ` 1672` ``` by (rule ab_semigroup_idem_mult_inf) ``` haftmann@35719 ` 1673` ``` from `A \ {}` obtain b B where "A = insert b B" by auto ``` haftmann@35719 ` 1674` ``` moreover with `finite A` have "finite B" by simp ``` haftmann@35719 ` 1675` ``` ultimately show ?thesis ``` haftmann@35719 ` 1676` ``` by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric]) ``` haftmann@35719 ` 1677` ``` (simp add: Inf_fold_inf) ``` haftmann@35719 ` 1678` ```qed ``` haftmann@35719 ` 1679` haftmann@35719 ` 1680` ```lemma Sup_fin_Sup: ``` haftmann@35719 ` 1681` ``` assumes "finite A" and "A \ {}" ``` haftmann@35719 ` 1682` ``` shows "\\<^bsub>fin\<^esub>A = Sup A" ``` haftmann@35719 ` 1683` ```proof - ``` haftmann@35719 ` 1684` ``` interpret ab_semigroup_idem_mult sup ``` haftmann@35719 ` 1685` ``` by (rule ab_semigroup_idem_mult_sup) ``` haftmann@35719 ` 1686` ``` from `A \ {}` obtain b B where "A = insert b B" by auto ``` haftmann@35719 ` 1687` ``` moreover with `finite A` have "finite B" by simp ``` haftmann@35719 ` 1688` ``` ultimately show ?thesis ``` haftmann@35719 ` 1689` ``` by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric]) ``` haftmann@35719 ` 1690` ``` (simp add: Sup_fold_sup) ``` haftmann@35719 ` 1691` ```qed ``` haftmann@35719 ` 1692` haftmann@35719 ` 1693` ```end ``` haftmann@35719 ` 1694` haftmann@22917 ` 1695` haftmann@22917 ` 1696` ```subsubsection {* Fold1 in linear orders with @{const min} and @{const max} *} ``` haftmann@22917 ` 1697` haftmann@22917 ` 1698` ```text{* ``` haftmann@22917 ` 1699` ``` As an application of @{text fold1} we define minimum ``` haftmann@22917 ` 1700` ``` and maximum in (not necessarily complete!) linear orders ``` haftmann@22917 ` 1701` ``` over (non-empty) sets by means of @{text fold1}. ``` haftmann@22917 ` 1702` ```*} ``` haftmann@22917 ` 1703` haftmann@24342 ` 1704` ```context linorder ``` haftmann@22917 ` 1705` ```begin ``` haftmann@22917 ` 1706` haftmann@26041 ` 1707` ```lemma ab_semigroup_idem_mult_min: ``` haftmann@26041 ` 1708` ``` "ab_semigroup_idem_mult min" ``` haftmann@28823 ` 1709` ``` proof qed (auto simp add: min_def) ``` haftmann@26041 ` 1710` haftmann@26041 ` 1711` ```lemma ab_semigroup_idem_mult_max: ``` haftmann@26041 ` 1712` ``` "ab_semigroup_idem_mult max" ``` haftmann@28823 ` 1713` ``` proof qed (auto simp add: max_def) ``` haftmann@26041 ` 1714` haftmann@26041 ` 1715` ```lemma max_lattice: ``` haftmann@35028 ` 1716` ``` "semilattice_inf (op \) (op >) max" ``` haftmann@32203 ` 1717` ``` by (fact min_max.dual_semilattice) ``` haftmann@26041 ` 1718` haftmann@26041 ` 1719` ```lemma dual_max: ``` haftmann@26041 ` 1720` ``` "ord.max (op \) = min" ``` haftmann@32642 ` 1721` ``` by (auto simp add: ord.max_def_raw min_def expand_fun_eq) ``` haftmann@26041 ` 1722` haftmann@26041 ` 1723` ```lemma dual_min: ``` haftmann@26041 ` 1724` ``` "ord.min (op \) = max" ``` haftmann@32642 ` 1725` ``` by (auto simp add: ord.min_def_raw max_def expand_fun_eq) ``` haftmann@26041 ` 1726` haftmann@26041 ` 1727` ```lemma strict_below_fold1_iff: ``` haftmann@26041 ` 1728` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1729` ``` shows "x < fold1 min A \ (\a\A. x < a)" ``` haftmann@26041 ` 1730` ```proof - ``` haftmann@29509 ` 1731` ``` interpret ab_semigroup_idem_mult min ``` haftmann@26041 ` 1732` ``` by (rule ab_semigroup_idem_mult_min) ``` haftmann@26041 ` 1733` ``` from assms show ?thesis ``` haftmann@26041 ` 1734` ``` by (induct rule: finite_ne_induct) ``` haftmann@26041 ` 1735` ``` (simp_all add: fold1_insert) ``` haftmann@26041 ` 1736` ```qed ``` haftmann@26041 ` 1737` haftmann@26041 ` 1738` ```lemma fold1_below_iff: ``` haftmann@26041 ` 1739` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1740` ``` shows "fold1 min A \ x \ (\a\A. a \ x)" ``` haftmann@26041 ` 1741` ```proof - ``` haftmann@29509 ` 1742` ``` interpret ab_semigroup_idem_mult min ``` haftmann@26041 ` 1743` ``` by (rule ab_semigroup_idem_mult_min) ``` haftmann@26041 ` 1744` ``` from assms show ?thesis ``` haftmann@26041 ` 1745` ``` by (induct rule: finite_ne_induct) ``` haftmann@26041 ` 1746` ``` (simp_all add: fold1_insert min_le_iff_disj) ``` haftmann@26041 ` 1747` ```qed ``` haftmann@26041 ` 1748` haftmann@26041 ` 1749` ```lemma fold1_strict_below_iff: ``` haftmann@26041 ` 1750` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1751` ``` shows "fold1 min A < x \ (\a\A. a < x)" ``` haftmann@26041 ` 1752` ```proof - ``` haftmann@29509 ` 1753` ``` interpret ab_semigroup_idem_mult min ``` haftmann@26041 ` 1754` ``` by (rule ab_semigroup_idem_mult_min) ``` haftmann@26041 ` 1755` ``` from assms show ?thesis ``` haftmann@26041 ` 1756` ``` by (induct rule: finite_ne_induct) ``` haftmann@26041 ` 1757` ``` (simp_all add: fold1_insert min_less_iff_disj) ``` haftmann@26041 ` 1758` ```qed ``` haftmann@26041 ` 1759` haftmann@26041 ` 1760` ```lemma fold1_antimono: ``` haftmann@26041 ` 1761` ``` assumes "A \ {}" and "A \ B" and "finite B" ``` haftmann@26041 ` 1762` ``` shows "fold1 min B \ fold1 min A" ``` haftmann@26041 ` 1763` ```proof cases ``` haftmann@26041 ` 1764` ``` assume "A = B" thus ?thesis by simp ``` haftmann@26041 ` 1765` ```next ``` haftmann@29509 ` 1766` ``` interpret ab_semigroup_idem_mult min ``` haftmann@26041 ` 1767` ``` by (rule ab_semigroup_idem_mult_min) ``` haftmann@26041 ` 1768` ``` assume "A \ B" ``` haftmann@26041 ` 1769` ``` have B: "B = A \ (B-A)" using `A \ B` by blast ``` haftmann@26041 ` 1770` ``` have "fold1 min B = fold1 min (A \ (B-A))" by(subst B)(rule refl) ``` haftmann@26041 ` 1771` ``` also have "\ = min (fold1 min A) (fold1 min (B-A))" ``` haftmann@26041 ` 1772` ``` proof - ``` haftmann@26041 ` 1773` ``` have "finite A" by(rule finite_subset[OF `A \ B` `finite B`]) ``` haftmann@26041 ` 1774` ``` moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *) ``` haftmann@26041 ` 1775` ``` moreover have "(B-A) \ {}" using prems by blast ``` haftmann@26041 ` 1776` ``` moreover have "A Int (B-A) = {}" using prems by blast ``` haftmann@26041 ` 1777` ``` ultimately show ?thesis using `A \ {}` by (rule_tac fold1_Un) ``` haftmann@26041 ` 1778` ``` qed ``` haftmann@26041 ` 1779` ``` also have "\ \ fold1 min A" by (simp add: min_le_iff_disj) ``` haftmann@26041 ` 1780` ``` finally show ?thesis . ``` haftmann@26041 ` 1781` ```qed ``` haftmann@26041 ` 1782` haftmann@22917 ` 1783` ```definition ``` haftmann@22917 ` 1784` ``` Min :: "'a set \ 'a" ``` haftmann@22917 ` 1785` ```where ``` haftmann@22917 ` 1786` ``` "Min = fold1 min" ``` haftmann@22917 ` 1787` haftmann@22917 ` 1788` ```definition ``` haftmann@22917 ` 1789` ``` Max :: "'a set \ 'a" ``` haftmann@22917 ` 1790` ```where ``` haftmann@22917 ` 1791` ``` "Max = fold1 max" ``` haftmann@22917 ` 1792` haftmann@22917 ` 1793` ```lemmas Min_singleton [simp] = fold1_singleton_def [OF Min_def] ``` haftmann@22917 ` 1794` ```lemmas Max_singleton [simp] = fold1_singleton_def [OF Max_def] ``` haftmann@26041 ` 1795` haftmann@26041 ` 1796` ```lemma Min_insert [simp]: ``` haftmann@26041 ` 1797` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1798` ``` shows "Min (insert x A) = min x (Min A)" ``` haftmann@26041 ` 1799` ```proof - ``` haftmann@29509 ` 1800` ``` interpret ab_semigroup_idem_mult min ``` haftmann@26041 ` 1801` ``` by (rule ab_semigroup_idem_mult_min) ``` haftmann@26041 ` 1802` ``` from assms show ?thesis by (rule fold1_insert_idem_def [OF Min_def]) ``` haftmann@26041 ` 1803` ```qed ``` haftmann@26041 ` 1804` haftmann@26041 ` 1805` ```lemma Max_insert [simp]: ``` haftmann@26041 ` 1806` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1807` ``` shows "Max (insert x A) = max x (Max A)" ``` haftmann@26041 ` 1808` ```proof - ``` haftmann@29509 ` 1809` ``` interpret ab_semigroup_idem_mult max ``` haftmann@26041 ` 1810` ``` by (rule ab_semigroup_idem_mult_max) ``` haftmann@26041 ` 1811` ``` from assms show ?thesis by (rule fold1_insert_idem_def [OF Max_def]) ``` haftmann@26041 ` 1812` ```qed ``` nipkow@15392 ` 1813` paulson@24427 ` 1814` ```lemma Min_in [simp]: ``` haftmann@26041 ` 1815` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1816` ``` shows "Min A \ A" ``` haftmann@26041 ` 1817` ```proof - ``` haftmann@29509 ` 1818` ``` interpret ab_semigroup_idem_mult min ``` haftmann@26041 ` 1819` ``` by (rule ab_semigroup_idem_mult_min) ``` haftmann@26041 ` 1820` ``` from assms fold1_in show ?thesis by (fastsimp simp: Min_def min_def) ``` haftmann@26041 ` 1821` ```qed ``` nipkow@15392 ` 1822` paulson@24427 ` 1823` ```lemma Max_in [simp]: ``` haftmann@26041 ` 1824` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1825` ``` shows "Max A \ A" ``` haftmann@26041 ` 1826` ```proof - ``` haftmann@29509 ` 1827` ``` interpret ab_semigroup_idem_mult max ``` haftmann@26041 ` 1828` ``` by (rule ab_semigroup_idem_mult_max) ``` haftmann@26041 ` 1829` ``` from assms fold1_in [of A] show ?thesis by (fastsimp simp: Max_def max_def) ``` haftmann@26041 ` 1830` ```qed ``` haftmann@26041 ` 1831` haftmann@26041 ` 1832` ```lemma Min_Un: ``` haftmann@26041 ` 1833` ``` assumes "finite A" and "A \ {}" and "finite B" and "B \ {}" ``` haftmann@26041 ` 1834` ``` shows "Min (A \ B) = min (Min A) (Min B)" ``` haftmann@26041 ` 1835` ```proof - ``` haftmann@29509 ` 1836` ``` interpret ab_semigroup_idem_mult min ``` haftmann@26041 ` 1837` ``` by (rule ab_semigroup_idem_mult_min) ``` haftmann@26041 ` 1838` ``` from assms show ?thesis ``` haftmann@26041 ` 1839` ``` by (simp add: Min_def fold1_Un2) ``` haftmann@26041 ` 1840` ```qed ``` haftmann@26041 ` 1841` haftmann@26041 ` 1842` ```lemma Max_Un: ``` haftmann@26041 ` 1843` ``` assumes "finite A" and "A \ {}" and "finite B" and "B \ {}" ``` haftmann@26041 ` 1844` ``` shows "Max (A \ B) = max (Max A) (Max B)" ``` haftmann@26041 ` 1845` ```proof - ``` haftmann@29509 ` 1846` ``` interpret ab_semigroup_idem_mult max ``` haftmann@26041 ` 1847` ``` by (rule ab_semigroup_idem_mult_max) ``` haftmann@26041 ` 1848` ``` from assms show ?thesis ``` haftmann@26041 ` 1849` ``` by (simp add: Max_def fold1_Un2) ``` haftmann@26041 ` 1850` ```qed ``` haftmann@26041 ` 1851` haftmann@26041 ` 1852` ```lemma hom_Min_commute: ``` haftmann@26041 ` 1853` ``` assumes "\x y. h (min x y) = min (h x) (h y)" ``` haftmann@26041 ` 1854` ``` and "finite N" and "N \ {}" ``` haftmann@26041 ` 1855` ``` shows "h (Min N) = Min (h ` N)" ``` haftmann@26041 ` 1856` ```proof - ``` haftmann@29509 ` 1857` ``` interpret ab_semigroup_idem_mult min ``` haftmann@26041 ` 1858` ``` by (rule ab_semigroup_idem_mult_min) ``` haftmann@26041 ` 1859` ``` from assms show ?thesis ``` haftmann@26041 ` 1860` ``` by (simp add: Min_def hom_fold1_commute) ``` haftmann@26041 ` 1861` ```qed ``` haftmann@26041 ` 1862` haftmann@26041 ` 1863` ```lemma hom_Max_commute: ``` haftmann@26041 ` 1864` ``` assumes "\x y. h (max x y) = max (h x) (h y)" ``` haftmann@26041 ` 1865` ``` and "finite N" and "N \ {}" ``` haftmann@26041 ` 1866` ``` shows "h (Max N) = Max (h ` N)" ``` haftmann@26041 ` 1867` ```proof - ``` haftmann@29509 ` 1868` ``` interpret ab_semigroup_idem_mult max ``` haftmann@26041 ` 1869` ``` by (rule ab_semigroup_idem_mult_max) ``` haftmann@26041 ` 1870` ``` from assms show ?thesis ``` haftmann@26041 ` 1871` ``` by (simp add: Max_def hom_fold1_commute [of h]) ``` haftmann@26041 ` 1872` ```qed ``` haftmann@26041 ` 1873` haftmann@26041 ` 1874` ```lemma Min_le [simp]: ``` haftmann@26757 ` 1875` ``` assumes "finite A" and "x \ A" ``` haftmann@26041 ` 1876` ``` shows "Min A \ x" ``` haftmann@32203 ` 1877` ``` using assms by (simp add: Min_def min_max.fold1_belowI) ``` haftmann@26041 ` 1878` haftmann@26041 ` 1879` ```lemma Max_ge [simp]: ``` haftmann@26757 ` 1880` ``` assumes "finite A" and "x \ A" ``` haftmann@26041 ` 1881` ``` shows "x \ Max A" ``` haftmann@26041 ` 1882` ```proof - ``` haftmann@35028 ` 1883` ``` interpret semilattice_inf "op \" "op >" max ``` haftmann@26041 ` 1884` ``` by (rule max_lattice) ``` haftmann@26041 ` 1885` ``` from assms show ?thesis by (simp add: Max_def fold1_belowI) ``` haftmann@26041 ` 1886` ```qed ``` haftmann@26041 ` 1887` haftmann@26041 ` 1888` ```lemma Min_ge_iff [simp, noatp]: ``` haftmann@26041 ` 1889` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1890` ``` shows "x \ Min A \ (\a\A. x \ a)" ``` haftmann@32203 ` 1891` ``` using assms by (simp add: Min_def min_max.below_fold1_iff) ``` haftmann@26041 ` 1892` haftmann@26041 ` 1893` ```lemma Max_le_iff [simp, noatp]: ``` haftmann@26041 ` 1894` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1895` ``` shows "Max A \ x \ (\a\A. a \ x)" ``` haftmann@26041 ` 1896` ```proof - ``` haftmann@35028 ` 1897` ``` interpret semilattice_inf "op \" "op >" max ``` haftmann@26041 ` 1898` ``` by (rule max_lattice) ``` haftmann@26041 ` 1899` ``` from assms show ?thesis by (simp add: Max_def below_fold1_iff) ``` haftmann@26041 ` 1900` ```qed ``` haftmann@26041 ` 1901` haftmann@26041 ` 1902` ```lemma Min_gr_iff [simp, noatp]: ``` haftmann@26041 ` 1903` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1904` ``` shows "x < Min A \ (\a\A. x < a)" ``` haftmann@32203 ` 1905` ``` using assms by (simp add: Min_def strict_below_fold1_iff) ``` haftmann@26041 ` 1906` haftmann@26041 ` 1907` ```lemma Max_less_iff [simp, noatp]: ``` haftmann@26041 ` 1908` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1909` ``` shows "Max A < x \ (\a\A. a < x)" ``` haftmann@26041 ` 1910` ```proof - ``` haftmann@32203 ` 1911` ``` interpret dual: linorder "op \" "op >" ``` haftmann@26041 ` 1912` ``` by (rule dual_linorder) ``` haftmann@26041 ` 1913` ``` from assms show ?thesis ``` haftmann@32203 ` 1914` ``` by (simp add: Max_def dual.strict_below_fold1_iff [folded dual.dual_max]) ``` haftmann@26041 ` 1915` ```qed ``` nipkow@18493 ` 1916` paulson@24286 ` 1917` ```lemma Min_le_iff [noatp]: ``` haftmann@26041 ` 1918` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1919` ``` shows "Min A \ x \ (\a\A. a \ x)" ``` haftmann@32203 ` 1920` ``` using assms by (simp add: Min_def fold1_below_iff) ``` nipkow@15497 ` 1921` paulson@24286 ` 1922` ```lemma Max_ge_iff [noatp]: ``` haftmann@26041 ` 1923` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1924` ``` shows "x \ Max A \ (\a\A. x \ a)" ``` haftmann@26041 ` 1925` ```proof - ``` haftmann@32203 ` 1926` ``` interpret dual: linorder "op \" "op >" ``` haftmann@26041 ` 1927` ``` by (rule dual_linorder) ``` haftmann@26041 ` 1928` ``` from assms show ?thesis ``` haftmann@32203 ` 1929` ``` by (simp add: Max_def dual.fold1_below_iff [folded dual.dual_max]) ``` haftmann@26041 ` 1930` ```qed ``` haftmann@22917 ` 1931` paulson@24286 ` 1932` ```lemma Min_less_iff [noatp]: ``` haftmann@26041 ` 1933` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1934` ``` shows "Min A < x \ (\a\A. a < x)" ``` haftmann@32203 ` 1935` ``` using assms by (simp add: Min_def fold1_strict_below_iff) ``` haftmann@22917 ` 1936` paulson@24286 ` 1937` ```lemma Max_gr_iff [noatp]: ``` haftmann@26041 ` 1938` ``` assumes "finite A" and "A \ {}" ``` haftmann@26041 ` 1939` ``` shows "x < Max A \ (\a\A. x < a)" ``` haftmann@26041 ` 1940` ```proof - ``` haftmann@32203 ` 1941` ``` interpret dual: linorder "op \" "op >" ``` haftmann@26041 ` 1942` ``` by (rule dual_linorder) ``` haftmann@26041 ` 1943` ``` from assms show ?thesis ``` haftmann@32203 ` 1944` ``` by (simp add: Max_def dual.fold1_strict_below_iff [folded dual.dual_max]) ``` haftmann@26041 ` 1945` ```qed ``` haftmann@26041 ` 1946` haftmann@30325 ` 1947` ```lemma Min_eqI: ``` haftmann@30325 ` 1948` ``` assumes "finite A" ``` haftmann@30325 ` 1949` ``` assumes "\y. y \ A \ y \ x" ``` haftmann@30325 ` 1950` ``` and "x \ A" ``` haftmann@30325 ` 1951` ``` shows "Min A = x" ``` haftmann@30325 ` 1952` ```proof (rule antisym) ``` haftmann@30325 ` 1953` ``` from `x \ A` have "A \ {}" by auto ``` haftmann@30325 ` 1954` ``` with assms show "Min A \ x" by simp ``` haftmann@30325 ` 1955` ```next ``` haftmann@30325 ` 1956` ``` from assms show "x \ Min A" by simp ``` haftmann@30325 ` 1957` ```qed ``` haftmann@30325 ` 1958` haftmann@30325 ` 1959` ```lemma Max_eqI: ``` haftmann@30325 ` 1960` ``` assumes "finite A" ``` haftmann@30325 ` 1961` ``` assumes "\y. y \ A \ y \ x" ``` haftmann@30325 ` 1962` ``` and "x \ A" ``` haftmann@30325 ` 1963` ``` shows "Max A = x" ``` haftmann@30325 ` 1964` ```proof (rule antisym) ``` haftmann@30325 ` 1965` ``` from `x \ A` have "A \ {}" by auto ``` haftmann@30325 ` 1966` ``` with assms show "Max A \ x" by simp ``` haftmann@30325 ` 1967` ```next ``` haftmann@30325 ` 1968` ``` from assms show "x \ Max A" by simp ``` haftmann@30325 ` 1969` ```qed ``` haftmann@30325 ` 1970` haftmann@26041 ` 1971` ```lemma Min_antimono: ``` haftmann@26041 ` 1972` ``` assumes "M \ N" and "M \ {}" and "finite N" ``` haftmann@26041 ` 1973` ``` shows "Min N \ Min M" ``` haftmann@32203 ` 1974` ``` using assms by (simp add: Min_def fold1_antimono) ``` haftmann@26041 ` 1975` haftmann@26041 ` 1976` ```lemma Max_mono: ``` haftmann@26041 ` 1977` ``` assumes "M \ N" and "M \ {}" and "finite N" ``` haftmann@26041 ` 1978` ``` shows "Max M \ Max N" ``` haftmann@26041 ` 1979` ```proof - ``` haftmann@32203 ` 1980` ``` interpret dual: linorder "op \" "op >" ``` haftmann@26041 ` 1981` ``` by (rule dual_linorder) ``` haftmann@26041 ` 1982` ``` from assms show ?thesis ``` haftmann@32203 ` 1983` ``` by (simp add: Max_def dual.fold1_antimono [folded dual.dual_max]) ``` haftmann@26041 ` 1984` ```qed ``` haftmann@22917 ` 1985` nipkow@32006 ` 1986` ```lemma finite_linorder_max_induct[consumes 1, case_names empty insert]: ``` krauss@26748 ` 1987` ``` "finite A \ P {} \ ``` nipkow@33434 ` 1988` ``` (!!b A. finite A \ ALL a:A. a < b \ P A \ P(insert b A)) ``` krauss@26748 ` 1989` ``` \ P A" ``` nipkow@32006 ` 1990` ```proof (induct rule: finite_psubset_induct) ``` krauss@26748 ` 1991` ``` fix A :: "'a set" ``` nipkow@32006 ` 1992` ``` assume IH: "!! B. finite B \ B < A \ P {} \ ``` nipkow@33434 ` 1993` ``` (!!b A. finite A \ (\a\A. a P A \ P (insert b A)) ``` krauss@26748 ` 1994` ``` \ P B" ``` krauss@26748 ` 1995` ``` and "finite A" and "P {}" ``` nipkow@33434 ` 1996` ``` and step: "!!b A. \finite A; \a\A. a < b; P A\ \ P (insert b A)" ``` krauss@26748 ` 1997` ``` show "P A" ``` haftmann@26757 ` 1998` ``` proof (cases "A = {}") ``` krauss@26748 ` 1999` ``` assume "A = {}" thus "P A" using `P {}` by simp ``` krauss@26748 ` 2000` ``` next ``` krauss@26748 ` 2001` ``` let ?B = "A - {Max A}" let ?A = "insert (Max A) ?B" ``` krauss@26748 ` 2002` ``` assume "A \ {}" ``` krauss@26748 ` 2003` ``` with `finite A` have "Max A : A" by auto ``` krauss@26748 ` 2004` ``` hence A: "?A = A" using insert_Diff_single insert_absorb by auto ``` krauss@26748 ` 2005` ``` moreover have "finite ?B" using `finite A` by simp ``` nipkow@33434 ` 2006` ``` ultimately have "P ?B" using `P {}` step IH[of ?B] by blast ``` nipkow@32006 ` 2007` ``` moreover have "\a\?B. a < Max A" using Max_ge [OF `finite A`] by fastsimp ``` nipkow@32006 ` 2008` ``` ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastsimp ``` krauss@26748 ` 2009` ``` qed ``` krauss@26748 ` 2010` ```qed ``` krauss@26748 ` 2011` nipkow@32006 ` 2012` ```lemma finite_linorder_min_induct[consumes 1, case_names empty insert]: ``` nipkow@33434 ` 2013` ``` "\finite A; P {}; \b A. \finite A; \a\A. b < a; P A\ \ P (insert b A)\ \ P A" ``` nipkow@32006 ` 2014` ```by(rule linorder.finite_linorder_max_induct[OF dual_linorder]) ``` nipkow@32006 ` 2015` haftmann@22917 ` 2016` ```end ``` haftmann@22917 ` 2017` haftmann@35028 ` 2018` ```context linordered_ab_semigroup_add ``` haftmann@22917 ` 2019` ```begin ``` haftmann@22917 ` 2020` haftmann@22917 ` 2021` ```lemma add_Min_commute: ``` haftmann@22917 ` 2022` ``` fixes k ``` haftmann@25062 ` 2023` ``` assumes "finite N" and "N \ {}" ``` haftmann@25062 ` 2024` ``` shows "k + Min N = Min {k + m | m. m \ N}" ``` haftmann@25062 ` 2025` ```proof - ``` haftmann@25062 ` 2026` ``` have "\x y. k + min x y = min (k + x) (k + y)" ``` haftmann@25062 ` 2027` ``` by (simp add: min_def not_le) ``` haftmann@25062 ` 2028` ``` (blast intro: antisym less_imp_le add_left_mono) ``` haftmann@25062 ` 2029` ``` with assms show ?thesis ``` haftmann@25062 ` 2030` ``` using hom_Min_commute [of "plus k" N] ``` haftmann@25062 ` 2031` ``` by simp (blast intro: arg_cong [where f = Min]) ``` haftmann@25062 ` 2032` ```qed ``` haftmann@22917 ` 2033` haftmann@22917 ` 2034` ```lemma add_Max_commute: ``` haftmann@22917 ` 2035` ``` fixes k ``` haftmann@25062 ` 2036` ``` assumes "finite N" and "N \ {}" ``` haftmann@25062 ` 2037` ``` shows "k + Max N = Max {k + m | m. m \ N}" ``` haftmann@25062 ` 2038` ```proof - ``` haftmann@25062 ` 2039` ``` have "\x y. k + max x y = max (k + x) (k + y)" ``` haftmann@25062 ` 2040` ``` by (simp add: max_def not_le) ``` haftmann@25062 ` 2041` ``` (blast intro: antisym less_imp_le add_left_mono) ``` haftmann@25062 ` 2042` ``` with assms show ?thesis ``` haftmann@25062 ` 2043` ``` using hom_Max_commute [of "plus k" N] ``` haftmann@25062 ` 2044` ``` by simp (blast intro: arg_cong [where f = Max]) ``` haftmann@25062 ` 2045` ```qed ``` haftmann@22917 ` 2046` haftmann@22917 ` 2047` ```end ``` haftmann@22917 ` 2048` haftmann@35034 ` 2049` ```context linordered_ab_group_add ``` haftmann@35034 ` 2050` ```begin ``` haftmann@35034 ` 2051` haftmann@35034 ` 2052` ```lemma minus_Max_eq_Min [simp]: ``` haftmann@35034 ` 2053` ``` "finite S \ S \ {} \ - (Max S) = Min (uminus ` S)" ``` haftmann@35034 ` 2054` ``` by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min) ``` haftmann@35034 ` 2055` haftmann@35034 ` 2056` ```lemma minus_Min_eq_Max [simp]: ``` haftmann@35034 ` 2057` ``` "finite S \ S \ {} \ - (Min S) = Max (uminus ` S)" ``` haftmann@35034 ` 2058` ``` by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max) ``` haftmann@35034 ` 2059` haftmann@35034 ` 2060` ```end ``` haftmann@35034 ` 2061` haftmann@25571 ` 2062` ```end ```