(* Title: HOL/Big_Operators.thy Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel with contributions by Jeremy Avigad *) header {* Big operators and finite (non-empty) sets *} theory Big_Operators imports Plain begin subsection {* Generic monoid operation over a set *} no_notation times (infixl "*" 70) no_notation Groups.one ("1") locale comm_monoid_big = comm_monoid + fixes F :: "('b \ 'a) \ 'b set \ 'a" assumes F_eq: "F g A = (if finite A then fold_image (op *) g 1 A else 1)" sublocale comm_monoid_big < folding_image proof qed (simp add: F_eq) context comm_monoid_big begin lemma infinite [simp]: "\ finite A \ F g A = 1" by (simp add: F_eq) lemma F_cong: assumes "A = B" "\x. x \ B \ h x = g x" shows "F h A = F g B" proof cases assume "finite A" with assms show ?thesis unfolding `A = B` by (simp cong: cong) next assume "\ finite A" then show ?thesis unfolding `A = B` by simp qed lemma If_cases: fixes P :: "'b \ bool" and g h :: "'b \ 'a" assumes fA: "finite A" shows "F (\x. if P x then h x else g x) A = F h (A \ {x. P x}) * F g (A \ - {x. P x})" proof- have a: "A = A \ {x. P x} \ A \ -{x. P x}" "(A \ {x. P x}) \ (A \ -{x. P x}) = {}" by blast+ from fA have f: "finite (A \ {x. P x})" "finite (A \ -{x. P x})" by auto let ?g = "\x. if P x then h x else g x" from union_disjoint[OF f a(2), of ?g] a(1) show ?thesis by (subst (1 2) F_cong) simp_all qed end text {* for ad-hoc proofs for @{const fold_image} *} lemma (in comm_monoid_add) comm_monoid_mult: "class.comm_monoid_mult (op +) 0" proof qed (auto intro: add_assoc add_commute) notation times (infixl "*" 70) notation Groups.one ("1") subsection {* Generalized summation over a set *} definition (in comm_monoid_add) setsum :: "('b \ 'a) => 'b set => 'a" where "setsum f A = (if finite A then fold_image (op +) f 0 A else 0)" sublocale comm_monoid_add < setsum!: comm_monoid_big "op +" 0 setsum proof qed (fact setsum_def) abbreviation Setsum ("\_" [1000] 999) where "\A == setsum (%x. x) A" text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is written @{text"\x\A. e"}. *} syntax "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10) syntax (xsymbols) "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\_\_. _)" [0, 51, 10] 10) syntax (HTML output) "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\_\_. _)" [0, 51, 10] 10) translations -- {* Beware of argument permutation! *} "SUM i:A. b" == "CONST setsum (%i. b) A" "\i\A. b" == "CONST setsum (%i. b) A" text{* Instead of @{term"\x\{x. P}. e"} we introduce the shorter @{text"\x|P. e"}. *} syntax "_qsetsum" :: "pttrn \ bool \ 'a \ 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10) syntax (xsymbols) "_qsetsum" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) syntax (HTML output) "_qsetsum" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) translations "SUM x|P. t" => "CONST setsum (%x. t) {x. P}" "\x|P. t" => "CONST setsum (%x. t) {x. P}" print_translation {* let fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] = if x <> y then raise Match else let val x' = Syntax_Trans.mark_bound x; val t' = subst_bound (x', t); val P' = subst_bound (x', P); in Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound x $ P' $ t' end | setsum_tr' _ = raise Match; in [(@{const_syntax setsum}, setsum_tr')] end *} lemma setsum_empty: "setsum f {} = 0" by (fact setsum.empty) lemma setsum_insert: "finite F ==> a \ F ==> setsum f (insert a F) = f a + setsum f F" by (fact setsum.insert) lemma setsum_infinite: "~ finite A ==> setsum f A = 0" by (fact setsum.infinite) lemma (in comm_monoid_add) setsum_reindex: assumes "inj_on f B" shows "setsum h (f ` B) = setsum (h \ f) B" proof - interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult) from assms show ?thesis by (auto simp add: setsum_def fold_image_reindex dest!:finite_imageD) qed lemma (in comm_monoid_add) setsum_reindex_id: "inj_on f B ==> setsum f B = setsum id (f ` B)" by (simp add: setsum_reindex) lemma (in comm_monoid_add) setsum_reindex_nonzero: assumes fS: "finite S" and nz: "\ x y. x \ S \ y \ S \ x \ y \ f x = f y \ h (f x) = 0" shows "setsum h (f ` S) = setsum (h o f) S" using nz proof(induct rule: finite_induct[OF fS]) case 1 thus ?case by simp next case (2 x F) {assume fxF: "f x \ f ` F" hence "\y \ F . f y = f x" by auto then obtain y where y: "y \ F" "f x = f y" by auto from "2.hyps" y have xy: "x \ y" by auto from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto also have "\ = setsum (h o f) (insert x F)" unfolding setsum.insert[OF `finite F` `x\F`] using h0 apply simp apply (rule "2.hyps"(3)) apply (rule_tac y="y" in "2.prems") apply simp_all done finally have ?case .} moreover {assume fxF: "f x \ f ` F" have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" using fxF "2.hyps" by simp also have "\ = setsum (h o f) (insert x F)" unfolding setsum.insert[OF `finite F` `x\F`] apply simp apply (rule cong [OF refl [of "op + (h (f x))"]]) apply (rule "2.hyps"(3)) apply (rule_tac y="y" in "2.prems") apply simp_all done finally have ?case .} ultimately show ?case by blast qed lemma (in comm_monoid_add) setsum_cong: "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" by (cases "finite A") (auto intro: setsum.cong) lemma (in comm_monoid_add) strong_setsum_cong [cong]: "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setsum (%x. f x) A = setsum (%x. g x) B" by (rule setsum_cong) (simp_all add: simp_implies_def) lemma (in comm_monoid_add) setsum_cong2: "\\x. x \ A \ f x = g x\ \ setsum f A = setsum g A" by (auto intro: setsum_cong) lemma (in comm_monoid_add) setsum_reindex_cong: "[|inj_on f A; B = f ` A; !!a. a:A \ g a = h (f a)|] ==> setsum h B = setsum g A" by (simp add: setsum_reindex) lemma (in comm_monoid_add) setsum_0[simp]: "setsum (%i. 0) A = 0" by (cases "finite A") (erule finite_induct, auto) lemma (in comm_monoid_add) setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0" by (simp add:setsum_cong) lemma (in comm_monoid_add) setsum_Un_Int: "finite A ==> finite B ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} by (fact setsum.union_inter) lemma (in comm_monoid_add) setsum_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" by (fact setsum.union_disjoint) lemma setsum_mono_zero_left: assumes fT: "finite T" and ST: "S \ T" and z: "\i \ T - S. f i = 0" shows "setsum f S = setsum f T" proof- have eq: "T = S \ (T - S)" using ST by blast have d: "S \ (T - S) = {}" using ST by blast from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) show ?thesis by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) qed lemma setsum_mono_zero_right: "finite T \ S \ T \ \i \ T - S. f i = 0 \ setsum f T = setsum f S" by(blast intro!: setsum_mono_zero_left[symmetric]) lemma setsum_mono_zero_cong_left: assumes fT: "finite T" and ST: "S \ T" and z: "\i \ T - S. g i = 0" and fg: "\x. x \ S \ f x = g x" shows "setsum f S = setsum g T" proof- have eq: "T = S \ (T - S)" using ST by blast have d: "S \ (T - S) = {}" using ST by blast from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) show ?thesis using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) qed lemma setsum_mono_zero_cong_right: assumes fT: "finite T" and ST: "S \ T" and z: "\i \ T - S. f i = 0" and fg: "\x. x \ S \ f x = g x" shows "setsum f T = setsum g S" using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto lemma setsum_delta: assumes fS: "finite S" shows "setsum (\k. if k=a then b k else 0) S = (if a \ S then b a else 0)" proof- let ?f = "(\k. if k=a then b k else 0)" {assume a: "a \ S" hence "\ k\ S. ?f k = 0" by simp hence ?thesis using a by simp} moreover {assume a: "a \ S" let ?A = "S - {a}" let ?B = "{a}" have eq: "S = ?A \ ?B" using a by blast have dj: "?A \ ?B = {}" by simp from fS have fAB: "finite ?A" "finite ?B" by auto have "setsum ?f S = setsum ?f ?A + setsum ?f ?B" using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] by simp then have ?thesis using a by simp} ultimately show ?thesis by blast qed lemma setsum_delta': assumes fS: "finite S" shows "setsum (\k. if a = k then b k else 0) S = (if a\ S then b a else 0)" using setsum_delta[OF fS, of a b, symmetric] by (auto intro: setsum_cong) lemma setsum_restrict_set: assumes fA: "finite A" shows "setsum f (A \ B) = setsum (\x. if x \ B then f x else 0) A" proof- from fA have fab: "finite (A \ B)" by auto have aba: "A \ B \ A" by blast let ?g = "\x. if x \ A\B then f x else 0" from setsum_mono_zero_left[OF fA aba, of ?g] show ?thesis by simp qed lemma setsum_cases: assumes fA: "finite A" shows "setsum (\x. if P x then f x else g x) A = setsum f (A \ {x. P x}) + setsum g (A \ - {x. P x})" using setsum.If_cases[OF fA] . (*But we can't get rid of finite I. If infinite, although the rhs is 0, the lhs need not be, since UNION I A could still be finite.*) lemma (in comm_monoid_add) setsum_UN_disjoint: assumes "finite I" and "ALL i:I. finite (A i)" and "ALL i:I. ALL j:I. i \ j --> A i Int A j = {}" shows "setsum f (UNION I A) = (\i\I. setsum f (A i))" proof - interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult) from assms show ?thesis by (simp add: setsum_def fold_image_UN_disjoint) qed text{*No need to assume that @{term C} is finite. If infinite, the rhs is directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*} lemma setsum_Union_disjoint: assumes "\A\C. finite A" "\A\C. \B\C. A \ B \ A Int B = {}" shows "setsum f (Union C) = setsum (setsum f) C" proof cases assume "finite C" from setsum_UN_disjoint[OF this assms] show ?thesis by (simp add: SUP_def) qed (force dest: finite_UnionD simp add: setsum_def) (*But we can't get rid of finite A. If infinite, although the lhs is 0, the rhs need not be, since SIGMA A B could still be finite.*) lemma (in comm_monoid_add) setsum_Sigma: assumes "finite A" and "ALL x:A. finite (B x)" shows "(\x\A. (\y\B x. f x y)) = (\(x,y)\(SIGMA x:A. B x). f x y)" proof - interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult) from assms show ?thesis by (simp add: setsum_def fold_image_Sigma split_def) qed text{*Here we can eliminate the finiteness assumptions, by cases.*} lemma setsum_cartesian_product: "(\x\A. (\y\B. f x y)) = (\(x,y) \ A <*> B. f x y)" apply (cases "finite A") apply (cases "finite B") apply (simp add: setsum_Sigma) apply (cases "A={}", simp) apply (simp) apply (auto simp add: setsum_def dest: finite_cartesian_productD1 finite_cartesian_productD2) done lemma (in comm_monoid_add) setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)" by (cases "finite A") (simp_all add: setsum.distrib) subsubsection {* Properties in more restricted classes of structures *} lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" apply (case_tac "finite A") prefer 2 apply (simp add: setsum_def) apply (erule rev_mp) apply (erule finite_induct, auto) done lemma setsum_eq_0_iff [simp]: "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" by (induct set: finite) auto lemma setsum_eq_Suc0_iff: "finite A \ (setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\b \ f b = 0))" apply(erule finite_induct) apply (auto simp add:add_is_1) done lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]] lemma setsum_Un_nat: "finite A ==> finite B ==> (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" -- {* For the natural numbers, we have subtraction. *} by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps) lemma setsum_Un: "finite A ==> finite B ==> (setsum f (A Un B) :: 'a :: ab_group_add) = setsum f A + setsum f B - setsum f (A Int B)" by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps) lemma (in comm_monoid_add) setsum_eq_general_reverses: assumes fS: "finite S" and fT: "finite T" and kh: "\y. y \ T \ k y \ S \ h (k y) = y" and hk: "\x. x \ S \ h x \ T \ k (h x) = x \ g (h x) = f x" shows "setsum f S = setsum g T" proof - interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult) show ?thesis apply (simp add: setsum_def fS fT) apply (rule fold_image_eq_general_inverses) apply (rule fS) apply (erule kh) apply (erule hk) done qed lemma (in comm_monoid_add) setsum_Un_zero: assumes fS: "finite S" and fT: "finite T" and I0: "\x \ S\T. f x = 0" shows "setsum f (S \ T) = setsum f S + setsum f T" proof - interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult) show ?thesis using fS fT apply (simp add: setsum_def) apply (rule fold_image_Un_one) using I0 by auto qed lemma setsum_UNION_zero: assumes fS: "finite S" and fSS: "\T \ S. finite T" and f0: "\T1 T2 x. T1\S \ T2\S \ T1 \ T2 \ x \ T1 \ x \ T2 \ f x = 0" shows "setsum f (\S) = setsum (\T. setsum f T) S" using fSS f0 proof(induct rule: finite_induct[OF fS]) case 1 thus ?case by simp next case (2 T F) then have fTF: "finite T" "\T\F. finite T" "finite F" and TF: "T \ F" and H: "setsum f (\ F) = setsum (setsum f) F" by auto from fTF have fUF: "finite (\F)" by auto from "2.prems" TF fTF show ?case by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f]) qed lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) = (if a:A then setsum f A - f a else setsum f A)" apply (case_tac "finite A") prefer 2 apply (simp add: setsum_def) apply (erule finite_induct) apply (auto simp add: insert_Diff_if) apply (drule_tac a = a in mk_disjoint_insert, auto) done lemma setsum_diff1: "finite A \ (setsum f (A - {a}) :: ('a::ab_group_add)) = (if a:A then setsum f A - f a else setsum f A)" by (erule finite_induct) (auto simp add: insert_Diff_if) lemma setsum_diff1'[rule_format]: "finite A \ a \ A \ (\ x \ A. f x) = f a + (\ x \ (A - {a}). f x)" apply (erule finite_induct[where F=A and P="% A. (a \ A \ (\ x \ A. f x) = f a + (\ x \ (A - {a}). f x))"]) apply (auto simp add: insert_Diff_if add_ac) done lemma setsum_diff1_ring: assumes "finite A" "a \ A" shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)" unfolding setsum_diff1'[OF assms] by auto (* By Jeremy Siek: *) lemma setsum_diff_nat: assumes "finite B" and "B \ A" shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)" using assms proof induct show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp next fix F x assume finF: "finite F" and xnotinF: "x \ F" and xFinA: "insert x F \ A" and IH: "F \ A \ setsum f (A - F) = setsum f A - setsum f F" from xnotinF xFinA have xinAF: "x \ (A - F)" by simp from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x" by (simp add: setsum_diff1_nat) from xFinA have "F \ A" by simp with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x" by simp from xnotinF have "A - insert x F = (A - F) - {x}" by auto with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x" by simp from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp qed lemma setsum_diff: assumes le: "finite A" "B \ A" shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))" proof - from le have finiteB: "finite B" using finite_subset by auto show ?thesis using finiteB le proof induct case empty thus ?case by auto next case (insert x F) thus ?case using le finiteB by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb) qed qed lemma setsum_mono: assumes le: "\i. i\K \ f (i::'a) \ ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))" shows "(\i\K. f i) \ (\i\K. g i)" proof (cases "finite K") case True thus ?thesis using le proof induct case empty thus ?case by simp next case insert thus ?case using add_mono by fastforce qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_strict_mono: fixes f :: "'a \ 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}" assumes "finite A" "A \ {}" and "!!x. x:A \ f x < g x" shows "setsum f A < setsum g A" using assms proof (induct rule: finite_ne_induct) case singleton thus ?case by simp next case insert thus ?case by (auto simp: add_strict_mono) qed lemma setsum_strict_mono_ex1: fixes f :: "'a \ 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}" assumes "finite A" and "ALL x:A. f x \ g x" and "EX a:A. f a < g a" shows "setsum f A < setsum g A" proof- from assms(3) obtain a where a: "a:A" "f a < g a" by blast have "setsum f A = setsum f ((A-{a}) \ {a})" by(simp add:insert_absorb[OF `a:A`]) also have "\ = setsum f (A-{a}) + setsum f {a}" using `finite A` by(subst setsum_Un_disjoint) auto also have "setsum f (A-{a}) \ setsum g (A-{a})" by(rule setsum_mono)(simp add: assms(2)) also have "setsum f {a} < setsum g {a}" using a by simp also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \ {a})" using `finite A` by(subst setsum_Un_disjoint[symmetric]) auto also have "\ = setsum g A" by(simp add:insert_absorb[OF `a:A`]) finally show ?thesis by (metis add_right_mono add_strict_left_mono) qed lemma setsum_negf: "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A" proof (cases "finite A") case True thus ?thesis by (induct set: finite) auto next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_subtractf: "setsum (%x. ((f x)::'a::ab_group_add) - g x) A = setsum f A - setsum g A" proof (cases "finite A") case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf) next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_nonneg: assumes nn: "\x\A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \ f x" shows "0 \ setsum f A" proof (cases "finite A") case True thus ?thesis using nn proof induct case empty then show ?case by simp next case (insert x F) then have "0 + 0 \ f x + setsum f F" by (blast intro: add_mono) with insert show ?case by simp qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_nonpos: assumes np: "\x\A. f x \ (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})" shows "setsum f A \ 0" proof (cases "finite A") case True thus ?thesis using np proof induct case empty then show ?case by simp next case (insert x F) then have "f x + setsum f F \ 0 + 0" by (blast intro: add_mono) with insert show ?case by simp qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_nonneg_leq_bound: fixes f :: "'a \ 'b::{ordered_ab_group_add}" assumes "finite s" "\i. i \ s \ f i \ 0" "(\i \ s. f i) = B" "i \ s" shows "f i \ B" proof - have "0 \ (\ i \ s - {i}. f i)" and "0 \ f i" using assms by (auto intro!: setsum_nonneg) moreover have "(\ i \ s - {i}. f i) + f i = B" using assms by (simp add: setsum_diff1) ultimately show ?thesis by auto qed lemma setsum_nonneg_0: fixes f :: "'a \ 'b::{ordered_ab_group_add}" assumes "finite s" and pos: "\ i. i \ s \ f i \ 0" and "(\ i \ s. f i) = 0" and i: "i \ s" shows "f i = 0" using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto lemma setsum_mono2: fixes f :: "'a \ 'b :: ordered_comm_monoid_add" assumes fin: "finite B" and sub: "A \ B" and nn: "\b. b \ B-A \ 0 \ f b" shows "setsum f A \ setsum f B" proof - have "setsum f A \ setsum f A + setsum f (B-A)" by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def) also have "\ = setsum f (A \ (B-A))" using fin finite_subset[OF sub fin] by (simp add:setsum_Un_disjoint del:Un_Diff_cancel) also have "A \ (B-A) = B" using sub by blast finally show ?thesis . qed lemma setsum_mono3: "finite B ==> A <= B ==> ALL x: B - A. 0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==> setsum f A <= setsum f B" apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)") apply (erule ssubst) apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)") apply simp apply (rule add_left_mono) apply (erule setsum_nonneg) apply (subst setsum_Un_disjoint [THEN sym]) apply (erule finite_subset, assumption) apply (rule finite_subset) prefer 2 apply assumption apply (auto simp add: sup_absorb2) done lemma setsum_right_distrib: fixes f :: "'a => ('b::semiring_0)" shows "r * setsum f A = setsum (%n. r * f n) A" proof (cases "finite A") case True thus ?thesis proof induct case empty thus ?case by simp next case (insert x A) thus ?case by (simp add: right_distrib) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_left_distrib: "setsum f A * (r::'a::semiring_0) = (\n\A. f n * r)" proof (cases "finite A") case True then show ?thesis proof induct case empty thus ?case by simp next case (insert x A) thus ?case by (simp add: left_distrib) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_divide_distrib: "setsum f A / (r::'a::field) = (\n\A. f n / r)" proof (cases "finite A") case True then show ?thesis proof induct case empty thus ?case by simp next case (insert x A) thus ?case by (simp add: add_divide_distrib) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_abs[iff]: fixes f :: "'a => ('b::ordered_ab_group_add_abs)" shows "abs (setsum f A) \ setsum (%i. abs(f i)) A" proof (cases "finite A") case True thus ?thesis proof induct case empty thus ?case by simp next case (insert x A) thus ?case by (auto intro: abs_triangle_ineq order_trans) qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_abs_ge_zero[iff]: fixes f :: "'a => ('b::ordered_ab_group_add_abs)" shows "0 \ setsum (%i. abs(f i)) A" proof (cases "finite A") case True thus ?thesis proof induct case empty thus ?case by simp next case (insert x A) thus ?case by auto qed next case False thus ?thesis by (simp add: setsum_def) qed lemma abs_setsum_abs[simp]: fixes f :: "'a => ('b::ordered_ab_group_add_abs)" shows "abs (\a\A. abs(f a)) = (\a\A. abs(f a))" proof (cases "finite A") case True thus ?thesis proof induct case empty thus ?case by simp next case (insert a A) hence "\\a\insert a A. \f a\\ = \\f a\ + (\a\A. \f a\)\" by simp also have "\ = \\f a\ + \\a\A. \f a\\\" using insert by simp also have "\ = \f a\ + \\a\A. \f a\\" by (simp del: abs_of_nonneg) also have "\ = (\a\insert a A. \f a\)" using insert by simp finally show ?case . qed next case False thus ?thesis by (simp add: setsum_def) qed lemma setsum_Plus: fixes A :: "'a set" and B :: "'b set" assumes fin: "finite A" "finite B" shows "setsum f (A <+> B) = setsum (f \ Inl) A + setsum (f \ Inr) B" proof - have "A <+> B = Inl ` A \ Inr ` B" by auto moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)" by auto moreover have "Inl ` A \ Inr ` B = ({} :: ('a + 'b) set)" by auto moreover have "inj_on (Inl :: 'a \ 'a + 'b) A" "inj_on (Inr :: 'b \ 'a + 'b) B" by(auto intro: inj_onI) ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex) qed text {* Commuting outer and inner summation *} lemma setsum_commute: "(\i\A. \j\B. f i j) = (\j\B. \i\A. f i j)" proof (simp add: setsum_cartesian_product) have "(\(x,y) \ A <*> B. f x y) = (\(y,x) \ (%(i, j). (j, i)) ` (A \ B). f x y)" (is "?s = _") apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on) apply (simp add: split_def) done also have "... = (\(y,x)\B \ A. f x y)" (is "_ = ?t") apply (simp add: swap_product) done finally show "?s = ?t" . qed lemma setsum_product: fixes f :: "'a => ('b::semiring_0)" shows "setsum f A * setsum g B = (\i\A. \j\B. f i * g j)" by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute) lemma setsum_mult_setsum_if_inj: fixes f :: "'a => ('b::semiring_0)" shows "inj_on (%(a,b). f a * g b) (A \ B) ==> setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}" by(auto simp: setsum_product setsum_cartesian_product intro!: setsum_reindex_cong[symmetric]) lemma setsum_constant [simp]: "(\x \ A. y) = of_nat(card A) * y" apply (cases "finite A") apply (erule finite_induct) apply (auto simp add: algebra_simps) done lemma setsum_bounded: assumes le: "\i. i\A \ f i \ (K::'a::{semiring_1, ordered_ab_semigroup_add})" shows "setsum f A \ of_nat(card A) * K" proof (cases "finite A") case True thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp next case False thus ?thesis by (simp add: setsum_def) qed subsubsection {* Cardinality as special case of @{const setsum} *} lemma card_eq_setsum: "card A = setsum (\x. 1) A" by (simp only: card_def setsum_def) lemma card_UN_disjoint: assumes "finite I" and "\i\I. finite (A i)" and "\i\I. \j\I. i \ j \ A i \ A j = {}" shows "card (UNION I A) = (\i\I. card(A i))" proof - have "(\i\I. card (A i)) = (\i\I. \x\A i. 1)" by simp with assms show ?thesis by (simp add: card_eq_setsum setsum_UN_disjoint del: setsum_constant) qed lemma card_Union_disjoint: "finite C ==> (ALL A:C. finite A) ==> (ALL A:C. ALL B:C. A \ B --> A Int B = {}) ==> card (Union C) = setsum card C" apply (frule card_UN_disjoint [of C id]) apply (simp_all add: SUP_def id_def) done text{*The image of a finite set can be expressed using @{term fold_image}.*} lemma image_eq_fold_image: "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A" proof (induct rule: finite_induct) case empty then show ?case by simp next interpret ab_semigroup_mult "op Un" proof qed auto case insert then show ?case by simp qed subsubsection {* Cardinality of products *} lemma card_SigmaI [simp]: "\ finite A; ALL a:A. finite (B a) \ \ card (SIGMA x: A. B x) = (\a\A. card (B a))" by(simp add: card_eq_setsum setsum_Sigma del:setsum_constant) (* lemma SigmaI_insert: "y \ A ==> (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \ (SIGMA x: A. B x))" by auto *) lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)" by (cases "finite A \ finite B") (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2) lemma card_cartesian_product_singleton: "card({x} <*> A) = card(A)" by (simp add: card_cartesian_product) subsection {* Generalized product over a set *} definition (in comm_monoid_mult) setprod :: "('b \ 'a) => 'b set => 'a" where "setprod f A = (if finite A then fold_image (op *) f 1 A else 1)" sublocale comm_monoid_mult < setprod!: comm_monoid_big "op *" 1 setprod proof qed (fact setprod_def) abbreviation Setprod ("\_" [1000] 999) where "\A == setprod (%x. x) A" syntax "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3PROD _:_. _)" [0, 51, 10] 10) syntax (xsymbols) "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\_\_. _)" [0, 51, 10] 10) syntax (HTML output) "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\_\_. _)" [0, 51, 10] 10) translations -- {* Beware of argument permutation! *} "PROD i:A. b" == "CONST setprod (%i. b) A" "\i\A. b" == "CONST setprod (%i. b) A" text{* Instead of @{term"\x\{x. P}. e"} we introduce the shorter @{text"\x|P. e"}. *} syntax "_qsetprod" :: "pttrn \ bool \ 'a \ 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10) syntax (xsymbols) "_qsetprod" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) syntax (HTML output) "_qsetprod" :: "pttrn \ bool \ 'a \ 'a" ("(3\_ | (_)./ _)" [0,0,10] 10) translations "PROD x|P. t" => "CONST setprod (%x. t) {x. P}" "\x|P. t" => "CONST setprod (%x. t) {x. P}" lemma setprod_empty: "setprod f {} = 1" by (fact setprod.empty) lemma setprod_insert: "[| finite A; a \ A |] ==> setprod f (insert a A) = f a * setprod f A" by (fact setprod.insert) lemma setprod_infinite: "~ finite A ==> setprod f A = 1" by (fact setprod.infinite) lemma setprod_reindex: "inj_on f B ==> setprod h (f ` B) = setprod (h \ f) B" by(auto simp: setprod_def fold_image_reindex dest!:finite_imageD) lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)" by (auto simp add: setprod_reindex) lemma setprod_cong: "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B" by(fastforce simp: setprod_def intro: fold_image_cong) lemma strong_setprod_cong[cong]: "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B" by(fastforce simp: simp_implies_def setprod_def intro: fold_image_cong) lemma setprod_reindex_cong: "inj_on f A ==> B = f ` A ==> g = h \ f ==> setprod h B = setprod g A" by (frule setprod_reindex, simp) lemma strong_setprod_reindex_cong: assumes i: "inj_on f A" and B: "B = f ` A" and eq: "\x. x \ A \ g x = (h \ f) x" shows "setprod h B = setprod g A" proof- have "setprod h B = setprod (h o f) A" by (simp add: B setprod_reindex[OF i, of h]) then show ?thesis apply simp apply (rule setprod_cong) apply simp by (simp add: eq) qed lemma setprod_Un_one: assumes fS: "finite S" and fT: "finite T" and I0: "\x \ S\T. f x = 1" shows "setprod f (S \ T) = setprod f S * setprod f T" using fS fT apply (simp add: setprod_def) apply (rule fold_image_Un_one) using I0 by auto lemma setprod_1: "setprod (%i. 1) A = 1" apply (case_tac "finite A") apply (erule finite_induct, auto simp add: mult_ac) done lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1" apply (subgoal_tac "setprod f F = setprod (%x. 1) F") apply (erule ssubst, rule setprod_1) apply (rule setprod_cong, auto) done lemma setprod_Un_Int: "finite A ==> finite B ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B" by(simp add: setprod_def fold_image_Un_Int[symmetric]) lemma setprod_Un_disjoint: "finite A ==> finite B ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B" by (subst setprod_Un_Int [symmetric], auto) lemma setprod_mono_one_left: assumes fT: "finite T" and ST: "S \ T" and z: "\i \ T - S. f i = 1" shows "setprod f S = setprod f T" proof- have eq: "T = S \ (T - S)" using ST by blast have d: "S \ (T - S) = {}" using ST by blast from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) show ?thesis by (simp add: setprod_Un_disjoint[OF f d, unfolded eq[symmetric]] setprod_1'[OF z]) qed lemmas setprod_mono_one_right = setprod_mono_one_left [THEN sym] lemma setprod_delta: assumes fS: "finite S" shows "setprod (\k. if k=a then b k else 1) S = (if a \ S then b a else 1)" proof- let ?f = "(\k. if k=a then b k else 1)" {assume a: "a \ S" hence "\ k\ S. ?f k = 1" by simp hence ?thesis using a by (simp add: setprod_1) } moreover {assume a: "a \ S" let ?A = "S - {a}" let ?B = "{a}" have eq: "S = ?A \ ?B" using a by blast have dj: "?A \ ?B = {}" by simp from fS have fAB: "finite ?A" "finite ?B" by auto have fA1: "setprod ?f ?A = 1" apply (rule setprod_1') by auto have "setprod ?f ?A * setprod ?f ?B = setprod ?f S" using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] by simp then have ?thesis using a by (simp add: fA1 cong: setprod_cong cong del: if_weak_cong)} ultimately show ?thesis by blast qed lemma setprod_delta': assumes fS: "finite S" shows "setprod (\k. if a = k then b k else 1) S = (if a\ S then b a else 1)" using setprod_delta[OF fS, of a b, symmetric] by (auto intro: setprod_cong) lemma setprod_UN_disjoint: "finite I ==> (ALL i:I. finite (A i)) ==> (ALL i:I. ALL j:I. i \ j --> A i Int A j = {}) ==> setprod f (UNION I A) = setprod (%i. setprod f (A i)) I" by (simp add: setprod_def fold_image_UN_disjoint) lemma setprod_Union_disjoint: assumes "\A\C. finite A" "\A\C. \B\C. A \ B \ A Int B = {}" shows "setprod f (Union C) = setprod (setprod f) C" proof cases assume "finite C" from setprod_UN_disjoint[OF this assms] show ?thesis by (simp add: SUP_def) qed (force dest: finite_UnionD simp add: setprod_def) lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==> (\x\A. (\y\ B x. f x y)) = (\(x,y)\(SIGMA x:A. B x). f x y)" by(simp add:setprod_def fold_image_Sigma split_def) text{*Here we can eliminate the finiteness assumptions, by cases.*} lemma setprod_cartesian_product: "(\x\A. (\y\ B. f x y)) = (\(x,y)\(A <*> B). f x y)" apply (cases "finite A") apply (cases "finite B") apply (simp add: setprod_Sigma) apply (cases "A={}", simp) apply (simp add: setprod_1) apply (auto simp add: setprod_def dest: finite_cartesian_productD1 finite_cartesian_productD2) done lemma setprod_timesf: "setprod (%x. f x * g x) A = (setprod f A * setprod g A)" by(simp add:setprod_def fold_image_distrib) subsubsection {* Properties in more restricted classes of structures *} lemma setprod_eq_1_iff [simp]: "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))" by (induct set: finite) auto lemma setprod_zero: "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0" apply (induct set: finite, force, clarsimp) apply (erule disjE, auto) done lemma setprod_nonneg [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) \ f x) --> 0 \ setprod f A" by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg) lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x) --> 0 < setprod f A" by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos) lemma setprod_zero_iff[simp]: "finite A ==> (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) = (EX x: A. f x = 0)" by (erule finite_induct, auto simp:no_zero_divisors) lemma setprod_pos_nat: "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0" using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric]) lemma setprod_pos_nat_iff[simp]: "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))" using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric]) lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \ 0) ==> (setprod f (A Un B) :: 'a ::{field}) = setprod f A * setprod f B / setprod f (A Int B)" by (subst setprod_Un_Int [symmetric], auto) lemma setprod_diff1: "finite A ==> f a \ 0 ==> (setprod f (A - {a}) :: 'a :: {field}) = (if a:A then setprod f A / f a else setprod f A)" by (erule finite_induct) (auto simp add: insert_Diff_if) lemma setprod_inversef: fixes f :: "'b \ 'a::field_inverse_zero" shows "finite A ==> setprod (inverse \ f) A = inverse (setprod f A)" by (erule finite_induct) auto lemma setprod_dividef: fixes f :: "'b \ 'a::field_inverse_zero" shows "finite A ==> setprod (%x. f x / g x) A = setprod f A / setprod g A" apply (subgoal_tac "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \ g) x) A") apply (erule ssubst) apply (subst divide_inverse) apply (subst setprod_timesf) apply (subst setprod_inversef, assumption+, rule refl) apply (rule setprod_cong, rule refl) apply (subst divide_inverse, auto) done lemma setprod_dvd_setprod [rule_format]: "(ALL x : A. f x dvd g x) \ setprod f A dvd setprod g A" apply (cases "finite A") apply (induct set: finite) apply (auto simp add: dvd_def) apply (rule_tac x = "k * ka" in exI) apply (simp add: algebra_simps) done lemma setprod_dvd_setprod_subset: "finite B \ A <= B \ setprod f A dvd setprod f B" apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)") apply (unfold dvd_def, blast) apply (subst setprod_Un_disjoint [symmetric]) apply (auto elim: finite_subset intro: setprod_cong) done lemma setprod_dvd_setprod_subset2: "finite B \ A <= B \ ALL x : A. (f x::'a::comm_semiring_1) dvd g x \ setprod f A dvd setprod g B" apply (rule dvd_trans) apply (rule setprod_dvd_setprod, erule (1) bspec) apply (erule (1) setprod_dvd_setprod_subset) done lemma dvd_setprod: "finite A \ i:A \ (f i ::'a::comm_semiring_1) dvd setprod f A" by (induct set: finite) (auto intro: dvd_mult) lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \ (d::'a::comm_semiring_1) dvd (SUM x : A. f x)" apply (cases "finite A") apply (induct set: finite) apply auto done lemma setprod_mono: fixes f :: "'a \ 'b\linordered_semidom" assumes "\i\A. 0 \ f i \ f i \ g i" shows "setprod f A \ setprod g A" proof (cases "finite A") case True hence ?thesis "setprod f A \ 0" using subset_refl[of A] proof (induct A rule: finite_subset_induct) case (insert a F) thus "setprod f (insert a F) \ setprod g (insert a F)" "0 \ setprod f (insert a F)" unfolding setprod_insert[OF insert(1,3)] using assms[rule_format,OF insert(2)] insert by (auto intro: mult_mono mult_nonneg_nonneg) qed auto thus ?thesis by simp qed auto lemma abs_setprod: fixes f :: "'a \ 'b\{linordered_field,abs}" shows "abs (setprod f A) = setprod (\x. abs (f x)) A" proof (cases "finite A") case True thus ?thesis by induct (auto simp add: field_simps abs_mult) qed auto lemma setprod_constant: "finite A ==> (\x\ A. (y::'a::{comm_monoid_mult})) = y^(card A)" apply (erule finite_induct) apply auto done lemma setprod_gen_delta: assumes fS: "finite S" 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)" proof- let ?f = "(\k. if k=a then b k else c)" {assume a: "a \ S" hence "\ k\ S. ?f k = c" by simp hence ?thesis using a setprod_constant[OF fS, of c] by (simp add: setprod_1 cong add: setprod_cong) } moreover {assume a: "a \ S" let ?A = "S - {a}" let ?B = "{a}" have eq: "S = ?A \ ?B" using a by blast have dj: "?A \ ?B = {}" by simp from fS have fAB: "finite ?A" "finite ?B" by auto have fA0:"setprod ?f ?A = setprod (\i. c) ?A" apply (rule setprod_cong) by auto have cA: "card ?A = card S - 1" using fS a by auto have fA1: "setprod ?f ?A = c ^ card ?A" unfolding fA0 apply (rule setprod_constant) using fS by auto have "setprod ?f ?A * setprod ?f ?B = setprod ?f S" using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] by simp then have ?thesis using a cA by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)} ultimately show ?thesis by blast qed subsection {* Versions of @{const inf} and @{const sup} on non-empty sets *} no_notation times (infixl "*" 70) no_notation Groups.one ("1") locale semilattice_big = semilattice + fixes F :: "'a set \ 'a" assumes F_eq: "finite A \ F A = fold1 (op *) A" sublocale semilattice_big < folding_one_idem proof qed (simp_all add: F_eq) notation times (infixl "*" 70) notation Groups.one ("1") context lattice begin definition Inf_fin :: "'a set \ 'a" ("\\<^bsub>fin\<^esub>_" [900] 900) where "Inf_fin = fold1 inf" definition Sup_fin :: "'a set \ 'a" ("\\<^bsub>fin\<^esub>_" [900] 900) where "Sup_fin = fold1 sup" end sublocale lattice < Inf_fin!: semilattice_big inf Inf_fin proof qed (simp add: Inf_fin_def) sublocale lattice < Sup_fin!: semilattice_big sup Sup_fin proof qed (simp add: Sup_fin_def) context semilattice_inf begin lemma ab_semigroup_idem_mult_inf: "class.ab_semigroup_idem_mult inf" proof qed (rule inf_assoc inf_commute inf_idem)+ lemma fold_inf_insert[simp]: "finite A \ Finite_Set.fold inf b (insert a A) = inf a (Finite_Set.fold inf b A)" by(rule comp_fun_idem.fold_insert_idem[OF ab_semigroup_idem_mult.comp_fun_idem[OF ab_semigroup_idem_mult_inf]]) lemma inf_le_fold_inf: "finite A \ ALL a:A. b \ a \ inf b c \ Finite_Set.fold inf c A" by (induct pred: finite) (auto intro: le_infI1) lemma fold_inf_le_inf: "finite A \ a \ A \ Finite_Set.fold inf b A \ inf a b" proof(induct arbitrary: a pred:finite) case empty thus ?case by simp next case (insert x A) show ?case proof cases assume "A = {}" thus ?thesis using insert by simp next assume "A \ {}" thus ?thesis using insert by (auto intro: le_infI2) qed qed lemma below_fold1_iff: assumes "finite A" "A \ {}" shows "x \ fold1 inf A \ (\a\A. x \ a)" proof - interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) show ?thesis using assms by (induct rule: finite_ne_induct) simp_all qed lemma fold1_belowI: assumes "finite A" and "a \ A" shows "fold1 inf A \ a" proof - from assms have "A \ {}" by auto from `finite A` `A \ {}` `a \ A` show ?thesis proof (induct rule: finite_ne_induct) case singleton thus ?case by simp next interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) case (insert x F) from insert(5) have "a = x \ a \ F" by simp thus ?case proof assume "a = x" thus ?thesis using insert by (simp add: mult_ac) next assume "a \ F" hence bel: "fold1 inf F \ a" by (rule insert) have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)" using insert by (simp add: mult_ac) also have "inf (fold1 inf F) a = fold1 inf F" using bel by (auto intro: antisym) also have "inf x \ = fold1 inf (insert x F)" using insert by (simp add: mult_ac) finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" . moreover have "inf (fold1 inf (insert x F)) a \ a" by simp ultimately show ?thesis by simp qed qed qed end context semilattice_sup begin lemma ab_semigroup_idem_mult_sup: "class.ab_semigroup_idem_mult sup" by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice) lemma fold_sup_insert[simp]: "finite A \ Finite_Set.fold sup b (insert a A) = sup a (Finite_Set.fold sup b A)" by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice) lemma fold_sup_le_sup: "finite A \ ALL a:A. a \ b \ Finite_Set.fold sup c A \ sup b c" by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice) lemma sup_le_fold_sup: "finite A \ a \ A \ sup a b \ Finite_Set.fold sup b A" by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice) end context lattice begin lemma Inf_le_Sup [simp]: "\ finite A; A \ {} \ \ \\<^bsub>fin\<^esub>A \ \\<^bsub>fin\<^esub>A" apply(unfold Sup_fin_def Inf_fin_def) apply(subgoal_tac "EX a. a:A") prefer 2 apply blast apply(erule exE) apply(rule order_trans) apply(erule (1) fold1_belowI) apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice]) done lemma sup_Inf_absorb [simp]: "finite A \ a \ A \ sup a (\\<^bsub>fin\<^esub>A) = a" apply(subst sup_commute) apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI) done lemma inf_Sup_absorb [simp]: "finite A \ a \ A \ inf a (\\<^bsub>fin\<^esub>A) = a" by (simp add: Sup_fin_def inf_absorb1 semilattice_inf.fold1_belowI [OF dual_semilattice]) end context distrib_lattice begin lemma sup_Inf1_distrib: assumes "finite A" and "A \ {}" shows "sup x (\\<^bsub>fin\<^esub>A) = \\<^bsub>fin\<^esub>{sup x a|a. a \ A}" proof - interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) from assms show ?thesis by (simp add: Inf_fin_def image_def hom_fold1_commute [where h="sup x", OF sup_inf_distrib1]) (rule arg_cong [where f="fold1 inf"], blast) qed lemma sup_Inf2_distrib: assumes A: "finite A" "A \ {}" and B: "finite B" "B \ {}" shows "sup (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B) = \\<^bsub>fin\<^esub>{sup a b|a b. a \ A \ b \ B}" using A proof (induct rule: finite_ne_induct) case singleton thus ?case by (simp add: sup_Inf1_distrib [OF B]) next interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) case (insert x A) have finB: "finite {sup x b |b. b \ B}" by(rule finite_surj[where f = "sup x", OF B(1)], auto) have finAB: "finite {sup a b |a b. a \ A \ b \ B}" proof - have "{sup a b |a b. a \ A \ b \ B} = (UN a:A. UN b:B. {sup a b})" by blast thus ?thesis by(simp add: insert(1) B(1)) qed have ne: "{sup a b |a b. a \ A \ b \ B} \ {}" using insert B by blast have "sup (\\<^bsub>fin\<^esub>(insert x A)) (\\<^bsub>fin\<^esub>B) = sup (inf x (\\<^bsub>fin\<^esub>A)) (\\<^bsub>fin\<^esub>B)" using insert by simp also have "\ = inf (sup x (\\<^bsub>fin\<^esub>B)) (sup (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2) also have "\ = inf (\\<^bsub>fin\<^esub>{sup x b|b. b \ B}) (\\<^bsub>fin\<^esub>{sup a b|a b. a \ A \ b \ B})" using insert by(simp add:sup_Inf1_distrib[OF B]) also have "\ = \\<^bsub>fin\<^esub>({sup x b |b. b \ B} \ {sup a b |a b. a \ A \ b \ B})" (is "_ = \\<^bsub>fin\<^esub>?M") using B insert by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne]) also have "?M = {sup a b |a b. a \ insert x A \ b \ B}" by blast finally show ?case . qed lemma inf_Sup1_distrib: assumes "finite A" and "A \ {}" shows "inf x (\\<^bsub>fin\<^esub>A) = \\<^bsub>fin\<^esub>{inf x a|a. a \ A}" proof - interpret ab_semigroup_idem_mult sup by (rule ab_semigroup_idem_mult_sup) from assms show ?thesis by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1]) (rule arg_cong [where f="fold1 sup"], blast) qed lemma inf_Sup2_distrib: assumes A: "finite A" "A \ {}" and B: "finite B" "B \ {}" shows "inf (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B) = \\<^bsub>fin\<^esub>{inf a b|a b. a \ A \ b \ B}" using A proof (induct rule: finite_ne_induct) case singleton thus ?case by(simp add: inf_Sup1_distrib [OF B]) next case (insert x A) have finB: "finite {inf x b |b. b \ B}" by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto) have finAB: "finite {inf a b |a b. a \ A \ b \ B}" proof - have "{inf a b |a b. a \ A \ b \ B} = (UN a:A. UN b:B. {inf a b})" by blast thus ?thesis by(simp add: insert(1) B(1)) qed have ne: "{inf a b |a b. a \ A \ b \ B} \ {}" using insert B by blast interpret ab_semigroup_idem_mult sup by (rule ab_semigroup_idem_mult_sup) have "inf (\\<^bsub>fin\<^esub>(insert x A)) (\\<^bsub>fin\<^esub>B) = inf (sup x (\\<^bsub>fin\<^esub>A)) (\\<^bsub>fin\<^esub>B)" using insert by simp also have "\ = sup (inf x (\\<^bsub>fin\<^esub>B)) (inf (\\<^bsub>fin\<^esub>A) (\\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2) also have "\ = sup (\\<^bsub>fin\<^esub>{inf x b|b. b \ B}) (\\<^bsub>fin\<^esub>{inf a b|a b. a \ A \ b \ B})" using insert by(simp add:inf_Sup1_distrib[OF B]) also have "\ = \\<^bsub>fin\<^esub>({inf x b |b. b \ B} \ {inf a b |a b. a \ A \ b \ B})" (is "_ = \\<^bsub>fin\<^esub>?M") using B insert by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne]) also have "?M = {inf a b |a b. a \ insert x A \ b \ B}" by blast finally show ?case . qed end context complete_lattice begin lemma Inf_fin_Inf: assumes "finite A" and "A \ {}" shows "\\<^bsub>fin\<^esub>A = Inf A" proof - interpret ab_semigroup_idem_mult inf by (rule ab_semigroup_idem_mult_inf) from `A \ {}` obtain b B where "A = {b} \ B" by auto moreover with `finite A` have "finite B" by simp ultimately show ?thesis by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric]) qed lemma Sup_fin_Sup: assumes "finite A" and "A \ {}" shows "\\<^bsub>fin\<^esub>A = Sup A" proof - interpret ab_semigroup_idem_mult sup by (rule ab_semigroup_idem_mult_sup) from `A \ {}` obtain b B where "A = {b} \ B" by auto moreover with `finite A` have "finite B" by simp ultimately show ?thesis by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric]) qed end subsection {* Versions of @{const min} and @{const max} on non-empty sets *} definition (in linorder) Min :: "'a set \ 'a" where "Min = fold1 min" definition (in linorder) Max :: "'a set \ 'a" where "Max = fold1 max" sublocale linorder < Min!: semilattice_big min Min proof qed (simp add: Min_def) sublocale linorder < Max!: semilattice_big max Max proof qed (simp add: Max_def) context linorder begin lemmas Min_singleton = Min.singleton lemmas Max_singleton = Max.singleton lemma Min_insert: assumes "finite A" and "A \ {}" shows "Min (insert x A) = min x (Min A)" using assms by simp lemma Max_insert: assumes "finite A" and "A \ {}" shows "Max (insert x A) = max x (Max A)" using assms by simp lemma Min_Un: assumes "finite A" and "A \ {}" and "finite B" and "B \ {}" shows "Min (A \ B) = min (Min A) (Min B)" using assms by (rule Min.union_idem) lemma Max_Un: assumes "finite A" and "A \ {}" and "finite B" and "B \ {}" shows "Max (A \ B) = max (Max A) (Max B)" using assms by (rule Max.union_idem) lemma hom_Min_commute: assumes "\x y. h (min x y) = min (h x) (h y)" and "finite N" and "N \ {}" shows "h (Min N) = Min (h ` N)" using assms by (rule Min.hom_commute) lemma hom_Max_commute: assumes "\x y. h (max x y) = max (h x) (h y)" and "finite N" and "N \ {}" shows "h (Max N) = Max (h ` N)" using assms by (rule Max.hom_commute) lemma ab_semigroup_idem_mult_min: "class.ab_semigroup_idem_mult min" proof qed (auto simp add: min_def) lemma ab_semigroup_idem_mult_max: "class.ab_semigroup_idem_mult max" proof qed (auto simp add: max_def) lemma max_lattice: "class.semilattice_inf max (op \) (op >)" by (fact min_max.dual_semilattice) lemma dual_max: "ord.max (op \) = min" by (auto simp add: ord.max_def_raw min_def fun_eq_iff) lemma dual_min: "ord.min (op \) = max" by (auto simp add: ord.min_def_raw max_def fun_eq_iff) lemma strict_below_fold1_iff: assumes "finite A" and "A \ {}" shows "x < fold1 min A \ (\a\A. x < a)" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms show ?thesis by (induct rule: finite_ne_induct) (simp_all add: fold1_insert) qed lemma fold1_below_iff: assumes "finite A" and "A \ {}" shows "fold1 min A \ x \ (\a\A. a \ x)" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms show ?thesis by (induct rule: finite_ne_induct) (simp_all add: fold1_insert min_le_iff_disj) qed lemma fold1_strict_below_iff: assumes "finite A" and "A \ {}" shows "fold1 min A < x \ (\a\A. a < x)" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms show ?thesis by (induct rule: finite_ne_induct) (simp_all add: fold1_insert min_less_iff_disj) qed lemma fold1_antimono: assumes "A \ {}" and "A \ B" and "finite B" shows "fold1 min B \ fold1 min A" proof cases assume "A = B" thus ?thesis by simp next interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) assume neq: "A \ B" have B: "B = A \ (B-A)" using `A \ B` by blast have "fold1 min B = fold1 min (A \ (B-A))" by(subst B)(rule refl) also have "\ = min (fold1 min A) (fold1 min (B-A))" proof - have "finite A" by(rule finite_subset[OF `A \ B` `finite B`]) moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) moreover have "(B-A) \ {}" using assms neq by blast moreover have "A Int (B-A) = {}" using assms by blast ultimately show ?thesis using `A \ {}` by (rule_tac fold1_Un) qed also have "\ \ fold1 min A" by (simp add: min_le_iff_disj) finally show ?thesis . qed lemma Min_in [simp]: assumes "finite A" and "A \ {}" shows "Min A \ A" proof - interpret ab_semigroup_idem_mult min by (rule ab_semigroup_idem_mult_min) from assms fold1_in show ?thesis by (fastforce simp: Min_def min_def) qed lemma Max_in [simp]: assumes "finite A" and "A \ {}" shows "Max A \ A" proof - interpret ab_semigroup_idem_mult max by (rule ab_semigroup_idem_mult_max) from assms fold1_in [of A] show ?thesis by (fastforce simp: Max_def max_def) qed lemma Min_le [simp]: assumes "finite A" and "x \ A" shows "Min A \ x" using assms by (simp add: Min_def min_max.fold1_belowI) lemma Max_ge [simp]: assumes "finite A" and "x \ A" shows "x \ Max A" by (simp add: Max_def semilattice_inf.fold1_belowI [OF max_lattice] assms) lemma Min_ge_iff [simp, no_atp]: assumes "finite A" and "A \ {}" shows "x \ Min A \ (\a\A. x \ a)" using assms by (simp add: Min_def min_max.below_fold1_iff) lemma Max_le_iff [simp, no_atp]: assumes "finite A" and "A \ {}" shows "Max A \ x \ (\a\A. a \ x)" by (simp add: Max_def semilattice_inf.below_fold1_iff [OF max_lattice] assms) lemma Min_gr_iff [simp, no_atp]: assumes "finite A" and "A \ {}" shows "x < Min A \ (\a\A. x < a)" using assms by (simp add: Min_def strict_below_fold1_iff) lemma Max_less_iff [simp, no_atp]: assumes "finite A" and "A \ {}" shows "Max A < x \ (\a\A. a < x)" by (simp add: Max_def linorder.dual_max [OF dual_linorder] linorder.strict_below_fold1_iff [OF dual_linorder] assms) lemma Min_le_iff [no_atp]: assumes "finite A" and "A \ {}" shows "Min A \ x \ (\a\A. a \ x)" using assms by (simp add: Min_def fold1_below_iff) lemma Max_ge_iff [no_atp]: assumes "finite A" and "A \ {}" shows "x \ Max A \ (\a\A. x \ a)" by (simp add: Max_def linorder.dual_max [OF dual_linorder] linorder.fold1_below_iff [OF dual_linorder] assms) lemma Min_less_iff [no_atp]: assumes "finite A" and "A \ {}" shows "Min A < x \ (\a\A. a < x)" using assms by (simp add: Min_def fold1_strict_below_iff) lemma Max_gr_iff [no_atp]: assumes "finite A" and "A \ {}" shows "x < Max A \ (\a\A. x < a)" by (simp add: Max_def linorder.dual_max [OF dual_linorder] linorder.fold1_strict_below_iff [OF dual_linorder] assms) lemma Min_eqI: assumes "finite A" assumes "\y. y \ A \ y \ x" and "x \ A" shows "Min A = x" proof (rule antisym) from `x \ A` have "A \ {}" by auto with assms show "Min A \ x" by simp next from assms show "x \ Min A" by simp qed lemma Max_eqI: assumes "finite A" assumes "\y. y \ A \ y \ x" and "x \ A" shows "Max A = x" proof (rule antisym) from `x \ A` have "A \ {}" by auto with assms show "Max A \ x" by simp next from assms show "x \ Max A" by simp qed lemma Min_antimono: assumes "M \ N" and "M \ {}" and "finite N" shows "Min N \ Min M" using assms by (simp add: Min_def fold1_antimono) lemma Max_mono: assumes "M \ N" and "M \ {}" and "finite N" shows "Max M \ Max N" by (simp add: Max_def linorder.dual_max [OF dual_linorder] linorder.fold1_antimono [OF dual_linorder] assms) lemma finite_linorder_max_induct[consumes 1, case_names empty insert]: assumes fin: "finite A" and empty: "P {}" and insert: "(!!b A. finite A \ ALL a:A. a < b \ P A \ P(insert b A))" shows "P A" using fin empty insert proof (induct rule: finite_psubset_induct) case (psubset A) have IH: "\B. \B < A; P {}; (\A b. \finite A; \a\A. a \ P (insert b A))\ \ P B" by fact have fin: "finite A" by fact have empty: "P {}" by fact have step: "\b A. \finite A; \a\A. a < b; P A\ \ P (insert b A)" by fact show "P A" proof (cases "A = {}") assume "A = {}" then show "P A" using `P {}` by simp next let ?B = "A - {Max A}" let ?A = "insert (Max A) ?B" have "finite ?B" using `finite A` by simp assume "A \ {}" with `finite A` have "Max A : A" by auto then have A: "?A = A" using insert_Diff_single insert_absorb by auto then have "P ?B" using `P {}` step IH[of ?B] by blast moreover have "\a\?B. a < Max A" using Max_ge [OF `finite A`] by fastforce ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastforce qed qed lemma finite_linorder_min_induct[consumes 1, case_names empty insert]: "\finite A; P {}; \b A. \finite A; \a\A. b < a; P A\ \ P (insert b A)\ \ P A" by(rule linorder.finite_linorder_max_induct[OF dual_linorder]) end context linordered_ab_semigroup_add begin lemma add_Min_commute: fixes k assumes "finite N" and "N \ {}" shows "k + Min N = Min {k + m | m. m \ N}" proof - have "\x y. k + min x y = min (k + x) (k + y)" by (simp add: min_def not_le) (blast intro: antisym less_imp_le add_left_mono) with assms show ?thesis using hom_Min_commute [of "plus k" N] by simp (blast intro: arg_cong [where f = Min]) qed lemma add_Max_commute: fixes k assumes "finite N" and "N \ {}" shows "k + Max N = Max {k + m | m. m \ N}" proof - have "\x y. k + max x y = max (k + x) (k + y)" by (simp add: max_def not_le) (blast intro: antisym less_imp_le add_left_mono) with assms show ?thesis using hom_Max_commute [of "plus k" N] by simp (blast intro: arg_cong [where f = Max]) qed end context linordered_ab_group_add begin lemma minus_Max_eq_Min [simp]: "finite S \ S \ {} \ - (Max S) = Min (uminus ` S)" by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min) lemma minus_Min_eq_Max [simp]: "finite S \ S \ {} \ - (Min S) = Max (uminus ` S)" by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max) end end