author | paulson <lp15@cam.ac.uk> |
Wed, 19 Aug 2015 19:18:19 +0100 | |
changeset 60974 | 6a6f15d8fbc4 |
parent 60758 | d8d85a8172b5 |
child 61032 | b57df8eecad6 |
permissions | -rw-r--r-- |
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(* Title: HOL/Groups_Big.thy |
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Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel |
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with contributions by Jeremy Avigad |
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*) |
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section \<open>Big sum and product over finite (non-empty) sets\<close> |
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theory Groups_Big |
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imports Finite_Set |
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begin |
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subsection \<open>Generic monoid operation over a set\<close> |
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no_notation times (infixl "*" 70) |
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no_notation Groups.one ("1") |
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locale comm_monoid_set = comm_monoid |
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begin |
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interpretation comp_fun_commute f |
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by default (simp add: fun_eq_iff left_commute) |
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interpretation comp?: comp_fun_commute "f \<circ> g" |
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by (fact comp_comp_fun_commute) |
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definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a" |
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where |
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eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A" |
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lemma infinite [simp]: |
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"\<not> finite A \<Longrightarrow> F g A = 1" |
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by (simp add: eq_fold) |
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lemma empty [simp]: |
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"F g {} = 1" |
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by (simp add: eq_fold) |
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lemma insert [simp]: |
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assumes "finite A" and "x \<notin> A" |
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shows "F g (insert x A) = g x * F g A" |
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using assms by (simp add: eq_fold) |
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lemma remove: |
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assumes "finite A" and "x \<in> A" |
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shows "F g A = g x * F g (A - {x})" |
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proof - |
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from \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B" |
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by (auto dest: mk_disjoint_insert) |
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moreover from \<open>finite A\<close> A have "finite B" by simp |
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ultimately show ?thesis by simp |
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qed |
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lemma insert_remove: |
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assumes "finite A" |
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shows "F g (insert x A) = g x * F g (A - {x})" |
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using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb) |
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lemma neutral: |
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assumes "\<forall>x\<in>A. g x = 1" |
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shows "F g A = 1" |
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using assms by (induct A rule: infinite_finite_induct) simp_all |
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lemma neutral_const [simp]: |
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"F (\<lambda>_. 1) A = 1" |
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by (simp add: neutral) |
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lemma union_inter: |
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assumes "finite A" and "finite B" |
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shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B" |
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-- \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close> |
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using assms proof (induct A) |
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case empty then show ?case by simp |
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next |
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case (insert x A) then show ?case |
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by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute) |
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qed |
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corollary union_inter_neutral: |
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assumes "finite A" and "finite B" |
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and I0: "\<forall>x \<in> A \<inter> B. g x = 1" |
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shows "F g (A \<union> B) = F g A * F g B" |
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using assms by (simp add: union_inter [symmetric] neutral) |
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corollary union_disjoint: |
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assumes "finite A" and "finite B" |
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assumes "A \<inter> B = {}" |
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shows "F g (A \<union> B) = F g A * F g B" |
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using assms by (simp add: union_inter_neutral) |
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lemma union_diff2: |
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assumes "finite A" and "finite B" |
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shows "F g (A \<union> B) = F g (A - B) * F g (B - A) * F g (A \<inter> B)" |
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proof - |
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have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B" |
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by auto |
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with assms show ?thesis by simp (subst union_disjoint, auto)+ |
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qed |
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lemma subset_diff: |
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assumes "B \<subseteq> A" and "finite A" |
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shows "F g A = F g (A - B) * F g B" |
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proof - |
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from assms have "finite (A - B)" by auto |
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moreover from assms have "finite B" by (rule finite_subset) |
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moreover from assms have "(A - B) \<inter> B = {}" by auto |
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ultimately have "F g (A - B \<union> B) = F g (A - B) * F g B" by (rule union_disjoint) |
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moreover from assms have "A \<union> B = A" by auto |
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ultimately show ?thesis by simp |
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qed |
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lemma setdiff_irrelevant: |
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assumes "finite A" |
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shows "F g (A - {x. g x = z}) = F g A" |
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using assms by (induct A) (simp_all add: insert_Diff_if) |
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lemma not_neutral_contains_not_neutral: |
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assumes "F g A \<noteq> z" |
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obtains a where "a \<in> A" and "g a \<noteq> z" |
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proof - |
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from assms have "\<exists>a\<in>A. g a \<noteq> z" |
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proof (induct A rule: infinite_finite_induct) |
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case (insert a A) |
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then show ?case by simp (rule, simp) |
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qed simp_all |
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with that show thesis by blast |
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qed |
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lemma reindex: |
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assumes "inj_on h A" |
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shows "F g (h ` A) = F (g \<circ> h) A" |
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proof (cases "finite A") |
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case True |
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with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc) |
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next |
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case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD) |
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with False show ?thesis by simp |
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qed |
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lemma cong: |
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assumes "A = B" |
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assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x" |
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shows "F g A = F h B" |
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using g_h unfolding \<open>A = B\<close> |
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by (induct B rule: infinite_finite_induct) auto |
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lemma strong_cong [cong]: |
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assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x" |
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shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B" |
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by (rule cong) (insert assms, simp_all add: simp_implies_def) |
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lemma reindex_cong: |
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assumes "inj_on l B" |
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assumes "A = l ` B" |
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assumes "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" |
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shows "F g A = F h B" |
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using assms by (simp add: reindex) |
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lemma UNION_disjoint: |
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assumes "finite I" and "\<forall>i\<in>I. finite (A i)" |
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and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" |
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shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I" |
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apply (insert assms) |
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apply (induct rule: finite_induct) |
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apply simp |
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apply atomize |
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166 |
apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i") |
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parents:
diff
changeset
|
167 |
prefer 2 apply blast |
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parents:
diff
changeset
|
168 |
apply (subgoal_tac "A x Int UNION Fa A = {}") |
1e7f2d296e19
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haftmann
parents:
diff
changeset
|
169 |
prefer 2 apply blast |
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haftmann
parents:
diff
changeset
|
170 |
apply (simp add: union_disjoint) |
1e7f2d296e19
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haftmann
parents:
diff
changeset
|
171 |
done |
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more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
172 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
173 |
lemma Union_disjoint: |
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haftmann
parents:
diff
changeset
|
174 |
assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}" |
57418 | 175 |
shows "F g (Union C) = (F \<circ> F) g C" |
54744
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parents:
diff
changeset
|
176 |
proof cases |
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haftmann
parents:
diff
changeset
|
177 |
assume "finite C" |
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diff
changeset
|
178 |
from UNION_disjoint [OF this assms] |
56166 | 179 |
show ?thesis by simp |
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diff
changeset
|
180 |
qed (auto dest: finite_UnionD intro: infinite) |
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parents:
diff
changeset
|
181 |
|
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parents:
diff
changeset
|
182 |
lemma distrib: |
1e7f2d296e19
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parents:
diff
changeset
|
183 |
"F (\<lambda>x. g x * h x) A = F g A * F h A" |
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more algebraic terminology for theories about big operators
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parents:
diff
changeset
|
184 |
using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
185 |
|
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more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
186 |
lemma Sigma: |
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haftmann
parents:
diff
changeset
|
187 |
"finite A \<Longrightarrow> \<forall>x\<in>A. finite (B x) \<Longrightarrow> F (\<lambda>x. F (g x) (B x)) A = F (split g) (SIGMA x:A. B x)" |
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parents:
diff
changeset
|
188 |
apply (subst Sigma_def) |
1e7f2d296e19
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haftmann
parents:
diff
changeset
|
189 |
apply (subst UNION_disjoint, assumption, simp) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
190 |
apply blast |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
191 |
apply (rule cong) |
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haftmann
parents:
diff
changeset
|
192 |
apply rule |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
193 |
apply (simp add: fun_eq_iff) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
194 |
apply (subst UNION_disjoint, simp, simp) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
195 |
apply blast |
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more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
196 |
apply (simp add: comp_def) |
1e7f2d296e19
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haftmann
parents:
diff
changeset
|
197 |
done |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
198 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
199 |
lemma related: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
200 |
assumes Re: "R 1 1" |
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parents:
diff
changeset
|
201 |
and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" |
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parents:
diff
changeset
|
202 |
and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
203 |
shows "R (F h S) (F g S)" |
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more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
204 |
using fS by (rule finite_subset_induct) (insert assms, auto) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
205 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
206 |
lemma mono_neutral_cong_left: |
1e7f2d296e19
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haftmann
parents:
diff
changeset
|
207 |
assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1" |
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parents:
diff
changeset
|
208 |
and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T" |
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haftmann
parents:
diff
changeset
|
209 |
proof- |
60758 | 210 |
have eq: "T = S \<union> (T - S)" using \<open>S \<subseteq> T\<close> by blast |
211 |
have d: "S \<inter> (T - S) = {}" using \<open>S \<subseteq> T\<close> by blast |
|
212 |
from \<open>finite T\<close> \<open>S \<subseteq> T\<close> have f: "finite S" "finite (T - S)" |
|
54744
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haftmann
parents:
diff
changeset
|
213 |
by (auto intro: finite_subset) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
214 |
show ?thesis using assms(4) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
215 |
by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)]) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
216 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
217 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
218 |
lemma mono_neutral_cong_right: |
1e7f2d296e19
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haftmann
parents:
diff
changeset
|
219 |
"\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk> |
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parents:
diff
changeset
|
220 |
\<Longrightarrow> F g T = F h S" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
221 |
by (auto intro!: mono_neutral_cong_left [symmetric]) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
222 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
223 |
lemma mono_neutral_left: |
1e7f2d296e19
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haftmann
parents:
diff
changeset
|
224 |
"\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
225 |
by (blast intro: mono_neutral_cong_left) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
226 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
227 |
lemma mono_neutral_right: |
1e7f2d296e19
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haftmann
parents:
diff
changeset
|
228 |
"\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
229 |
by (blast intro!: mono_neutral_left [symmetric]) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
230 |
|
57129
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|
231 |
lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T" |
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parents:
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diff
changeset
|
232 |
by (auto simp: bij_betw_def reindex) |
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parents:
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diff
changeset
|
233 |
|
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parents:
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diff
changeset
|
234 |
lemma reindex_bij_witness: |
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parents:
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diff
changeset
|
235 |
assumes witness: |
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parents:
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diff
changeset
|
236 |
"\<And>a. a \<in> S \<Longrightarrow> i (j a) = a" |
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parents:
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diff
changeset
|
237 |
"\<And>a. a \<in> S \<Longrightarrow> j a \<in> T" |
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parents:
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diff
changeset
|
238 |
"\<And>b. b \<in> T \<Longrightarrow> j (i b) = b" |
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hoelzl
parents:
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diff
changeset
|
239 |
"\<And>b. b \<in> T \<Longrightarrow> i b \<in> S" |
7edb7550663e
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parents:
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diff
changeset
|
240 |
assumes eq: |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
241 |
"\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
242 |
shows "F g S = F h T" |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
243 |
proof - |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
244 |
have "bij_betw j S T" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
245 |
using bij_betw_byWitness[where A=S and f=j and f'=i and A'=T] witness by auto |
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hoelzl
parents:
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diff
changeset
|
246 |
moreover have "F g S = F (\<lambda>x. h (j x)) S" |
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hoelzl
parents:
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diff
changeset
|
247 |
by (intro cong) (auto simp: eq) |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
248 |
ultimately show ?thesis |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
249 |
by (simp add: reindex_bij_betw) |
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parents:
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diff
changeset
|
250 |
qed |
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hoelzl
parents:
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diff
changeset
|
251 |
|
7edb7550663e
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parents:
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diff
changeset
|
252 |
lemma reindex_bij_betw_not_neutral: |
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parents:
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diff
changeset
|
253 |
assumes fin: "finite S'" "finite T'" |
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hoelzl
parents:
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diff
changeset
|
254 |
assumes bij: "bij_betw h (S - S') (T - T')" |
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hoelzl
parents:
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diff
changeset
|
255 |
assumes nn: |
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hoelzl
parents:
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diff
changeset
|
256 |
"\<And>a. a \<in> S' \<Longrightarrow> g (h a) = z" |
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hoelzl
parents:
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diff
changeset
|
257 |
"\<And>b. b \<in> T' \<Longrightarrow> g b = z" |
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hoelzl
parents:
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diff
changeset
|
258 |
shows "F (\<lambda>x. g (h x)) S = F g T" |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
259 |
proof - |
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hoelzl
parents:
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diff
changeset
|
260 |
have [simp]: "finite S \<longleftrightarrow> finite T" |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
261 |
using bij_betw_finite[OF bij] fin by auto |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
262 |
|
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56545
diff
changeset
|
263 |
show ?thesis |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
264 |
proof cases |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
265 |
assume "finite S" |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
266 |
with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')" |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
267 |
by (intro mono_neutral_cong_right) auto |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
268 |
also have "\<dots> = F g (T - T')" |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
269 |
using bij by (rule reindex_bij_betw) |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
270 |
also have "\<dots> = F g T" |
60758 | 271 |
using nn \<open>finite S\<close> by (intro mono_neutral_cong_left) auto |
57129
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hoelzl
parents:
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diff
changeset
|
272 |
finally show ?thesis . |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
273 |
qed simp |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
274 |
qed |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
275 |
|
57418 | 276 |
lemma reindex_nontrivial: |
277 |
assumes "finite A" |
|
278 |
and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = 1" |
|
279 |
shows "F g (h ` A) = F (g \<circ> h) A" |
|
280 |
proof (subst reindex_bij_betw_not_neutral [symmetric]) |
|
281 |
show "bij_betw h (A - {x \<in> A. (g \<circ> h) x = 1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = 1})" |
|
282 |
using nz by (auto intro!: inj_onI simp: bij_betw_def) |
|
60758 | 283 |
qed (insert \<open>finite A\<close>, auto) |
57418 | 284 |
|
57129
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parents:
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diff
changeset
|
285 |
lemma reindex_bij_witness_not_neutral: |
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hoelzl
parents:
56545
diff
changeset
|
286 |
assumes fin: "finite S'" "finite T'" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
287 |
assumes witness: |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
288 |
"\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
289 |
"\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
290 |
"\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
291 |
"\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
292 |
assumes nn: |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
293 |
"\<And>a. a \<in> S' \<Longrightarrow> g a = z" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
294 |
"\<And>b. b \<in> T' \<Longrightarrow> h b = z" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
295 |
assumes eq: |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
296 |
"\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
297 |
shows "F g S = F h T" |
7edb7550663e
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hoelzl
parents:
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diff
changeset
|
298 |
proof - |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
299 |
have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))" |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
300 |
using witness by (intro bij_betw_byWitness[where f'=i]) auto |
7edb7550663e
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hoelzl
parents:
56545
diff
changeset
|
301 |
have F_eq: "F g S = F (\<lambda>x. h (j x)) S" |
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56545
diff
changeset
|
302 |
by (intro cong) (auto simp: eq) |
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56545
diff
changeset
|
303 |
show ?thesis |
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56545
diff
changeset
|
304 |
unfolding F_eq using fin nn eq |
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56545
diff
changeset
|
305 |
by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto |
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56545
diff
changeset
|
306 |
qed |
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56545
diff
changeset
|
307 |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
308 |
lemma delta: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
309 |
assumes fS: "finite S" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
310 |
shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
311 |
proof- |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
312 |
let ?f = "(\<lambda>k. if k=a then b k else 1)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
313 |
{ assume a: "a \<notin> S" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
314 |
hence "\<forall>k\<in>S. ?f k = 1" by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
315 |
hence ?thesis using a by simp } |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
316 |
moreover |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
317 |
{ assume a: "a \<in> S" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
318 |
let ?A = "S - {a}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
319 |
let ?B = "{a}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
320 |
have eq: "S = ?A \<union> ?B" using a by blast |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
321 |
have dj: "?A \<inter> ?B = {}" by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
322 |
from fS have fAB: "finite ?A" "finite ?B" by auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
323 |
have "F ?f S = F ?f ?A * F ?f ?B" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
324 |
using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]] |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
325 |
by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
326 |
then have ?thesis using a by simp } |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
327 |
ultimately show ?thesis by blast |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
328 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
329 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
330 |
lemma delta': |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
331 |
assumes fS: "finite S" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
332 |
shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
333 |
using delta [OF fS, of a b, symmetric] by (auto intro: cong) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
334 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
335 |
lemma If_cases: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
336 |
fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
337 |
assumes fA: "finite A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
338 |
shows "F (\<lambda>x. if P x then h x else g x) A = |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
339 |
F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
340 |
proof - |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
341 |
have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
342 |
"(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
343 |
by blast+ |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
344 |
from fA |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
345 |
have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
346 |
let ?g = "\<lambda>x. if P x then h x else g x" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
347 |
from union_disjoint [OF f a(2), of ?g] a(1) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
348 |
show ?thesis |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
349 |
by (subst (1 2) cong) simp_all |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
350 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
351 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
352 |
lemma cartesian_product: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
353 |
"F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
354 |
apply (rule sym) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
355 |
apply (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
356 |
apply (cases "finite B") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
357 |
apply (simp add: Sigma) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
358 |
apply (cases "A={}", simp) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
359 |
apply simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
360 |
apply (auto intro: infinite dest: finite_cartesian_productD2) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
361 |
apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
362 |
done |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
363 |
|
57418 | 364 |
lemma inter_restrict: |
365 |
assumes "finite A" |
|
366 |
shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else 1) A" |
|
367 |
proof - |
|
368 |
let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else 1" |
|
369 |
have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else 1) = 1" |
|
370 |
by simp |
|
371 |
moreover have "A \<inter> B \<subseteq> A" by blast |
|
60758 | 372 |
ultimately have "F ?g (A \<inter> B) = F ?g A" using \<open>finite A\<close> |
57418 | 373 |
by (intro mono_neutral_left) auto |
374 |
then show ?thesis by simp |
|
375 |
qed |
|
376 |
||
377 |
lemma inter_filter: |
|
378 |
"finite A \<Longrightarrow> F g {x \<in> A. P x} = F (\<lambda>x. if P x then g x else 1) A" |
|
379 |
by (simp add: inter_restrict [symmetric, of A "{x. P x}" g, simplified mem_Collect_eq] Int_def) |
|
380 |
||
381 |
lemma Union_comp: |
|
382 |
assumes "\<forall>A \<in> B. finite A" |
|
383 |
and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = 1" |
|
384 |
shows "F g (\<Union>B) = (F \<circ> F) g B" |
|
385 |
using assms proof (induct B rule: infinite_finite_induct) |
|
386 |
case (infinite A) |
|
387 |
then have "\<not> finite (\<Union>A)" by (blast dest: finite_UnionD) |
|
388 |
with infinite show ?case by simp |
|
389 |
next |
|
390 |
case empty then show ?case by simp |
|
391 |
next |
|
392 |
case (insert A B) |
|
393 |
then have "finite A" "finite B" "finite (\<Union>B)" "A \<notin> B" |
|
394 |
and "\<forall>x\<in>A \<inter> \<Union>B. g x = 1" |
|
395 |
and H: "F g (\<Union>B) = (F o F) g B" by auto |
|
396 |
then have "F g (A \<union> \<Union>B) = F g A * F g (\<Union>B)" |
|
397 |
by (simp add: union_inter_neutral) |
|
60758 | 398 |
with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case |
57418 | 399 |
by (simp add: H) |
400 |
qed |
|
401 |
||
402 |
lemma commute: |
|
403 |
"F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B" |
|
404 |
unfolding cartesian_product |
|
405 |
by (rule reindex_bij_witness [where i = "\<lambda>(i, j). (j, i)" and j = "\<lambda>(i, j). (j, i)"]) auto |
|
406 |
||
407 |
lemma commute_restrict: |
|
408 |
"finite A \<Longrightarrow> finite B \<Longrightarrow> |
|
409 |
F (\<lambda>x. F (g x) {y. y \<in> B \<and> R x y}) A = F (\<lambda>y. F (\<lambda>x. g x y) {x. x \<in> A \<and> R x y}) B" |
|
410 |
by (simp add: inter_filter) (rule commute) |
|
411 |
||
412 |
lemma Plus: |
|
413 |
fixes A :: "'b set" and B :: "'c set" |
|
414 |
assumes fin: "finite A" "finite B" |
|
415 |
shows "F g (A <+> B) = F (g \<circ> Inl) A * F (g \<circ> Inr) B" |
|
416 |
proof - |
|
417 |
have "A <+> B = Inl ` A \<union> Inr ` B" by auto |
|
418 |
moreover from fin have "finite (Inl ` A :: ('b + 'c) set)" "finite (Inr ` B :: ('b + 'c) set)" |
|
419 |
by auto |
|
420 |
moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('b + 'c) set)" by auto |
|
421 |
moreover have "inj_on (Inl :: 'b \<Rightarrow> 'b + 'c) A" "inj_on (Inr :: 'c \<Rightarrow> 'b + 'c) B" |
|
422 |
by (auto intro: inj_onI) |
|
423 |
ultimately show ?thesis using fin |
|
424 |
by (simp add: union_disjoint reindex) |
|
425 |
qed |
|
426 |
||
58195 | 427 |
lemma same_carrier: |
428 |
assumes "finite C" |
|
429 |
assumes subset: "A \<subseteq> C" "B \<subseteq> C" |
|
430 |
assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 1" |
|
431 |
shows "F g A = F h B \<longleftrightarrow> F g C = F h C" |
|
432 |
proof - |
|
60758 | 433 |
from \<open>finite C\<close> subset have |
58195 | 434 |
"finite A" and "finite B" and "finite (C - A)" and "finite (C - B)" |
435 |
by (auto elim: finite_subset) |
|
436 |
from subset have [simp]: "A - (C - A) = A" by auto |
|
437 |
from subset have [simp]: "B - (C - B) = B" by auto |
|
438 |
from subset have "C = A \<union> (C - A)" by auto |
|
439 |
then have "F g C = F g (A \<union> (C - A))" by simp |
|
440 |
also have "\<dots> = F g (A - (C - A)) * F g (C - A - A) * F g (A \<inter> (C - A))" |
|
60758 | 441 |
using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2) |
58195 | 442 |
finally have P: "F g C = F g A" using trivial by simp |
443 |
from subset have "C = B \<union> (C - B)" by auto |
|
444 |
then have "F h C = F h (B \<union> (C - B))" by simp |
|
445 |
also have "\<dots> = F h (B - (C - B)) * F h (C - B - B) * F h (B \<inter> (C - B))" |
|
60758 | 446 |
using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2) |
58195 | 447 |
finally have Q: "F h C = F h B" using trivial by simp |
448 |
from P Q show ?thesis by simp |
|
449 |
qed |
|
450 |
||
451 |
lemma same_carrierI: |
|
452 |
assumes "finite C" |
|
453 |
assumes subset: "A \<subseteq> C" "B \<subseteq> C" |
|
454 |
assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 1" |
|
455 |
assumes "F g C = F h C" |
|
456 |
shows "F g A = F h B" |
|
457 |
using assms same_carrier [of C A B] by simp |
|
458 |
||
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
459 |
end |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
460 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
461 |
notation times (infixl "*" 70) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
462 |
notation Groups.one ("1") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
463 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
464 |
|
60758 | 465 |
subsection \<open>Generalized summation over a set\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
466 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
467 |
context comm_monoid_add |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
468 |
begin |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
469 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
470 |
definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
471 |
where |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
472 |
"setsum = comm_monoid_set.F plus 0" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
473 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
474 |
sublocale setsum!: comm_monoid_set plus 0 |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
475 |
where |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
476 |
"comm_monoid_set.F plus 0 = setsum" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
477 |
proof - |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
478 |
show "comm_monoid_set plus 0" .. |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
479 |
then interpret setsum!: comm_monoid_set plus 0 . |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
480 |
from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
481 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
482 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
483 |
abbreviation |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
484 |
Setsum ("\<Sum>_" [1000] 999) where |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
485 |
"\<Sum>A \<equiv> setsum (%x. x) A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
486 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
487 |
end |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
488 |
|
60758 | 489 |
text\<open>Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is |
490 |
written @{text"\<Sum>x\<in>A. e"}.\<close> |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
491 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
492 |
syntax |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
493 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_./ _)" [0, 51, 10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
494 |
syntax (xsymbols) |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
495 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
496 |
syntax (HTML output) |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
497 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
498 |
|
60758 | 499 |
translations -- \<open>Beware of argument permutation!\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
500 |
"SUM i:A. b" == "CONST setsum (%i. b) A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
501 |
"\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
502 |
|
60758 | 503 |
text\<open>Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter |
504 |
@{text"\<Sum>x|P. e"}.\<close> |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
505 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
506 |
syntax |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
507 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
508 |
syntax (xsymbols) |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
509 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Sum>_ | (_)./ _)" [0,0,10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
510 |
syntax (HTML output) |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
511 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Sum>_ | (_)./ _)" [0,0,10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
512 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
513 |
translations |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
514 |
"SUM x|P. t" => "CONST setsum (%x. t) {x. P}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
515 |
"\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
516 |
|
60758 | 517 |
print_translation \<open> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
518 |
let |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
519 |
fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] = |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
520 |
if x <> y then raise Match |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
521 |
else |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
522 |
let |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
523 |
val x' = Syntax_Trans.mark_bound_body (x, Tx); |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
524 |
val t' = subst_bound (x', t); |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
525 |
val P' = subst_bound (x', P); |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
526 |
in |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
527 |
Syntax.const @{syntax_const "_qsetsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t' |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
528 |
end |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
529 |
| setsum_tr' _ = raise Match; |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
530 |
in [(@{const_syntax setsum}, K setsum_tr')] end |
60758 | 531 |
\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
532 |
|
60758 | 533 |
text \<open>TODO generalization candidates\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
534 |
|
57418 | 535 |
lemma setsum_image_gen: |
536 |
assumes fS: "finite S" |
|
537 |
shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)" |
|
538 |
proof- |
|
539 |
{ fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto } |
|
540 |
hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S" |
|
541 |
by simp |
|
542 |
also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)" |
|
543 |
by (rule setsum.commute_restrict [OF fS finite_imageI [OF fS]]) |
|
544 |
finally show ?thesis . |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
545 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
546 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
547 |
|
60758 | 548 |
subsubsection \<open>Properties in more restricted classes of structures\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
549 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
550 |
lemma setsum_Un: "finite A ==> finite B ==> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
551 |
(setsum f (A Un B) :: 'a :: ab_group_add) = |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
552 |
setsum f A + setsum f B - setsum f (A Int B)" |
57418 | 553 |
by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
554 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
555 |
lemma setsum_Un2: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
556 |
assumes "finite (A \<union> B)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
557 |
shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
558 |
proof - |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
559 |
have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
560 |
by auto |
57418 | 561 |
with assms show ?thesis by simp (subst setsum.union_disjoint, auto)+ |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
562 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
563 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
564 |
lemma setsum_diff1: "finite A \<Longrightarrow> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
565 |
(setsum f (A - {a}) :: ('a::ab_group_add)) = |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
566 |
(if a:A then setsum f A - f a else setsum f A)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
567 |
by (erule finite_induct) (auto simp add: insert_Diff_if) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
568 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
569 |
lemma setsum_diff: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
570 |
assumes le: "finite A" "B \<subseteq> A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
571 |
shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
572 |
proof - |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
573 |
from le have finiteB: "finite B" using finite_subset by auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
574 |
show ?thesis using finiteB le |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
575 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
576 |
case empty |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
577 |
thus ?case by auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
578 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
579 |
case (insert x F) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
580 |
thus ?case using le finiteB |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
581 |
by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
582 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
583 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
584 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
585 |
lemma setsum_mono: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
586 |
assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
587 |
shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
588 |
proof (cases "finite K") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
589 |
case True |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
590 |
thus ?thesis using le |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
591 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
592 |
case empty |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
593 |
thus ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
594 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
595 |
case insert |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
596 |
thus ?case using add_mono by fastforce |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
597 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
598 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
599 |
case False then show ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
600 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
601 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
602 |
lemma setsum_strict_mono: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
603 |
fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
604 |
assumes "finite A" "A \<noteq> {}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
605 |
and "!!x. x:A \<Longrightarrow> f x < g x" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
606 |
shows "setsum f A < setsum g A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
607 |
using assms |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
608 |
proof (induct rule: finite_ne_induct) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
609 |
case singleton thus ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
610 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
611 |
case insert thus ?case by (auto simp: add_strict_mono) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
612 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
613 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
614 |
lemma setsum_strict_mono_ex1: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
615 |
fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
616 |
assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
617 |
shows "setsum f A < setsum g A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
618 |
proof- |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
619 |
from assms(3) obtain a where a: "a:A" "f a < g a" by blast |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
620 |
have "setsum f A = setsum f ((A-{a}) \<union> {a})" |
60758 | 621 |
by(simp add:insert_absorb[OF \<open>a:A\<close>]) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
622 |
also have "\<dots> = setsum f (A-{a}) + setsum f {a}" |
60758 | 623 |
using \<open>finite A\<close> by(subst setsum.union_disjoint) auto |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
624 |
also have "setsum f (A-{a}) \<le> setsum g (A-{a})" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
625 |
by(rule setsum_mono)(simp add: assms(2)) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
626 |
also have "setsum f {a} < setsum g {a}" using a by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
627 |
also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})" |
60758 | 628 |
using \<open>finite A\<close> by(subst setsum.union_disjoint[symmetric]) auto |
629 |
also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF \<open>a:A\<close>]) |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
630 |
finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
631 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
632 |
|
59416
fde2659085e1
generalized sum_diff_distrib to setsum_subtractf_nat
hoelzl
parents:
59010
diff
changeset
|
633 |
lemma setsum_negf: "(\<Sum>x\<in>A. - f x::'a::ab_group_add) = - (\<Sum>x\<in>A. f x)" |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
634 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
635 |
case True thus ?thesis by (induct set: finite) auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
636 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
637 |
case False thus ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
638 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
639 |
|
59416
fde2659085e1
generalized sum_diff_distrib to setsum_subtractf_nat
hoelzl
parents:
59010
diff
changeset
|
640 |
lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x::'a::ab_group_add) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)" |
57418 | 641 |
using setsum.distrib [of f "- g" A] by (simp add: setsum_negf) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
642 |
|
59416
fde2659085e1
generalized sum_diff_distrib to setsum_subtractf_nat
hoelzl
parents:
59010
diff
changeset
|
643 |
lemma setsum_subtractf_nat: |
fde2659085e1
generalized sum_diff_distrib to setsum_subtractf_nat
hoelzl
parents:
59010
diff
changeset
|
644 |
"(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x::nat) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)" |
fde2659085e1
generalized sum_diff_distrib to setsum_subtractf_nat
hoelzl
parents:
59010
diff
changeset
|
645 |
by (induction A rule: infinite_finite_induct) (auto simp: setsum_mono) |
fde2659085e1
generalized sum_diff_distrib to setsum_subtractf_nat
hoelzl
parents:
59010
diff
changeset
|
646 |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
647 |
lemma setsum_nonneg: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
648 |
assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
649 |
shows "0 \<le> setsum f A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
650 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
651 |
case True thus ?thesis using nn |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
652 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
653 |
case empty then show ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
654 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
655 |
case (insert x F) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
656 |
then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
657 |
with insert show ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
658 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
659 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
660 |
case False thus ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
661 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
662 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
663 |
lemma setsum_nonpos: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
664 |
assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
665 |
shows "setsum f A \<le> 0" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
666 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
667 |
case True thus ?thesis using np |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
668 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
669 |
case empty then show ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
670 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
671 |
case (insert x F) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
672 |
then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
673 |
with insert show ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
674 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
675 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
676 |
case False thus ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
677 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
678 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
679 |
lemma setsum_nonneg_leq_bound: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
680 |
fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
681 |
assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
682 |
shows "f i \<le> B" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
683 |
proof - |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
684 |
have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
685 |
using assms by (auto intro!: setsum_nonneg) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
686 |
moreover |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
687 |
have "(\<Sum> i \<in> s - {i}. f i) + f i = B" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
688 |
using assms by (simp add: setsum_diff1) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
689 |
ultimately show ?thesis by auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
690 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
691 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
692 |
lemma setsum_nonneg_0: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
693 |
fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
694 |
assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
695 |
and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
696 |
shows "f i = 0" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
697 |
using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
698 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
699 |
lemma setsum_mono2: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
700 |
fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
701 |
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
702 |
shows "setsum f A \<le> setsum f B" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
703 |
proof - |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
704 |
have "setsum f A \<le> setsum f A + setsum f (B-A)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
705 |
by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
706 |
also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin] |
57418 | 707 |
by (simp add: setsum.union_disjoint del:Un_Diff_cancel) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
708 |
also have "A \<union> (B-A) = B" using sub by blast |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
709 |
finally show ?thesis . |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
710 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
711 |
|
57418 | 712 |
lemma setsum_le_included: |
713 |
fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add" |
|
714 |
assumes "finite s" "finite t" |
|
715 |
and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)" |
|
716 |
shows "setsum f s \<le> setsum g t" |
|
717 |
proof - |
|
718 |
have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s" |
|
719 |
proof (rule setsum_mono) |
|
720 |
fix y assume "y \<in> s" |
|
721 |
with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto |
|
722 |
with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y") |
|
723 |
using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro] |
|
724 |
by (auto intro!: setsum_mono2) |
|
725 |
qed |
|
726 |
also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)" |
|
727 |
using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg) |
|
728 |
also have "... \<le> setsum g t" |
|
729 |
using assms by (auto simp: setsum_image_gen[symmetric]) |
|
730 |
finally show ?thesis . |
|
731 |
qed |
|
732 |
||
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
733 |
lemma setsum_mono3: "finite B ==> A <= B ==> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
734 |
ALL x: B - A. |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
735 |
0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
736 |
setsum f A <= setsum f B" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
737 |
apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
738 |
apply (erule ssubst) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
739 |
apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
740 |
apply simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
741 |
apply (rule add_left_mono) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
742 |
apply (erule setsum_nonneg) |
57418 | 743 |
apply (subst setsum.union_disjoint [THEN sym]) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
744 |
apply (erule finite_subset, assumption) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
745 |
apply (rule finite_subset) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
746 |
prefer 2 |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
747 |
apply assumption |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
748 |
apply (auto simp add: sup_absorb2) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
749 |
done |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
750 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
751 |
lemma setsum_right_distrib: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
752 |
fixes f :: "'a => ('b::semiring_0)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
753 |
shows "r * setsum f A = setsum (%n. r * f n) A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
754 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
755 |
case True |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
756 |
thus ?thesis |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
757 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
758 |
case empty thus ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
759 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
760 |
case (insert x A) thus ?case by (simp add: distrib_left) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
761 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
762 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
763 |
case False thus ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
764 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
765 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
766 |
lemma setsum_left_distrib: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
767 |
"setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
768 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
769 |
case True |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
770 |
then show ?thesis |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
771 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
772 |
case empty thus ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
773 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
774 |
case (insert x A) thus ?case by (simp add: distrib_right) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
775 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
776 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
777 |
case False thus ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
778 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
779 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
780 |
lemma setsum_divide_distrib: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
781 |
"setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
782 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
783 |
case True |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
784 |
then show ?thesis |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
785 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
786 |
case empty thus ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
787 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
788 |
case (insert x A) thus ?case by (simp add: add_divide_distrib) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
789 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
790 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
791 |
case False thus ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
792 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
793 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
794 |
lemma setsum_abs[iff]: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
795 |
fixes f :: "'a => ('b::ordered_ab_group_add_abs)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
796 |
shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
797 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
798 |
case True |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
799 |
thus ?thesis |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
800 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
801 |
case empty thus ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
802 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
803 |
case (insert x A) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
804 |
thus ?case by (auto intro: abs_triangle_ineq order_trans) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
805 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
806 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
807 |
case False thus ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
808 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
809 |
|
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
810 |
lemma setsum_abs_ge_zero[iff]: |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
811 |
fixes f :: "'a => ('b::ordered_ab_group_add_abs)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
812 |
shows "0 \<le> setsum (%i. abs(f i)) A" |
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
813 |
by (simp add: setsum_nonneg) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
814 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
815 |
lemma abs_setsum_abs[simp]: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
816 |
fixes f :: "'a => ('b::ordered_ab_group_add_abs)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
817 |
shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
818 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
819 |
case True |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
820 |
thus ?thesis |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
821 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
822 |
case empty thus ?case by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
823 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
824 |
case (insert a A) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
825 |
hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
826 |
also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>" using insert by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
827 |
also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
828 |
by (simp del: abs_of_nonneg) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
829 |
also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
830 |
finally show ?case . |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
831 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
832 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
833 |
case False thus ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
834 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
835 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
836 |
lemma setsum_diff1_ring: assumes "finite A" "a \<in> A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
837 |
shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)" |
57418 | 838 |
unfolding setsum.remove [OF assms] by auto |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
839 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
840 |
lemma setsum_product: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
841 |
fixes f :: "'a => ('b::semiring_0)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
842 |
shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)" |
57418 | 843 |
by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
844 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
845 |
lemma setsum_mult_setsum_if_inj: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
846 |
fixes f :: "'a => ('b::semiring_0)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
847 |
shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
848 |
setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}" |
57418 | 849 |
by(auto simp: setsum_product setsum.cartesian_product |
850 |
intro!: setsum.reindex_cong[symmetric]) |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
851 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
852 |
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
853 |
apply (case_tac "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
854 |
prefer 2 apply simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
855 |
apply (erule rev_mp) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
856 |
apply (erule finite_induct, auto) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
857 |
done |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
858 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
859 |
lemma setsum_eq_0_iff [simp]: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
860 |
"finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
861 |
by (induct set: finite) auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
862 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
863 |
lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
864 |
setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
865 |
apply(erule finite_induct) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
866 |
apply (auto simp add:add_is_1) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
867 |
done |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
868 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
869 |
lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]] |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
870 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
871 |
lemma setsum_Un_nat: "finite A ==> finite B ==> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
872 |
(setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" |
60758 | 873 |
-- \<open>For the natural numbers, we have subtraction.\<close> |
57418 | 874 |
by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
875 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
876 |
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) = |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
877 |
(if a:A then setsum f A - f a else setsum f A)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
878 |
apply (case_tac "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
879 |
prefer 2 apply simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
880 |
apply (erule finite_induct) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
881 |
apply (auto simp add: insert_Diff_if) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
882 |
apply (drule_tac a = a in mk_disjoint_insert, auto) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
883 |
done |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
884 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
885 |
lemma setsum_diff_nat: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
886 |
assumes "finite B" and "B \<subseteq> A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
887 |
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
888 |
using assms |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
889 |
proof induct |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
890 |
show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
891 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
892 |
fix F x assume finF: "finite F" and xnotinF: "x \<notin> F" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
893 |
and xFinA: "insert x F \<subseteq> A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
894 |
and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
895 |
from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
896 |
from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
897 |
by (simp add: setsum_diff1_nat) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
898 |
from xFinA have "F \<subseteq> A" by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
899 |
with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
900 |
with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
901 |
by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
902 |
from xnotinF have "A - insert x F = (A - F) - {x}" by auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
903 |
with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
904 |
by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
905 |
from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
906 |
with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
907 |
by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
908 |
thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
909 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
910 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
911 |
lemma setsum_comp_morphism: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
912 |
assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
913 |
shows "setsum (h \<circ> g) A = h (setsum g A)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
914 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
915 |
case False then show ?thesis by (simp add: assms) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
916 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
917 |
case True then show ?thesis by (induct A) (simp_all add: assms) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
918 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
919 |
|
59010 | 920 |
lemma (in comm_semiring_1) dvd_setsum: |
921 |
"(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A" |
|
922 |
by (induct A rule: infinite_finite_induct) simp_all |
|
923 |
||
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
924 |
lemma setsum_pos2: |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
925 |
assumes "finite I" "i \<in> I" "0 < f i" "(\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i)" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
926 |
shows "(0::'a::{ordered_ab_group_add,comm_monoid_add}) < setsum f I" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
927 |
proof - |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
928 |
have "0 \<le> setsum f (I-{i})" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
929 |
using assms by (simp add: setsum_nonneg) |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
930 |
also have "... < setsum f (I-{i}) + f i" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
931 |
using assms by auto |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
932 |
also have "... = setsum f I" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
933 |
using assms by (simp add: setsum.remove) |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
934 |
finally show ?thesis . |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
935 |
qed |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
936 |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
937 |
|
60758 | 938 |
subsubsection \<open>Cardinality as special case of @{const setsum}\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
939 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
940 |
lemma card_eq_setsum: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
941 |
"card A = setsum (\<lambda>x. 1) A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
942 |
proof - |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
943 |
have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
944 |
by (simp add: fun_eq_iff) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
945 |
then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
946 |
by (rule arg_cong) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
947 |
then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
948 |
by (blast intro: fun_cong) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
949 |
then show ?thesis by (simp add: card.eq_fold setsum.eq_fold) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
950 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
951 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
952 |
lemma setsum_constant [simp]: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
953 |
"(\<Sum>x \<in> A. y) = of_nat (card A) * y" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
954 |
apply (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
955 |
apply (erule finite_induct) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
956 |
apply (auto simp add: algebra_simps) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
957 |
done |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
958 |
|
59615
fdfdf89a83a6
A few new lemmas and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
59416
diff
changeset
|
959 |
lemma setsum_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A" |
fdfdf89a83a6
A few new lemmas and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
59416
diff
changeset
|
960 |
using setsum.distrib[of f "\<lambda>_. 1" A] |
fdfdf89a83a6
A few new lemmas and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
59416
diff
changeset
|
961 |
by simp |
58349 | 962 |
|
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
963 |
lemma setsum_bounded_above: |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
964 |
assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
965 |
shows "setsum f A \<le> of_nat (card A) * K" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
966 |
proof (cases "finite A") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
967 |
case True |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
968 |
thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
969 |
next |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
970 |
case False thus ?thesis by simp |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
971 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
972 |
|
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
973 |
lemma setsum_bounded_above_strict: |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
974 |
assumes "\<And>i. i\<in>A \<Longrightarrow> f i < (K::'a::{ordered_cancel_ab_semigroup_add,semiring_1})" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
975 |
"card A > 0" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
976 |
shows "setsum f A < of_nat (card A) * K" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
977 |
using assms setsum_strict_mono[where A=A and g = "%x. K"] |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
978 |
by (simp add: card_gt_0_iff) |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
979 |
|
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
980 |
lemma setsum_bounded_below: |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
981 |
assumes le: "\<And>i. i\<in>A \<Longrightarrow> (K::'a::{semiring_1, ordered_ab_semigroup_add}) \<le> f i" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
982 |
shows "of_nat (card A) * K \<le> setsum f A" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
983 |
proof (cases "finite A") |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
984 |
case True |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
985 |
thus ?thesis using le setsum_mono[where K=A and f = "%x. K"] by simp |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
986 |
next |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
987 |
case False thus ?thesis by simp |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
988 |
qed |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
989 |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
990 |
lemma card_UN_disjoint: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
991 |
assumes "finite I" and "\<forall>i\<in>I. finite (A i)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
992 |
and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
993 |
shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
994 |
proof - |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
995 |
have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp |
57418 | 996 |
with assms show ?thesis by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
997 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
998 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
999 |
lemma card_Union_disjoint: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1000 |
"finite C ==> (ALL A:C. finite A) ==> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1001 |
(ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1002 |
==> card (Union C) = setsum card C" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1003 |
apply (frule card_UN_disjoint [of C id]) |
56166 | 1004 |
apply simp_all |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1005 |
done |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1006 |
|
57418 | 1007 |
lemma setsum_multicount_gen: |
1008 |
assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)" |
|
1009 |
shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r") |
|
1010 |
proof- |
|
1011 |
have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto |
|
1012 |
also have "\<dots> = ?r" unfolding setsum.commute_restrict [OF assms(1-2)] |
|
1013 |
using assms(3) by auto |
|
1014 |
finally show ?thesis . |
|
1015 |
qed |
|
1016 |
||
1017 |
lemma setsum_multicount: |
|
1018 |
assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)" |
|
1019 |
shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r") |
|
1020 |
proof- |
|
1021 |
have "?l = setsum (\<lambda>i. k) T" by (rule setsum_multicount_gen) (auto simp: assms) |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
1022 |
also have "\<dots> = ?r" by (simp add: mult.commute) |
57418 | 1023 |
finally show ?thesis by auto |
1024 |
qed |
|
1025 |
||
58437 | 1026 |
lemma (in ordered_comm_monoid_add) setsum_pos: |
1027 |
"finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < setsum f I" |
|
1028 |
by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos) |
|
1029 |
||
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1030 |
|
60758 | 1031 |
subsubsection \<open>Cardinality of products\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1032 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1033 |
lemma card_SigmaI [simp]: |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1034 |
"\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1035 |
\<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))" |
57418 | 1036 |
by(simp add: card_eq_setsum setsum.Sigma del:setsum_constant) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1037 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1038 |
(* |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1039 |
lemma SigmaI_insert: "y \<notin> A ==> |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1040 |
(SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1041 |
by auto |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1042 |
*) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1043 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1044 |
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1045 |
by (cases "finite A \<and> finite B") |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1046 |
(auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1047 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1048 |
lemma card_cartesian_product_singleton: "card({x} <*> A) = card(A)" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1049 |
by (simp add: card_cartesian_product) |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1050 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1051 |
|
60758 | 1052 |
subsection \<open>Generalized product over a set\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1053 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1054 |
context comm_monoid_mult |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1055 |
begin |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1056 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1057 |
definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1058 |
where |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1059 |
"setprod = comm_monoid_set.F times 1" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1060 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1061 |
sublocale setprod!: comm_monoid_set times 1 |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1062 |
where |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1063 |
"comm_monoid_set.F times 1 = setprod" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1064 |
proof - |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1065 |
show "comm_monoid_set times 1" .. |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1066 |
then interpret setprod!: comm_monoid_set times 1 . |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1067 |
from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1068 |
qed |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1069 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1070 |
abbreviation |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1071 |
Setprod ("\<Prod>_" [1000] 999) where |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1072 |
"\<Prod>A \<equiv> setprod (\<lambda>x. x) A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1073 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1074 |
end |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1075 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1076 |
syntax |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
1077 |
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(4PROD _:_./ _)" [0, 51, 10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1078 |
syntax (xsymbols) |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
1079 |
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1080 |
syntax (HTML output) |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
1081 |
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1082 |
|
60758 | 1083 |
translations -- \<open>Beware of argument permutation!\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1084 |
"PROD i:A. b" == "CONST setprod (%i. b) A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1085 |
"\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1086 |
|
60758 | 1087 |
text\<open>Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter |
1088 |
@{text"\<Prod>x|P. e"}.\<close> |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1089 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1090 |
syntax |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
1091 |
"_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(4PROD _ |/ _./ _)" [0,0,10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1092 |
syntax (xsymbols) |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
1093 |
"_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Prod>_ | (_)./ _)" [0,0,10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1094 |
syntax (HTML output) |
60494
e726f88232d3
correccted the pretty-printing specs for setsum and setprod
paulson <lp15@cam.ac.uk>
parents:
60429
diff
changeset
|
1095 |
"_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Prod>_ | (_)./ _)" [0,0,10] 10) |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1096 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1097 |
translations |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1098 |
"PROD x|P. t" => "CONST setprod (%x. t) {x. P}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1099 |
"\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1100 |
|
59010 | 1101 |
context comm_monoid_mult |
1102 |
begin |
|
1103 |
||
1104 |
lemma setprod_dvd_setprod: |
|
1105 |
"(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A" |
|
1106 |
proof (induct A rule: infinite_finite_induct) |
|
1107 |
case infinite then show ?case by (auto intro: dvdI) |
|
1108 |
next |
|
1109 |
case empty then show ?case by (auto intro: dvdI) |
|
1110 |
next |
|
1111 |
case (insert a A) then |
|
1112 |
have "f a dvd g a" and "setprod f A dvd setprod g A" by simp_all |
|
1113 |
then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s" by (auto elim!: dvdE) |
|
1114 |
then have "g a * setprod g A = f a * setprod f A * (r * s)" by (simp add: ac_simps) |
|
1115 |
with insert.hyps show ?case by (auto intro: dvdI) |
|
1116 |
qed |
|
1117 |
||
1118 |
lemma setprod_dvd_setprod_subset: |
|
1119 |
"finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B" |
|
1120 |
by (auto simp add: setprod.subset_diff ac_simps intro: dvdI) |
|
1121 |
||
1122 |
end |
|
1123 |
||
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1124 |
|
60758 | 1125 |
subsubsection \<open>Properties in more restricted classes of structures\<close> |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1126 |
|
59010 | 1127 |
context comm_semiring_1 |
1128 |
begin |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1129 |
|
59010 | 1130 |
lemma dvd_setprod_eqI [intro]: |
1131 |
assumes "finite A" and "a \<in> A" and "b = f a" |
|
1132 |
shows "b dvd setprod f A" |
|
1133 |
proof - |
|
60758 | 1134 |
from \<open>finite A\<close> have "setprod f (insert a (A - {a})) = f a * setprod f (A - {a})" |
59010 | 1135 |
by (intro setprod.insert) auto |
60758 | 1136 |
also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A" by blast |
59010 | 1137 |
finally have "setprod f A = f a * setprod f (A - {a})" . |
60758 | 1138 |
with \<open>b = f a\<close> show ?thesis by simp |
59010 | 1139 |
qed |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1140 |
|
59010 | 1141 |
lemma dvd_setprodI [intro]: |
1142 |
assumes "finite A" and "a \<in> A" |
|
1143 |
shows "f a dvd setprod f A" |
|
1144 |
using assms by auto |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1145 |
|
59010 | 1146 |
lemma setprod_zero: |
1147 |
assumes "finite A" and "\<exists>a\<in>A. f a = 0" |
|
1148 |
shows "setprod f A = 0" |
|
1149 |
using assms proof (induct A) |
|
1150 |
case empty then show ?case by simp |
|
1151 |
next |
|
1152 |
case (insert a A) |
|
1153 |
then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp |
|
1154 |
then have "f a * setprod f A = 0" by rule (simp_all add: insert) |
|
1155 |
with insert show ?case by simp |
|
1156 |
qed |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1157 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1158 |
lemma setprod_dvd_setprod_subset2: |
59010 | 1159 |
assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a" |
1160 |
shows "setprod f A dvd setprod g B" |
|
1161 |
proof - |
|
1162 |
from assms have "setprod f A dvd setprod g A" |
|
1163 |
by (auto intro: setprod_dvd_setprod) |
|
1164 |
moreover from assms have "setprod g A dvd setprod g B" |
|
1165 |
by (auto intro: setprod_dvd_setprod_subset) |
|
1166 |
ultimately show ?thesis by (rule dvd_trans) |
|
1167 |
qed |
|
1168 |
||
1169 |
end |
|
1170 |
||
1171 |
lemma setprod_zero_iff [simp]: |
|
1172 |
assumes "finite A" |
|
59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59615
diff
changeset
|
1173 |
shows "setprod f A = (0::'a::semidom) \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)" |
59010 | 1174 |
using assms by (induct A) (auto simp: no_zero_divisors) |
1175 |
||
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1176 |
lemma (in semidom_divide) setprod_diff1: |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1177 |
assumes "finite A" and "f a \<noteq> 0" |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60353
diff
changeset
|
1178 |
shows "setprod f (A - {a}) = (if a \<in> A then setprod f A div f a else setprod f A)" |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1179 |
proof (cases "a \<notin> A") |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1180 |
case True then show ?thesis by simp |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1181 |
next |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1182 |
case False with assms show ?thesis |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1183 |
proof (induct A rule: finite_induct) |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1184 |
case empty then show ?case by simp |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1185 |
next |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1186 |
case (insert b B) |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1187 |
then show ?case |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1188 |
proof (cases "a = b") |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1189 |
case True with insert show ?thesis by simp |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1190 |
next |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1191 |
case False with insert have "a \<in> B" by simp |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1192 |
def C \<equiv> "B - {a}" |
60758 | 1193 |
with \<open>finite B\<close> \<open>a \<in> B\<close> |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1194 |
have *: "B = insert a C" "finite C" "a \<notin> C" by auto |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1195 |
with insert show ?thesis by (auto simp add: insert_commute ac_simps) |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1196 |
qed |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1197 |
qed |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
59867
diff
changeset
|
1198 |
qed |
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1199 |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59833
diff
changeset
|
1200 |
lemma (in field) setprod_inversef: |
59010 | 1201 |
"finite A \<Longrightarrow> setprod (inverse \<circ> f) A = inverse (setprod f A)" |
1202 |
by (induct A rule: finite_induct) simp_all |
|
1203 |
||
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59833
diff
changeset
|
1204 |
lemma (in field) setprod_dividef: |
59010 | 1205 |
"finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A" |
1206 |
using setprod_inversef [of A g] by (simp add: divide_inverse setprod.distrib) |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1207 |
|
59010 | 1208 |
lemma setprod_Un: |
1209 |
fixes f :: "'b \<Rightarrow> 'a :: field" |
|
1210 |
assumes "finite A" and "finite B" |
|
1211 |
and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0" |
|
1212 |
shows "setprod f (A \<union> B) = setprod f A * setprod f B / setprod f (A \<inter> B)" |
|
1213 |
proof - |
|
1214 |
from assms have "setprod f A * setprod f B = setprod f (A \<union> B) * setprod f (A \<inter> B)" |
|
1215 |
by (simp add: setprod.union_inter [symmetric, of A B]) |
|
1216 |
with assms show ?thesis by simp |
|
1217 |
qed |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1218 |
|
59010 | 1219 |
lemma (in linordered_semidom) setprod_nonneg: |
1220 |
"(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A" |
|
1221 |
by (induct A rule: infinite_finite_induct) simp_all |
|
1222 |
||
1223 |
lemma (in linordered_semidom) setprod_pos: |
|
1224 |
"(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A" |
|
1225 |
by (induct A rule: infinite_finite_induct) simp_all |
|
1226 |
||
1227 |
lemma (in linordered_semidom) setprod_mono: |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1228 |
assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i" |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1229 |
shows "setprod f A \<le> setprod g A" |
59010 | 1230 |
using assms by (induct A rule: infinite_finite_induct) |
1231 |
(auto intro!: setprod_nonneg mult_mono) |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1232 |
|
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1233 |
lemma (in linordered_semidom) setprod_mono_strict: |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1234 |
assumes"finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1235 |
shows "setprod f A < setprod g A" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1236 |
using assms |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1237 |
apply (induct A rule: finite_induct) |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1238 |
apply (simp add: ) |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1239 |
apply (force intro: mult_strict_mono' setprod_nonneg) |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1240 |
done |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1241 |
|
59010 | 1242 |
lemma (in linordered_field) abs_setprod: |
1243 |
"\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)" |
|
1244 |
by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult) |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1245 |
|
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1246 |
lemma setprod_eq_1_iff [simp]: |
59010 | 1247 |
"finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = (1::nat))" |
1248 |
by (induct A rule: finite_induct) simp_all |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1249 |
|
59010 | 1250 |
lemma setprod_pos_nat_iff [simp]: |
1251 |
"finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > (0::nat))" |
|
1252 |
using setprod_zero_iff by (simp del:neq0_conv add:neq0_conv [symmetric]) |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1253 |
|
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1254 |
lemma setsum_nonneg_eq_0_iff: |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1255 |
fixes f :: "'a \<Rightarrow> 'b::ordered_ab_group_add" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1256 |
shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)" |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1257 |
apply (induct set: finite, simp) |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1258 |
apply (simp add: add_nonneg_eq_0_iff setsum_nonneg) |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1259 |
done |
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
1260 |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
diff
changeset
|
1261 |
end |