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author | haftmann |

Tue, 11 Jan 2022 06:48:02 +0000 | |

changeset 74979 | 4d77dd3019d1 |

parent 74978 | 1f30aa91f496 |

child 74982 | a10873b3c7d4 |

earlier availability of lifting

--- a/src/HOL/Equiv_Relations.thy Tue Jan 11 06:47:47 2022 +0000 +++ b/src/HOL/Equiv_Relations.thy Tue Jan 11 06:48:02 2022 +0000 @@ -5,7 +5,7 @@ section \<open>Equivalence Relations in Higher-Order Set Theory\<close> theory Equiv_Relations - imports Groups_Big + imports BNF_Least_Fixpoint begin subsection \<open>Equivalence relations -- set version\<close> @@ -348,60 +348,6 @@ by (auto simp: Union_quotient dest: quotient_disj) qed (use assms in blast) -lemma card_quotient_disjoint: - assumes "finite A" "inj_on (\<lambda>x. {x} // r) A" - shows "card (A//r) = card A" -proof - - have "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> r `` {j} \<noteq> r `` {i}" - using assms by (fastforce simp add: quotient_def inj_on_def) - with assms show ?thesis - by (simp add: quotient_def card_UN_disjoint) -qed - -text \<open>By Jakub Kądziołka:\<close> - -lemma sum_fun_comp: - assumes "finite S" "finite R" "g ` S \<subseteq> R" - shows "(\<Sum>x \<in> S. f (g x)) = (\<Sum>y \<in> R. of_nat (card {x \<in> S. g x = y}) * f y)" -proof - - let ?r = "relation_of (\<lambda>p q. g p = g q) S" - have eqv: "equiv S ?r" - unfolding relation_of_def by (auto intro: comp_equivI) - have finite: "C \<in> S//?r \<Longrightarrow> finite C" for C - by (fact finite_equiv_class[OF \<open>finite S\<close> equiv_type[OF \<open>equiv S ?r\<close>]]) - have disjoint: "A \<in> S//?r \<Longrightarrow> B \<in> S//?r \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}" for A B - using eqv quotient_disj by blast - - let ?cls = "\<lambda>y. {x \<in> S. y = g x}" - have quot_as_img: "S//?r = ?cls ` g ` S" - by (auto simp add: relation_of_def quotient_def) - have cls_inj: "inj_on ?cls (g ` S)" - by (auto intro: inj_onI) - - have rest_0: "(\<Sum>y \<in> R - g ` S. of_nat (card (?cls y)) * f y) = 0" - proof - - have "of_nat (card (?cls y)) * f y = 0" if asm: "y \<in> R - g ` S" for y - proof - - from asm have *: "?cls y = {}" by auto - show ?thesis unfolding * by simp - qed - thus ?thesis by simp - qed - - have "(\<Sum>x \<in> S. f (g x)) = (\<Sum>C \<in> S//?r. \<Sum>x \<in> C. f (g x))" - using eqv finite disjoint - by (simp flip: sum.Union_disjoint[simplified] add: Union_quotient) - also have "... = (\<Sum>y \<in> g ` S. \<Sum>x \<in> ?cls y. f (g x))" - unfolding quot_as_img by (simp add: sum.reindex[OF cls_inj]) - also have "... = (\<Sum>y \<in> g ` S. \<Sum>x \<in> ?cls y. f y)" - by auto - also have "... = (\<Sum>y \<in> g ` S. of_nat (card (?cls y)) * f y)" - by (simp flip: sum_constant) - also have "... = (\<Sum>y \<in> R. of_nat (card (?cls y)) * f y)" - using rest_0 by (simp add: sum.subset_diff[OF \<open>g ` S \<subseteq> R\<close> \<open>finite R\<close>]) - finally show ?thesis - by (simp add: eq_commute) -qed subsection \<open>Projection\<close>

--- a/src/HOL/Groups_Big.thy Tue Jan 11 06:47:47 2022 +0000 +++ b/src/HOL/Groups_Big.thy Tue Jan 11 06:48:02 2022 +0000 @@ -8,7 +8,7 @@ section \<open>Big sum and product over finite (non-empty) sets\<close> theory Groups_Big - imports Power + imports Power Equiv_Relations begin subsection \<open>Generic monoid operation over a set\<close> @@ -1259,6 +1259,16 @@ using card_Un_le nat_add_left_cancel_le by (force intro: order_trans) qed auto +lemma card_quotient_disjoint: + assumes "finite A" "inj_on (\<lambda>x. {x} // r) A" + shows "card (A//r) = card A" +proof - + have "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> r `` {j} \<noteq> r `` {i}" + using assms by (fastforce simp add: quotient_def inj_on_def) + with assms show ?thesis + by (simp add: quotient_def card_UN_disjoint) +qed + lemma sum_multicount_gen: assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)" shows "sum (\<lambda>i. (card {j\<in>t. R i j})) s = sum k t" @@ -1303,6 +1313,52 @@ qed qed simp +text \<open>By Jakub Kądziołka:\<close> + +lemma sum_fun_comp: + assumes "finite S" "finite R" "g ` S \<subseteq> R" + shows "(\<Sum>x \<in> S. f (g x)) = (\<Sum>y \<in> R. of_nat (card {x \<in> S. g x = y}) * f y)" +proof - + let ?r = "relation_of (\<lambda>p q. g p = g q) S" + have eqv: "equiv S ?r" + unfolding relation_of_def by (auto intro: comp_equivI) + have finite: "C \<in> S//?r \<Longrightarrow> finite C" for C + by (fact finite_equiv_class[OF `finite S` equiv_type[OF `equiv S ?r`]]) + have disjoint: "A \<in> S//?r \<Longrightarrow> B \<in> S//?r \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}" for A B + using eqv quotient_disj by blast + + let ?cls = "\<lambda>y. {x \<in> S. y = g x}" + have quot_as_img: "S//?r = ?cls ` g ` S" + by (auto simp add: relation_of_def quotient_def) + have cls_inj: "inj_on ?cls (g ` S)" + by (auto intro: inj_onI) + + have rest_0: "(\<Sum>y \<in> R - g ` S. of_nat (card (?cls y)) * f y) = 0" + proof - + have "of_nat (card (?cls y)) * f y = 0" if asm: "y \<in> R - g ` S" for y + proof - + from asm have *: "?cls y = {}" by auto + show ?thesis unfolding * by simp + qed + thus ?thesis by simp + qed + + have "(\<Sum>x \<in> S. f (g x)) = (\<Sum>C \<in> S//?r. \<Sum>x \<in> C. f (g x))" + using eqv finite disjoint + by (simp flip: sum.Union_disjoint[simplified] add: Union_quotient) + also have "... = (\<Sum>y \<in> g ` S. \<Sum>x \<in> ?cls y. f (g x))" + unfolding quot_as_img by (simp add: sum.reindex[OF cls_inj]) + also have "... = (\<Sum>y \<in> g ` S. \<Sum>x \<in> ?cls y. f y)" + by auto + also have "... = (\<Sum>y \<in> g ` S. of_nat (card (?cls y)) * f y)" + by (simp flip: sum_constant) + also have "... = (\<Sum>y \<in> R. of_nat (card (?cls y)) * f y)" + using rest_0 by (simp add: sum.subset_diff[OF \<open>g ` S \<subseteq> R\<close> \<open>finite R\<close>]) + finally show ?thesis + by (simp add: eq_commute) +qed + + subsubsection \<open>Cardinality of products\<close>

--- a/src/HOL/Int.thy Tue Jan 11 06:47:47 2022 +0000 +++ b/src/HOL/Int.thy Tue Jan 11 06:48:02 2022 +0000 @@ -6,7 +6,7 @@ section \<open>The Integers as Equivalence Classes over Pairs of Natural Numbers\<close> theory Int - imports Equiv_Relations Power Quotient Fun_Def + imports Quotient Groups_Big Fun_Def begin subsection \<open>Definition of integers as a quotient type\<close>

--- a/src/HOL/Lattices_Big.thy Tue Jan 11 06:47:47 2022 +0000 +++ b/src/HOL/Lattices_Big.thy Tue Jan 11 06:48:02 2022 +0000 @@ -6,7 +6,7 @@ section \<open>Big infimum (minimum) and supremum (maximum) over finite (non-empty) sets\<close> theory Lattices_Big - imports Option + imports Groups_Big Option begin subsection \<open>Generic lattice operations over a set\<close>

--- a/src/HOL/Lifting_Set.thy Tue Jan 11 06:47:47 2022 +0000 +++ b/src/HOL/Lifting_Set.thy Tue Jan 11 06:48:02 2022 +0000 @@ -5,7 +5,7 @@ section \<open>Setup for Lifting/Transfer for the set type\<close> theory Lifting_Set -imports Lifting +imports Lifting Groups_Big begin subsection \<open>Relator and predicator properties\<close>