author | haftmann |
Mon, 04 Jun 2007 15:43:30 +0200 | |
changeset 23234 | b78bce9a0bcc |
parent 23087 | ad7244663431 |
child 23277 | aa158e145ea3 |
permissions | -rw-r--r-- |
12396 | 1 |
(* Title: HOL/Finite_Set.thy |
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ID: $Id$ |
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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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||
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header {* Finite sets *} |
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||
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theory Finite_Set |
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imports Divides |
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begin |
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|
15392 | 13 |
subsection {* Definition and basic properties *} |
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|
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inductive2 finite :: "'a set => bool" |
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where |
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emptyI [simp, intro!]: "finite {}" |
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| insertI [simp, intro!]: "finite A ==> finite (insert a A)" |
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|
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lemma ex_new_if_finite: -- "does not depend on def of finite at all" |
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assumes "\<not> finite (UNIV :: 'a set)" and "finite A" |
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shows "\<exists>a::'a. a \<notin> A" |
|
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proof - |
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from prems have "A \<noteq> UNIV" by blast |
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thus ?thesis by blast |
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qed |
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lemma finite_induct [case_names empty insert, induct set: finite]: |
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"finite F ==> |
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P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F" |
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-- {* Discharging @{text "x \<notin> F"} entails extra work. *} |
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proof - |
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13421 | 33 |
assume "P {}" and |
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insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)" |
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assume "finite F" |
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thus "P F" |
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proof induct |
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show "P {}" . |
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fix x F assume F: "finite F" and P: "P F" |
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show "P (insert x F)" |
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proof cases |
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assume "x \<in> F" |
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hence "insert x F = F" by (rule insert_absorb) |
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with P show ?thesis by (simp only:) |
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next |
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assume "x \<notin> F" |
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from F this P show ?thesis by (rule insert) |
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qed |
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qed |
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qed |
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||
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lemma finite_ne_induct[case_names singleton insert, consumes 2]: |
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assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow> |
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\<lbrakk> \<And>x. P{x}; |
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\<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk> |
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\<Longrightarrow> P F" |
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using fin |
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proof induct |
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case empty thus ?case by simp |
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next |
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case (insert x F) |
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show ?case |
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proof cases |
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assume "F = {}" thus ?thesis using insert(4) by simp |
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next |
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assume "F \<noteq> {}" thus ?thesis using insert by blast |
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qed |
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qed |
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||
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lemma finite_subset_induct [consumes 2, case_names empty insert]: |
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"finite F ==> F \<subseteq> A ==> |
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P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==> |
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P F" |
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proof - |
|
13421 | 75 |
assume "P {}" and insert: |
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"!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)" |
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assume "finite F" |
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thus "F \<subseteq> A ==> P F" |
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proof induct |
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show "P {}" . |
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fix x F assume "finite F" and "x \<notin> F" |
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and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A" |
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show "P (insert x F)" |
|
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proof (rule insert) |
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from i show "x \<in> A" by blast |
|
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from i have "F \<subseteq> A" by blast |
|
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with P show "P F" . |
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qed |
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89 |
qed |
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qed |
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||
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text{* Finite sets are the images of initial segments of natural numbers: *} |
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||
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lemma finite_imp_nat_seg_image_inj_on: |
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assumes fin: "finite A" |
|
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shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}" |
|
15392 | 97 |
using fin |
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proof induct |
|
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case empty |
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15510 | 100 |
show ?case |
101 |
proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp |
|
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qed |
|
15392 | 103 |
next |
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case (insert a A) |
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have notinA: "a \<notin> A" . |
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from insert.hyps obtain n f |
|
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where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast |
|
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hence "insert a A = f(n:=a) ` {i. i < Suc n}" |
|
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"inj_on (f(n:=a)) {i. i < Suc n}" using notinA |
|
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by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq) |
|
15392 | 111 |
thus ?case by blast |
112 |
qed |
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||
114 |
lemma nat_seg_image_imp_finite: |
|
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"!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A" |
|
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proof (induct n) |
|
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case 0 thus ?case by simp |
|
118 |
next |
|
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case (Suc n) |
|
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let ?B = "f ` {i. i < n}" |
|
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have finB: "finite ?B" by(rule Suc.hyps[OF refl]) |
|
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show ?case |
|
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proof cases |
|
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assume "\<exists>k<n. f n = f k" |
|
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hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq) |
|
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thus ?thesis using finB by simp |
|
127 |
next |
|
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assume "\<not>(\<exists> k<n. f n = f k)" |
|
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hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq) |
|
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thus ?thesis using finB by simp |
|
131 |
qed |
|
132 |
qed |
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||
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lemma finite_conv_nat_seg_image: |
|
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"finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})" |
|
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by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on) |
15392 | 137 |
|
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subsubsection{* Finiteness and set theoretic constructions *} |
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139 |
||
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)" |
141 |
-- {* The union of two finite sets is finite. *} |
|
22262 | 142 |
by (induct set: finite) simp_all |
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|
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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A" |
|
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-- {* Every subset of a finite set is finite. *} |
|
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proof - |
|
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assume "finite B" |
|
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thus "!!A. A \<subseteq> B ==> finite A" |
|
149 |
proof induct |
|
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case empty |
|
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thus ?case by simp |
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152 |
next |
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case (insert x F A) |
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have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" . |
155 |
show "finite A" |
|
156 |
proof cases |
|
157 |
assume x: "x \<in> A" |
|
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with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff) |
|
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with r have "finite (A - {x})" . |
|
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hence "finite (insert x (A - {x}))" .. |
|
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also have "insert x (A - {x}) = A" by (rule insert_Diff) |
|
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finally show ?thesis . |
|
163 |
next |
|
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show "A \<subseteq> F ==> ?thesis" . |
|
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assume "x \<notin> A" |
|
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with A show "A \<subseteq> F" by (simp add: subset_insert_iff) |
|
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qed |
|
168 |
qed |
|
169 |
qed |
|
170 |
||
18423 | 171 |
lemma finite_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}" |
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using finite_subset[of "{x \<in> A. P x}" "A"] by blast |
173 |
||
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lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)" |
175 |
by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI) |
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176 |
||
177 |
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)" |
|
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-- {* The converse obviously fails. *} |
|
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by (blast intro: finite_subset) |
|
180 |
||
181 |
lemma finite_insert [simp]: "finite (insert a A) = finite A" |
|
182 |
apply (subst insert_is_Un) |
|
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apply (simp only: finite_Un, blast) |
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done |
185 |
||
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lemma finite_Union[simp, intro]: |
187 |
"\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)" |
|
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by (induct rule:finite_induct) simp_all |
|
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||
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lemma finite_empty_induct: |
191 |
"finite A ==> |
|
192 |
P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}" |
|
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proof - |
|
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assume "finite A" |
|
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and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})" |
|
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have "P (A - A)" |
|
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proof - |
|
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fix c b :: "'a set" |
|
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presume c: "finite c" and b: "finite b" |
|
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and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})" |
|
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from c show "c \<subseteq> b ==> P (b - c)" |
|
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proof induct |
|
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case empty |
|
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from P1 show ?case by simp |
|
205 |
next |
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case (insert x F) |
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have "P (b - F - {x})" |
208 |
proof (rule P2) |
|
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from _ b show "finite (b - F)" by (rule finite_subset) blast |
|
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from insert show "x \<in> b - F" by simp |
|
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from insert show "P (b - F)" by simp |
|
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qed |
|
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also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric]) |
|
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finally show ?case . |
|
215 |
qed |
|
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next |
|
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show "A \<subseteq> A" .. |
|
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qed |
|
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thus "P {}" by simp |
|
220 |
qed |
|
221 |
||
222 |
lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)" |
|
223 |
by (rule Diff_subset [THEN finite_subset]) |
|
224 |
||
225 |
lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)" |
|
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apply (subst Diff_insert) |
|
227 |
apply (case_tac "a : A - B") |
|
228 |
apply (rule finite_insert [symmetric, THEN trans]) |
|
14208 | 229 |
apply (subst insert_Diff, simp_all) |
12396 | 230 |
done |
231 |
||
19870 | 232 |
lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A" |
233 |
by simp |
|
234 |
||
12396 | 235 |
|
15392 | 236 |
text {* Image and Inverse Image over Finite Sets *} |
13825 | 237 |
|
238 |
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)" |
|
239 |
-- {* The image of a finite set is finite. *} |
|
22262 | 240 |
by (induct set: finite) simp_all |
13825 | 241 |
|
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lemma finite_surj: "finite A ==> B <= f ` A ==> finite B" |
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243 |
apply (frule finite_imageI) |
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apply (erule finite_subset, assumption) |
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done |
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246 |
|
13825 | 247 |
lemma finite_range_imageI: |
248 |
"finite (range g) ==> finite (range (%x. f (g x)))" |
|
14208 | 249 |
apply (drule finite_imageI, simp) |
13825 | 250 |
done |
251 |
||
12396 | 252 |
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" |
253 |
proof - |
|
254 |
have aux: "!!A. finite (A - {}) = finite A" by simp |
|
255 |
fix B :: "'a set" |
|
256 |
assume "finite B" |
|
257 |
thus "!!A. f`A = B ==> inj_on f A ==> finite A" |
|
258 |
apply induct |
|
259 |
apply simp |
|
260 |
apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})") |
|
261 |
apply clarify |
|
262 |
apply (simp (no_asm_use) add: inj_on_def) |
|
14208 | 263 |
apply (blast dest!: aux [THEN iffD1], atomize) |
12396 | 264 |
apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) |
14208 | 265 |
apply (frule subsetD [OF equalityD2 insertI1], clarify) |
12396 | 266 |
apply (rule_tac x = xa in bexI) |
267 |
apply (simp_all add: inj_on_image_set_diff) |
|
268 |
done |
|
269 |
qed (rule refl) |
|
270 |
||
271 |
||
13825 | 272 |
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}" |
273 |
-- {* The inverse image of a singleton under an injective function |
|
274 |
is included in a singleton. *} |
|
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apply (auto simp add: inj_on_def) |
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276 |
apply (blast intro: the_equality [symmetric]) |
13825 | 277 |
done |
278 |
||
279 |
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)" |
|
280 |
-- {* The inverse image of a finite set under an injective function |
|
281 |
is finite. *} |
|
22262 | 282 |
apply (induct set: finite) |
21575 | 283 |
apply simp_all |
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apply (subst vimage_insert) |
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|
285 |
apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton]) |
13825 | 286 |
done |
287 |
||
288 |
||
15392 | 289 |
text {* The finite UNION of finite sets *} |
12396 | 290 |
|
291 |
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" |
|
22262 | 292 |
by (induct set: finite) simp_all |
12396 | 293 |
|
294 |
text {* |
|
295 |
Strengthen RHS to |
|
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296 |
@{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}? |
12396 | 297 |
|
298 |
We'd need to prove |
|
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299 |
@{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"} |
12396 | 300 |
by induction. *} |
301 |
||
302 |
lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" |
|
303 |
by (blast intro: finite_UN_I finite_subset) |
|
304 |
||
305 |
||
17022 | 306 |
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)" |
307 |
by (simp add: Plus_def) |
|
308 |
||
15392 | 309 |
text {* Sigma of finite sets *} |
12396 | 310 |
|
311 |
lemma finite_SigmaI [simp]: |
|
312 |
"finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" |
|
313 |
by (unfold Sigma_def) (blast intro!: finite_UN_I) |
|
314 |
||
15402 | 315 |
lemma finite_cartesian_product: "[| finite A; finite B |] ==> |
316 |
finite (A <*> B)" |
|
317 |
by (rule finite_SigmaI) |
|
318 |
||
12396 | 319 |
lemma finite_Prod_UNIV: |
320 |
"finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)" |
|
321 |
apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") |
|
322 |
apply (erule ssubst) |
|
14208 | 323 |
apply (erule finite_SigmaI, auto) |
12396 | 324 |
done |
325 |
||
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326 |
lemma finite_cartesian_productD1: |
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|
327 |
"[| finite (A <*> B); B \<noteq> {} |] ==> finite A" |
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apply (auto simp add: finite_conv_nat_seg_image) |
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apply (drule_tac x=n in spec) |
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apply (drule_tac x="fst o f" in spec) |
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apply (auto simp add: o_def) |
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prefer 2 apply (force dest!: equalityD2) |
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|
333 |
apply (drule equalityD1) |
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|
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apply (rename_tac y x) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
335 |
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
336 |
prefer 2 apply force |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
337 |
apply clarify |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
338 |
apply (rule_tac x=k in image_eqI, auto) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
339 |
done |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
340 |
|
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
341 |
lemma finite_cartesian_productD2: |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
342 |
"[| finite (A <*> B); A \<noteq> {} |] ==> finite B" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
343 |
apply (auto simp add: finite_conv_nat_seg_image) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
344 |
apply (drule_tac x=n in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
345 |
apply (drule_tac x="snd o f" in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
346 |
apply (auto simp add: o_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
347 |
prefer 2 apply (force dest!: equalityD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
348 |
apply (drule equalityD1) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
349 |
apply (rename_tac x y) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
350 |
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
351 |
prefer 2 apply force |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
352 |
apply clarify |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
353 |
apply (rule_tac x=k in image_eqI, auto) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
354 |
done |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
355 |
|
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
356 |
|
15392 | 357 |
text {* The powerset of a finite set *} |
12396 | 358 |
|
359 |
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A" |
|
360 |
proof |
|
361 |
assume "finite (Pow A)" |
|
362 |
with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast |
|
363 |
thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp |
|
364 |
next |
|
365 |
assume "finite A" |
|
366 |
thus "finite (Pow A)" |
|
367 |
by induct (simp_all add: finite_UnI finite_imageI Pow_insert) |
|
368 |
qed |
|
369 |
||
15392 | 370 |
|
371 |
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A" |
|
372 |
by(blast intro: finite_subset[OF subset_Pow_Union]) |
|
373 |
||
374 |
||
12396 | 375 |
lemma finite_converse [iff]: "finite (r^-1) = finite r" |
376 |
apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r") |
|
377 |
apply simp |
|
378 |
apply (rule iffI) |
|
379 |
apply (erule finite_imageD [unfolded inj_on_def]) |
|
380 |
apply (simp split add: split_split) |
|
381 |
apply (erule finite_imageI) |
|
14208 | 382 |
apply (simp add: converse_def image_def, auto) |
12396 | 383 |
apply (rule bexI) |
384 |
prefer 2 apply assumption |
|
385 |
apply simp |
|
386 |
done |
|
387 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset
|
388 |
|
15392 | 389 |
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi |
390 |
Ehmety) *} |
|
12396 | 391 |
|
392 |
lemma finite_Field: "finite r ==> finite (Field r)" |
|
393 |
-- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *} |
|
22262 | 394 |
apply (induct set: finite) |
12396 | 395 |
apply (auto simp add: Field_def Domain_insert Range_insert) |
396 |
done |
|
397 |
||
398 |
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r" |
|
399 |
apply clarify |
|
400 |
apply (erule trancl_induct) |
|
401 |
apply (auto simp add: Field_def) |
|
402 |
done |
|
403 |
||
404 |
lemma finite_trancl: "finite (r^+) = finite r" |
|
405 |
apply auto |
|
406 |
prefer 2 |
|
407 |
apply (rule trancl_subset_Field2 [THEN finite_subset]) |
|
408 |
apply (rule finite_SigmaI) |
|
409 |
prefer 3 |
|
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
13595
diff
changeset
|
410 |
apply (blast intro: r_into_trancl' finite_subset) |
12396 | 411 |
apply (auto simp add: finite_Field) |
412 |
done |
|
413 |
||
414 |
||
15392 | 415 |
subsection {* A fold functional for finite sets *} |
416 |
||
417 |
text {* The intended behaviour is |
|
15480 | 418 |
@{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"} |
15392 | 419 |
if @{text f} is associative-commutative. For an application of @{text fold} |
420 |
se the definitions of sums and products over finite sets. |
|
421 |
*} |
|
422 |
||
22262 | 423 |
inductive2 |
424 |
foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a => bool" |
|
425 |
for f :: "'a => 'a => 'a" |
|
426 |
and g :: "'b => 'a" |
|
427 |
and z :: 'a |
|
428 |
where |
|
429 |
emptyI [intro]: "foldSet f g z {} z" |
|
430 |
| insertI [intro]: |
|
431 |
"\<lbrakk> x \<notin> A; foldSet f g z A y \<rbrakk> |
|
432 |
\<Longrightarrow> foldSet f g z (insert x A) (f (g x) y)" |
|
433 |
||
434 |
inductive_cases2 empty_foldSetE [elim!]: "foldSet f g z {} x" |
|
15392 | 435 |
|
436 |
constdefs |
|
21733 | 437 |
fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a" |
22262 | 438 |
"fold f g z A == THE x. foldSet f g z A x" |
15392 | 439 |
|
15498 | 440 |
text{*A tempting alternative for the definiens is |
22262 | 441 |
@{term "if finite A then THE x. foldSet f g e A x else e"}. |
15498 | 442 |
It allows the removal of finiteness assumptions from the theorems |
443 |
@{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}. |
|
444 |
The proofs become ugly, with @{text rule_format}. It is not worth the effort.*} |
|
445 |
||
446 |
||
15392 | 447 |
lemma Diff1_foldSet: |
22262 | 448 |
"foldSet f g z (A - {x}) y ==> x: A ==> foldSet f g z A (f (g x) y)" |
15392 | 449 |
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto) |
450 |
||
22262 | 451 |
lemma foldSet_imp_finite: "foldSet f g z A x==> finite A" |
15392 | 452 |
by (induct set: foldSet) auto |
453 |
||
22262 | 454 |
lemma finite_imp_foldSet: "finite A ==> EX x. foldSet f g z A x" |
455 |
by (induct set: finite) auto |
|
15392 | 456 |
|
457 |
||
458 |
subsubsection {* Commutative monoids *} |
|
15480 | 459 |
|
22917 | 460 |
(*FIXME integrate with Orderings.thy/OrderedGroup.thy*) |
15392 | 461 |
locale ACf = |
462 |
fixes f :: "'a => 'a => 'a" (infixl "\<cdot>" 70) |
|
463 |
assumes commute: "x \<cdot> y = y \<cdot> x" |
|
464 |
and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" |
|
22917 | 465 |
begin |
466 |
||
467 |
lemma left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)" |
|
15392 | 468 |
proof - |
469 |
have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute) |
|
470 |
also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc) |
|
471 |
also have "z \<cdot> x = x \<cdot> z" by (simp only: commute) |
|
472 |
finally show ?thesis . |
|
473 |
qed |
|
474 |
||
22917 | 475 |
lemmas AC = assoc commute left_commute |
476 |
||
477 |
end |
|
478 |
||
479 |
locale ACe = ACf + |
|
480 |
fixes e :: 'a |
|
481 |
assumes ident [simp]: "x \<cdot> e = x" |
|
482 |
begin |
|
483 |
||
484 |
lemma left_ident [simp]: "e \<cdot> x = x" |
|
15392 | 485 |
proof - |
486 |
have "x \<cdot> e = x" by (rule ident) |
|
487 |
thus ?thesis by (subst commute) |
|
488 |
qed |
|
489 |
||
22917 | 490 |
end |
491 |
||
492 |
locale ACIf = ACf + |
|
493 |
assumes idem: "x \<cdot> x = x" |
|
494 |
begin |
|
495 |
||
496 |
lemma idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y" |
|
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
497 |
proof - |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
498 |
have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
499 |
also have "\<dots> = x \<cdot> y" by(simp add:idem) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
500 |
finally show ?thesis . |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
501 |
qed |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
502 |
|
22917 | 503 |
lemmas ACI = AC idem idem2 |
504 |
||
505 |
end |
|
506 |
||
15402 | 507 |
|
15392 | 508 |
subsubsection{*From @{term foldSet} to @{term fold}*} |
509 |
||
15510 | 510 |
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})" |
19868 | 511 |
by (auto simp add: less_Suc_eq) |
15510 | 512 |
|
513 |
lemma insert_image_inj_on_eq: |
|
514 |
"[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; |
|
515 |
inj_on h {i. i < Suc m}|] |
|
516 |
==> A = h ` {i. i < m}" |
|
517 |
apply (auto simp add: image_less_Suc inj_on_def) |
|
518 |
apply (blast intro: less_trans) |
|
519 |
done |
|
520 |
||
521 |
lemma insert_inj_onE: |
|
522 |
assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" |
|
523 |
and inj_on: "inj_on h {i::nat. i<n}" |
|
524 |
shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n" |
|
525 |
proof (cases n) |
|
526 |
case 0 thus ?thesis using aA by auto |
|
527 |
next |
|
528 |
case (Suc m) |
|
529 |
have nSuc: "n = Suc m" . |
|
530 |
have mlessn: "m<n" by (simp add: nSuc) |
|
15532 | 531 |
from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE) |
15520 | 532 |
let ?hm = "swap k m h" |
533 |
have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn |
|
534 |
by (simp add: inj_on_swap_iff inj_on) |
|
15510 | 535 |
show ?thesis |
15520 | 536 |
proof (intro exI conjI) |
537 |
show "inj_on ?hm {i. i < m}" using inj_hm |
|
15510 | 538 |
by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on) |
15520 | 539 |
show "m<n" by (rule mlessn) |
540 |
show "A = ?hm ` {i. i < m}" |
|
541 |
proof (rule insert_image_inj_on_eq) |
|
542 |
show "inj_on (swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp |
|
543 |
show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) |
|
544 |
show "insert (?hm m) A = ?hm ` {i. i < Suc m}" |
|
545 |
using aA hkeq nSuc klessn |
|
546 |
by (auto simp add: swap_def image_less_Suc fun_upd_image |
|
547 |
less_Suc_eq inj_on_image_set_diff [OF inj_on]) |
|
15479 | 548 |
qed |
549 |
qed |
|
550 |
qed |
|
551 |
||
15392 | 552 |
lemma (in ACf) foldSet_determ_aux: |
15510 | 553 |
"!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n}; |
22262 | 554 |
foldSet f g z A x; foldSet f g z A x' \<rbrakk> |
15392 | 555 |
\<Longrightarrow> x' = x" |
15510 | 556 |
proof (induct n rule: less_induct) |
557 |
case (less n) |
|
558 |
have IH: "!!m h A x x'. |
|
559 |
\<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m}; |
|
22262 | 560 |
foldSet f g z A x; foldSet f g z A x'\<rbrakk> \<Longrightarrow> x' = x" . |
561 |
have Afoldx: "foldSet f g z A x" and Afoldx': "foldSet f g z A x'" |
|
15510 | 562 |
and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" . |
563 |
show ?case |
|
564 |
proof (rule foldSet.cases [OF Afoldx]) |
|
22262 | 565 |
assume "A = {}" and "x = z" |
15510 | 566 |
with Afoldx' show "x' = x" by blast |
15392 | 567 |
next |
15510 | 568 |
fix B b u |
22262 | 569 |
assume AbB: "A = insert b B" and x: "x = g b \<cdot> u" |
570 |
and notinB: "b \<notin> B" and Bu: "foldSet f g z B u" |
|
15510 | 571 |
show "x'=x" |
572 |
proof (rule foldSet.cases [OF Afoldx']) |
|
22262 | 573 |
assume "A = {}" and "x' = z" |
15510 | 574 |
with AbB show "x' = x" by blast |
15392 | 575 |
next |
15510 | 576 |
fix C c v |
22262 | 577 |
assume AcC: "A = insert c C" and x': "x' = g c \<cdot> v" |
578 |
and notinC: "c \<notin> C" and Cv: "foldSet f g z C v" |
|
15510 | 579 |
from A AbB have Beq: "insert b B = h`{i. i<n}" by simp |
580 |
from insert_inj_onE [OF Beq notinB injh] |
|
581 |
obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" |
|
582 |
and Beq: "B = hB ` {i. i < mB}" |
|
583 |
and lessB: "mB < n" by auto |
|
584 |
from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp |
|
585 |
from insert_inj_onE [OF Ceq notinC injh] |
|
586 |
obtain hC mC where inj_onC: "inj_on hC {i. i < mC}" |
|
587 |
and Ceq: "C = hC ` {i. i < mC}" |
|
588 |
and lessC: "mC < n" by auto |
|
589 |
show "x'=x" |
|
15392 | 590 |
proof cases |
15510 | 591 |
assume "b=c" |
592 |
then moreover have "B = C" using AbB AcC notinB notinC by auto |
|
593 |
ultimately show ?thesis using Bu Cv x x' IH[OF lessC Ceq inj_onC] |
|
594 |
by auto |
|
15392 | 595 |
next |
596 |
assume diff: "b \<noteq> c" |
|
597 |
let ?D = "B - {c}" |
|
598 |
have B: "B = insert c ?D" and C: "C = insert b ?D" |
|
15510 | 599 |
using AbB AcC notinB notinC diff by(blast elim!:equalityE)+ |
15402 | 600 |
have "finite A" by(rule foldSet_imp_finite[OF Afoldx]) |
15510 | 601 |
with AbB have "finite ?D" by simp |
22262 | 602 |
then obtain d where Dfoldd: "foldSet f g z ?D d" |
17589 | 603 |
using finite_imp_foldSet by iprover |
15506 | 604 |
moreover have cinB: "c \<in> B" using B by auto |
22262 | 605 |
ultimately have "foldSet f g z B (g c \<cdot> d)" |
15392 | 606 |
by(rule Diff1_foldSet) |
15510 | 607 |
hence "g c \<cdot> d = u" by (rule IH [OF lessB Beq inj_onB Bu]) |
608 |
moreover have "g b \<cdot> d = v" |
|
609 |
proof (rule IH[OF lessC Ceq inj_onC Cv]) |
|
22262 | 610 |
show "foldSet f g z C (g b \<cdot> d)" using C notinB Dfoldd |
15392 | 611 |
by fastsimp |
612 |
qed |
|
15510 | 613 |
ultimately show ?thesis using x x' by (auto simp: AC) |
15392 | 614 |
qed |
615 |
qed |
|
616 |
qed |
|
617 |
qed |
|
618 |
||
619 |
||
620 |
lemma (in ACf) foldSet_determ: |
|
22262 | 621 |
"foldSet f g z A x ==> foldSet f g z A y ==> y = x" |
15510 | 622 |
apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) |
623 |
apply (blast intro: foldSet_determ_aux [rule_format]) |
|
15392 | 624 |
done |
625 |
||
22262 | 626 |
lemma (in ACf) fold_equality: "foldSet f g z A y ==> fold f g z A = y" |
15392 | 627 |
by (unfold fold_def) (blast intro: foldSet_determ) |
628 |
||
629 |
text{* The base case for @{text fold}: *} |
|
630 |
||
15480 | 631 |
lemma fold_empty [simp]: "fold f g z {} = z" |
15392 | 632 |
by (unfold fold_def) blast |
633 |
||
634 |
lemma (in ACf) fold_insert_aux: "x \<notin> A ==> |
|
22262 | 635 |
(foldSet f g z (insert x A) v) = |
636 |
(EX y. foldSet f g z A y & v = f (g x) y)" |
|
15392 | 637 |
apply auto |
638 |
apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE]) |
|
639 |
apply (fastsimp dest: foldSet_imp_finite) |
|
640 |
apply (blast intro: foldSet_determ) |
|
641 |
done |
|
642 |
||
643 |
text{* The recursion equation for @{text fold}: *} |
|
644 |
||
645 |
lemma (in ACf) fold_insert[simp]: |
|
15480 | 646 |
"finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)" |
15392 | 647 |
apply (unfold fold_def) |
648 |
apply (simp add: fold_insert_aux) |
|
649 |
apply (rule the_equality) |
|
650 |
apply (auto intro: finite_imp_foldSet |
|
651 |
cong add: conj_cong simp add: fold_def [symmetric] fold_equality) |
|
652 |
done |
|
653 |
||
15535 | 654 |
lemma (in ACf) fold_rec: |
655 |
assumes fin: "finite A" and a: "a:A" |
|
656 |
shows "fold f g z A = f (g a) (fold f g z (A - {a}))" |
|
657 |
proof- |
|
658 |
have A: "A = insert a (A - {a})" using a by blast |
|
659 |
hence "fold f g z A = fold f g z (insert a (A - {a}))" by simp |
|
660 |
also have "\<dots> = f (g a) (fold f g z (A - {a}))" |
|
661 |
by(rule fold_insert) (simp add:fin)+ |
|
662 |
finally show ?thesis . |
|
663 |
qed |
|
664 |
||
15392 | 665 |
|
15480 | 666 |
text{* A simplified version for idempotent functions: *} |
667 |
||
15509 | 668 |
lemma (in ACIf) fold_insert_idem: |
15480 | 669 |
assumes finA: "finite A" |
15508 | 670 |
shows "fold f g z (insert a A) = g a \<cdot> fold f g z A" |
15480 | 671 |
proof cases |
672 |
assume "a \<in> A" |
|
673 |
then obtain B where A: "A = insert a B" and disj: "a \<notin> B" |
|
674 |
by(blast dest: mk_disjoint_insert) |
|
675 |
show ?thesis |
|
676 |
proof - |
|
677 |
from finA A have finB: "finite B" by(blast intro: finite_subset) |
|
678 |
have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp |
|
679 |
also have "\<dots> = (g a) \<cdot> (fold f g z B)" |
|
15506 | 680 |
using finB disj by simp |
15480 | 681 |
also have "\<dots> = g a \<cdot> fold f g z A" |
682 |
using A finB disj by(simp add:idem assoc[symmetric]) |
|
683 |
finally show ?thesis . |
|
684 |
qed |
|
685 |
next |
|
686 |
assume "a \<notin> A" |
|
687 |
with finA show ?thesis by simp |
|
688 |
qed |
|
689 |
||
15484 | 690 |
lemma (in ACIf) foldI_conv_id: |
691 |
"finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)" |
|
15509 | 692 |
by(erule finite_induct)(simp_all add: fold_insert_idem del: fold_insert) |
15484 | 693 |
|
15392 | 694 |
subsubsection{*Lemmas about @{text fold}*} |
695 |
||
696 |
lemma (in ACf) fold_commute: |
|
15487 | 697 |
"finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)" |
22262 | 698 |
apply (induct set: finite) |
21575 | 699 |
apply simp |
15487 | 700 |
apply (simp add: left_commute [of x]) |
15392 | 701 |
done |
702 |
||
703 |
lemma (in ACf) fold_nest_Un_Int: |
|
704 |
"finite A ==> finite B |
|
15480 | 705 |
==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)" |
22262 | 706 |
apply (induct set: finite) |
21575 | 707 |
apply simp |
15392 | 708 |
apply (simp add: fold_commute Int_insert_left insert_absorb) |
709 |
done |
|
710 |
||
711 |
lemma (in ACf) fold_nest_Un_disjoint: |
|
712 |
"finite A ==> finite B ==> A Int B = {} |
|
15480 | 713 |
==> fold f g z (A Un B) = fold f g (fold f g z B) A" |
15392 | 714 |
by (simp add: fold_nest_Un_Int) |
715 |
||
716 |
lemma (in ACf) fold_reindex: |
|
15487 | 717 |
assumes fin: "finite A" |
718 |
shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A" |
|
15506 | 719 |
using fin apply induct |
15392 | 720 |
apply simp |
721 |
apply simp |
|
722 |
done |
|
723 |
||
724 |
lemma (in ACe) fold_Un_Int: |
|
725 |
"finite A ==> finite B ==> |
|
726 |
fold f g e A \<cdot> fold f g e B = |
|
727 |
fold f g e (A Un B) \<cdot> fold f g e (A Int B)" |
|
22262 | 728 |
apply (induct set: finite, simp) |
15392 | 729 |
apply (simp add: AC insert_absorb Int_insert_left) |
730 |
done |
|
731 |
||
732 |
corollary (in ACe) fold_Un_disjoint: |
|
733 |
"finite A ==> finite B ==> A Int B = {} ==> |
|
734 |
fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B" |
|
735 |
by (simp add: fold_Un_Int) |
|
736 |
||
737 |
lemma (in ACe) fold_UN_disjoint: |
|
738 |
"\<lbrakk> finite I; ALL i:I. finite (A i); |
|
739 |
ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk> |
|
740 |
\<Longrightarrow> fold f g e (UNION I A) = |
|
741 |
fold f (%i. fold f g e (A i)) e I" |
|
22262 | 742 |
apply (induct set: finite, simp, atomize) |
15392 | 743 |
apply (subgoal_tac "ALL i:F. x \<noteq> i") |
744 |
prefer 2 apply blast |
|
745 |
apply (subgoal_tac "A x Int UNION F A = {}") |
|
746 |
prefer 2 apply blast |
|
747 |
apply (simp add: fold_Un_disjoint) |
|
748 |
done |
|
749 |
||
15506 | 750 |
text{*Fusion theorem, as described in |
751 |
Graham Hutton's paper, |
|
752 |
A Tutorial on the Universality and Expressiveness of Fold, |
|
753 |
JFP 9:4 (355-372), 1999.*} |
|
754 |
lemma (in ACf) fold_fusion: |
|
755 |
includes ACf g |
|
756 |
shows |
|
757 |
"finite A ==> |
|
758 |
(!!x y. h (g x y) = f x (h y)) ==> |
|
759 |
h (fold g j w A) = fold f j (h w) A" |
|
22262 | 760 |
by (induct set: finite) simp_all |
15506 | 761 |
|
15392 | 762 |
lemma (in ACf) fold_cong: |
15480 | 763 |
"finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A" |
764 |
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C") |
|
15392 | 765 |
apply simp |
766 |
apply (erule finite_induct, simp) |
|
767 |
apply (simp add: subset_insert_iff, clarify) |
|
768 |
apply (subgoal_tac "finite C") |
|
769 |
prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) |
|
770 |
apply (subgoal_tac "C = insert x (C - {x})") |
|
771 |
prefer 2 apply blast |
|
772 |
apply (erule ssubst) |
|
773 |
apply (drule spec) |
|
774 |
apply (erule (1) notE impE) |
|
775 |
apply (simp add: Ball_def del: insert_Diff_single) |
|
776 |
done |
|
777 |
||
778 |
lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==> |
|
779 |
fold f (%x. fold f (g x) e (B x)) e A = |
|
780 |
fold f (split g) e (SIGMA x:A. B x)" |
|
781 |
apply (subst Sigma_def) |
|
15506 | 782 |
apply (subst fold_UN_disjoint, assumption, simp) |
15392 | 783 |
apply blast |
784 |
apply (erule fold_cong) |
|
15506 | 785 |
apply (subst fold_UN_disjoint, simp, simp) |
15392 | 786 |
apply blast |
15506 | 787 |
apply simp |
15392 | 788 |
done |
789 |
||
790 |
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow> |
|
791 |
fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)" |
|
15506 | 792 |
apply (erule finite_induct, simp) |
15392 | 793 |
apply (simp add:AC) |
794 |
done |
|
795 |
||
796 |
||
22917 | 797 |
text{* Interpretation of locales -- see OrderedGroup.thy *} |
798 |
||
799 |
interpretation AC_add: ACe ["op +" "0::'a::comm_monoid_add"] |
|
800 |
by unfold_locales (auto intro: add_assoc add_commute) |
|
801 |
||
802 |
interpretation AC_mult: ACe ["op *" "1::'a::comm_monoid_mult"] |
|
803 |
by unfold_locales (auto intro: mult_assoc mult_commute) |
|
804 |
||
805 |
||
15402 | 806 |
subsection {* Generalized summation over a set *} |
807 |
||
808 |
constdefs |
|
809 |
setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add" |
|
810 |
"setsum f A == if finite A then fold (op +) f 0 A else 0" |
|
811 |
||
19535 | 812 |
abbreviation |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21249
diff
changeset
|
813 |
Setsum ("\<Sum>_" [1000] 999) where |
19535 | 814 |
"\<Sum>A == setsum (%x. x) A" |
815 |
||
15402 | 816 |
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is |
817 |
written @{text"\<Sum>x\<in>A. e"}. *} |
|
818 |
||
819 |
syntax |
|
17189 | 820 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10) |
15402 | 821 |
syntax (xsymbols) |
17189 | 822 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) |
15402 | 823 |
syntax (HTML output) |
17189 | 824 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) |
15402 | 825 |
|
826 |
translations -- {* Beware of argument permutation! *} |
|
827 |
"SUM i:A. b" == "setsum (%i. b) A" |
|
828 |
"\<Sum>i\<in>A. b" == "setsum (%i. b) A" |
|
829 |
||
830 |
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter |
|
831 |
@{text"\<Sum>x|P. e"}. *} |
|
832 |
||
833 |
syntax |
|
17189 | 834 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10) |
15402 | 835 |
syntax (xsymbols) |
17189 | 836 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10) |
15402 | 837 |
syntax (HTML output) |
17189 | 838 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10) |
15402 | 839 |
|
840 |
translations |
|
841 |
"SUM x|P. t" => "setsum (%x. t) {x. P}" |
|
842 |
"\<Sum>x|P. t" => "setsum (%x. t) {x. P}" |
|
843 |
||
844 |
print_translation {* |
|
845 |
let |
|
19535 | 846 |
fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = |
847 |
if x<>y then raise Match |
|
848 |
else let val x' = Syntax.mark_bound x |
|
849 |
val t' = subst_bound(x',t) |
|
850 |
val P' = subst_bound(x',P) |
|
851 |
in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end |
|
852 |
in [("setsum", setsum_tr')] end |
|
15402 | 853 |
*} |
854 |
||
19535 | 855 |
|
15402 | 856 |
lemma setsum_empty [simp]: "setsum f {} = 0" |
857 |
by (simp add: setsum_def) |
|
858 |
||
859 |
lemma setsum_insert [simp]: |
|
860 |
"finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F" |
|
15765 | 861 |
by (simp add: setsum_def) |
15402 | 862 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
863 |
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
864 |
by (simp add: setsum_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
865 |
|
15402 | 866 |
lemma setsum_reindex: |
867 |
"inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B" |
|
15765 | 868 |
by(auto simp add: setsum_def AC_add.fold_reindex dest!:finite_imageD) |
15402 | 869 |
|
870 |
lemma setsum_reindex_id: |
|
871 |
"inj_on f B ==> setsum f B = setsum id (f ` B)" |
|
872 |
by (auto simp add: setsum_reindex) |
|
873 |
||
874 |
lemma setsum_cong: |
|
875 |
"A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" |
|
15765 | 876 |
by(fastsimp simp: setsum_def intro: AC_add.fold_cong) |
15402 | 877 |
|
16733
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
878 |
lemma strong_setsum_cong[cong]: |
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
879 |
"A = B ==> (!!x. x:B =simp=> f x = g x) |
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
880 |
==> setsum (%x. f x) A = setsum (%x. g x) B" |
16632
ad2895beef79
Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents:
16550
diff
changeset
|
881 |
by(fastsimp simp: simp_implies_def setsum_def intro: AC_add.fold_cong) |
ad2895beef79
Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents:
16550
diff
changeset
|
882 |
|
15554 | 883 |
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"; |
884 |
by (rule setsum_cong[OF refl], auto); |
|
885 |
||
15402 | 886 |
lemma setsum_reindex_cong: |
15554 | 887 |
"[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] |
15402 | 888 |
==> setsum h B = setsum g A" |
889 |
by (simp add: setsum_reindex cong: setsum_cong) |
|
890 |
||
15542 | 891 |
lemma setsum_0[simp]: "setsum (%i. 0) A = 0" |
15402 | 892 |
apply (clarsimp simp: setsum_def) |
15765 | 893 |
apply (erule finite_induct, auto) |
15402 | 894 |
done |
895 |
||
15543 | 896 |
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0" |
897 |
by(simp add:setsum_cong) |
|
15402 | 898 |
|
899 |
lemma setsum_Un_Int: "finite A ==> finite B ==> |
|
900 |
setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" |
|
901 |
-- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} |
|
15765 | 902 |
by(simp add: setsum_def AC_add.fold_Un_Int [symmetric]) |
15402 | 903 |
|
904 |
lemma setsum_Un_disjoint: "finite A ==> finite B |
|
905 |
==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" |
|
906 |
by (subst setsum_Un_Int [symmetric], auto) |
|
907 |
||
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
908 |
(*But we can't get rid of finite I. If infinite, although the rhs is 0, |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
909 |
the lhs need not be, since UNION I A could still be finite.*) |
15402 | 910 |
lemma setsum_UN_disjoint: |
911 |
"finite I ==> (ALL i:I. finite (A i)) ==> |
|
912 |
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==> |
|
913 |
setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))" |
|
15765 | 914 |
by(simp add: setsum_def AC_add.fold_UN_disjoint cong: setsum_cong) |
15402 | 915 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
916 |
text{*No need to assume that @{term C} is finite. If infinite, the rhs is |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
917 |
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*} |
15402 | 918 |
lemma setsum_Union_disjoint: |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
919 |
"[| (ALL A:C. finite A); |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
920 |
(ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
921 |
==> setsum f (Union C) = setsum (setsum f) C" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
922 |
apply (cases "finite C") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
923 |
prefer 2 apply (force dest: finite_UnionD simp add: setsum_def) |
15402 | 924 |
apply (frule setsum_UN_disjoint [of C id f]) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
925 |
apply (unfold Union_def id_def, assumption+) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
926 |
done |
15402 | 927 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
928 |
(*But we can't get rid of finite A. If infinite, although the lhs is 0, |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
929 |
the rhs need not be, since SIGMA A B could still be finite.*) |
15402 | 930 |
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==> |
17189 | 931 |
(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)" |
15765 | 932 |
by(simp add:setsum_def AC_add.fold_Sigma split_def cong:setsum_cong) |
15402 | 933 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
934 |
text{*Here we can eliminate the finiteness assumptions, by cases.*} |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
935 |
lemma setsum_cartesian_product: |
17189 | 936 |
"(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)" |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
937 |
apply (cases "finite A") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
938 |
apply (cases "finite B") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
939 |
apply (simp add: setsum_Sigma) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
940 |
apply (cases "A={}", simp) |
15543 | 941 |
apply (simp) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
942 |
apply (auto simp add: setsum_def |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
943 |
dest: finite_cartesian_productD1 finite_cartesian_productD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
944 |
done |
15402 | 945 |
|
946 |
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)" |
|
15765 | 947 |
by(simp add:setsum_def AC_add.fold_distrib) |
15402 | 948 |
|
949 |
||
950 |
subsubsection {* Properties in more restricted classes of structures *} |
|
951 |
||
952 |
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" |
|
953 |
apply (case_tac "finite A") |
|
954 |
prefer 2 apply (simp add: setsum_def) |
|
955 |
apply (erule rev_mp) |
|
956 |
apply (erule finite_induct, auto) |
|
957 |
done |
|
958 |
||
959 |
lemma setsum_eq_0_iff [simp]: |
|
960 |
"finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" |
|
22262 | 961 |
by (induct set: finite) auto |
15402 | 962 |
|
963 |
lemma setsum_Un_nat: "finite A ==> finite B ==> |
|
964 |
(setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" |
|
965 |
-- {* For the natural numbers, we have subtraction. *} |
|
966 |
by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps) |
|
967 |
||
968 |
lemma setsum_Un: "finite A ==> finite B ==> |
|
969 |
(setsum f (A Un B) :: 'a :: ab_group_add) = |
|
970 |
setsum f A + setsum f B - setsum f (A Int B)" |
|
971 |
by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps) |
|
972 |
||
973 |
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) = |
|
974 |
(if a:A then setsum f A - f a else setsum f A)" |
|
975 |
apply (case_tac "finite A") |
|
976 |
prefer 2 apply (simp add: setsum_def) |
|
977 |
apply (erule finite_induct) |
|
978 |
apply (auto simp add: insert_Diff_if) |
|
979 |
apply (drule_tac a = a in mk_disjoint_insert, auto) |
|
980 |
done |
|
981 |
||
982 |
lemma setsum_diff1: "finite A \<Longrightarrow> |
|
983 |
(setsum f (A - {a}) :: ('a::ab_group_add)) = |
|
984 |
(if a:A then setsum f A - f a else setsum f A)" |
|
985 |
by (erule finite_induct) (auto simp add: insert_Diff_if) |
|
986 |
||
15552
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
987 |
lemma setsum_diff1'[rule_format]: "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)" |
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
988 |
apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"]) |
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
989 |
apply (auto simp add: insert_Diff_if add_ac) |
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
990 |
done |
8ab8e425410b
added setsum_diff1' which holds in more general cases than setsum_diff1
obua
parents:
15543
diff
changeset
|
991 |
|
15402 | 992 |
(* By Jeremy Siek: *) |
993 |
||
994 |
lemma setsum_diff_nat: |
|
19535 | 995 |
assumes "finite B" |
996 |
and "B \<subseteq> A" |
|
997 |
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)" |
|
998 |
using prems |
|
999 |
proof induct |
|
15402 | 1000 |
show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp |
1001 |
next |
|
1002 |
fix F x assume finF: "finite F" and xnotinF: "x \<notin> F" |
|
1003 |
and xFinA: "insert x F \<subseteq> A" |
|
1004 |
and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F" |
|
1005 |
from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp |
|
1006 |
from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x" |
|
1007 |
by (simp add: setsum_diff1_nat) |
|
1008 |
from xFinA have "F \<subseteq> A" by simp |
|
1009 |
with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp |
|
1010 |
with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x" |
|
1011 |
by simp |
|
1012 |
from xnotinF have "A - insert x F = (A - F) - {x}" by auto |
|
1013 |
with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x" |
|
1014 |
by simp |
|
1015 |
from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp |
|
1016 |
with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" |
|
1017 |
by simp |
|
1018 |
thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp |
|
1019 |
qed |
|
1020 |
||
1021 |
lemma setsum_diff: |
|
1022 |
assumes le: "finite A" "B \<subseteq> A" |
|
1023 |
shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))" |
|
1024 |
proof - |
|
1025 |
from le have finiteB: "finite B" using finite_subset by auto |
|
1026 |
show ?thesis using finiteB le |
|
21575 | 1027 |
proof induct |
19535 | 1028 |
case empty |
1029 |
thus ?case by auto |
|
1030 |
next |
|
1031 |
case (insert x F) |
|
1032 |
thus ?case using le finiteB |
|
1033 |
by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb) |
|
15402 | 1034 |
qed |
19535 | 1035 |
qed |
15402 | 1036 |
|
1037 |
lemma setsum_mono: |
|
1038 |
assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))" |
|
1039 |
shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)" |
|
1040 |
proof (cases "finite K") |
|
1041 |
case True |
|
1042 |
thus ?thesis using le |
|
19535 | 1043 |
proof induct |
15402 | 1044 |
case empty |
1045 |
thus ?case by simp |
|
1046 |
next |
|
1047 |
case insert |
|
19535 | 1048 |
thus ?case using add_mono by fastsimp |
15402 | 1049 |
qed |
1050 |
next |
|
1051 |
case False |
|
1052 |
thus ?thesis |
|
1053 |
by (simp add: setsum_def) |
|
1054 |
qed |
|
1055 |
||
15554 | 1056 |
lemma setsum_strict_mono: |
19535 | 1057 |
fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}" |
1058 |
assumes "finite A" "A \<noteq> {}" |
|
1059 |
and "!!x. x:A \<Longrightarrow> f x < g x" |
|
1060 |
shows "setsum f A < setsum g A" |
|
1061 |
using prems |
|
15554 | 1062 |
proof (induct rule: finite_ne_induct) |
1063 |
case singleton thus ?case by simp |
|
1064 |
next |
|
1065 |
case insert thus ?case by (auto simp: add_strict_mono) |
|
1066 |
qed |
|
1067 |
||
15535 | 1068 |
lemma setsum_negf: |
19535 | 1069 |
"setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A" |
15535 | 1070 |
proof (cases "finite A") |
22262 | 1071 |
case True thus ?thesis by (induct set: finite) auto |
15535 | 1072 |
next |
1073 |
case False thus ?thesis by (simp add: setsum_def) |
|
1074 |
qed |
|
15402 | 1075 |
|
15535 | 1076 |
lemma setsum_subtractf: |
19535 | 1077 |
"setsum (%x. ((f x)::'a::ab_group_add) - g x) A = |
1078 |
setsum f A - setsum g A" |
|
15535 | 1079 |
proof (cases "finite A") |
1080 |
case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf) |
|
1081 |
next |
|
1082 |
case False thus ?thesis by (simp add: setsum_def) |
|
1083 |
qed |
|
15402 | 1084 |
|
15535 | 1085 |
lemma setsum_nonneg: |
19535 | 1086 |
assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x" |
1087 |
shows "0 \<le> setsum f A" |
|
15535 | 1088 |
proof (cases "finite A") |
1089 |
case True thus ?thesis using nn |
|
21575 | 1090 |
proof induct |
19535 | 1091 |
case empty then show ?case by simp |
1092 |
next |
|
1093 |
case (insert x F) |
|
1094 |
then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono) |
|
1095 |
with insert show ?case by simp |
|
1096 |
qed |
|
15535 | 1097 |
next |
1098 |
case False thus ?thesis by (simp add: setsum_def) |
|
1099 |
qed |
|
15402 | 1100 |
|
15535 | 1101 |
lemma setsum_nonpos: |
19535 | 1102 |
assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})" |
1103 |
shows "setsum f A \<le> 0" |
|
15535 | 1104 |
proof (cases "finite A") |
1105 |
case True thus ?thesis using np |
|
21575 | 1106 |
proof induct |
19535 | 1107 |
case empty then show ?case by simp |
1108 |
next |
|
1109 |
case (insert x F) |
|
1110 |
then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono) |
|
1111 |
with insert show ?case by simp |
|
1112 |
qed |
|
15535 | 1113 |
next |
1114 |
case False thus ?thesis by (simp add: setsum_def) |
|
1115 |
qed |
|
15402 | 1116 |
|
15539 | 1117 |
lemma setsum_mono2: |
1118 |
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}" |
|
1119 |
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b" |
|
1120 |
shows "setsum f A \<le> setsum f B" |
|
1121 |
proof - |
|
1122 |
have "setsum f A \<le> setsum f A + setsum f (B-A)" |
|
1123 |
by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def) |
|
1124 |
also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin] |
|
1125 |
by (simp add:setsum_Un_disjoint del:Un_Diff_cancel) |
|
1126 |
also have "A \<union> (B-A) = B" using sub by blast |
|
1127 |
finally show ?thesis . |
|
1128 |
qed |
|
15542 | 1129 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1130 |
lemma setsum_mono3: "finite B ==> A <= B ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1131 |
ALL x: B - A. |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1132 |
0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1133 |
setsum f A <= setsum f B" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1134 |
apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1135 |
apply (erule ssubst) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1136 |
apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1137 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1138 |
apply (rule add_left_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1139 |
apply (erule setsum_nonneg) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1140 |
apply (subst setsum_Un_disjoint [THEN sym]) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1141 |
apply (erule finite_subset, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1142 |
apply (rule finite_subset) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1143 |
prefer 2 |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1144 |
apply assumption |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1145 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1146 |
apply (rule setsum_cong) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1147 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1148 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1149 |
|
19279 | 1150 |
lemma setsum_right_distrib: |
22934
64ecb3d6790a
generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents:
22917
diff
changeset
|
1151 |
fixes f :: "'a => ('b::semiring_0)" |
15402 | 1152 |
shows "r * setsum f A = setsum (%n. r * f n) A" |
1153 |
proof (cases "finite A") |
|
1154 |
case True |
|
1155 |
thus ?thesis |
|
21575 | 1156 |
proof induct |
15402 | 1157 |
case empty thus ?case by simp |
1158 |
next |
|
1159 |
case (insert x A) thus ?case by (simp add: right_distrib) |
|
1160 |
qed |
|
1161 |
next |
|
1162 |
case False thus ?thesis by (simp add: setsum_def) |
|
1163 |
qed |
|
1164 |
||
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1165 |
lemma setsum_left_distrib: |
22934
64ecb3d6790a
generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents:
22917
diff
changeset
|
1166 |
"setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)" |
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1167 |
proof (cases "finite A") |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1168 |
case True |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1169 |
then show ?thesis |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1170 |
proof induct |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1171 |
case empty thus ?case by simp |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1172 |
next |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1173 |
case (insert x A) thus ?case by (simp add: left_distrib) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1174 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1175 |
next |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1176 |
case False thus ?thesis by (simp add: setsum_def) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1177 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1178 |
|
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1179 |
lemma setsum_divide_distrib: |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1180 |
"setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)" |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1181 |
proof (cases "finite A") |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1182 |
case True |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1183 |
then show ?thesis |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1184 |
proof induct |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1185 |
case empty thus ?case by simp |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1186 |
next |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1187 |
case (insert x A) thus ?case by (simp add: add_divide_distrib) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1188 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1189 |
next |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1190 |
case False thus ?thesis by (simp add: setsum_def) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1191 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1192 |
|
15535 | 1193 |
lemma setsum_abs[iff]: |
15402 | 1194 |
fixes f :: "'a => ('b::lordered_ab_group_abs)" |
1195 |
shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A" |
|
15535 | 1196 |
proof (cases "finite A") |
1197 |
case True |
|
1198 |
thus ?thesis |
|
21575 | 1199 |
proof induct |
15535 | 1200 |
case empty thus ?case by simp |
1201 |
next |
|
1202 |
case (insert x A) |
|
1203 |
thus ?case by (auto intro: abs_triangle_ineq order_trans) |
|
1204 |
qed |
|
15402 | 1205 |
next |
15535 | 1206 |
case False thus ?thesis by (simp add: setsum_def) |
15402 | 1207 |
qed |
1208 |
||
15535 | 1209 |
lemma setsum_abs_ge_zero[iff]: |
15402 | 1210 |
fixes f :: "'a => ('b::lordered_ab_group_abs)" |
1211 |
shows "0 \<le> setsum (%i. abs(f i)) A" |
|
15535 | 1212 |
proof (cases "finite A") |
1213 |
case True |
|
1214 |
thus ?thesis |
|
21575 | 1215 |
proof induct |
15535 | 1216 |
case empty thus ?case by simp |
1217 |
next |
|
21733 | 1218 |
case (insert x A) thus ?case by (auto simp: add_nonneg_nonneg) |
15535 | 1219 |
qed |
15402 | 1220 |
next |
15535 | 1221 |
case False thus ?thesis by (simp add: setsum_def) |
15402 | 1222 |
qed |
1223 |
||
15539 | 1224 |
lemma abs_setsum_abs[simp]: |
1225 |
fixes f :: "'a => ('b::lordered_ab_group_abs)" |
|
1226 |
shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))" |
|
1227 |
proof (cases "finite A") |
|
1228 |
case True |
|
1229 |
thus ?thesis |
|
21575 | 1230 |
proof induct |
15539 | 1231 |
case empty thus ?case by simp |
1232 |
next |
|
1233 |
case (insert a A) |
|
1234 |
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 |
|
1235 |
also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>" using insert by simp |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1236 |
also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16760
diff
changeset
|
1237 |
by (simp del: abs_of_nonneg) |
15539 | 1238 |
also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp |
1239 |
finally show ?case . |
|
1240 |
qed |
|
1241 |
next |
|
1242 |
case False thus ?thesis by (simp add: setsum_def) |
|
1243 |
qed |
|
1244 |
||
15402 | 1245 |
|
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1246 |
text {* Commuting outer and inner summation *} |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1247 |
|
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1248 |
lemma swap_inj_on: |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1249 |
"inj_on (%(i, j). (j, i)) (A \<times> B)" |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1250 |
by (unfold inj_on_def) fast |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1251 |
|
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1252 |
lemma swap_product: |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1253 |
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A" |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1254 |
by (simp add: split_def image_def) blast |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1255 |
|
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1256 |
lemma setsum_commute: |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1257 |
"(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)" |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1258 |
proof (simp add: setsum_cartesian_product) |
17189 | 1259 |
have "(\<Sum>(x,y) \<in> A <*> B. f x y) = |
1260 |
(\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)" |
|
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1261 |
(is "?s = _") |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1262 |
apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1263 |
apply (simp add: split_def) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1264 |
done |
17189 | 1265 |
also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)" |
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1266 |
(is "_ = ?t") |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1267 |
apply (simp add: swap_product) |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1268 |
done |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1269 |
finally show "?s = ?t" . |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1270 |
qed |
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1271 |
|
19279 | 1272 |
lemma setsum_product: |
22934
64ecb3d6790a
generalize setsum lemmas from semiring_0_cancel to semiring_0
huffman
parents:
22917
diff
changeset
|
1273 |
fixes f :: "'a => ('b::semiring_0)" |
19279 | 1274 |
shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)" |
1275 |
by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute) |
|
1276 |
||
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
17085
diff
changeset
|
1277 |
|
15402 | 1278 |
subsection {* Generalized product over a set *} |
1279 |
||
1280 |
constdefs |
|
1281 |
setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult" |
|
1282 |
"setprod f A == if finite A then fold (op *) f 1 A else 1" |
|
1283 |
||
19535 | 1284 |
abbreviation |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21249
diff
changeset
|
1285 |
Setprod ("\<Prod>_" [1000] 999) where |
19535 | 1286 |
"\<Prod>A == setprod (%x. x) A" |
1287 |
||
15402 | 1288 |
syntax |
17189 | 1289 |
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3PROD _:_. _)" [0, 51, 10] 10) |
15402 | 1290 |
syntax (xsymbols) |
17189 | 1291 |
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10) |
15402 | 1292 |
syntax (HTML output) |
17189 | 1293 |
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10) |
16550 | 1294 |
|
1295 |
translations -- {* Beware of argument permutation! *} |
|
1296 |
"PROD i:A. b" == "setprod (%i. b) A" |
|
1297 |
"\<Prod>i\<in>A. b" == "setprod (%i. b) A" |
|
1298 |
||
1299 |
text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter |
|
1300 |
@{text"\<Prod>x|P. e"}. *} |
|
1301 |
||
1302 |
syntax |
|
17189 | 1303 |
"_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10) |
16550 | 1304 |
syntax (xsymbols) |
17189 | 1305 |
"_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10) |
16550 | 1306 |
syntax (HTML output) |
17189 | 1307 |
"_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10) |
16550 | 1308 |
|
15402 | 1309 |
translations |
16550 | 1310 |
"PROD x|P. t" => "setprod (%x. t) {x. P}" |
1311 |
"\<Prod>x|P. t" => "setprod (%x. t) {x. P}" |
|
1312 |
||
15402 | 1313 |
|
1314 |
lemma setprod_empty [simp]: "setprod f {} = 1" |
|
1315 |
by (auto simp add: setprod_def) |
|
1316 |
||
1317 |
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==> |
|
1318 |
setprod f (insert a A) = f a * setprod f A" |
|
19931
fb32b43e7f80
Restructured locales with predicates: import is now an interpretation.
ballarin
parents:
19870
diff
changeset
|
1319 |
by (simp add: setprod_def) |
15402 | 1320 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1321 |
lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1322 |
by (simp add: setprod_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1323 |
|
15402 | 1324 |
lemma setprod_reindex: |
1325 |
"inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B" |
|
15765 | 1326 |
by(auto simp: setprod_def AC_mult.fold_reindex dest!:finite_imageD) |
15402 | 1327 |
|
1328 |
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)" |
|
1329 |
by (auto simp add: setprod_reindex) |
|
1330 |
||
1331 |
lemma setprod_cong: |
|
1332 |
"A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B" |
|
15765 | 1333 |
by(fastsimp simp: setprod_def intro: AC_mult.fold_cong) |
15402 | 1334 |
|
16632
ad2895beef79
Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents:
16550
diff
changeset
|
1335 |
lemma strong_setprod_cong: |
ad2895beef79
Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents:
16550
diff
changeset
|
1336 |
"A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B" |
ad2895beef79
Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents:
16550
diff
changeset
|
1337 |
by(fastsimp simp: simp_implies_def setprod_def intro: AC_mult.fold_cong) |
ad2895beef79
Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents:
16550
diff
changeset
|
1338 |
|
15402 | 1339 |
lemma setprod_reindex_cong: "inj_on f A ==> |
1340 |
B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A" |
|
1341 |
by (frule setprod_reindex, simp) |
|
1342 |
||
1343 |
||
1344 |
lemma setprod_1: "setprod (%i. 1) A = 1" |
|
1345 |
apply (case_tac "finite A") |
|
1346 |
apply (erule finite_induct, auto simp add: mult_ac) |
|
1347 |
done |
|
1348 |
||
1349 |
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1" |
|
1350 |
apply (subgoal_tac "setprod f F = setprod (%x. 1) F") |
|
1351 |
apply (erule ssubst, rule setprod_1) |
|
1352 |
apply (rule setprod_cong, auto) |
|
1353 |
done |
|
1354 |
||
1355 |
lemma setprod_Un_Int: "finite A ==> finite B |
|
1356 |
==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B" |
|
15765 | 1357 |
by(simp add: setprod_def AC_mult.fold_Un_Int[symmetric]) |
15402 | 1358 |
|
1359 |
lemma setprod_Un_disjoint: "finite A ==> finite B |
|
1360 |
==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B" |
|
1361 |
by (subst setprod_Un_Int [symmetric], auto) |
|
1362 |
||
1363 |
lemma setprod_UN_disjoint: |
|
1364 |
"finite I ==> (ALL i:I. finite (A i)) ==> |
|
1365 |
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==> |
|
1366 |
setprod f (UNION I A) = setprod (%i. setprod f (A i)) I" |
|
15765 | 1367 |
by(simp add: setprod_def AC_mult.fold_UN_disjoint cong: setprod_cong) |
15402 | 1368 |
|
1369 |
lemma setprod_Union_disjoint: |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1370 |
"[| (ALL A:C. finite A); |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1371 |
(ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1372 |
==> setprod f (Union C) = setprod (setprod f) C" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1373 |
apply (cases "finite C") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1374 |
prefer 2 apply (force dest: finite_UnionD simp add: setprod_def) |
15402 | 1375 |
apply (frule setprod_UN_disjoint [of C id f]) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1376 |
apply (unfold Union_def id_def, assumption+) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1377 |
done |
15402 | 1378 |
|
1379 |
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==> |
|
16550 | 1380 |
(\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) = |
17189 | 1381 |
(\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)" |
15765 | 1382 |
by(simp add:setprod_def AC_mult.fold_Sigma split_def cong:setprod_cong) |
15402 | 1383 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1384 |
text{*Here we can eliminate the finiteness assumptions, by cases.*} |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1385 |
lemma setprod_cartesian_product: |
17189 | 1386 |
"(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)" |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1387 |
apply (cases "finite A") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1388 |
apply (cases "finite B") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1389 |
apply (simp add: setprod_Sigma) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1390 |
apply (cases "A={}", simp) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1391 |
apply (simp add: setprod_1) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1392 |
apply (auto simp add: setprod_def |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1393 |
dest: finite_cartesian_productD1 finite_cartesian_productD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1394 |
done |
15402 | 1395 |
|
1396 |
lemma setprod_timesf: |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1397 |
"setprod (%x. f x * g x) A = (setprod f A * setprod g A)" |
15765 | 1398 |
by(simp add:setprod_def AC_mult.fold_distrib) |
15402 | 1399 |
|
1400 |
||
1401 |
subsubsection {* Properties in more restricted classes of structures *} |
|
1402 |
||
1403 |
lemma setprod_eq_1_iff [simp]: |
|
1404 |
"finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))" |
|
22262 | 1405 |
by (induct set: finite) auto |
15402 | 1406 |
|
1407 |
lemma setprod_zero: |
|
1408 |
"finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0" |
|
22262 | 1409 |
apply (induct set: finite, force, clarsimp) |
15402 | 1410 |
apply (erule disjE, auto) |
1411 |
done |
|
1412 |
||
1413 |
lemma setprod_nonneg [rule_format]: |
|
1414 |
"(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A" |
|
1415 |
apply (case_tac "finite A") |
|
22262 | 1416 |
apply (induct set: finite, force, clarsimp) |
15402 | 1417 |
apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force) |
1418 |
apply (rule mult_mono, assumption+) |
|
1419 |
apply (auto simp add: setprod_def) |
|
1420 |
done |
|
1421 |
||
1422 |
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x) |
|
1423 |
--> 0 < setprod f A" |
|
1424 |
apply (case_tac "finite A") |
|
22262 | 1425 |
apply (induct set: finite, force, clarsimp) |
15402 | 1426 |
apply (subgoal_tac "0 * 0 < f x * setprod f F", force) |
1427 |
apply (rule mult_strict_mono, assumption+) |
|
1428 |
apply (auto simp add: setprod_def) |
|
1429 |
done |
|
1430 |
||
1431 |
lemma setprod_nonzero [rule_format]: |
|
1432 |
"(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==> |
|
1433 |
finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0" |
|
1434 |
apply (erule finite_induct, auto) |
|
1435 |
done |
|
1436 |
||
1437 |
lemma setprod_zero_eq: |
|
1438 |
"(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==> |
|
1439 |
finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)" |
|
1440 |
apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast) |
|
1441 |
done |
|
1442 |
||
1443 |
lemma setprod_nonzero_field: |
|
1444 |
"finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0" |
|
1445 |
apply (rule setprod_nonzero, auto) |
|
1446 |
done |
|
1447 |
||
1448 |
lemma setprod_zero_eq_field: |
|
1449 |
"finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)" |
|
1450 |
apply (rule setprod_zero_eq, auto) |
|
1451 |
done |
|
1452 |
||
1453 |
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==> |
|
1454 |
(setprod f (A Un B) :: 'a ::{field}) |
|
1455 |
= setprod f A * setprod f B / setprod f (A Int B)" |
|
1456 |
apply (subst setprod_Un_Int [symmetric], auto) |
|
1457 |
apply (subgoal_tac "finite (A Int B)") |
|
1458 |
apply (frule setprod_nonzero_field [of "A Int B" f], assumption) |
|
1459 |
apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self) |
|
1460 |
done |
|
1461 |
||
1462 |
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==> |
|
1463 |
(setprod f (A - {a}) :: 'a :: {field}) = |
|
1464 |
(if a:A then setprod f A / f a else setprod f A)" |
|
1465 |
apply (erule finite_induct) |
|
1466 |
apply (auto simp add: insert_Diff_if) |
|
1467 |
apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a") |
|
1468 |
apply (erule ssubst) |
|
1469 |
apply (subst times_divide_eq_right [THEN sym]) |
|
1470 |
apply (auto simp add: mult_ac times_divide_eq_right divide_self) |
|
1471 |
done |
|
1472 |
||
1473 |
lemma setprod_inversef: "finite A ==> |
|
1474 |
ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==> |
|
1475 |
setprod (inverse \<circ> f) A = inverse (setprod f A)" |
|
1476 |
apply (erule finite_induct) |
|
1477 |
apply (simp, simp) |
|
1478 |
done |
|
1479 |
||
1480 |
lemma setprod_dividef: |
|
1481 |
"[|finite A; |
|
1482 |
\<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|] |
|
1483 |
==> setprod (%x. f x / g x) A = setprod f A / setprod g A" |
|
1484 |
apply (subgoal_tac |
|
1485 |
"setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A") |
|
1486 |
apply (erule ssubst) |
|
1487 |
apply (subst divide_inverse) |
|
1488 |
apply (subst setprod_timesf) |
|
1489 |
apply (subst setprod_inversef, assumption+, rule refl) |
|
1490 |
apply (rule setprod_cong, rule refl) |
|
1491 |
apply (subst divide_inverse, auto) |
|
1492 |
done |
|
1493 |
||
12396 | 1494 |
subsection {* Finite cardinality *} |
1495 |
||
15402 | 1496 |
text {* This definition, although traditional, is ugly to work with: |
1497 |
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}. |
|
1498 |
But now that we have @{text setsum} things are easy: |
|
12396 | 1499 |
*} |
1500 |
||
1501 |
constdefs |
|
1502 |
card :: "'a set => nat" |
|
15402 | 1503 |
"card A == setsum (%x. 1::nat) A" |
12396 | 1504 |
|
1505 |
lemma card_empty [simp]: "card {} = 0" |
|
15402 | 1506 |
by (simp add: card_def) |
1507 |
||
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1508 |
lemma card_infinite [simp]: "~ finite A ==> card A = 0" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1509 |
by (simp add: card_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1510 |
|
15402 | 1511 |
lemma card_eq_setsum: "card A = setsum (%x. 1) A" |
1512 |
by (simp add: card_def) |
|
12396 | 1513 |
|
1514 |
lemma card_insert_disjoint [simp]: |
|
1515 |
"finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)" |
|
15765 | 1516 |
by(simp add: card_def) |
15402 | 1517 |
|
1518 |
lemma card_insert_if: |
|
1519 |
"finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))" |
|
1520 |
by (simp add: insert_absorb) |
|
12396 | 1521 |
|
1522 |
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})" |
|
1523 |
apply auto |
|
15506 | 1524 |
apply (drule_tac a = x in mk_disjoint_insert, clarify, auto) |
12396 | 1525 |
done |
1526 |
||
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1527 |
lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1528 |
by auto |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1529 |
|
12396 | 1530 |
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A" |
14302 | 1531 |
apply(rule_tac t = A in insert_Diff [THEN subst], assumption) |
1532 |
apply(simp del:insert_Diff_single) |
|
1533 |
done |
|
12396 | 1534 |
|
1535 |
lemma card_Diff_singleton: |
|
1536 |
"finite A ==> x: A ==> card (A - {x}) = card A - 1" |
|
1537 |
by (simp add: card_Suc_Diff1 [symmetric]) |
|
1538 |
||
1539 |
lemma card_Diff_singleton_if: |
|
1540 |
"finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)" |
|
1541 |
by (simp add: card_Diff_singleton) |
|
1542 |
||
1543 |
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))" |
|
1544 |
by (simp add: card_insert_if card_Suc_Diff1) |
|
1545 |
||
1546 |
lemma card_insert_le: "finite A ==> card A <= card (insert x A)" |
|
1547 |
by (simp add: card_insert_if) |
|
1548 |
||
15402 | 1549 |
lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B" |
15539 | 1550 |
by (simp add: card_def setsum_mono2) |
15402 | 1551 |
|
12396 | 1552 |
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)" |
22262 | 1553 |
apply (induct set: finite, simp, clarify) |
12396 | 1554 |
apply (subgoal_tac "finite A & A - {x} <= F") |
14208 | 1555 |
prefer 2 apply (blast intro: finite_subset, atomize) |
12396 | 1556 |
apply (drule_tac x = "A - {x}" in spec) |
1557 |
apply (simp add: card_Diff_singleton_if split add: split_if_asm) |
|
14208 | 1558 |
apply (case_tac "card A", auto) |
12396 | 1559 |
done |
1560 |
||
1561 |
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B" |
|
1562 |
apply (simp add: psubset_def linorder_not_le [symmetric]) |
|
1563 |
apply (blast dest: card_seteq) |
|
1564 |
done |
|
1565 |
||
1566 |
lemma card_Un_Int: "finite A ==> finite B |
|
1567 |
==> card A + card B = card (A Un B) + card (A Int B)" |
|
15402 | 1568 |
by(simp add:card_def setsum_Un_Int) |
12396 | 1569 |
|
1570 |
lemma card_Un_disjoint: "finite A ==> finite B |
|
1571 |
==> A Int B = {} ==> card (A Un B) = card A + card B" |
|
1572 |
by (simp add: card_Un_Int) |
|
1573 |
||
1574 |
lemma card_Diff_subset: |
|
15402 | 1575 |
"finite B ==> B <= A ==> card (A - B) = card A - card B" |
1576 |
by(simp add:card_def setsum_diff_nat) |
|
12396 | 1577 |
|
1578 |
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A" |
|
1579 |
apply (rule Suc_less_SucD) |
|
1580 |
apply (simp add: card_Suc_Diff1) |
|
1581 |
done |
|
1582 |
||
1583 |
lemma card_Diff2_less: |
|
1584 |
"finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A" |
|
1585 |
apply (case_tac "x = y") |
|
1586 |
apply (simp add: card_Diff1_less) |
|
1587 |
apply (rule less_trans) |
|
1588 |
prefer 2 apply (auto intro!: card_Diff1_less) |
|
1589 |
done |
|
1590 |
||
1591 |
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A" |
|
1592 |
apply (case_tac "x : A") |
|
1593 |
apply (simp_all add: card_Diff1_less less_imp_le) |
|
1594 |
done |
|
1595 |
||
1596 |
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B" |
|
14208 | 1597 |
by (erule psubsetI, blast) |
12396 | 1598 |
|
14889 | 1599 |
lemma insert_partition: |
15402 | 1600 |
"\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk> |
1601 |
\<Longrightarrow> x \<inter> \<Union> F = {}" |
|
14889 | 1602 |
by auto |
1603 |
||
19793 | 1604 |
text{* main cardinality theorem *} |
14889 | 1605 |
lemma card_partition [rule_format]: |
1606 |
"finite C ==> |
|
1607 |
finite (\<Union> C) --> |
|
1608 |
(\<forall>c\<in>C. card c = k) --> |
|
1609 |
(\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) --> |
|
1610 |
k * card(C) = card (\<Union> C)" |
|
1611 |
apply (erule finite_induct, simp) |
|
1612 |
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition |
|
1613 |
finite_subset [of _ "\<Union> (insert x F)"]) |
|
1614 |
done |
|
1615 |
||
12396 | 1616 |
|
19793 | 1617 |
text{*The form of a finite set of given cardinality*} |
1618 |
||
1619 |
lemma card_eq_SucD: |
|
1620 |
assumes cardeq: "card A = Suc k" and fin: "finite A" |
|
1621 |
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k" |
|
1622 |
proof - |
|
1623 |
have "card A \<noteq> 0" using cardeq by auto |
|
1624 |
then obtain b where b: "b \<in> A" using fin by auto |
|
1625 |
show ?thesis |
|
1626 |
proof (intro exI conjI) |
|
1627 |
show "A = insert b (A-{b})" using b by blast |
|
1628 |
show "b \<notin> A - {b}" by blast |
|
1629 |
show "card (A - {b}) = k" by (simp add: fin cardeq b card_Diff_singleton) |
|
1630 |
qed |
|
1631 |
qed |
|
1632 |
||
1633 |
||
1634 |
lemma card_Suc_eq: |
|
1635 |
"finite A ==> |
|
1636 |
(card A = Suc k) = (\<exists>b B. A = insert b B & b \<notin> B & card B = k)" |
|
1637 |
by (auto dest!: card_eq_SucD) |
|
1638 |
||
1639 |
lemma card_1_eq: |
|
1640 |
"finite A ==> (card A = Suc 0) = (\<exists>x. A = {x})" |
|
1641 |
by (auto dest!: card_eq_SucD) |
|
1642 |
||
1643 |
lemma card_2_eq: |
|
1644 |
"finite A ==> (card A = Suc(Suc 0)) = (\<exists>x y. x\<noteq>y & A = {x,y})" |
|
1645 |
by (auto dest!: card_eq_SucD, blast) |
|
1646 |
||
1647 |
||
15539 | 1648 |
lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y" |
1649 |
apply (cases "finite A") |
|
1650 |
apply (erule finite_induct) |
|
1651 |
apply (auto simp add: ring_distrib add_ac) |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1652 |
done |
15402 | 1653 |
|
21199
2d83f93c3580
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents:
19984
diff
changeset
|
1654 |
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{recpower, comm_monoid_mult})) = y^(card A)" |
15402 | 1655 |
apply (erule finite_induct) |
1656 |
apply (auto simp add: power_Suc) |
|
1657 |
done |
|
1658 |
||
15542 | 1659 |
lemma setsum_bounded: |
1660 |
assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{comm_semiring_1_cancel, pordered_ab_semigroup_add})" |
|
1661 |
shows "setsum f A \<le> of_nat(card A) * K" |
|
1662 |
proof (cases "finite A") |
|
1663 |
case True |
|
1664 |
thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp |
|
1665 |
next |
|
1666 |
case False thus ?thesis by (simp add: setsum_def) |
|
1667 |
qed |
|
1668 |
||
15402 | 1669 |
|
1670 |
subsubsection {* Cardinality of unions *} |
|
1671 |
||
1672 |
lemma card_UN_disjoint: |
|
1673 |
"finite I ==> (ALL i:I. finite (A i)) ==> |
|
1674 |
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==> |
|
1675 |
card (UNION I A) = (\<Sum>i\<in>I. card(A i))" |
|
15539 | 1676 |
apply (simp add: card_def del: setsum_constant) |
15402 | 1677 |
apply (subgoal_tac |
1678 |
"setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I") |
|
15539 | 1679 |
apply (simp add: setsum_UN_disjoint del: setsum_constant) |
1680 |
apply (simp cong: setsum_cong) |
|
15402 | 1681 |
done |
1682 |
||
1683 |
lemma card_Union_disjoint: |
|
1684 |
"finite C ==> (ALL A:C. finite A) ==> |
|
1685 |
(ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==> |
|
1686 |
card (Union C) = setsum card C" |
|
1687 |
apply (frule card_UN_disjoint [of C id]) |
|
1688 |
apply (unfold Union_def id_def, assumption+) |
|
1689 |
done |
|
1690 |
||
12396 | 1691 |
subsubsection {* Cardinality of image *} |
1692 |
||
15447 | 1693 |
text{*The image of a finite set can be expressed using @{term fold}.*} |
1694 |
lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A" |
|
1695 |
apply (erule finite_induct, simp) |
|
1696 |
apply (subst ACf.fold_insert) |
|
1697 |
apply (auto simp add: ACf_def) |
|
1698 |
done |
|
1699 |
||
12396 | 1700 |
lemma card_image_le: "finite A ==> card (f ` A) <= card A" |
22262 | 1701 |
apply (induct set: finite) |
21575 | 1702 |
apply simp |
12396 | 1703 |
apply (simp add: le_SucI finite_imageI card_insert_if) |
1704 |
done |
|
1705 |
||
15402 | 1706 |
lemma card_image: "inj_on f A ==> card (f ` A) = card A" |
15539 | 1707 |
by(simp add:card_def setsum_reindex o_def del:setsum_constant) |
12396 | 1708 |
|
1709 |
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A" |
|
1710 |
by (simp add: card_seteq card_image) |
|
1711 |
||
15111 | 1712 |
lemma eq_card_imp_inj_on: |
1713 |
"[| finite A; card(f ` A) = card A |] ==> inj_on f A" |
|
21575 | 1714 |
apply (induct rule:finite_induct) |
1715 |
apply simp |
|
15111 | 1716 |
apply(frule card_image_le[where f = f]) |
1717 |
apply(simp add:card_insert_if split:if_splits) |
|
1718 |
done |
|
1719 |
||
1720 |
lemma inj_on_iff_eq_card: |
|
1721 |
"finite A ==> inj_on f A = (card(f ` A) = card A)" |
|
1722 |
by(blast intro: card_image eq_card_imp_inj_on) |
|
1723 |
||
12396 | 1724 |
|
15402 | 1725 |
lemma card_inj_on_le: |
1726 |
"[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B" |
|
1727 |
apply (subgoal_tac "finite A") |
|
1728 |
apply (force intro: card_mono simp add: card_image [symmetric]) |
|
1729 |
apply (blast intro: finite_imageD dest: finite_subset) |
|
1730 |
done |
|
1731 |
||
1732 |
lemma card_bij_eq: |
|
1733 |
"[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A; |
|
1734 |
finite A; finite B |] ==> card A = card B" |
|
1735 |
by (auto intro: le_anti_sym card_inj_on_le) |
|
1736 |
||
1737 |
||
1738 |
subsubsection {* Cardinality of products *} |
|
1739 |
||
1740 |
(* |
|
1741 |
lemma SigmaI_insert: "y \<notin> A ==> |
|
1742 |
(SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))" |
|
1743 |
by auto |
|
1744 |
*) |
|
1745 |
||
1746 |
lemma card_SigmaI [simp]: |
|
1747 |
"\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk> |
|
1748 |
\<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))" |
|
15539 | 1749 |
by(simp add:card_def setsum_Sigma del:setsum_constant) |
15402 | 1750 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1751 |
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1752 |
apply (cases "finite A") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1753 |
apply (cases "finite B") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1754 |
apply (auto simp add: card_eq_0_iff |
15539 | 1755 |
dest: finite_cartesian_productD1 finite_cartesian_productD2) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1756 |
done |
15402 | 1757 |
|
1758 |
lemma card_cartesian_product_singleton: "card({x} <*> A) = card(A)" |
|
15539 | 1759 |
by (simp add: card_cartesian_product) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1760 |
|
15402 | 1761 |
|
1762 |
||
12396 | 1763 |
subsubsection {* Cardinality of the Powerset *} |
1764 |
||
1765 |
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *) |
|
22262 | 1766 |
apply (induct set: finite) |
12396 | 1767 |
apply (simp_all add: Pow_insert) |
14208 | 1768 |
apply (subst card_Un_disjoint, blast) |
1769 |
apply (blast intro: finite_imageI, blast) |
|
12396 | 1770 |
apply (subgoal_tac "inj_on (insert x) (Pow F)") |
1771 |
apply (simp add: card_image Pow_insert) |
|
1772 |
apply (unfold inj_on_def) |
|
1773 |
apply (blast elim!: equalityE) |
|
1774 |
done |
|
1775 |
||
15392 | 1776 |
text {* Relates to equivalence classes. Based on a theorem of |
1777 |
F. Kammüller's. *} |
|
12396 | 1778 |
|
1779 |
lemma dvd_partition: |
|
15392 | 1780 |
"finite (Union C) ==> |
12396 | 1781 |
ALL c : C. k dvd card c ==> |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset
|
1782 |
(ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==> |
12396 | 1783 |
k dvd card (Union C)" |
15392 | 1784 |
apply(frule finite_UnionD) |
1785 |
apply(rotate_tac -1) |
|
22262 | 1786 |
apply (induct set: finite, simp_all, clarify) |
12396 | 1787 |
apply (subst card_Un_disjoint) |
1788 |
apply (auto simp add: dvd_add disjoint_eq_subset_Compl) |
|
1789 |
done |
|
1790 |
||
1791 |
||
15392 | 1792 |
subsection{* A fold functional for non-empty sets *} |
1793 |
||
1794 |
text{* Does not require start value. *} |
|
12396 | 1795 |
|
22262 | 1796 |
inductive2 |
1797 |
fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool" |
|
1798 |
for f :: "'a => 'a => 'a" |
|
1799 |
where |
|
15506 | 1800 |
fold1Set_insertI [intro]: |
22262 | 1801 |
"\<lbrakk> foldSet f id a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x" |
12396 | 1802 |
|
15392 | 1803 |
constdefs |
1804 |
fold1 :: "('a => 'a => 'a) => 'a set => 'a" |
|
22262 | 1805 |
"fold1 f A == THE x. fold1Set f A x" |
15506 | 1806 |
|
1807 |
lemma fold1Set_nonempty: |
|
22917 | 1808 |
"fold1Set f A x \<Longrightarrow> A \<noteq> {}" |
1809 |
by(erule fold1Set.cases, simp_all) |
|
15392 | 1810 |
|
22262 | 1811 |
inductive_cases2 empty_fold1SetE [elim!]: "fold1Set f {} x" |
1812 |
||
1813 |
inductive_cases2 insert_fold1SetE [elim!]: "fold1Set f (insert a X) x" |
|
1814 |
||
1815 |
||
1816 |
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)" |
|
15506 | 1817 |
by (blast intro: foldSet.intros elim: foldSet.cases) |
15392 | 1818 |
|
22917 | 1819 |
lemma fold1_singleton [simp]: "fold1 f {a} = a" |
15508 | 1820 |
by (unfold fold1_def) blast |
12396 | 1821 |
|
15508 | 1822 |
lemma finite_nonempty_imp_fold1Set: |
22262 | 1823 |
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x" |
15508 | 1824 |
apply (induct A rule: finite_induct) |
1825 |
apply (auto dest: finite_imp_foldSet [of _ f id]) |
|
1826 |
done |
|
15506 | 1827 |
|
1828 |
text{*First, some lemmas about @{term foldSet}.*} |
|
15392 | 1829 |
|
15508 | 1830 |
lemma (in ACf) foldSet_insert_swap: |
22262 | 1831 |
assumes fold: "foldSet f id b A y" |
1832 |
shows "b \<notin> A \<Longrightarrow> foldSet f id z (insert b A) (z \<cdot> y)" |
|
15508 | 1833 |
using fold |
1834 |
proof (induct rule: foldSet.induct) |
|
1835 |
case emptyI thus ?case by (force simp add: fold_insert_aux commute) |
|
1836 |
next |
|
22262 | 1837 |
case (insertI x A y) |
1838 |
have "foldSet f (\<lambda>u. u) z (insert x (insert b A)) (x \<cdot> (z \<cdot> y))" |
|
15521 | 1839 |
using insertI by force --{*how does @{term id} get unfolded?*} |
15508 | 1840 |
thus ?case by (simp add: insert_commute AC) |
1841 |
qed |
|
1842 |
||
1843 |
lemma (in ACf) foldSet_permute_diff: |
|
22262 | 1844 |
assumes fold: "foldSet f id b A x" |
1845 |
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> foldSet f id a (insert b (A-{a})) x" |
|
15508 | 1846 |
using fold |
1847 |
proof (induct rule: foldSet.induct) |
|
1848 |
case emptyI thus ?case by simp |
|
1849 |
next |
|
22262 | 1850 |
case (insertI x A y) |
15521 | 1851 |
have "a = x \<or> a \<in> A" using insertI by simp |
1852 |
thus ?case |
|
1853 |
proof |
|
1854 |
assume "a = x" |
|
1855 |
with insertI show ?thesis |
|
1856 |
by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap) |
|
1857 |
next |
|
1858 |
assume ainA: "a \<in> A" |
|
22262 | 1859 |
hence "foldSet f id a (insert x (insert b (A - {a}))) (x \<cdot> y)" |
15521 | 1860 |
using insertI by (force simp: id_def) |
1861 |
moreover |
|
1862 |
have "insert x (insert b (A - {a})) = insert b (insert x A - {a})" |
|
1863 |
using ainA insertI by blast |
|
1864 |
ultimately show ?thesis by (simp add: id_def) |
|
15508 | 1865 |
qed |
1866 |
qed |
|
1867 |
||
1868 |
lemma (in ACf) fold1_eq_fold: |
|
1869 |
"[|finite A; a \<notin> A|] ==> fold1 f (insert a A) = fold f id a A" |
|
1870 |
apply (simp add: fold1_def fold_def) |
|
1871 |
apply (rule the_equality) |
|
1872 |
apply (best intro: foldSet_determ theI dest: finite_imp_foldSet [of _ f id]) |
|
1873 |
apply (rule sym, clarify) |
|
1874 |
apply (case_tac "Aa=A") |
|
1875 |
apply (best intro: the_equality foldSet_determ) |
|
22262 | 1876 |
apply (subgoal_tac "foldSet f id a A x") |
15508 | 1877 |
apply (best intro: the_equality foldSet_determ) |
1878 |
apply (subgoal_tac "insert aa (Aa - {a}) = A") |
|
1879 |
prefer 2 apply (blast elim: equalityE) |
|
1880 |
apply (auto dest: foldSet_permute_diff [where a=a]) |
|
1881 |
done |
|
1882 |
||
15521 | 1883 |
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)" |
1884 |
apply safe |
|
1885 |
apply simp |
|
1886 |
apply (drule_tac x=x in spec) |
|
1887 |
apply (drule_tac x="A-{x}" in spec, auto) |
|
15508 | 1888 |
done |
1889 |
||
15521 | 1890 |
lemma (in ACf) fold1_insert: |
1891 |
assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A" |
|
1892 |
shows "fold1 f (insert x A) = f x (fold1 f A)" |
|
1893 |
proof - |
|
1894 |
from nonempty obtain a A' where "A = insert a A' & a ~: A'" |
|
1895 |
by (auto simp add: nonempty_iff) |
|
1896 |
with A show ?thesis |
|
1897 |
by (simp add: insert_commute [of x] fold1_eq_fold eq_commute) |
|
1898 |
qed |
|
1899 |
||
15509 | 1900 |
lemma (in ACIf) fold1_insert_idem [simp]: |
15521 | 1901 |
assumes nonempty: "A \<noteq> {}" and A: "finite A" |
1902 |
shows "fold1 f (insert x A) = f x (fold1 f A)" |
|
1903 |
proof - |
|
1904 |
from nonempty obtain a A' where A': "A = insert a A' & a ~: A'" |
|
1905 |
by (auto simp add: nonempty_iff) |
|
1906 |
show ?thesis |
|
1907 |
proof cases |
|
1908 |
assume "a = x" |
|
1909 |
thus ?thesis |
|
1910 |
proof cases |
|
1911 |
assume "A' = {}" |
|
1912 |
with prems show ?thesis by (simp add: idem) |
|
1913 |
next |
|
1914 |
assume "A' \<noteq> {}" |
|
1915 |
with prems show ?thesis |
|
1916 |
by (simp add: fold1_insert assoc [symmetric] idem) |
|
1917 |
qed |
|
1918 |
next |
|
1919 |
assume "a \<noteq> x" |
|
1920 |
with prems show ?thesis |
|
1921 |
by (simp add: insert_commute fold1_eq_fold fold_insert_idem) |
|
1922 |
qed |
|
1923 |
qed |
|
15506 | 1924 |
|
22917 | 1925 |
lemma (in ACIf) hom_fold1_commute: |
1926 |
assumes hom: "!!x y. h(f x y) = f (h x) (h y)" |
|
1927 |
and N: "finite N" "N \<noteq> {}" shows "h(fold1 f N) = fold1 f (h ` N)" |
|
1928 |
using N proof (induct rule: finite_ne_induct) |
|
1929 |
case singleton thus ?case by simp |
|
1930 |
next |
|
1931 |
case (insert n N) |
|
1932 |
then have "h(fold1 f (insert n N)) = h(f n (fold1 f N))" by simp |
|
1933 |
also have "\<dots> = f (h n) (h(fold1 f N))" by(rule hom) |
|
1934 |
also have "h(fold1 f N) = fold1 f (h ` N)" by(rule insert) |
|
1935 |
also have "f (h n) \<dots> = fold1 f (insert (h n) (h ` N))" |
|
1936 |
using insert by(simp) |
|
1937 |
also have "insert (h n) (h ` N) = h ` insert n N" by simp |
|
1938 |
finally show ?case . |
|
1939 |
qed |
|
1940 |
||
15506 | 1941 |
|
15508 | 1942 |
text{* Now the recursion rules for definitions: *} |
1943 |
||
22917 | 1944 |
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a" |
15508 | 1945 |
by(simp add:fold1_singleton) |
1946 |
||
1947 |
lemma (in ACf) fold1_insert_def: |
|
22917 | 1948 |
"\<lbrakk> g = fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x \<cdot> (g A)" |
15508 | 1949 |
by(simp add:fold1_insert) |
1950 |
||
15509 | 1951 |
lemma (in ACIf) fold1_insert_idem_def: |
22917 | 1952 |
"\<lbrakk> g = fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x \<cdot> (g A)" |
15509 | 1953 |
by(simp add:fold1_insert_idem) |
15508 | 1954 |
|
1955 |
subsubsection{* Determinacy for @{term fold1Set} *} |
|
1956 |
||
1957 |
text{*Not actually used!!*} |
|
12396 | 1958 |
|
15506 | 1959 |
lemma (in ACf) foldSet_permute: |
22262 | 1960 |
"[|foldSet f id b (insert a A) x; a \<notin> A; b \<notin> A|] |
1961 |
==> foldSet f id a (insert b A) x" |
|
15506 | 1962 |
apply (case_tac "a=b") |
1963 |
apply (auto dest: foldSet_permute_diff) |
|
1964 |
done |
|
15376 | 1965 |
|
15506 | 1966 |
lemma (in ACf) fold1Set_determ: |
22262 | 1967 |
"fold1Set f A x ==> fold1Set f A y ==> y = x" |
15506 | 1968 |
proof (clarify elim!: fold1Set.cases) |
1969 |
fix A x B y a b |
|
22262 | 1970 |
assume Ax: "foldSet f id a A x" |
1971 |
assume By: "foldSet f id b B y" |
|
15506 | 1972 |
assume anotA: "a \<notin> A" |
1973 |
assume bnotB: "b \<notin> B" |
|
1974 |
assume eq: "insert a A = insert b B" |
|
1975 |
show "y=x" |
|
1976 |
proof cases |
|
1977 |
assume same: "a=b" |
|
1978 |
hence "A=B" using anotA bnotB eq by (blast elim!: equalityE) |
|
1979 |
thus ?thesis using Ax By same by (blast intro: foldSet_determ) |
|
15392 | 1980 |
next |
15506 | 1981 |
assume diff: "a\<noteq>b" |
1982 |
let ?D = "B - {a}" |
|
1983 |
have B: "B = insert a ?D" and A: "A = insert b ?D" |
|
1984 |
and aB: "a \<in> B" and bA: "b \<in> A" |
|
1985 |
using eq anotA bnotB diff by (blast elim!:equalityE)+ |
|
1986 |
with aB bnotB By |
|
22262 | 1987 |
have "foldSet f id a (insert b ?D) y" |
15506 | 1988 |
by (auto intro: foldSet_permute simp add: insert_absorb) |
1989 |
moreover |
|
22262 | 1990 |
have "foldSet f id a (insert b ?D) x" |
15506 | 1991 |
by (simp add: A [symmetric] Ax) |
1992 |
ultimately show ?thesis by (blast intro: foldSet_determ) |
|
15392 | 1993 |
qed |
12396 | 1994 |
qed |
1995 |
||
22262 | 1996 |
lemma (in ACf) fold1Set_equality: "fold1Set f A y ==> fold1 f A = y" |
15506 | 1997 |
by (unfold fold1_def) (blast intro: fold1Set_determ) |
1998 |
||
1999 |
declare |
|
2000 |
empty_foldSetE [rule del] foldSet.intros [rule del] |
|
2001 |
empty_fold1SetE [rule del] insert_fold1SetE [rule del] |
|
19931
fb32b43e7f80
Restructured locales with predicates: import is now an interpretation.
ballarin
parents:
19870
diff
changeset
|
2002 |
-- {* No more proofs involve these relations. *} |
15376 | 2003 |
|
22917 | 2004 |
|
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2005 |
subsubsection{* Semi-Lattices *} |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2006 |
|
22917 | 2007 |
locale ACIfSL = ord + ACIf + |
2008 |
assumes below_def: "x \<sqsubseteq> y \<longleftrightarrow> x \<cdot> y = x" |
|
2009 |
and strict_below_def: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y" |
|
2010 |
begin |
|
2011 |
||
2012 |
lemma below_refl [simp]: "x \<^loc>\<le> x" |
|
2013 |
by (simp add: below_def idem) |
|
2014 |
||
2015 |
lemma below_antisym: |
|
2016 |
assumes xy: "x \<^loc>\<le> y" and yx: "y \<^loc>\<le> x" |
|
2017 |
shows "x = y" |
|
2018 |
using xy [unfolded below_def, symmetric] |
|
2019 |
yx [unfolded below_def commute] |
|
2020 |
by (rule trans) |
|
2021 |
||
2022 |
lemma below_trans: |
|
2023 |
assumes xy: "x \<^loc>\<le> y" and yz: "y \<^loc>\<le> z" |
|
2024 |
shows "x \<^loc>\<le> z" |
|
2025 |
proof - |
|
2026 |
from xy have x_xy: "x \<cdot> y = x" by (simp add: below_def) |
|
2027 |
from yz have y_yz: "y \<cdot> z = y" by (simp add: below_def) |
|
2028 |
from y_yz have "x \<cdot> y \<cdot> z = x \<cdot> y" by (simp add: assoc) |
|
2029 |
with x_xy have "x \<cdot> y \<cdot> z = x" by simp |
|
2030 |
moreover from x_xy have "x \<cdot> z = x \<cdot> y \<cdot> z" by simp |
|
2031 |
ultimately have "x \<cdot> z = x" by simp |
|
2032 |
then show ?thesis by (simp add: below_def) |
|
2033 |
qed |
|
2034 |
||
2035 |
lemma below_f_conv [simp]: "x \<sqsubseteq> y \<cdot> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)" |
|
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2036 |
proof |
15500 | 2037 |
assume "x \<sqsubseteq> y \<cdot> z" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2038 |
hence xyzx: "x \<cdot> (y \<cdot> z) = x" by(simp add: below_def) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2039 |
have "x \<cdot> y = x" |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2040 |
proof - |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2041 |
have "x \<cdot> y = (x \<cdot> (y \<cdot> z)) \<cdot> y" by(rule subst[OF xyzx], rule refl) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2042 |
also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2043 |
also have "\<dots> = x" by(rule xyzx) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2044 |
finally show ?thesis . |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2045 |
qed |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2046 |
moreover have "x \<cdot> z = x" |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2047 |
proof - |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2048 |
have "x \<cdot> z = (x \<cdot> (y \<cdot> z)) \<cdot> z" by(rule subst[OF xyzx], rule refl) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2049 |
also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2050 |
also have "\<dots> = x" by(rule xyzx) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2051 |
finally show ?thesis . |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2052 |
qed |
15500 | 2053 |
ultimately show "x \<sqsubseteq> y \<and> x \<sqsubseteq> z" by(simp add: below_def) |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2054 |
next |
15500 | 2055 |
assume a: "x \<sqsubseteq> y \<and> x \<sqsubseteq> z" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2056 |
hence y: "x \<cdot> y = x" and z: "x \<cdot> z = x" by(simp_all add: below_def) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2057 |
have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by(simp add:assoc) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2058 |
also have "x \<cdot> y = x" using a by(simp_all add: below_def) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2059 |
also have "x \<cdot> z = x" using a by(simp_all add: below_def) |
15500 | 2060 |
finally show "x \<sqsubseteq> y \<cdot> z" by(simp_all add: below_def) |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2061 |
qed |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2062 |
|
22917 | 2063 |
end |
2064 |
||
2065 |
interpretation ACIfSL < order |
|
2066 |
by unfold_locales |
|
2067 |
(simp add: strict_below_def, auto intro: below_refl below_trans below_antisym) |
|
2068 |
||
2069 |
locale ACIfSLlin = ACIfSL + |
|
2070 |
assumes lin: "x\<cdot>y \<in> {x,y}" |
|
2071 |
begin |
|
2072 |
||
2073 |
lemma above_f_conv: |
|
15500 | 2074 |
"x \<cdot> y \<sqsubseteq> z = (x \<sqsubseteq> z \<or> y \<sqsubseteq> z)" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2075 |
proof |
15500 | 2076 |
assume a: "x \<cdot> y \<sqsubseteq> z" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2077 |
have "x \<cdot> y = x \<or> x \<cdot> y = y" using lin[of x y] by simp |
15500 | 2078 |
thus "x \<sqsubseteq> z \<or> y \<sqsubseteq> z" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2079 |
proof |
15500 | 2080 |
assume "x \<cdot> y = x" hence "x \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis .. |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2081 |
next |
15500 | 2082 |
assume "x \<cdot> y = y" hence "y \<sqsubseteq> z" by(rule subst)(rule a) thus ?thesis .. |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2083 |
qed |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2084 |
next |
15500 | 2085 |
assume "x \<sqsubseteq> z \<or> y \<sqsubseteq> z" |
2086 |
thus "x \<cdot> y \<sqsubseteq> z" |
|
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2087 |
proof |
15500 | 2088 |
assume a: "x \<sqsubseteq> z" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2089 |
have "(x \<cdot> y) \<cdot> z = (x \<cdot> z) \<cdot> y" by(simp add:ACI) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2090 |
also have "x \<cdot> z = x" using a by(simp add:below_def) |
15500 | 2091 |
finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def) |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2092 |
next |
15500 | 2093 |
assume a: "y \<sqsubseteq> z" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2094 |
have "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" by(simp add:ACI) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2095 |
also have "y \<cdot> z = y" using a by(simp add:below_def) |
15500 | 2096 |
finally show "x \<cdot> y \<sqsubseteq> z" by(simp add:below_def) |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2097 |
qed |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2098 |
qed |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2099 |
|
22917 | 2100 |
lemma strict_below_f_conv[simp]: "x \<sqsubset> y \<cdot> z = (x \<sqsubset> y \<and> x \<sqsubset> z)" |
18493 | 2101 |
apply(simp add: strict_below_def) |
2102 |
using lin[of y z] by (auto simp:below_def ACI) |
|
2103 |
||
22917 | 2104 |
lemma strict_above_f_conv: |
19931
fb32b43e7f80
Restructured locales with predicates: import is now an interpretation.
ballarin
parents:
19870
diff
changeset
|
2105 |
"x \<cdot> y \<sqsubset> z = (x \<sqsubset> z \<or> y \<sqsubset> z)" |
18493 | 2106 |
apply(simp add: strict_below_def above_f_conv) |
2107 |
using lin[of y z] lin[of x z] by (auto simp:below_def ACI) |
|
2108 |
||
22917 | 2109 |
end |
2110 |
||
2111 |
interpretation ACIfSLlin < linorder |
|
2112 |
by unfold_locales |
|
2113 |
(insert lin [simplified insert_iff], simp add: below_def commute) |
|
2114 |
||
18493 | 2115 |
|
15502 | 2116 |
subsubsection{* Lemmas about @{text fold1} *} |
15484 | 2117 |
|
2118 |
lemma (in ACf) fold1_Un: |
|
2119 |
assumes A: "finite A" "A \<noteq> {}" |
|
2120 |
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow> |
|
2121 |
fold1 f (A Un B) = f (fold1 f A) (fold1 f B)" |
|
2122 |
using A |
|
2123 |
proof(induct rule:finite_ne_induct) |
|
2124 |
case singleton thus ?case by(simp add:fold1_insert) |
|
2125 |
next |
|
2126 |
case insert thus ?case by (simp add:fold1_insert assoc) |
|
2127 |
qed |
|
2128 |
||
2129 |
lemma (in ACIf) fold1_Un2: |
|
2130 |
assumes A: "finite A" "A \<noteq> {}" |
|
2131 |
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> |
|
2132 |
fold1 f (A Un B) = f (fold1 f A) (fold1 f B)" |
|
2133 |
using A |
|
2134 |
proof(induct rule:finite_ne_induct) |
|
15509 | 2135 |
case singleton thus ?case by(simp add:fold1_insert_idem) |
15484 | 2136 |
next |
15509 | 2137 |
case insert thus ?case by (simp add:fold1_insert_idem assoc) |
15484 | 2138 |
qed |
2139 |
||
2140 |
lemma (in ACf) fold1_in: |
|
2141 |
assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}" |
|
2142 |
shows "fold1 f A \<in> A" |
|
2143 |
using A |
|
2144 |
proof (induct rule:finite_ne_induct) |
|
15506 | 2145 |
case singleton thus ?case by simp |
15484 | 2146 |
next |
2147 |
case insert thus ?case using elem by (force simp add:fold1_insert) |
|
2148 |
qed |
|
2149 |
||
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2150 |
lemma (in ACIfSL) below_fold1_iff: |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2151 |
assumes A: "finite A" "A \<noteq> {}" |
15500 | 2152 |
shows "x \<sqsubseteq> fold1 f A = (\<forall>a\<in>A. x \<sqsubseteq> a)" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2153 |
using A |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2154 |
by(induct rule:finite_ne_induct) simp_all |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2155 |
|
18493 | 2156 |
lemma (in ACIfSLlin) strict_below_fold1_iff: |
2157 |
"finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> x \<sqsubset> fold1 f A = (\<forall>a\<in>A. x \<sqsubset> a)" |
|
2158 |
by(induct rule:finite_ne_induct) simp_all |
|
2159 |
||
2160 |
||
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2161 |
lemma (in ACIfSL) fold1_belowI: |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2162 |
assumes A: "finite A" "A \<noteq> {}" |
15500 | 2163 |
shows "a \<in> A \<Longrightarrow> fold1 f A \<sqsubseteq> a" |
15484 | 2164 |
using A |
2165 |
proof (induct rule:finite_ne_induct) |
|
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2166 |
case singleton thus ?case by simp |
15484 | 2167 |
next |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2168 |
case (insert x F) |
15517 | 2169 |
from insert(5) have "a = x \<or> a \<in> F" by simp |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2170 |
thus ?case |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2171 |
proof |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2172 |
assume "a = x" thus ?thesis using insert by(simp add:below_def ACI) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2173 |
next |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2174 |
assume "a \<in> F" |
15508 | 2175 |
hence bel: "fold1 f F \<sqsubseteq> a" by(rule insert) |
2176 |
have "fold1 f (insert x F) \<cdot> a = x \<cdot> (fold1 f F \<cdot> a)" |
|
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2177 |
using insert by(simp add:below_def ACI) |
15508 | 2178 |
also have "fold1 f F \<cdot> a = fold1 f F" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2179 |
using bel by(simp add:below_def ACI) |
15508 | 2180 |
also have "x \<cdot> \<dots> = fold1 f (insert x F)" |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2181 |
using insert by(simp add:below_def ACI) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2182 |
finally show ?thesis by(simp add:below_def) |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2183 |
qed |
15484 | 2184 |
qed |
2185 |
||
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2186 |
lemma (in ACIfSLlin) fold1_below_iff: |
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2187 |
assumes A: "finite A" "A \<noteq> {}" |
15500 | 2188 |
shows "fold1 f A \<sqsubseteq> x = (\<exists>a\<in>A. a \<sqsubseteq> x)" |
15484 | 2189 |
using A |
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2190 |
by(induct rule:finite_ne_induct)(simp_all add:above_f_conv) |
15484 | 2191 |
|
18493 | 2192 |
lemma (in ACIfSLlin) fold1_strict_below_iff: |
2193 |
assumes A: "finite A" "A \<noteq> {}" |
|
2194 |
shows "fold1 f A \<sqsubset> x = (\<exists>a\<in>A. a \<sqsubset> x)" |
|
2195 |
using A |
|
2196 |
by(induct rule:finite_ne_induct)(simp_all add:strict_above_f_conv) |
|
2197 |
||
18423 | 2198 |
lemma (in ACIfSLlin) fold1_antimono: |
2199 |
assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B" |
|
2200 |
shows "fold1 f B \<sqsubseteq> fold1 f A" |
|
2201 |
proof(cases) |
|
2202 |
assume "A = B" thus ?thesis by simp |
|
2203 |
next |
|
2204 |
assume "A \<noteq> B" |
|
2205 |
have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast |
|
2206 |
have "fold1 f B = fold1 f (A \<union> (B-A))" by(subst B)(rule refl) |
|
2207 |
also have "\<dots> = f (fold1 f A) (fold1 f (B-A))" |
|
2208 |
proof - |
|
2209 |
have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`]) |
|
18493 | 2210 |
moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *) |
18423 | 2211 |
moreover have "(B-A) \<noteq> {}" using prems by blast |
2212 |
moreover have "A Int (B-A) = {}" using prems by blast |
|
2213 |
ultimately show ?thesis using `A \<noteq> {}` by(rule_tac fold1_Un) |
|
2214 |
qed |
|
2215 |
also have "\<dots> \<sqsubseteq> fold1 f A" by(simp add: above_f_conv) |
|
2216 |
finally show ?thesis . |
|
2217 |
qed |
|
2218 |
||
2219 |
||
22917 | 2220 |
subsubsection {* Fold1 in lattices with @{const inf} and @{const sup} *} |
2221 |
||
2222 |
text{* |
|
2223 |
As an application of @{text fold1} we define infimum |
|
2224 |
and supremum in (not necessarily complete!) lattices |
|
2225 |
over (non-empty) sets by means of @{text fold1}. |
|
2226 |
*} |
|
2227 |
||
2228 |
lemma (in lower_semilattice) ACf_inf: "ACf (op \<sqinter>)" |
|
2229 |
by (blast intro: ACf.intro inf_commute inf_assoc) |
|
2230 |
||
2231 |
lemma (in upper_semilattice) ACf_sup: "ACf (op \<squnion>)" |
|
2232 |
by (blast intro: ACf.intro sup_commute sup_assoc) |
|
2233 |
||
2234 |
lemma (in lower_semilattice) ACIf_inf: "ACIf (op \<sqinter>)" |
|
15500 | 2235 |
apply(rule ACIf.intro) |
2236 |
apply(rule ACf_inf) |
|
2237 |
apply(rule ACIf_axioms.intro) |
|
2238 |
apply(rule inf_idem) |
|
2239 |
done |
|
2240 |
||
22917 | 2241 |
lemma (in upper_semilattice) ACIf_sup: "ACIf (op \<squnion>)" |
15500 | 2242 |
apply(rule ACIf.intro) |
2243 |
apply(rule ACf_sup) |
|
2244 |
apply(rule ACIf_axioms.intro) |
|
2245 |
apply(rule sup_idem) |
|
2246 |
done |
|
2247 |
||
22917 | 2248 |
lemma (in lower_semilattice) ACIfSL_inf: "ACIfSL (op \<^loc>\<le>) (op \<^loc><) (op \<sqinter>)" |
15500 | 2249 |
apply(rule ACIfSL.intro) |
19931
fb32b43e7f80
Restructured locales with predicates: import is now an interpretation.
ballarin
parents:
19870
diff
changeset
|
2250 |
apply(rule ACIf.intro) |
15500 | 2251 |
apply(rule ACf_inf) |
2252 |
apply(rule ACIf.axioms[OF ACIf_inf]) |
|
2253 |
apply(rule ACIfSL_axioms.intro) |
|
2254 |
apply(rule iffI) |
|
21733 | 2255 |
apply(blast intro: antisym inf_le1 inf_le2 inf_greatest refl) |
15500 | 2256 |
apply(erule subst) |
2257 |
apply(rule inf_le2) |
|
22917 | 2258 |
apply(rule less_le) |
15500 | 2259 |
done |
2260 |
||
22917 | 2261 |
lemma (in upper_semilattice) ACIfSL_sup: "ACIfSL (%x y. y \<^loc>\<le> x) (%x y. y \<^loc>< x) (op \<squnion>)" |
15500 | 2262 |
apply(rule ACIfSL.intro) |
19931
fb32b43e7f80
Restructured locales with predicates: import is now an interpretation.
ballarin
parents:
19870
diff
changeset
|
2263 |
apply(rule ACIf.intro) |
15500 | 2264 |
apply(rule ACf_sup) |
2265 |
apply(rule ACIf.axioms[OF ACIf_sup]) |
|
2266 |
apply(rule ACIfSL_axioms.intro) |
|
2267 |
apply(rule iffI) |
|
21733 | 2268 |
apply(blast intro: antisym sup_ge1 sup_ge2 sup_least refl) |
15500 | 2269 |
apply(erule subst) |
2270 |
apply(rule sup_ge2) |
|
22917 | 2271 |
apply(simp add: neq_commute less_le) |
15500 | 2272 |
done |
2273 |
||
22917 | 2274 |
locale Lattice = lattice -- {* we do not pollute the @{text lattice} clas *} |
2275 |
begin |
|
2276 |
||
2277 |
definition |
|
2278 |
Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) |
|
2279 |
where |
|
2280 |
"Inf = fold1 (op \<sqinter>)" |
|
2281 |
||
2282 |
definition |
|
2283 |
Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) |
|
2284 |
where |
|
2285 |
"Sup = fold1 (op \<squnion>)" |
|
2286 |
||
2287 |
end |
|
2288 |
||
2289 |
locale Distrib_Lattice = distrib_lattice + Lattice |
|
15500 | 2290 |
|
15780 | 2291 |
lemma (in Lattice) Inf_le_Sup[simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Squnion>A" |
15500 | 2292 |
apply(unfold Sup_def Inf_def) |
2293 |
apply(subgoal_tac "EX a. a:A") |
|
2294 |
prefer 2 apply blast |
|
2295 |
apply(erule exE) |
|
22388 | 2296 |
apply(rule order_trans) |
15500 | 2297 |
apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_inf]) |
2298 |
apply(erule (2) ACIfSL.fold1_belowI[OF ACIfSL_sup]) |
|
2299 |
done |
|
2300 |
||
15780 | 2301 |
lemma (in Lattice) sup_Inf_absorb[simp]: |
15504 | 2302 |
"\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<squnion> \<Sqinter>A) = a" |
15512
ed1fa4617f52
Extracted generic lattice stuff to new Lattice_Locales.thy
nipkow
parents:
15510
diff
changeset
|
2303 |
apply(subst sup_commute) |
21733 | 2304 |
apply(simp add:Inf_def sup_absorb2 ACIfSL.fold1_belowI[OF ACIfSL_inf]) |
15504 | 2305 |
done |
2306 |
||
15780 | 2307 |
lemma (in Lattice) inf_Sup_absorb[simp]: |
15504 | 2308 |
"\<lbrakk> finite A; A \<noteq> {}; a \<in> A \<rbrakk> \<Longrightarrow> (a \<sqinter> \<Squnion>A) = a" |
21733 | 2309 |
by(simp add:Sup_def inf_absorb1 ACIfSL.fold1_belowI[OF ACIfSL_sup]) |
15504 | 2310 |
|
18423 | 2311 |
lemma (in Distrib_Lattice) sup_Inf1_distrib: |
2312 |
"finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (x \<squnion> \<Sqinter>A) = \<Sqinter>{x \<squnion> a|a. a \<in> A}" |
|
2313 |
apply(simp add:Inf_def image_def |
|
2314 |
ACIf.hom_fold1_commute[OF ACIf_inf, where h="sup x", OF sup_inf_distrib1]) |
|
2315 |
apply(rule arg_cong, blast) |
|
2316 |
done |
|
2317 |
||
2318 |
||
15512
ed1fa4617f52
Extracted generic lattice stuff to new Lattice_Locales.thy
nipkow
parents:
15510
diff
changeset
|
2319 |
lemma (in Distrib_Lattice) sup_Inf2_distrib: |
15500 | 2320 |
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}" |
2321 |
shows "(\<Sqinter>A \<squnion> \<Sqinter>B) = \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}" |
|
2322 |
using A |
|
2323 |
proof (induct rule: finite_ne_induct) |
|
2324 |
case singleton thus ?case |
|
22917 | 2325 |
by (simp add: sup_Inf1_distrib[OF B] fold1_singleton_def[OF Inf_def]) |
15500 | 2326 |
next |
2327 |
case (insert x A) |
|
2328 |
have finB: "finite {x \<squnion> b |b. b \<in> B}" |
|
21733 | 2329 |
by(rule finite_surj[where f = "%b. x \<squnion> b", OF B(1)], auto) |
15500 | 2330 |
have finAB: "finite {a \<squnion> b |a b. a \<in> A \<and> b \<in> B}" |
2331 |
proof - |
|
2332 |
have "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<squnion> b})" |
|
2333 |
by blast |
|
15517 | 2334 |
thus ?thesis by(simp add: insert(1) B(1)) |
15500 | 2335 |
qed |
2336 |
have ne: "{a \<squnion> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast |
|
2337 |
have "\<Sqinter>(insert x A) \<squnion> \<Sqinter>B = (x \<sqinter> \<Sqinter>A) \<squnion> \<Sqinter>B" |
|
22917 | 2338 |
using insert |
2339 |
thm ACIf.fold1_insert_idem_def |
|
2340 |
by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_inf Inf_def]) |
|
15500 | 2341 |
also have "\<dots> = (x \<squnion> \<Sqinter>B) \<sqinter> (\<Sqinter>A \<squnion> \<Sqinter>B)" by(rule sup_inf_distrib2) |
2342 |
also have "\<dots> = \<Sqinter>{x \<squnion> b|b. b \<in> B} \<sqinter> \<Sqinter>{a \<squnion> b|a b. a \<in> A \<and> b \<in> B}" |
|
2343 |
using insert by(simp add:sup_Inf1_distrib[OF B]) |
|
2344 |
also have "\<dots> = \<Sqinter>({x\<squnion>b |b. b \<in> B} \<union> {a\<squnion>b |a b. a \<in> A \<and> b \<in> B})" |
|
2345 |
(is "_ = \<Sqinter>?M") |
|
2346 |
using B insert |
|
2347 |
by(simp add:Inf_def ACIf.fold1_Un2[OF ACIf_inf finB _ finAB ne]) |
|
2348 |
also have "?M = {a \<squnion> b |a b. a \<in> insert x A \<and> b \<in> B}" |
|
2349 |
by blast |
|
2350 |
finally show ?case . |
|
2351 |
qed |
|
2352 |
||
15484 | 2353 |
|
18423 | 2354 |
lemma (in Distrib_Lattice) inf_Sup1_distrib: |
2355 |
"finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (x \<sqinter> \<Squnion>A) = \<Squnion>{x \<sqinter> a|a. a \<in> A}" |
|
2356 |
apply(simp add:Sup_def image_def |
|
2357 |
ACIf.hom_fold1_commute[OF ACIf_sup, where h="inf x", OF inf_sup_distrib1]) |
|
2358 |
apply(rule arg_cong, blast) |
|
2359 |
done |
|
2360 |
||
2361 |
||
2362 |
lemma (in Distrib_Lattice) inf_Sup2_distrib: |
|
2363 |
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}" |
|
2364 |
shows "(\<Squnion>A \<sqinter> \<Squnion>B) = \<Squnion>{a \<sqinter> b|a b. a \<in> A \<and> b \<in> B}" |
|
2365 |
using A |
|
2366 |
proof (induct rule: finite_ne_induct) |
|
2367 |
case singleton thus ?case |
|
2368 |
by(simp add: inf_Sup1_distrib[OF B] fold1_singleton_def[OF Sup_def]) |
|
2369 |
next |
|
2370 |
case (insert x A) |
|
2371 |
have finB: "finite {x \<sqinter> b |b. b \<in> B}" |
|
21733 | 2372 |
by(rule finite_surj[where f = "%b. x \<sqinter> b", OF B(1)], auto) |
18423 | 2373 |
have finAB: "finite {a \<sqinter> b |a b. a \<in> A \<and> b \<in> B}" |
2374 |
proof - |
|
2375 |
have "{a \<sqinter> b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {a \<sqinter> b})" |
|
2376 |
by blast |
|
2377 |
thus ?thesis by(simp add: insert(1) B(1)) |
|
2378 |
qed |
|
2379 |
have ne: "{a \<sqinter> b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast |
|
2380 |
have "\<Squnion>(insert x A) \<sqinter> \<Squnion>B = (x \<squnion> \<Squnion>A) \<sqinter> \<Squnion>B" |
|
2381 |
using insert by(simp add:ACIf.fold1_insert_idem_def[OF ACIf_sup Sup_def]) |
|
2382 |
also have "\<dots> = (x \<sqinter> \<Squnion>B) \<squnion> (\<Squnion>A \<sqinter> \<Squnion>B)" by(rule inf_sup_distrib2) |
|
2383 |
also have "\<dots> = \<Squnion>{x \<sqinter> b|b. b \<in> B} \<squnion> \<Squnion>{a \<sqinter> b|a b. a \<in> A \<and> b \<in> B}" |
|
2384 |
using insert by(simp add:inf_Sup1_distrib[OF B]) |
|
2385 |
also have "\<dots> = \<Squnion>({x\<sqinter>b |b. b \<in> B} \<union> {a\<sqinter>b |a b. a \<in> A \<and> b \<in> B})" |
|
2386 |
(is "_ = \<Squnion>?M") |
|
2387 |
using B insert |
|
2388 |
by(simp add:Sup_def ACIf.fold1_Un2[OF ACIf_sup finB _ finAB ne]) |
|
2389 |
also have "?M = {a \<sqinter> b |a b. a \<in> insert x A \<and> b \<in> B}" |
|
2390 |
by blast |
|
2391 |
finally show ?case . |
|
2392 |
qed |
|
2393 |
||
22917 | 2394 |
text {* |
2395 |
Infimum and supremum in complete lattices may also |
|
2396 |
be characterized by @{const fold1}: |
|
2397 |
*} |
|
2398 |
||
2399 |
lemma (in complete_lattice) Inf_fold1: |
|
22941 | 2400 |
"finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Sqinter>A = fold1 (op \<sqinter>) A" |
22917 | 2401 |
by (induct A set: finite) |
22941 | 2402 |
(simp_all add: Inf_insert_simp ACIf.fold1_insert_idem [OF ACIf_inf]) |
22917 | 2403 |
|
2404 |
lemma (in complete_lattice) Sup_fold1: |
|
23234 | 2405 |
"finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Squnion>A = fold1 (op \<squnion>) A" |
22917 | 2406 |
by (induct A set: finite) |
22941 | 2407 |
(simp_all add: Sup_insert_simp ACIf.fold1_insert_idem [OF ACIf_sup]) |
22917 | 2408 |
|
2409 |
||
2410 |
subsubsection {* Fold1 in linear orders with @{const min} and @{const max} *} |
|
2411 |
||
2412 |
text{* |
|
2413 |
As an application of @{text fold1} we define minimum |
|
2414 |
and maximum in (not necessarily complete!) linear orders |
|
2415 |
over (non-empty) sets by means of @{text fold1}. |
|
2416 |
*} |
|
2417 |
||
23234 | 2418 |
locale Linorder = linorder -- {* we do not pollute the @{text linorder} class *} |
22917 | 2419 |
begin |
2420 |
||
2421 |
definition |
|
2422 |
Min :: "'a set \<Rightarrow> 'a" |
|
2423 |
where |
|
2424 |
"Min = fold1 min" |
|
2425 |
||
2426 |
definition |
|
2427 |
Max :: "'a set \<Rightarrow> 'a" |
|
2428 |
where |
|
2429 |
"Max = fold1 max" |
|
2430 |
||
2431 |
text {* recall: @{term min} and @{term max} behave like @{const inf} and @{const sup} *} |
|
2432 |
||
2433 |
lemma ACIf_min: "ACIf min" |
|
2434 |
by (rule lower_semilattice.ACIf_inf, |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2435 |
rule lattice.axioms, |
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2436 |
rule distrib_lattice.axioms, |
22917 | 2437 |
rule distrib_lattice_min_max) |
2438 |
||
2439 |
lemma ACf_min: "ACf min" |
|
2440 |
by (rule lower_semilattice.ACf_inf, |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2441 |
rule lattice.axioms, |
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2442 |
rule distrib_lattice.axioms, |
22917 | 2443 |
rule distrib_lattice_min_max) |
2444 |
||
2445 |
lemma ACIfSL_min: "ACIfSL (op \<^loc>\<le>) (op \<^loc><) min" |
|
2446 |
by (rule lower_semilattice.ACIfSL_inf, |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2447 |
rule lattice.axioms, |
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2448 |
rule distrib_lattice.axioms, |
22917 | 2449 |
rule distrib_lattice_min_max) |
2450 |
||
2451 |
lemma ACIfSLlin_min: "ACIfSLlin (op \<^loc>\<le>) (op \<^loc><) min" |
|
2452 |
by (rule ACIfSLlin.intro, |
|
2453 |
rule lower_semilattice.ACIfSL_inf, |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2454 |
rule lattice.axioms, |
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2455 |
rule distrib_lattice.axioms, |
22917 | 2456 |
rule distrib_lattice_min_max) |
2457 |
(unfold_locales, simp add: min_def) |
|
2458 |
||
2459 |
lemma ACIf_max: "ACIf max" |
|
2460 |
by (rule upper_semilattice.ACIf_sup, |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2461 |
rule lattice.axioms, |
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2462 |
rule distrib_lattice.axioms, |
22917 | 2463 |
rule distrib_lattice_min_max) |
2464 |
||
2465 |
lemma ACf_max: "ACf max" |
|
2466 |
by (rule upper_semilattice.ACf_sup, |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2467 |
rule lattice.axioms, |
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2468 |
rule distrib_lattice.axioms, |
22917 | 2469 |
rule distrib_lattice_min_max) |
2470 |
||
2471 |
lemma ACIfSL_max: "ACIfSL (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x) max" |
|
2472 |
by (rule upper_semilattice.ACIfSL_sup, |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2473 |
rule lattice.axioms, |
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2474 |
rule distrib_lattice.axioms, |
22917 | 2475 |
rule distrib_lattice_min_max) |
2476 |
||
2477 |
lemma ACIfSLlin_max: "ACIfSLlin (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x) max" |
|
2478 |
by (rule ACIfSLlin.intro, |
|
2479 |
rule upper_semilattice.ACIfSL_sup, |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2480 |
rule lattice.axioms, |
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2481 |
rule distrib_lattice.axioms, |
22917 | 2482 |
rule distrib_lattice_min_max) |
2483 |
(unfold_locales, simp add: max_def) |
|
2484 |
||
2485 |
lemmas Min_singleton [simp] = fold1_singleton_def [OF Min_def] |
|
2486 |
lemmas Max_singleton [simp] = fold1_singleton_def [OF Max_def] |
|
2487 |
lemmas Min_insert [simp] = ACIf.fold1_insert_idem_def [OF ACIf_min Min_def] |
|
2488 |
lemmas Max_insert [simp] = ACIf.fold1_insert_idem_def [OF ACIf_max Max_def] |
|
15392 | 2489 |
|
2490 |
lemma Min_in [simp]: |
|
15484 | 2491 |
shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A" |
22917 | 2492 |
using ACf.fold1_in [OF ACf_min] |
2493 |
by (fastsimp simp: Min_def min_def) |
|
15392 | 2494 |
|
2495 |
lemma Max_in [simp]: |
|
15484 | 2496 |
shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A" |
22917 | 2497 |
using ACf.fold1_in [OF ACf_max] |
2498 |
by (fastsimp simp: Max_def max_def) |
|
2499 |
||
2500 |
lemma Min_antimono: "\<lbrakk> M \<subseteq> N; M \<noteq> {}; finite N \<rbrakk> \<Longrightarrow> Min N \<^loc>\<le> Min M" |
|
2501 |
by (simp add: Min_def ACIfSLlin.fold1_antimono [OF ACIfSLlin_min]) |
|
2502 |
||
2503 |
lemma Max_mono: "\<lbrakk> M \<subseteq> N; M \<noteq> {}; finite N \<rbrakk> \<Longrightarrow> Max M \<^loc>\<le> Max N" |
|
2504 |
by (simp add: Max_def ACIfSLlin.fold1_antimono [OF ACIfSLlin_max]) |
|
2505 |
||
2506 |
lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<^loc>\<le> x" |
|
2507 |
by (simp add: Min_def ACIfSL.fold1_belowI [OF ACIfSL_min]) |
|
2508 |
||
2509 |
lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<^loc>\<le> Max A" |
|
2510 |
by (simp add: Max_def ACIfSL.fold1_belowI [OF ACIfSL_max]) |
|
2511 |
||
2512 |
lemma Min_ge_iff [simp]: |
|
2513 |
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> x \<^loc>\<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<^loc>\<le> a)" |
|
2514 |
by (simp add: Min_def ACIfSL.below_fold1_iff [OF ACIfSL_min]) |
|
2515 |
||
2516 |
lemma Max_le_iff [simp]: |
|
2517 |
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Max A \<^loc>\<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<^loc>\<le> x)" |
|
2518 |
by (simp add: Max_def ACIfSL.below_fold1_iff [OF ACIfSL_max]) |
|
2519 |
||
2520 |
lemma Min_gr_iff [simp]: |
|
2521 |
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> x \<^loc>< Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<^loc>< a)" |
|
2522 |
by (simp add: Min_def ACIfSLlin.strict_below_fold1_iff [OF ACIfSLlin_min]) |
|
2523 |
||
2524 |
lemma Max_less_iff [simp]: |
|
2525 |
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Max A \<^loc>< x \<longleftrightarrow> (\<forall>a\<in>A. a \<^loc>< x)" |
|
2526 |
by (simp add: Max_def ACIfSLlin.strict_below_fold1_iff [OF ACIfSLlin_max]) |
|
18493 | 2527 |
|
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2528 |
lemma Min_le_iff: |
22917 | 2529 |
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Min A \<^loc>\<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<^loc>\<le> x)" |
2530 |
by (simp add: Min_def ACIfSLlin.fold1_below_iff [OF ACIfSLlin_min]) |
|
15497
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2531 |
|
53bca254719a
Added semi-lattice locales and reorganized fold1 lemmas
nipkow
parents:
15487
diff
changeset
|
2532 |
lemma Max_ge_iff: |
22917 | 2533 |
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> x \<^loc>\<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<^loc>\<le> a)" |
2534 |
by (simp add: Max_def ACIfSLlin.fold1_below_iff [OF ACIfSLlin_max]) |
|
2535 |
||
2536 |
lemma Min_less_iff: |
|
2537 |
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> Min A \<^loc>< x \<longleftrightarrow> (\<exists>a\<in>A. a \<^loc>< x)" |
|
2538 |
by (simp add: Min_def ACIfSLlin.fold1_strict_below_iff [OF ACIfSLlin_min]) |
|
2539 |
||
2540 |
lemma Max_gr_iff: |
|
2541 |
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> x \<^loc>< Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<^loc>< a)" |
|
2542 |
by (simp add: Max_def ACIfSLlin.fold1_strict_below_iff [OF ACIfSLlin_max]) |
|
18493 | 2543 |
|
18423 | 2544 |
lemma Min_Un: "\<lbrakk>finite A; A \<noteq> {}; finite B; B \<noteq> {}\<rbrakk> |
2545 |
\<Longrightarrow> Min (A \<union> B) = min (Min A) (Min B)" |
|
22917 | 2546 |
by (simp add: Min_def ACIf.fold1_Un2 [OF ACIf_min]) |
18423 | 2547 |
|
2548 |
lemma Max_Un: "\<lbrakk>finite A; A \<noteq> {}; finite B; B \<noteq> {}\<rbrakk> |
|
2549 |
\<Longrightarrow> Max (A \<union> B) = max (Max A) (Max B)" |
|
22917 | 2550 |
by (simp add: Max_def ACIf.fold1_Un2 [OF ACIf_max]) |
18423 | 2551 |
|
2552 |
lemma hom_Min_commute: |
|
22917 | 2553 |
"(\<And>x y. h (min x y) = min (h x) (h y)) |
2554 |
\<Longrightarrow> finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> h (Min N) = Min (h ` N)" |
|
2555 |
by (simp add: Min_def ACIf.hom_fold1_commute [OF ACIf_min]) |
|
18423 | 2556 |
|
2557 |
lemma hom_Max_commute: |
|
22917 | 2558 |
"(\<And>x y. h (max x y) = max (h x) (h y)) |
2559 |
\<Longrightarrow> finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> h (Max N) = Max (h ` N)" |
|
2560 |
by (simp add: Max_def ACIf.hom_fold1_commute [OF ACIf_max]) |
|
2561 |
||
2562 |
end |
|
2563 |
||
2564 |
locale Linorder_ab_semigroup_add = Linorder + pordered_ab_semigroup_add |
|
2565 |
begin |
|
2566 |
||
2567 |
lemma add_Min_commute: |
|
2568 |
fixes k |
|
2569 |
shows "finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> k \<^loc>+ Min N = Min {k \<^loc>+ m | m. m \<in> N}" |
|
2570 |
apply (subgoal_tac "\<And>x y. k \<^loc>+ min x y = min (k \<^loc>+ x) (k \<^loc>+ y)") |
|
2571 |
using hom_Min_commute [of "(op \<^loc>+) k" N] |
|
2572 |
apply simp apply (rule arg_cong [where f = Min]) apply blast |
|
2573 |
apply (simp add: min_def not_le) |
|
2574 |
apply (blast intro: antisym less_imp_le add_left_mono) |
|
2575 |
done |
|
2576 |
||
2577 |
lemma add_Max_commute: |
|
2578 |
fixes k |
|
2579 |
shows "finite N \<Longrightarrow> N \<noteq> {} \<Longrightarrow> k \<^loc>+ Max N = Max {k \<^loc>+ m | m. m \<in> N}" |
|
2580 |
apply (subgoal_tac "\<And>x y. k \<^loc>+ max x y = max (k \<^loc>+ x) (k \<^loc>+ y)") |
|
2581 |
using hom_Max_commute [of "(op \<^loc>+) k" N] |
|
2582 |
apply simp apply (rule arg_cong [where f = Max]) apply blast |
|
2583 |
apply (simp add: max_def not_le) |
|
2584 |
apply (blast intro: antisym less_imp_le add_left_mono) |
|
2585 |
done |
|
2586 |
||
2587 |
end |
|
2588 |
||
2589 |
definition |
|
2590 |
Min :: "'a set \<Rightarrow> 'a\<Colon>linorder" |
|
2591 |
where |
|
2592 |
"Min = fold1 min" |
|
2593 |
||
2594 |
definition |
|
2595 |
Max :: "'a set \<Rightarrow> 'a\<Colon>linorder" |
|
2596 |
where |
|
2597 |
"Max = fold1 max" |
|
2598 |
||
2599 |
lemma Linorder_Min: |
|
2600 |
"Linorder.Min (op \<le>) = Min" |
|
2601 |
proof |
|
2602 |
fix A :: "'a set" |
|
2603 |
show "Linorder.Min (op \<le>) A = Min A" |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2604 |
by (simp add: Min_def Linorder.Min_def [OF Linorder.intro, OF linorder_axioms] |
23087 | 2605 |
ord_class.min) |
22917 | 2606 |
qed |
2607 |
||
2608 |
lemma Linorder_Max: |
|
2609 |
"Linorder.Max (op \<le>) = Max" |
|
2610 |
proof |
|
2611 |
fix A :: "'a set" |
|
2612 |
show "Linorder.Max (op \<le>) A = Max A" |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2613 |
by (simp add: Max_def Linorder.Max_def [OF Linorder.intro, OF linorder_axioms] |
23087 | 2614 |
ord_class.max) |
22917 | 2615 |
qed |
2616 |
||
23234 | 2617 |
(*FIXME: temporary solution - doesn't work properly*) |
23087 | 2618 |
interpretation [folded ord_class.min ord_class.max, unfolded Linorder_Min Linorder_Max]: |
22917 | 2619 |
Linorder_ab_semigroup_add ["op \<le> \<Colon> 'a\<Colon>{linorder, pordered_ab_semigroup_add} \<Rightarrow> 'a \<Rightarrow> bool" "op <" "op +"] |
2620 |
by (rule Linorder_ab_semigroup_add.intro, |
|
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2621 |
rule Linorder.intro, rule linorder_axioms, rule pordered_ab_semigroup_add_axioms) |
22917 | 2622 |
hide const Min Max |
2623 |
||
23087 | 2624 |
interpretation [folded ord_class.min ord_class.max, unfolded Linorder_Min Linorder_Max]: |
22917 | 2625 |
Linorder ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <"] |
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2626 |
by (rule Linorder.intro, rule linorder_axioms) |
22917 | 2627 |
declare Min_singleton [simp] |
2628 |
declare Max_singleton [simp] |
|
2629 |
declare Min_insert [simp] |
|
2630 |
declare Max_insert [simp] |
|
2631 |
declare Min_in [simp] |
|
2632 |
declare Max_in [simp] |
|
2633 |
declare Min_le [simp] |
|
2634 |
declare Max_ge [simp] |
|
2635 |
declare Min_ge_iff [simp] |
|
2636 |
declare Max_le_iff [simp] |
|
2637 |
declare Min_gr_iff [simp] |
|
2638 |
declare Max_less_iff [simp] |
|
2639 |
declare Max_less_iff [simp] |
|
18423 | 2640 |
|
2641 |
||
22388 | 2642 |
subsection {* Class @{text finite} *} |
2643 |
||
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2644 |
setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*} |
22473 | 2645 |
class finite (attach UNIV) = type + |
22388 | 2646 |
assumes finite: "finite UNIV" |
23018
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2647 |
setup {* Sign.parent_path *} |
1d29bc31b0cb
no special treatment in naming of locale predicates stemming form classes
haftmann
parents:
22941
diff
changeset
|
2648 |
hide const finite |
17022 | 2649 |
|
2650 |
lemma finite_set: "finite (A::'a::finite set)" |
|
22388 | 2651 |
by (rule finite_subset [OF subset_UNIV finite]) |
2652 |
||
2653 |
lemma univ_unit: |
|
2654 |
"UNIV = {()}" by auto |
|
17022 | 2655 |
|
2656 |
instance unit :: finite |
|
2657 |
proof |
|
2658 |
have "finite {()}" by simp |
|
22388 | 2659 |
also note univ_unit [symmetric] |
17022 | 2660 |
finally show "finite (UNIV :: unit set)" . |
2661 |
qed |
|
2662 |
||
22388 | 2663 |
lemmas [code func] = univ_unit |
2664 |
||
2665 |
lemma univ_bool: |
|
2666 |
"UNIV = {False, True}" by auto |
|
2667 |
||
17022 | 2668 |
instance bool :: finite |
2669 |
proof |
|
22388 | 2670 |
have "finite {False, True}" by simp |
2671 |
also note univ_bool [symmetric] |
|
17022 | 2672 |
finally show "finite (UNIV :: bool set)" . |
2673 |
qed |
|
2674 |
||
22388 | 2675 |
lemmas [code func] = univ_bool |
17022 | 2676 |
|
2677 |
instance * :: (finite, finite) finite |
|
2678 |
proof |
|
2679 |
show "finite (UNIV :: ('a \<times> 'b) set)" |
|
2680 |
proof (rule finite_Prod_UNIV) |
|
2681 |
show "finite (UNIV :: 'a set)" by (rule finite) |
|
2682 |
show "finite (UNIV :: 'b set)" by (rule finite) |
|
2683 |
qed |
|
2684 |
qed |
|
2685 |
||
22388 | 2686 |
lemma univ_prod [code func]: |
2687 |
"UNIV = (UNIV \<Colon> 'a\<Colon>finite set) \<times> (UNIV \<Colon> 'b\<Colon>finite set)" |
|
2688 |
unfolding UNIV_Times_UNIV .. |
|
2689 |
||
17022 | 2690 |
instance "+" :: (finite, finite) finite |
2691 |
proof |
|
2692 |
have a: "finite (UNIV :: 'a set)" by (rule finite) |
|
2693 |
have b: "finite (UNIV :: 'b set)" by (rule finite) |
|
2694 |
from a b have "finite ((UNIV :: 'a set) <+> (UNIV :: 'b set))" |
|
2695 |
by (rule finite_Plus) |
|
2696 |
thus "finite (UNIV :: ('a + 'b) set)" by simp |
|
2697 |
qed |
|
2698 |
||
22388 | 2699 |
lemma univ_sum [code func]: |
2700 |
"UNIV = (UNIV \<Colon> 'a\<Colon>finite set) <+> (UNIV \<Colon> 'b\<Colon>finite set)" |
|
2701 |
unfolding UNIV_Plus_UNIV .. |
|
17022 | 2702 |
|
22398 | 2703 |
lemma insert_None_conv_UNIV: "insert None (range Some) = UNIV" |
2704 |
by (rule set_ext, case_tac x, auto) |
|
2705 |
||
2706 |
instance option :: (finite) finite |
|
2707 |
proof |
|
2708 |
have "finite (UNIV :: 'a set)" by (rule finite) |
|
2709 |
hence "finite (insert None (Some ` (UNIV :: 'a set)))" by simp |
|
2710 |
also have "insert None (Some ` (UNIV :: 'a set)) = UNIV" |
|
2711 |
by (rule insert_None_conv_UNIV) |
|
2712 |
finally show "finite (UNIV :: 'a option set)" . |
|
2713 |
qed |
|
2714 |
||
2715 |
lemma univ_option [code func]: |
|
2716 |
"UNIV = insert (None \<Colon> 'a\<Colon>finite option) (image Some UNIV)" |
|
2717 |
unfolding insert_None_conv_UNIV .. |
|
2718 |
||
17022 | 2719 |
instance set :: (finite) finite |
2720 |
proof |
|
2721 |
have "finite (UNIV :: 'a set)" by (rule finite) |
|
2722 |
hence "finite (Pow (UNIV :: 'a set))" |
|
2723 |
by (rule finite_Pow_iff [THEN iffD2]) |
|
2724 |
thus "finite (UNIV :: 'a set set)" by simp |
|
2725 |
qed |
|
2726 |
||
22388 | 2727 |
lemma univ_set [code func]: |
2728 |
"UNIV = Pow (UNIV \<Colon> 'a\<Colon>finite set)" unfolding Pow_UNIV .. |
|
2729 |
||
17022 | 2730 |
lemma inj_graph: "inj (%f. {(x, y). y = f x})" |
22388 | 2731 |
by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq) |
17022 | 2732 |
|
21215
7c9337a0e30a
made locale partial_order compatible with axclass order
haftmann
parents:
21199
diff
changeset
|
2733 |
instance "fun" :: (finite, finite) finite |
17022 | 2734 |
proof |
2735 |
show "finite (UNIV :: ('a => 'b) set)" |
|
2736 |
proof (rule finite_imageD) |
|
2737 |
let ?graph = "%f::'a => 'b. {(x, y). y = f x}" |
|
2738 |
show "finite (range ?graph)" by (rule finite_set) |
|
2739 |
show "inj ?graph" by (rule inj_graph) |
|
2740 |
qed |
|
2741 |
qed |
|
2742 |
||
22388 | 2743 |
|
22425 | 2744 |
subsection {* Equality and order on functions *} |
22388 | 2745 |
|
2746 |
instance "fun" :: (finite, eq) eq .. |
|
2747 |
||
2748 |
lemma eq_fun [code func]: |
|
2749 |
"f = g \<longleftrightarrow> (\<forall>x\<Colon>'a\<Colon>finite \<in> UNIV. (f x \<Colon> 'b\<Colon>eq) = g x)" |
|
2750 |
unfolding expand_fun_eq by auto |
|
2751 |
||
22425 | 2752 |
lemma order_fun [code func]: |
2753 |
"f \<le> g \<longleftrightarrow> (\<forall>x\<Colon>'a\<Colon>finite \<in> UNIV. (f x \<Colon> 'b\<Colon>order) \<le> g x)" |
|
2754 |
"f < g \<longleftrightarrow> f \<le> g \<and> (\<exists>x\<Colon>'a\<Colon>finite \<in> UNIV. (f x \<Colon> 'b\<Colon>order) < g x)" |
|
2755 |
unfolding le_fun_def less_fun_def less_le |
|
2756 |
by (auto simp add: expand_fun_eq) |
|
2757 |
||
15042 | 2758 |
end |