author | wenzelm |
Fri, 19 Jul 2002 18:44:36 +0200 | |
changeset 13400 | dbb608cd11c4 |
parent 13390 | c48c634605f1 |
child 13421 | 8fcdf4a26468 |
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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License: GPL (GNU GENERAL PUBLIC LICENSE) |
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*) |
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header {* Finite sets *} |
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theory Finite_Set = Divides + Power + Inductive + SetInterval: |
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subsection {* Collection of finite sets *} |
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consts Finites :: "'a set set" |
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inductive Finites |
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intros |
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emptyI [simp, intro!]: "{} : Finites" |
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insertI [simp, intro!]: "A : Finites ==> insert a A : Finites" |
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syntax |
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finite :: "'a set => bool" |
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translations |
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"finite A" == "A : Finites" |
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axclass finite \<subseteq> type |
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finite: "finite UNIV" |
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lemma finite_induct [case_names empty insert, induct set: Finites]: |
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"finite F ==> |
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P {} ==> (!!F x. 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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assume "P {}" and insert: "!!F x. 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 F x 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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lemma finite_subset_induct [consumes 2, case_names empty insert]: |
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"finite F ==> F \<subseteq> A ==> |
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P {} ==> (!!F a. 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 - |
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assume "P {}" and insert: "!!F a. 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 F x 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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qed |
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qed |
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)" |
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-- {* The union of two finite sets is finite. *} |
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by (induct set: Finites) simp_all |
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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" |
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proof induct |
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case empty |
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thus ?case by simp |
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next |
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case (insert F x A) |
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have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" . |
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show "finite A" |
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proof cases |
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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 . |
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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 |
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qed |
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qed |
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lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)" |
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by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI) |
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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) |
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lemma finite_insert [simp]: "finite (insert a A) = finite A" |
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apply (subst insert_is_Un) |
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apply (simp only: finite_Un) |
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apply blast |
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done |
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lemma finite_imageI: "finite F ==> finite (h ` F)" |
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-- {* The image of a finite set is finite. *} |
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by (induct set: Finites) simp_all |
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lemma finite_range_imageI: |
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"finite (range g) ==> finite (range (%x. f (g x)))" |
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apply (drule finite_imageI) |
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apply simp |
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done |
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lemma finite_empty_induct: |
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"finite A ==> |
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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 |
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next |
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case (insert F x) |
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have "P (b - F - {x})" |
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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 . |
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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 |
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qed |
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lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)" |
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by (rule Diff_subset [THEN finite_subset]) |
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lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)" |
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apply (subst Diff_insert) |
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apply (case_tac "a : A - B") |
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apply (rule finite_insert [symmetric, THEN trans]) |
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apply (subst insert_Diff) |
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apply simp_all |
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done |
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lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" |
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proof - |
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have aux: "!!A. finite (A - {}) = finite A" by simp |
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fix B :: "'a set" |
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assume "finite B" |
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thus "!!A. f`A = B ==> inj_on f A ==> finite A" |
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apply induct |
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apply simp |
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apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})") |
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apply clarify |
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apply (simp (no_asm_use) add: inj_on_def) |
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apply (blast dest!: aux [THEN iffD1]) |
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apply atomize |
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apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) |
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apply (frule subsetD [OF equalityD2 insertI1]) |
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apply clarify |
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apply (rule_tac x = xa in bexI) |
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apply (simp_all add: inj_on_image_set_diff) |
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done |
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qed (rule refl) |
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subsubsection {* The finite UNION of finite sets *} |
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lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" |
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by (induct set: Finites) simp_all |
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text {* |
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Strengthen RHS to |
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@{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x ~= {}})"}? |
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We'd need to prove |
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@{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x ~= {}}"} |
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by induction. *} |
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lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" |
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by (blast intro: finite_UN_I finite_subset) |
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subsubsection {* Sigma of finite sets *} |
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lemma finite_SigmaI [simp]: |
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"finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" |
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by (unfold Sigma_def) (blast intro!: finite_UN_I) |
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lemma finite_Prod_UNIV: |
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"finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)" |
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apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") |
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apply (erule ssubst) |
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apply (erule finite_SigmaI) |
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apply auto |
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done |
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instance unit :: finite |
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proof |
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have "finite {()}" by simp |
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also have "{()} = UNIV" by auto |
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finally show "finite (UNIV :: unit set)" . |
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qed |
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instance * :: (finite, finite) finite |
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proof |
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show "finite (UNIV :: ('a \<times> 'b) set)" |
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proof (rule finite_Prod_UNIV) |
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show "finite (UNIV :: 'a set)" by (rule finite) |
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show "finite (UNIV :: 'b set)" by (rule finite) |
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qed |
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qed |
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subsubsection {* The powerset of a finite set *} |
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lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A" |
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proof |
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assume "finite (Pow A)" |
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with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast |
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thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp |
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next |
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assume "finite A" |
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thus "finite (Pow A)" |
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by induct (simp_all add: finite_UnI finite_imageI Pow_insert) |
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qed |
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lemma finite_converse [iff]: "finite (r^-1) = finite r" |
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apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r") |
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apply simp |
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apply (rule iffI) |
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apply (erule finite_imageD [unfolded inj_on_def]) |
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apply (simp split add: split_split) |
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apply (erule finite_imageI) |
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apply (simp add: converse_def image_def) |
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apply auto |
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apply (rule bexI) |
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prefer 2 apply assumption |
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apply simp |
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done |
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lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..k(}" |
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by (induct k) (simp_all add: lessThan_Suc) |
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lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}" |
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by (induct k) (simp_all add: atMost_Suc) |
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lemma bounded_nat_set_is_finite: |
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"(ALL i:N. i < (n::nat)) ==> finite N" |
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-- {* A bounded set of natural numbers is finite. *} |
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apply (rule finite_subset) |
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apply (rule_tac [2] finite_lessThan) |
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apply auto |
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done |
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subsubsection {* Finiteness of transitive closure *} |
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text {* (Thanks to Sidi Ehmety) *} |
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lemma finite_Field: "finite r ==> finite (Field r)" |
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-- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *} |
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apply (induct set: Finites) |
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apply (auto simp add: Field_def Domain_insert Range_insert) |
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done |
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lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r" |
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apply clarify |
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apply (erule trancl_induct) |
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apply (auto simp add: Field_def) |
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done |
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lemma finite_trancl: "finite (r^+) = finite r" |
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apply auto |
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prefer 2 |
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apply (rule trancl_subset_Field2 [THEN finite_subset]) |
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apply (rule finite_SigmaI) |
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prefer 3 |
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apply (blast intro: r_into_trancl finite_subset) |
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apply (auto simp add: finite_Field) |
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done |
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subsection {* Finite cardinality *} |
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text {* |
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This definition, although traditional, is ugly to work with: @{text |
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"card A == LEAST n. EX f. A = {f i | i. i < n}"}. Therefore we have |
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switched to an inductive one: |
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*} |
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consts cardR :: "('a set \<times> nat) set" |
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inductive cardR |
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intros |
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EmptyI: "({}, 0) : cardR" |
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InsertI: "(A, n) : cardR ==> a \<notin> A ==> (insert a A, Suc n) : cardR" |
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constdefs |
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card :: "'a set => nat" |
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"card A == THE n. (A, n) : cardR" |
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inductive_cases cardR_emptyE: "({}, n) : cardR" |
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inductive_cases cardR_insertE: "(insert a A,n) : cardR" |
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lemma cardR_SucD: "(A, n) : cardR ==> a : A --> (EX m. n = Suc m)" |
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by (induct set: cardR) simp_all |
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lemma cardR_determ_aux1: |
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"(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A - {a}, n) : cardR)" |
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apply (induct set: cardR) |
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apply auto |
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apply (simp add: insert_Diff_if) |
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apply auto |
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apply (drule cardR_SucD) |
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apply (blast intro!: cardR.intros) |
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done |
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lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \<notin> A ==> (A, m) : cardR" |
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by (drule cardR_determ_aux1) auto |
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lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)" |
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apply (induct set: cardR) |
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apply (safe elim!: cardR_emptyE cardR_insertE) |
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apply (rename_tac B b m) |
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apply (case_tac "a = b") |
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apply (subgoal_tac "A = B") |
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prefer 2 apply (blast elim: equalityE) |
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apply blast |
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apply (subgoal_tac "EX C. A = insert b C & B = insert a C") |
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prefer 2 |
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apply (rule_tac x = "A Int B" in exI) |
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apply (blast elim: equalityE) |
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apply (frule_tac A = B in cardR_SucD) |
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apply (blast intro!: cardR.intros dest!: cardR_determ_aux2) |
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done |
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lemma cardR_imp_finite: "(A, n) : cardR ==> finite A" |
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by (induct set: cardR) simp_all |
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lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR" |
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by (induct set: Finites) (auto intro!: cardR.intros) |
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lemma card_equality: "(A,n) : cardR ==> card A = n" |
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by (unfold card_def) (blast intro: cardR_determ) |
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lemma card_empty [simp]: "card {} = 0" |
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by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE) |
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lemma card_insert_disjoint [simp]: |
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"finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)" |
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proof - |
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assume x: "x \<notin> A" |
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hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)" |
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apply (auto intro!: cardR.intros) |
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apply (rule_tac A1 = A in finite_imp_cardR [THEN exE]) |
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apply (force dest: cardR_imp_finite) |
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apply (blast intro!: cardR.intros intro: cardR_determ) |
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done |
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assume "finite A" |
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thus ?thesis |
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apply (simp add: card_def aux) |
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apply (rule the_equality) |
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apply (auto intro: finite_imp_cardR |
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cong: conj_cong simp: card_def [symmetric] card_equality) |
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done |
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qed |
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lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})" |
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apply auto |
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apply (drule_tac a = x in mk_disjoint_insert) |
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apply clarify |
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apply (rotate_tac -1) |
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apply auto |
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done |
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lemma card_insert_if: |
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"finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))" |
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by (simp add: insert_absorb) |
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lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A" |
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apply (rule_tac t = A in insert_Diff [THEN subst]) |
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apply assumption |
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apply simp |
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done |
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lemma card_Diff_singleton: |
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414 |
"finite A ==> x: A ==> card (A - {x}) = card A - 1" |
|
415 |
by (simp add: card_Suc_Diff1 [symmetric]) |
|
416 |
||
417 |
lemma card_Diff_singleton_if: |
|
418 |
"finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)" |
|
419 |
by (simp add: card_Diff_singleton) |
|
420 |
||
421 |
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))" |
|
422 |
by (simp add: card_insert_if card_Suc_Diff1) |
|
423 |
||
424 |
lemma card_insert_le: "finite A ==> card A <= card (insert x A)" |
|
425 |
by (simp add: card_insert_if) |
|
426 |
||
427 |
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)" |
|
428 |
apply (induct set: Finites) |
|
429 |
apply simp |
|
430 |
apply clarify |
|
431 |
apply (subgoal_tac "finite A & A - {x} <= F") |
|
432 |
prefer 2 apply (blast intro: finite_subset) |
|
433 |
apply atomize |
|
434 |
apply (drule_tac x = "A - {x}" in spec) |
|
435 |
apply (simp add: card_Diff_singleton_if split add: split_if_asm) |
|
436 |
apply (case_tac "card A") |
|
437 |
apply auto |
|
438 |
done |
|
439 |
||
440 |
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B" |
|
441 |
apply (simp add: psubset_def linorder_not_le [symmetric]) |
|
442 |
apply (blast dest: card_seteq) |
|
443 |
done |
|
444 |
||
445 |
lemma card_mono: "finite B ==> A <= B ==> card A <= card B" |
|
446 |
apply (case_tac "A = B") |
|
447 |
apply simp |
|
448 |
apply (simp add: linorder_not_less [symmetric]) |
|
449 |
apply (blast dest: card_seteq intro: order_less_imp_le) |
|
450 |
done |
|
451 |
||
452 |
lemma card_Un_Int: "finite A ==> finite B |
|
453 |
==> card A + card B = card (A Un B) + card (A Int B)" |
|
454 |
apply (induct set: Finites) |
|
455 |
apply simp |
|
456 |
apply (simp add: insert_absorb Int_insert_left) |
|
457 |
done |
|
458 |
||
459 |
lemma card_Un_disjoint: "finite A ==> finite B |
|
460 |
==> A Int B = {} ==> card (A Un B) = card A + card B" |
|
461 |
by (simp add: card_Un_Int) |
|
462 |
||
463 |
lemma card_Diff_subset: |
|
464 |
"finite A ==> B <= A ==> card A - card B = card (A - B)" |
|
465 |
apply (subgoal_tac "(A - B) Un B = A") |
|
466 |
prefer 2 apply blast |
|
467 |
apply (rule add_right_cancel [THEN iffD1]) |
|
468 |
apply (rule card_Un_disjoint [THEN subst]) |
|
469 |
apply (erule_tac [4] ssubst) |
|
470 |
prefer 3 apply blast |
|
471 |
apply (simp_all add: add_commute not_less_iff_le |
|
472 |
add_diff_inverse card_mono finite_subset) |
|
473 |
done |
|
474 |
||
475 |
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A" |
|
476 |
apply (rule Suc_less_SucD) |
|
477 |
apply (simp add: card_Suc_Diff1) |
|
478 |
done |
|
479 |
||
480 |
lemma card_Diff2_less: |
|
481 |
"finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A" |
|
482 |
apply (case_tac "x = y") |
|
483 |
apply (simp add: card_Diff1_less) |
|
484 |
apply (rule less_trans) |
|
485 |
prefer 2 apply (auto intro!: card_Diff1_less) |
|
486 |
done |
|
487 |
||
488 |
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A" |
|
489 |
apply (case_tac "x : A") |
|
490 |
apply (simp_all add: card_Diff1_less less_imp_le) |
|
491 |
done |
|
492 |
||
493 |
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B" |
|
494 |
apply (erule psubsetI) |
|
495 |
apply blast |
|
496 |
done |
|
497 |
||
498 |
||
499 |
subsubsection {* Cardinality of image *} |
|
500 |
||
501 |
lemma card_image_le: "finite A ==> card (f ` A) <= card A" |
|
502 |
apply (induct set: Finites) |
|
503 |
apply simp |
|
504 |
apply (simp add: le_SucI finite_imageI card_insert_if) |
|
505 |
done |
|
506 |
||
507 |
lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A" |
|
508 |
apply (induct set: Finites) |
|
509 |
apply simp_all |
|
510 |
apply atomize |
|
511 |
apply safe |
|
512 |
apply (unfold inj_on_def) |
|
513 |
apply blast |
|
514 |
apply (subst card_insert_disjoint) |
|
515 |
apply (erule finite_imageI) |
|
516 |
apply blast |
|
517 |
apply blast |
|
518 |
done |
|
519 |
||
520 |
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A" |
|
521 |
by (simp add: card_seteq card_image) |
|
522 |
||
523 |
||
524 |
subsubsection {* Cardinality of the Powerset *} |
|
525 |
||
526 |
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *) |
|
527 |
apply (induct set: Finites) |
|
528 |
apply (simp_all add: Pow_insert) |
|
529 |
apply (subst card_Un_disjoint) |
|
530 |
apply blast |
|
531 |
apply (blast intro: finite_imageI) |
|
532 |
apply blast |
|
533 |
apply (subgoal_tac "inj_on (insert x) (Pow F)") |
|
534 |
apply (simp add: card_image Pow_insert) |
|
535 |
apply (unfold inj_on_def) |
|
536 |
apply (blast elim!: equalityE) |
|
537 |
done |
|
538 |
||
539 |
text {* |
|
540 |
\medskip Relates to equivalence classes. Based on a theorem of |
|
541 |
F. Kammüller's. The @{prop "finite C"} premise is redundant. |
|
542 |
*} |
|
543 |
||
544 |
lemma dvd_partition: |
|
545 |
"finite C ==> finite (Union C) ==> |
|
546 |
ALL c : C. k dvd card c ==> |
|
547 |
(ALL c1: C. ALL c2: C. c1 ~= c2 --> c1 Int c2 = {}) ==> |
|
548 |
k dvd card (Union C)" |
|
549 |
apply (induct set: Finites) |
|
550 |
apply simp_all |
|
551 |
apply clarify |
|
552 |
apply (subst card_Un_disjoint) |
|
553 |
apply (auto simp add: dvd_add disjoint_eq_subset_Compl) |
|
554 |
done |
|
555 |
||
556 |
||
557 |
subsection {* A fold functional for finite sets *} |
|
558 |
||
559 |
text {* |
|
560 |
For @{text n} non-negative we have @{text "fold f e {x1, ..., xn} = |
|
561 |
f x1 (... (f xn e))"} where @{text f} is at least left-commutative. |
|
562 |
*} |
|
563 |
||
564 |
consts |
|
565 |
foldSet :: "('b => 'a => 'a) => 'a => ('b set \<times> 'a) set" |
|
566 |
||
567 |
inductive "foldSet f e" |
|
568 |
intros |
|
569 |
emptyI [intro]: "({}, e) : foldSet f e" |
|
570 |
insertI [intro]: "x \<notin> A ==> (A, y) : foldSet f e ==> (insert x A, f x y) : foldSet f e" |
|
571 |
||
572 |
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e" |
|
573 |
||
574 |
constdefs |
|
575 |
fold :: "('b => 'a => 'a) => 'a => 'b set => 'a" |
|
576 |
"fold f e A == THE x. (A, x) : foldSet f e" |
|
577 |
||
578 |
lemma Diff1_foldSet: "(A - {x}, y) : foldSet f e ==> x: A ==> (A, f x y) : foldSet f e" |
|
579 |
apply (erule insert_Diff [THEN subst], rule foldSet.intros) |
|
580 |
apply auto |
|
581 |
done |
|
582 |
||
583 |
lemma foldSet_imp_finite [simp]: "(A, x) : foldSet f e ==> finite A" |
|
584 |
by (induct set: foldSet) auto |
|
585 |
||
586 |
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f e" |
|
587 |
by (induct set: Finites) auto |
|
588 |
||
589 |
||
590 |
subsubsection {* Left-commutative operations *} |
|
591 |
||
592 |
locale LC = |
|
593 |
fixes f :: "'b => 'a => 'a" (infixl "\<cdot>" 70) |
|
594 |
assumes left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)" |
|
595 |
||
596 |
lemma (in LC) foldSet_determ_aux: |
|
597 |
"ALL A x. card A < n --> (A, x) : foldSet f e --> |
|
598 |
(ALL y. (A, y) : foldSet f e --> y = x)" |
|
599 |
apply (induct n) |
|
600 |
apply (auto simp add: less_Suc_eq) |
|
601 |
apply (erule foldSet.cases) |
|
602 |
apply blast |
|
603 |
apply (erule foldSet.cases) |
|
604 |
apply blast |
|
605 |
apply clarify |
|
606 |
txt {* force simplification of @{text "card A < card (insert ...)"}. *} |
|
607 |
apply (erule rev_mp) |
|
608 |
apply (simp add: less_Suc_eq_le) |
|
609 |
apply (rule impI) |
|
610 |
apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb") |
|
611 |
apply (subgoal_tac "Aa = Ab") |
|
612 |
prefer 2 apply (blast elim!: equalityE) |
|
613 |
apply blast |
|
614 |
txt {* case @{prop "xa \<notin> xb"}. *} |
|
615 |
apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab") |
|
616 |
prefer 2 apply (blast elim!: equalityE) |
|
617 |
apply clarify |
|
618 |
apply (subgoal_tac "Aa = insert xb Ab - {xa}") |
|
619 |
prefer 2 apply blast |
|
620 |
apply (subgoal_tac "card Aa <= card Ab") |
|
621 |
prefer 2 |
|
622 |
apply (rule Suc_le_mono [THEN subst]) |
|
623 |
apply (simp add: card_Suc_Diff1) |
|
624 |
apply (rule_tac A1 = "Aa - {xb}" and f1 = f in finite_imp_foldSet [THEN exE]) |
|
625 |
apply (blast intro: foldSet_imp_finite finite_Diff) |
|
626 |
apply (frule (1) Diff1_foldSet) |
|
627 |
apply (subgoal_tac "ya = f xb x") |
|
628 |
prefer 2 apply (blast del: equalityCE) |
|
629 |
apply (subgoal_tac "(Ab - {xa}, x) : foldSet f e") |
|
630 |
prefer 2 apply simp |
|
631 |
apply (subgoal_tac "yb = f xa x") |
|
632 |
prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet) |
|
633 |
apply (simp (no_asm_simp) add: left_commute) |
|
634 |
done |
|
635 |
||
636 |
lemma (in LC) foldSet_determ: "(A, x) : foldSet f e ==> (A, y) : foldSet f e ==> y = x" |
|
637 |
by (blast intro: foldSet_determ_aux [rule_format]) |
|
638 |
||
639 |
lemma (in LC) fold_equality: "(A, y) : foldSet f e ==> fold f e A = y" |
|
640 |
by (unfold fold_def) (blast intro: foldSet_determ) |
|
641 |
||
642 |
lemma fold_empty [simp]: "fold f e {} = e" |
|
643 |
by (unfold fold_def) blast |
|
644 |
||
645 |
lemma (in LC) fold_insert_aux: "x \<notin> A ==> |
|
646 |
((insert x A, v) : foldSet f e) = |
|
647 |
(EX y. (A, y) : foldSet f e & v = f x y)" |
|
648 |
apply auto |
|
649 |
apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE]) |
|
650 |
apply (fastsimp dest: foldSet_imp_finite) |
|
651 |
apply (blast intro: foldSet_determ) |
|
652 |
done |
|
653 |
||
654 |
lemma (in LC) fold_insert: |
|
655 |
"finite A ==> x \<notin> A ==> fold f e (insert x A) = f x (fold f e A)" |
|
656 |
apply (unfold fold_def) |
|
657 |
apply (simp add: fold_insert_aux) |
|
658 |
apply (rule the_equality) |
|
659 |
apply (auto intro: finite_imp_foldSet |
|
660 |
cong add: conj_cong simp add: fold_def [symmetric] fold_equality) |
|
661 |
done |
|
662 |
||
663 |
lemma (in LC) fold_commute: "finite A ==> (!!e. f x (fold f e A) = fold f (f x e) A)" |
|
664 |
apply (induct set: Finites) |
|
665 |
apply simp |
|
666 |
apply (simp add: left_commute fold_insert) |
|
667 |
done |
|
668 |
||
669 |
lemma (in LC) fold_nest_Un_Int: |
|
670 |
"finite A ==> finite B |
|
671 |
==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)" |
|
672 |
apply (induct set: Finites) |
|
673 |
apply simp |
|
674 |
apply (simp add: fold_insert fold_commute Int_insert_left insert_absorb) |
|
675 |
done |
|
676 |
||
677 |
lemma (in LC) fold_nest_Un_disjoint: |
|
678 |
"finite A ==> finite B ==> A Int B = {} |
|
679 |
==> fold f e (A Un B) = fold f (fold f e B) A" |
|
680 |
by (simp add: fold_nest_Un_Int) |
|
681 |
||
682 |
declare foldSet_imp_finite [simp del] |
|
683 |
empty_foldSetE [rule del] foldSet.intros [rule del] |
|
684 |
-- {* Delete rules to do with @{text foldSet} relation. *} |
|
685 |
||
686 |
||
687 |
||
688 |
subsubsection {* Commutative monoids *} |
|
689 |
||
690 |
text {* |
|
691 |
We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"} |
|
692 |
instead of @{text "'b => 'a => 'a"}. |
|
693 |
*} |
|
694 |
||
695 |
locale ACe = |
|
696 |
fixes f :: "'a => 'a => 'a" (infixl "\<cdot>" 70) |
|
697 |
and e :: 'a |
|
698 |
assumes ident [simp]: "x \<cdot> e = x" |
|
699 |
and commute: "x \<cdot> y = y \<cdot> x" |
|
700 |
and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" |
|
701 |
||
702 |
lemma (in ACe) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)" |
|
703 |
proof - |
|
704 |
have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute) |
|
705 |
also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc) |
|
706 |
also have "z \<cdot> x = x \<cdot> z" by (simp only: commute) |
|
707 |
finally show ?thesis . |
|
708 |
qed |
|
709 |
||
12718 | 710 |
lemmas (in ACe) AC = assoc commute left_commute |
12396 | 711 |
|
12693 | 712 |
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x" |
12396 | 713 |
proof - |
714 |
have "x \<cdot> e = x" by (rule ident) |
|
715 |
thus ?thesis by (subst commute) |
|
716 |
qed |
|
717 |
||
718 |
lemma (in ACe) fold_Un_Int: |
|
719 |
"finite A ==> finite B ==> |
|
720 |
fold f e A \<cdot> fold f e B = fold f e (A Un B) \<cdot> fold f e (A Int B)" |
|
721 |
apply (induct set: Finites) |
|
722 |
apply simp |
|
13400 | 723 |
apply (simp add: AC insert_absorb Int_insert_left |
724 |
LC.fold_insert [OF LC.intro, OF LC_axioms.intro]) |
|
12396 | 725 |
done |
726 |
||
727 |
lemma (in ACe) fold_Un_disjoint: |
|
728 |
"finite A ==> finite B ==> A Int B = {} ==> |
|
729 |
fold f e (A Un B) = fold f e A \<cdot> fold f e B" |
|
730 |
by (simp add: fold_Un_Int) |
|
731 |
||
732 |
lemma (in ACe) fold_Un_disjoint2: |
|
733 |
"finite A ==> finite B ==> A Int B = {} ==> |
|
734 |
fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B" |
|
735 |
proof - |
|
736 |
assume b: "finite B" |
|
737 |
assume "finite A" |
|
738 |
thus "A Int B = {} ==> |
|
739 |
fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B" |
|
740 |
proof induct |
|
741 |
case empty |
|
742 |
thus ?case by simp |
|
743 |
next |
|
744 |
case (insert F x) |
|
745 |
have "fold (f \<circ> g) e (insert x F \<union> B) = fold (f \<circ> g) e (insert x (F \<union> B))" |
|
746 |
by simp |
|
747 |
also have "... = (f \<circ> g) x (fold (f \<circ> g) e (F \<union> B))" |
|
13400 | 748 |
by (rule LC.fold_insert [OF LC.intro]) |
13365 | 749 |
(insert b insert, auto simp add: left_commute intro: LC_axioms.intro) |
12396 | 750 |
also from insert have "fold (f \<circ> g) e (F \<union> B) = |
751 |
fold (f \<circ> g) e F \<cdot> fold (f \<circ> g) e B" by blast |
|
752 |
also have "(f \<circ> g) x ... = (f \<circ> g) x (fold (f \<circ> g) e F) \<cdot> fold (f \<circ> g) e B" |
|
753 |
by (simp add: AC) |
|
754 |
also have "(f \<circ> g) x (fold (f \<circ> g) e F) = fold (f \<circ> g) e (insert x F)" |
|
13400 | 755 |
by (rule LC.fold_insert [OF LC.intro, symmetric]) (insert b insert, |
13365 | 756 |
auto simp add: left_commute intro: LC_axioms.intro) |
12396 | 757 |
finally show ?case . |
758 |
qed |
|
759 |
qed |
|
760 |
||
761 |
||
762 |
subsection {* Generalized summation over a set *} |
|
763 |
||
764 |
constdefs |
|
765 |
setsum :: "('a => 'b) => 'a set => 'b::plus_ac0" |
|
766 |
"setsum f A == if finite A then fold (op + o f) 0 A else 0" |
|
767 |
||
768 |
syntax |
|
769 |
"_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\<Sum>_:_. _" [0, 51, 10] 10) |
|
770 |
syntax (xsymbols) |
|
771 |
"_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\<Sum>_\<in>_. _" [0, 51, 10] 10) |
|
772 |
translations |
|
773 |
"\<Sum>i:A. b" == "setsum (%i. b) A" -- {* Beware of argument permutation! *} |
|
774 |
||
775 |
||
776 |
lemma setsum_empty [simp]: "setsum f {} = 0" |
|
777 |
by (simp add: setsum_def) |
|
778 |
||
779 |
lemma setsum_insert [simp]: |
|
780 |
"finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F" |
|
13365 | 781 |
by (simp add: setsum_def |
13400 | 782 |
LC.fold_insert [OF LC.intro, OF LC_axioms.intro] plus_ac0_left_commute) |
12396 | 783 |
|
784 |
lemma setsum_0: "setsum (\<lambda>i. 0) A = 0" |
|
785 |
apply (case_tac "finite A") |
|
786 |
prefer 2 apply (simp add: setsum_def) |
|
787 |
apply (erule finite_induct) |
|
788 |
apply auto |
|
789 |
done |
|
790 |
||
791 |
lemma setsum_eq_0_iff [simp]: |
|
792 |
"finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" |
|
793 |
by (induct set: Finites) auto |
|
794 |
||
795 |
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" |
|
796 |
apply (case_tac "finite A") |
|
797 |
prefer 2 apply (simp add: setsum_def) |
|
798 |
apply (erule rev_mp) |
|
799 |
apply (erule finite_induct) |
|
800 |
apply auto |
|
801 |
done |
|
802 |
||
803 |
lemma card_eq_setsum: "finite A ==> card A = setsum (\<lambda>x. 1) A" |
|
804 |
-- {* Could allow many @{text "card"} proofs to be simplified. *} |
|
805 |
by (induct set: Finites) auto |
|
806 |
||
807 |
lemma setsum_Un_Int: "finite A ==> finite B |
|
808 |
==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" |
|
809 |
-- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} |
|
810 |
apply (induct set: Finites) |
|
811 |
apply simp |
|
812 |
apply (simp add: plus_ac0 Int_insert_left insert_absorb) |
|
813 |
done |
|
814 |
||
815 |
lemma setsum_Un_disjoint: "finite A ==> finite B |
|
816 |
==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" |
|
817 |
apply (subst setsum_Un_Int [symmetric]) |
|
818 |
apply auto |
|
819 |
done |
|
820 |
||
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12718
diff
changeset
|
821 |
lemma setsum_UN_disjoint: |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12718
diff
changeset
|
822 |
fixes f :: "'a => 'b::plus_ac0" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12718
diff
changeset
|
823 |
shows |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12718
diff
changeset
|
824 |
"finite I ==> (ALL i:I. finite (A i)) ==> |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12718
diff
changeset
|
825 |
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==> |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12718
diff
changeset
|
826 |
setsum f (UNION I A) = setsum (\<lambda>i. setsum f (A i)) I" |
12396 | 827 |
apply (induct set: Finites) |
828 |
apply simp |
|
829 |
apply atomize |
|
830 |
apply (subgoal_tac "ALL i:F. x \<noteq> i") |
|
831 |
prefer 2 apply blast |
|
832 |
apply (subgoal_tac "A x Int UNION F A = {}") |
|
833 |
prefer 2 apply blast |
|
834 |
apply (simp add: setsum_Un_disjoint) |
|
835 |
done |
|
836 |
||
837 |
lemma setsum_addf: "setsum (\<lambda>x. f x + g x) A = (setsum f A + setsum g A)" |
|
838 |
apply (case_tac "finite A") |
|
839 |
prefer 2 apply (simp add: setsum_def) |
|
840 |
apply (erule finite_induct) |
|
841 |
apply auto |
|
842 |
apply (simp add: plus_ac0) |
|
843 |
done |
|
844 |
||
845 |
lemma setsum_Un: "finite A ==> finite B ==> |
|
846 |
(setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" |
|
847 |
-- {* For the natural numbers, we have subtraction. *} |
|
848 |
apply (subst setsum_Un_Int [symmetric]) |
|
849 |
apply auto |
|
850 |
done |
|
851 |
||
852 |
lemma setsum_diff1: "(setsum f (A - {a}) :: nat) = |
|
853 |
(if a:A then setsum f A - f a else setsum f A)" |
|
854 |
apply (case_tac "finite A") |
|
855 |
prefer 2 apply (simp add: setsum_def) |
|
856 |
apply (erule finite_induct) |
|
857 |
apply (auto simp add: insert_Diff_if) |
|
858 |
apply (drule_tac a = a in mk_disjoint_insert) |
|
859 |
apply auto |
|
860 |
done |
|
861 |
||
862 |
lemma setsum_cong: |
|
863 |
"A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" |
|
864 |
apply (case_tac "finite B") |
|
865 |
prefer 2 apply (simp add: setsum_def) |
|
866 |
apply simp |
|
867 |
apply (subgoal_tac "ALL C. C <= B --> (ALL x:C. f x = g x) --> setsum f C = setsum g C") |
|
868 |
apply simp |
|
869 |
apply (erule finite_induct) |
|
870 |
apply simp |
|
871 |
apply (simp add: subset_insert_iff) |
|
872 |
apply clarify |
|
873 |
apply (subgoal_tac "finite C") |
|
874 |
prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) |
|
875 |
apply (subgoal_tac "C = insert x (C - {x})") |
|
876 |
prefer 2 apply blast |
|
877 |
apply (erule ssubst) |
|
878 |
apply (drule spec) |
|
879 |
apply (erule (1) notE impE) |
|
880 |
apply (simp add: Ball_def) |
|
881 |
done |
|
882 |
||
883 |
||
884 |
text {* |
|
885 |
\medskip Basic theorem about @{text "choose"}. By Florian |
|
886 |
Kammüller, tidied by LCP. |
|
887 |
*} |
|
888 |
||
889 |
lemma card_s_0_eq_empty: |
|
890 |
"finite A ==> card {B. B \<subseteq> A & card B = 0} = 1" |
|
891 |
apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) |
|
892 |
apply (simp cong add: rev_conj_cong) |
|
893 |
done |
|
894 |
||
895 |
lemma choose_deconstruct: "finite M ==> x \<notin> M |
|
896 |
==> {s. s <= insert x M & card(s) = Suc k} |
|
897 |
= {s. s <= M & card(s) = Suc k} Un |
|
898 |
{s. EX t. t <= M & card(t) = k & s = insert x t}" |
|
899 |
apply safe |
|
900 |
apply (auto intro: finite_subset [THEN card_insert_disjoint]) |
|
901 |
apply (drule_tac x = "xa - {x}" in spec) |
|
902 |
apply (subgoal_tac "x ~: xa") |
|
903 |
apply auto |
|
904 |
apply (erule rev_mp, subst card_Diff_singleton) |
|
905 |
apply (auto intro: finite_subset) |
|
906 |
done |
|
907 |
||
908 |
lemma card_inj_on_le: |
|
909 |
"finite A ==> finite B ==> f ` A \<subseteq> B ==> inj_on f A ==> card A <= card B" |
|
910 |
by (auto intro: card_mono simp add: card_image [symmetric]) |
|
911 |
||
912 |
lemma card_bij_eq: "finite A ==> finite B ==> |
|
913 |
f ` A \<subseteq> B ==> inj_on f A ==> g ` B \<subseteq> A ==> inj_on g B ==> card A = card B" |
|
914 |
by (auto intro: le_anti_sym card_inj_on_le) |
|
915 |
||
916 |
lemma constr_bij: "finite A ==> x \<notin> A ==> |
|
917 |
card {B. EX C. C <= A & card(C) = k & B = insert x C} = |
|
918 |
card {B. B <= A & card(B) = k}" |
|
919 |
apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq) |
|
920 |
apply (rule_tac B = "Pow (insert x A) " in finite_subset) |
|
921 |
apply (rule_tac [3] B = "Pow (A) " in finite_subset) |
|
922 |
apply fast+ |
|
923 |
txt {* arity *} |
|
924 |
apply (auto elim!: equalityE simp add: inj_on_def) |
|
925 |
apply (subst Diff_insert0) |
|
926 |
apply auto |
|
927 |
done |
|
928 |
||
929 |
text {* |
|
930 |
Main theorem: combinatorial statement about number of subsets of a set. |
|
931 |
*} |
|
932 |
||
933 |
lemma n_sub_lemma: |
|
934 |
"!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)" |
|
935 |
apply (induct k) |
|
936 |
apply (simp add: card_s_0_eq_empty) |
|
937 |
apply atomize |
|
938 |
apply (rotate_tac -1, erule finite_induct) |
|
939 |
apply (simp_all (no_asm_simp) cong add: conj_cong add: card_s_0_eq_empty choose_deconstruct) |
|
940 |
apply (subst card_Un_disjoint) |
|
941 |
prefer 4 apply (force simp add: constr_bij) |
|
942 |
prefer 3 apply force |
|
943 |
prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] |
|
944 |
finite_subset [of _ "Pow (insert x F)", standard]) |
|
945 |
apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) |
|
946 |
done |
|
947 |
||
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12718
diff
changeset
|
948 |
theorem n_subsets: "finite A ==> card {B. B <= A & card B = k} = (card A choose k)" |
12396 | 949 |
by (simp add: n_sub_lemma) |
950 |
||
951 |
end |