--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Finite_Set.thy Thu Dec 06 00:38:55 2001 +0100
@@ -0,0 +1,947 @@
+(* Title: HOL/Finite_Set.thy
+ ID: $Id$
+ Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+*)
+
+header {* Finite sets *}
+
+theory Finite_Set = Divides + Power + Inductive + SetInterval:
+
+subsection {* Collection of finite sets *}
+
+consts Finites :: "'a set set"
+
+inductive Finites
+ intros
+ emptyI [simp, intro!]: "{} : Finites"
+ insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
+
+syntax
+ finite :: "'a set => bool"
+translations
+ "finite A" == "A : Finites"
+
+axclass finite \<subseteq> type
+ finite: "finite UNIV"
+
+
+lemma finite_induct [case_names empty insert, induct set: Finites]:
+ "finite F ==>
+ P {} ==> (!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
+ -- {* Discharging @{text "x \<notin> F"} entails extra work. *}
+proof -
+ assume "P {}" and insert: "!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
+ assume "finite F"
+ thus "P F"
+ proof induct
+ show "P {}" .
+ fix F x assume F: "finite F" and P: "P F"
+ show "P (insert x F)"
+ proof cases
+ assume "x \<in> F"
+ hence "insert x F = F" by (rule insert_absorb)
+ with P show ?thesis by (simp only:)
+ next
+ assume "x \<notin> F"
+ from F this P show ?thesis by (rule insert)
+ qed
+ qed
+qed
+
+lemma finite_subset_induct [consumes 2, case_names empty insert]:
+ "finite F ==> F \<subseteq> A ==>
+ P {} ==> (!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
+ P F"
+proof -
+ assume "P {}" and insert: "!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
+ assume "finite F"
+ thus "F \<subseteq> A ==> P F"
+ proof induct
+ show "P {}" .
+ fix F x assume "finite F" and "x \<notin> F"
+ and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
+ show "P (insert x F)"
+ proof (rule insert)
+ from i show "x \<in> A" by blast
+ from i have "F \<subseteq> A" by blast
+ with P show "P F" .
+ qed
+ qed
+qed
+
+lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
+ -- {* The union of two finite sets is finite. *}
+ by (induct set: Finites) simp_all
+
+lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
+ -- {* Every subset of a finite set is finite. *}
+proof -
+ assume "finite B"
+ thus "!!A. A \<subseteq> B ==> finite A"
+ proof induct
+ case empty
+ thus ?case by simp
+ next
+ case (insert F x A)
+ have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
+ show "finite A"
+ proof cases
+ assume x: "x \<in> A"
+ with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
+ with r have "finite (A - {x})" .
+ hence "finite (insert x (A - {x}))" ..
+ also have "insert x (A - {x}) = A" by (rule insert_Diff)
+ finally show ?thesis .
+ next
+ show "A \<subseteq> F ==> ?thesis" .
+ assume "x \<notin> A"
+ with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
+ qed
+ qed
+qed
+
+lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
+ by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
+
+lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
+ -- {* The converse obviously fails. *}
+ by (blast intro: finite_subset)
+
+lemma finite_insert [simp]: "finite (insert a A) = finite A"
+ apply (subst insert_is_Un)
+ apply (simp only: finite_Un)
+ apply blast
+ done
+
+lemma finite_imageI: "finite F ==> finite (h ` F)"
+ -- {* The image of a finite set is finite. *}
+ by (induct set: Finites) simp_all
+
+lemma finite_range_imageI:
+ "finite (range g) ==> finite (range (%x. f (g x)))"
+ apply (drule finite_imageI)
+ apply simp
+ done
+
+lemma finite_empty_induct:
+ "finite A ==>
+ P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
+proof -
+ assume "finite A"
+ and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
+ have "P (A - A)"
+ proof -
+ fix c b :: "'a set"
+ presume c: "finite c" and b: "finite b"
+ and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
+ from c show "c \<subseteq> b ==> P (b - c)"
+ proof induct
+ case empty
+ from P1 show ?case by simp
+ next
+ case (insert F x)
+ have "P (b - F - {x})"
+ proof (rule P2)
+ from _ b show "finite (b - F)" by (rule finite_subset) blast
+ from insert show "x \<in> b - F" by simp
+ from insert show "P (b - F)" by simp
+ qed
+ also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
+ finally show ?case .
+ qed
+ next
+ show "A \<subseteq> A" ..
+ qed
+ thus "P {}" by simp
+qed
+
+lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
+ by (rule Diff_subset [THEN finite_subset])
+
+lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
+ apply (subst Diff_insert)
+ apply (case_tac "a : A - B")
+ apply (rule finite_insert [symmetric, THEN trans])
+ apply (subst insert_Diff)
+ apply simp_all
+ done
+
+
+lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
+proof -
+ have aux: "!!A. finite (A - {}) = finite A" by simp
+ fix B :: "'a set"
+ assume "finite B"
+ thus "!!A. f`A = B ==> inj_on f A ==> finite A"
+ apply induct
+ apply simp
+ apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
+ apply clarify
+ apply (simp (no_asm_use) add: inj_on_def)
+ apply (blast dest!: aux [THEN iffD1])
+ apply atomize
+ apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
+ apply (frule subsetD [OF equalityD2 insertI1])
+ apply clarify
+ apply (rule_tac x = xa in bexI)
+ apply (simp_all add: inj_on_image_set_diff)
+ done
+qed (rule refl)
+
+
+subsubsection {* The finite UNION of finite sets *}
+
+lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
+ by (induct set: Finites) simp_all
+
+text {*
+ Strengthen RHS to
+ @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x ~= {}})"}?
+
+ We'd need to prove
+ @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x ~= {}}"}
+ by induction. *}
+
+lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
+ by (blast intro: finite_UN_I finite_subset)
+
+
+subsubsection {* Sigma of finite sets *}
+
+lemma finite_SigmaI [simp]:
+ "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
+ by (unfold Sigma_def) (blast intro!: finite_UN_I)
+
+lemma finite_Prod_UNIV:
+ "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
+ apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
+ apply (erule ssubst)
+ apply (erule finite_SigmaI)
+ apply auto
+ done
+
+instance unit :: finite
+proof
+ have "finite {()}" by simp
+ also have "{()} = UNIV" by auto
+ finally show "finite (UNIV :: unit set)" .
+qed
+
+instance * :: (finite, finite) finite
+proof
+ show "finite (UNIV :: ('a \<times> 'b) set)"
+ proof (rule finite_Prod_UNIV)
+ show "finite (UNIV :: 'a set)" by (rule finite)
+ show "finite (UNIV :: 'b set)" by (rule finite)
+ qed
+qed
+
+
+subsubsection {* The powerset of a finite set *}
+
+lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
+proof
+ assume "finite (Pow A)"
+ with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
+ thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
+next
+ assume "finite A"
+ thus "finite (Pow A)"
+ by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
+qed
+
+lemma finite_converse [iff]: "finite (r^-1) = finite r"
+ apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
+ apply simp
+ apply (rule iffI)
+ apply (erule finite_imageD [unfolded inj_on_def])
+ apply (simp split add: split_split)
+ apply (erule finite_imageI)
+ apply (simp add: converse_def image_def)
+ apply auto
+ apply (rule bexI)
+ prefer 2 apply assumption
+ apply simp
+ done
+
+lemma finite_lessThan [iff]: (fixes k :: nat) "finite {..k(}"
+ by (induct k) (simp_all add: lessThan_Suc)
+
+lemma finite_atMost [iff]: (fixes k :: nat) "finite {..k}"
+ by (induct k) (simp_all add: atMost_Suc)
+
+lemma bounded_nat_set_is_finite:
+ "(ALL i:N. i < (n::nat)) ==> finite N"
+ -- {* A bounded set of natural numbers is finite. *}
+ apply (rule finite_subset)
+ apply (rule_tac [2] finite_lessThan)
+ apply auto
+ done
+
+
+subsubsection {* Finiteness of transitive closure *}
+
+text {* (Thanks to Sidi Ehmety) *}
+
+lemma finite_Field: "finite r ==> finite (Field r)"
+ -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
+ apply (induct set: Finites)
+ apply (auto simp add: Field_def Domain_insert Range_insert)
+ done
+
+lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
+ apply clarify
+ apply (erule trancl_induct)
+ apply (auto simp add: Field_def)
+ done
+
+lemma finite_trancl: "finite (r^+) = finite r"
+ apply auto
+ prefer 2
+ apply (rule trancl_subset_Field2 [THEN finite_subset])
+ apply (rule finite_SigmaI)
+ prefer 3
+ apply (blast intro: r_into_trancl finite_subset)
+ apply (auto simp add: finite_Field)
+ done
+
+
+subsection {* Finite cardinality *}
+
+text {*
+ This definition, although traditional, is ugly to work with: @{text
+ "card A == LEAST n. EX f. A = {f i | i. i < n}"}. Therefore we have
+ switched to an inductive one:
+*}
+
+consts cardR :: "('a set \<times> nat) set"
+
+inductive cardR
+ intros
+ EmptyI: "({}, 0) : cardR"
+ InsertI: "(A, n) : cardR ==> a \<notin> A ==> (insert a A, Suc n) : cardR"
+
+constdefs
+ card :: "'a set => nat"
+ "card A == THE n. (A, n) : cardR"
+
+inductive_cases cardR_emptyE: "({}, n) : cardR"
+inductive_cases cardR_insertE: "(insert a A,n) : cardR"
+
+lemma cardR_SucD: "(A, n) : cardR ==> a : A --> (EX m. n = Suc m)"
+ by (induct set: cardR) simp_all
+
+lemma cardR_determ_aux1:
+ "(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A - {a}, n) : cardR)"
+ apply (induct set: cardR)
+ apply auto
+ apply (simp add: insert_Diff_if)
+ apply auto
+ apply (drule cardR_SucD)
+ apply (blast intro!: cardR.intros)
+ done
+
+lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \<notin> A ==> (A, m) : cardR"
+ by (drule cardR_determ_aux1) auto
+
+lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)"
+ apply (induct set: cardR)
+ apply (safe elim!: cardR_emptyE cardR_insertE)
+ apply (rename_tac B b m)
+ apply (case_tac "a = b")
+ apply (subgoal_tac "A = B")
+ prefer 2 apply (blast elim: equalityE)
+ apply blast
+ apply (subgoal_tac "EX C. A = insert b C & B = insert a C")
+ prefer 2
+ apply (rule_tac x = "A Int B" in exI)
+ apply (blast elim: equalityE)
+ apply (frule_tac A = B in cardR_SucD)
+ apply (blast intro!: cardR.intros dest!: cardR_determ_aux2)
+ done
+
+lemma cardR_imp_finite: "(A, n) : cardR ==> finite A"
+ by (induct set: cardR) simp_all
+
+lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR"
+ by (induct set: Finites) (auto intro!: cardR.intros)
+
+lemma card_equality: "(A,n) : cardR ==> card A = n"
+ by (unfold card_def) (blast intro: cardR_determ)
+
+lemma card_empty [simp]: "card {} = 0"
+ by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE)
+
+lemma card_insert_disjoint [simp]:
+ "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
+proof -
+ assume x: "x \<notin> A"
+ hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)"
+ apply (auto intro!: cardR.intros)
+ apply (rule_tac A1 = A in finite_imp_cardR [THEN exE])
+ apply (force dest: cardR_imp_finite)
+ apply (blast intro!: cardR.intros intro: cardR_determ)
+ done
+ assume "finite A"
+ thus ?thesis
+ apply (simp add: card_def aux)
+ apply (rule the_equality)
+ apply (auto intro: finite_imp_cardR
+ cong: conj_cong simp: card_def [symmetric] card_equality)
+ done
+qed
+
+lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
+ apply auto
+ apply (drule_tac a = x in mk_disjoint_insert)
+ apply clarify
+ apply (rotate_tac -1)
+ apply auto
+ done
+
+lemma card_insert_if:
+ "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
+ by (simp add: insert_absorb)
+
+lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
+ apply (rule_tac t = A in insert_Diff [THEN subst])
+ apply assumption
+ apply simp
+ done
+
+lemma card_Diff_singleton:
+ "finite A ==> x: A ==> card (A - {x}) = card A - 1"
+ by (simp add: card_Suc_Diff1 [symmetric])
+
+lemma card_Diff_singleton_if:
+ "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
+ by (simp add: card_Diff_singleton)
+
+lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
+ by (simp add: card_insert_if card_Suc_Diff1)
+
+lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
+ by (simp add: card_insert_if)
+
+lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
+ apply (induct set: Finites)
+ apply simp
+ apply clarify
+ apply (subgoal_tac "finite A & A - {x} <= F")
+ prefer 2 apply (blast intro: finite_subset)
+ apply atomize
+ apply (drule_tac x = "A - {x}" in spec)
+ apply (simp add: card_Diff_singleton_if split add: split_if_asm)
+ apply (case_tac "card A")
+ apply auto
+ done
+
+lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
+ apply (simp add: psubset_def linorder_not_le [symmetric])
+ apply (blast dest: card_seteq)
+ done
+
+lemma card_mono: "finite B ==> A <= B ==> card A <= card B"
+ apply (case_tac "A = B")
+ apply simp
+ apply (simp add: linorder_not_less [symmetric])
+ apply (blast dest: card_seteq intro: order_less_imp_le)
+ done
+
+lemma card_Un_Int: "finite A ==> finite B
+ ==> card A + card B = card (A Un B) + card (A Int B)"
+ apply (induct set: Finites)
+ apply simp
+ apply (simp add: insert_absorb Int_insert_left)
+ done
+
+lemma card_Un_disjoint: "finite A ==> finite B
+ ==> A Int B = {} ==> card (A Un B) = card A + card B"
+ by (simp add: card_Un_Int)
+
+lemma card_Diff_subset:
+ "finite A ==> B <= A ==> card A - card B = card (A - B)"
+ apply (subgoal_tac "(A - B) Un B = A")
+ prefer 2 apply blast
+ apply (rule add_right_cancel [THEN iffD1])
+ apply (rule card_Un_disjoint [THEN subst])
+ apply (erule_tac [4] ssubst)
+ prefer 3 apply blast
+ apply (simp_all add: add_commute not_less_iff_le
+ add_diff_inverse card_mono finite_subset)
+ done
+
+lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
+ apply (rule Suc_less_SucD)
+ apply (simp add: card_Suc_Diff1)
+ done
+
+lemma card_Diff2_less:
+ "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
+ apply (case_tac "x = y")
+ apply (simp add: card_Diff1_less)
+ apply (rule less_trans)
+ prefer 2 apply (auto intro!: card_Diff1_less)
+ done
+
+lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
+ apply (case_tac "x : A")
+ apply (simp_all add: card_Diff1_less less_imp_le)
+ done
+
+lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
+ apply (erule psubsetI)
+ apply blast
+ done
+
+
+subsubsection {* Cardinality of image *}
+
+lemma card_image_le: "finite A ==> card (f ` A) <= card A"
+ apply (induct set: Finites)
+ apply simp
+ apply (simp add: le_SucI finite_imageI card_insert_if)
+ done
+
+lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A"
+ apply (induct set: Finites)
+ apply simp_all
+ apply atomize
+ apply safe
+ apply (unfold inj_on_def)
+ apply blast
+ apply (subst card_insert_disjoint)
+ apply (erule finite_imageI)
+ apply blast
+ apply blast
+ done
+
+lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
+ by (simp add: card_seteq card_image)
+
+
+subsubsection {* Cardinality of the Powerset *}
+
+lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *)
+ apply (induct set: Finites)
+ apply (simp_all add: Pow_insert)
+ apply (subst card_Un_disjoint)
+ apply blast
+ apply (blast intro: finite_imageI)
+ apply blast
+ apply (subgoal_tac "inj_on (insert x) (Pow F)")
+ apply (simp add: card_image Pow_insert)
+ apply (unfold inj_on_def)
+ apply (blast elim!: equalityE)
+ done
+
+text {*
+ \medskip Relates to equivalence classes. Based on a theorem of
+ F. Kammüller's. The @{prop "finite C"} premise is redundant.
+*}
+
+lemma dvd_partition:
+ "finite C ==> finite (Union C) ==>
+ ALL c : C. k dvd card c ==>
+ (ALL c1: C. ALL c2: C. c1 ~= c2 --> c1 Int c2 = {}) ==>
+ k dvd card (Union C)"
+ apply (induct set: Finites)
+ apply simp_all
+ apply clarify
+ apply (subst card_Un_disjoint)
+ apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
+ done
+
+
+subsection {* A fold functional for finite sets *}
+
+text {*
+ For @{text n} non-negative we have @{text "fold f e {x1, ..., xn} =
+ f x1 (... (f xn e))"} where @{text f} is at least left-commutative.
+*}
+
+consts
+ foldSet :: "('b => 'a => 'a) => 'a => ('b set \<times> 'a) set"
+
+inductive "foldSet f e"
+ intros
+ emptyI [intro]: "({}, e) : foldSet f e"
+ insertI [intro]: "x \<notin> A ==> (A, y) : foldSet f e ==> (insert x A, f x y) : foldSet f e"
+
+inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e"
+
+constdefs
+ fold :: "('b => 'a => 'a) => 'a => 'b set => 'a"
+ "fold f e A == THE x. (A, x) : foldSet f e"
+
+lemma Diff1_foldSet: "(A - {x}, y) : foldSet f e ==> x: A ==> (A, f x y) : foldSet f e"
+ apply (erule insert_Diff [THEN subst], rule foldSet.intros)
+ apply auto
+ done
+
+lemma foldSet_imp_finite [simp]: "(A, x) : foldSet f e ==> finite A"
+ by (induct set: foldSet) auto
+
+lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f e"
+ by (induct set: Finites) auto
+
+
+subsubsection {* Left-commutative operations *}
+
+locale LC =
+ fixes f :: "'b => 'a => 'a" (infixl "\<cdot>" 70)
+ assumes left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
+
+lemma (in LC) foldSet_determ_aux:
+ "ALL A x. card A < n --> (A, x) : foldSet f e -->
+ (ALL y. (A, y) : foldSet f e --> y = x)"
+ apply (induct n)
+ apply (auto simp add: less_Suc_eq)
+ apply (erule foldSet.cases)
+ apply blast
+ apply (erule foldSet.cases)
+ apply blast
+ apply clarify
+ txt {* force simplification of @{text "card A < card (insert ...)"}. *}
+ apply (erule rev_mp)
+ apply (simp add: less_Suc_eq_le)
+ apply (rule impI)
+ apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb")
+ apply (subgoal_tac "Aa = Ab")
+ prefer 2 apply (blast elim!: equalityE)
+ apply blast
+ txt {* case @{prop "xa \<notin> xb"}. *}
+ apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab")
+ prefer 2 apply (blast elim!: equalityE)
+ apply clarify
+ apply (subgoal_tac "Aa = insert xb Ab - {xa}")
+ prefer 2 apply blast
+ apply (subgoal_tac "card Aa <= card Ab")
+ prefer 2
+ apply (rule Suc_le_mono [THEN subst])
+ apply (simp add: card_Suc_Diff1)
+ apply (rule_tac A1 = "Aa - {xb}" and f1 = f in finite_imp_foldSet [THEN exE])
+ apply (blast intro: foldSet_imp_finite finite_Diff)
+ apply (frule (1) Diff1_foldSet)
+ apply (subgoal_tac "ya = f xb x")
+ prefer 2 apply (blast del: equalityCE)
+ apply (subgoal_tac "(Ab - {xa}, x) : foldSet f e")
+ prefer 2 apply simp
+ apply (subgoal_tac "yb = f xa x")
+ prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet)
+ apply (simp (no_asm_simp) add: left_commute)
+ done
+
+lemma (in LC) foldSet_determ: "(A, x) : foldSet f e ==> (A, y) : foldSet f e ==> y = x"
+ by (blast intro: foldSet_determ_aux [rule_format])
+
+lemma (in LC) fold_equality: "(A, y) : foldSet f e ==> fold f e A = y"
+ by (unfold fold_def) (blast intro: foldSet_determ)
+
+lemma fold_empty [simp]: "fold f e {} = e"
+ by (unfold fold_def) blast
+
+lemma (in LC) fold_insert_aux: "x \<notin> A ==>
+ ((insert x A, v) : foldSet f e) =
+ (EX y. (A, y) : foldSet f e & v = f x y)"
+ apply auto
+ apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
+ apply (fastsimp dest: foldSet_imp_finite)
+ apply (blast intro: foldSet_determ)
+ done
+
+lemma (in LC) fold_insert:
+ "finite A ==> x \<notin> A ==> fold f e (insert x A) = f x (fold f e A)"
+ apply (unfold fold_def)
+ apply (simp add: fold_insert_aux)
+ apply (rule the_equality)
+ apply (auto intro: finite_imp_foldSet
+ cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
+ done
+
+lemma (in LC) fold_commute: "finite A ==> (!!e. f x (fold f e A) = fold f (f x e) A)"
+ apply (induct set: Finites)
+ apply simp
+ apply (simp add: left_commute fold_insert)
+ done
+
+lemma (in LC) fold_nest_Un_Int:
+ "finite A ==> finite B
+ ==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)"
+ apply (induct set: Finites)
+ apply simp
+ apply (simp add: fold_insert fold_commute Int_insert_left insert_absorb)
+ done
+
+lemma (in LC) fold_nest_Un_disjoint:
+ "finite A ==> finite B ==> A Int B = {}
+ ==> fold f e (A Un B) = fold f (fold f e B) A"
+ by (simp add: fold_nest_Un_Int)
+
+declare foldSet_imp_finite [simp del]
+ empty_foldSetE [rule del] foldSet.intros [rule del]
+ -- {* Delete rules to do with @{text foldSet} relation. *}
+
+
+
+subsubsection {* Commutative monoids *}
+
+text {*
+ We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
+ instead of @{text "'b => 'a => 'a"}.
+*}
+
+locale ACe =
+ fixes f :: "'a => 'a => 'a" (infixl "\<cdot>" 70)
+ and e :: 'a
+ assumes ident [simp]: "x \<cdot> e = x"
+ and commute: "x \<cdot> y = y \<cdot> x"
+ and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
+
+lemma (in ACe) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
+proof -
+ have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
+ also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
+ also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
+ finally show ?thesis .
+qed
+
+lemma (in ACe)
+ AC: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" "x \<cdot> y = y \<cdot> x" "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
+ by (rule assoc, rule commute, rule left_commute) (* FIXME localize "lemmas" (!??) *)
+
+lemma (in ACe [simp]) left_ident: "e \<cdot> x = x"
+proof -
+ have "x \<cdot> e = x" by (rule ident)
+ thus ?thesis by (subst commute)
+qed
+
+lemma (in ACe) fold_Un_Int:
+ "finite A ==> finite B ==>
+ fold f e A \<cdot> fold f e B = fold f e (A Un B) \<cdot> fold f e (A Int B)"
+ apply (induct set: Finites)
+ apply simp
+ apply (simp add: AC fold_insert insert_absorb Int_insert_left)
+ done
+
+lemma (in ACe) fold_Un_disjoint:
+ "finite A ==> finite B ==> A Int B = {} ==>
+ fold f e (A Un B) = fold f e A \<cdot> fold f e B"
+ by (simp add: fold_Un_Int)
+
+lemma (in ACe) fold_Un_disjoint2:
+ "finite A ==> finite B ==> A Int B = {} ==>
+ fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
+proof -
+ assume b: "finite B"
+ assume "finite A"
+ thus "A Int B = {} ==>
+ fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B"
+ proof induct
+ case empty
+ thus ?case by simp
+ next
+ case (insert F x)
+ have "fold (f \<circ> g) e (insert x F \<union> B) = fold (f \<circ> g) e (insert x (F \<union> B))"
+ by simp
+ also have "... = (f \<circ> g) x (fold (f \<circ> g) e (F \<union> B))"
+ by (rule fold_insert) (insert b insert, auto simp add: left_commute) (* FIXME import of fold_insert (!?) *)
+ also from insert have "fold (f \<circ> g) e (F \<union> B) =
+ fold (f \<circ> g) e F \<cdot> fold (f \<circ> g) e B" by blast
+ also have "(f \<circ> g) x ... = (f \<circ> g) x (fold (f \<circ> g) e F) \<cdot> fold (f \<circ> g) e B"
+ by (simp add: AC)
+ also have "(f \<circ> g) x (fold (f \<circ> g) e F) = fold (f \<circ> g) e (insert x F)"
+ by (rule fold_insert [symmetric]) (insert b insert, auto simp add: left_commute)
+ finally show ?case .
+ qed
+qed
+
+
+subsection {* Generalized summation over a set *}
+
+constdefs
+ setsum :: "('a => 'b) => 'a set => 'b::plus_ac0"
+ "setsum f A == if finite A then fold (op + o f) 0 A else 0"
+
+syntax
+ "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\<Sum>_:_. _" [0, 51, 10] 10)
+syntax (xsymbols)
+ "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\<Sum>_\<in>_. _" [0, 51, 10] 10)
+translations
+ "\<Sum>i:A. b" == "setsum (%i. b) A" -- {* Beware of argument permutation! *}
+
+
+lemma setsum_empty [simp]: "setsum f {} = 0"
+ by (simp add: setsum_def)
+
+lemma setsum_insert [simp]:
+ "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
+ by (simp add: setsum_def fold_insert plus_ac0_left_commute)
+
+lemma setsum_0: "setsum (\<lambda>i. 0) A = 0"
+ apply (case_tac "finite A")
+ prefer 2 apply (simp add: setsum_def)
+ apply (erule finite_induct)
+ apply auto
+ done
+
+lemma setsum_eq_0_iff [simp]:
+ "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
+ by (induct set: Finites) auto
+
+lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
+ apply (case_tac "finite A")
+ prefer 2 apply (simp add: setsum_def)
+ apply (erule rev_mp)
+ apply (erule finite_induct)
+ apply auto
+ done
+
+lemma card_eq_setsum: "finite A ==> card A = setsum (\<lambda>x. 1) A"
+ -- {* Could allow many @{text "card"} proofs to be simplified. *}
+ by (induct set: Finites) auto
+
+lemma setsum_Un_Int: "finite A ==> finite B
+ ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
+ -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
+ apply (induct set: Finites)
+ apply simp
+ apply (simp add: plus_ac0 Int_insert_left insert_absorb)
+ done
+
+lemma setsum_Un_disjoint: "finite A ==> finite B
+ ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
+ apply (subst setsum_Un_Int [symmetric])
+ apply auto
+ done
+
+lemma setsum_UN_disjoint: (fixes f :: "'a => 'b::plus_ac0")
+ "finite I ==> (ALL i:I. finite (A i)) ==>
+ (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
+ setsum f (UNION I A) = setsum (\<lambda>i. setsum f (A i)) I"
+ apply (induct set: Finites)
+ apply simp
+ apply atomize
+ apply (subgoal_tac "ALL i:F. x \<noteq> i")
+ prefer 2 apply blast
+ apply (subgoal_tac "A x Int UNION F A = {}")
+ prefer 2 apply blast
+ apply (simp add: setsum_Un_disjoint)
+ done
+
+lemma setsum_addf: "setsum (\<lambda>x. f x + g x) A = (setsum f A + setsum g A)"
+ apply (case_tac "finite A")
+ prefer 2 apply (simp add: setsum_def)
+ apply (erule finite_induct)
+ apply auto
+ apply (simp add: plus_ac0)
+ done
+
+lemma setsum_Un: "finite A ==> finite B ==>
+ (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
+ -- {* For the natural numbers, we have subtraction. *}
+ apply (subst setsum_Un_Int [symmetric])
+ apply auto
+ done
+
+lemma setsum_diff1: "(setsum f (A - {a}) :: nat) =
+ (if a:A then setsum f A - f a else setsum f A)"
+ apply (case_tac "finite A")
+ prefer 2 apply (simp add: setsum_def)
+ apply (erule finite_induct)
+ apply (auto simp add: insert_Diff_if)
+ apply (drule_tac a = a in mk_disjoint_insert)
+ apply auto
+ done
+
+lemma setsum_cong:
+ "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
+ apply (case_tac "finite B")
+ prefer 2 apply (simp add: setsum_def)
+ apply simp
+ apply (subgoal_tac "ALL C. C <= B --> (ALL x:C. f x = g x) --> setsum f C = setsum g C")
+ apply simp
+ apply (erule finite_induct)
+ apply simp
+ apply (simp add: subset_insert_iff)
+ apply clarify
+ apply (subgoal_tac "finite C")
+ prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
+ apply (subgoal_tac "C = insert x (C - {x})")
+ prefer 2 apply blast
+ apply (erule ssubst)
+ apply (drule spec)
+ apply (erule (1) notE impE)
+ apply (simp add: Ball_def)
+ done
+
+
+text {*
+ \medskip Basic theorem about @{text "choose"}. By Florian
+ Kammüller, tidied by LCP.
+*}
+
+lemma card_s_0_eq_empty:
+ "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
+ apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
+ apply (simp cong add: rev_conj_cong)
+ done
+
+lemma choose_deconstruct: "finite M ==> x \<notin> M
+ ==> {s. s <= insert x M & card(s) = Suc k}
+ = {s. s <= M & card(s) = Suc k} Un
+ {s. EX t. t <= M & card(t) = k & s = insert x t}"
+ apply safe
+ apply (auto intro: finite_subset [THEN card_insert_disjoint])
+ apply (drule_tac x = "xa - {x}" in spec)
+ apply (subgoal_tac "x ~: xa")
+ apply auto
+ apply (erule rev_mp, subst card_Diff_singleton)
+ apply (auto intro: finite_subset)
+ done
+
+lemma card_inj_on_le:
+ "finite A ==> finite B ==> f ` A \<subseteq> B ==> inj_on f A ==> card A <= card B"
+ by (auto intro: card_mono simp add: card_image [symmetric])
+
+lemma card_bij_eq: "finite A ==> finite B ==>
+ f ` A \<subseteq> B ==> inj_on f A ==> g ` B \<subseteq> A ==> inj_on g B ==> card A = card B"
+ by (auto intro: le_anti_sym card_inj_on_le)
+
+lemma constr_bij: "finite A ==> x \<notin> A ==>
+ card {B. EX C. C <= A & card(C) = k & B = insert x C} =
+ card {B. B <= A & card(B) = k}"
+ apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
+ apply (rule_tac B = "Pow (insert x A) " in finite_subset)
+ apply (rule_tac [3] B = "Pow (A) " in finite_subset)
+ apply fast+
+ txt {* arity *}
+ apply (auto elim!: equalityE simp add: inj_on_def)
+ apply (subst Diff_insert0)
+ apply auto
+ done
+
+text {*
+ Main theorem: combinatorial statement about number of subsets of a set.
+*}
+
+lemma n_sub_lemma:
+ "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
+ apply (induct k)
+ apply (simp add: card_s_0_eq_empty)
+ apply atomize
+ apply (rotate_tac -1, erule finite_induct)
+ apply (simp_all (no_asm_simp) cong add: conj_cong add: card_s_0_eq_empty choose_deconstruct)
+ apply (subst card_Un_disjoint)
+ prefer 4 apply (force simp add: constr_bij)
+ prefer 3 apply force
+ prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
+ finite_subset [of _ "Pow (insert x F)", standard])
+ apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
+ done
+
+theorem n_subsets: "finite A ==> card {B. B <= A & card(B) = k} = (card A choose k)"
+ by (simp add: n_sub_lemma)
+
+end