--- a/src/HOL/Finite_Set.thy Wed Feb 07 17:26:49 2007 +0100
+++ b/src/HOL/Finite_Set.thy Wed Feb 07 17:28:09 2007 +0100
@@ -12,14 +12,10 @@
subsection {* Definition and basic properties *}
-consts Finites :: "'a set set"
-abbreviation
- "finite A == A : Finites"
-
-inductive Finites
- intros
- emptyI [simp, intro!]: "{} : Finites"
- insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
+inductive2 finite :: "'a set => bool"
+ where
+ emptyI [simp, intro!]: "finite {}"
+ | insertI [simp, intro!]: "finite A ==> finite (insert a A)"
axclass finite \<subseteq> type
finite: "finite UNIV"
@@ -32,7 +28,7 @@
thus ?thesis by blast
qed
-lemma finite_induct [case_names empty insert, induct set: Finites]:
+lemma finite_induct [case_names empty insert, induct set: finite]:
"finite F ==>
P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
-- {* Discharging @{text "x \<notin> F"} entails extra work. *}
@@ -146,7 +142,7 @@
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
+ by (induct set: finite) simp_all
lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
-- {* Every subset of a finite set is finite. *}
@@ -244,7 +240,7 @@
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
-- {* The image of a finite set is finite. *}
- by (induct set: Finites) simp_all
+ by (induct set: finite) simp_all
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
apply (frule finite_imageI)
@@ -286,7 +282,7 @@
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
-- {* The inverse image of a finite set under an injective function
is finite. *}
- apply (induct set: Finites)
+ apply (induct set: finite)
apply simp_all
apply (subst vimage_insert)
apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
@@ -296,7 +292,7 @@
text {* 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
+ by (induct set: finite) simp_all
text {*
Strengthen RHS to
@@ -398,7 +394,7 @@
lemma finite_Field: "finite r ==> finite (Field r)"
-- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
- apply (induct set: Finites)
+ apply (induct set: finite)
apply (auto simp add: Field_def Domain_insert Range_insert)
done
@@ -427,38 +423,39 @@
se the definitions of sums and products over finite sets.
*}
-consts
- foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
-
-inductive "foldSet f g z"
-intros
-emptyI [intro]: "({}, z) : foldSet f g z"
-insertI [intro]:
- "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
- \<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
-
-inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
+inductive2
+ foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a => bool"
+ for f :: "'a => 'a => 'a"
+ and g :: "'b => 'a"
+ and z :: 'a
+where
+ emptyI [intro]: "foldSet f g z {} z"
+| insertI [intro]:
+ "\<lbrakk> x \<notin> A; foldSet f g z A y \<rbrakk>
+ \<Longrightarrow> foldSet f g z (insert x A) (f (g x) y)"
+
+inductive_cases2 empty_foldSetE [elim!]: "foldSet f g z {} x"
constdefs
fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
- "fold f g z A == THE x. (A, x) : foldSet f g z"
+ "fold f g z A == THE x. foldSet f g z A x"
text{*A tempting alternative for the definiens is
-@{term "if finite A then THE x. (A, x) : foldSet f g e else e"}.
+@{term "if finite A then THE x. foldSet f g e A x else e"}.
It allows the removal of finiteness assumptions from the theorems
@{text fold_commute}, @{text fold_reindex} and @{text fold_distrib}.
The proofs become ugly, with @{text rule_format}. It is not worth the effort.*}
lemma Diff1_foldSet:
- "(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
+ "foldSet f g z (A - {x}) y ==> x: A ==> foldSet f g z A (f (g x) y)"
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
-lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
+lemma foldSet_imp_finite: "foldSet f g z A x==> finite A"
by (induct set: foldSet) auto
-lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
- by (induct set: Finites) auto
+lemma finite_imp_foldSet: "finite A ==> EX x. foldSet f g z A x"
+ by (induct set: finite) auto
subsubsection {* Commutative monoids *}
@@ -554,33 +551,31 @@
lemma (in ACf) foldSet_determ_aux:
"!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; inj_on h {i. i<n};
- (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
+ foldSet f g z A x; foldSet f g z A x' \<rbrakk>
\<Longrightarrow> x' = x"
proof (induct n rule: less_induct)
case (less n)
have IH: "!!m h A x x'.
\<lbrakk>m<n; A = h ` {i. i<m}; inj_on h {i. i<m};
- (A,x) \<in> foldSet f g z; (A, x') \<in> foldSet f g z\<rbrakk> \<Longrightarrow> x' = x" .
- have Afoldx: "(A,x) \<in> foldSet f g z" and Afoldx': "(A,x') \<in> foldSet f g z"
+ foldSet f g z A x; foldSet f g z A x'\<rbrakk> \<Longrightarrow> x' = x" .
+ have Afoldx: "foldSet f g z A x" and Afoldx': "foldSet f g z A x'"
and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" .
show ?case
proof (rule foldSet.cases [OF Afoldx])
- assume "(A, x) = ({}, z)"
+ assume "A = {}" and "x = z"
with Afoldx' show "x' = x" by blast
next
fix B b u
- assume "(A,x) = (insert b B, g b \<cdot> u)" and notinB: "b \<notin> B"
- and Bu: "(B,u) \<in> foldSet f g z"
- hence AbB: "A = insert b B" and x: "x = g b \<cdot> u" by auto
+ assume AbB: "A = insert b B" and x: "x = g b \<cdot> u"
+ and notinB: "b \<notin> B" and Bu: "foldSet f g z B u"
show "x'=x"
proof (rule foldSet.cases [OF Afoldx'])
- assume "(A, x') = ({}, z)"
+ assume "A = {}" and "x' = z"
with AbB show "x' = x" by blast
next
fix C c v
- assume "(A,x') = (insert c C, g c \<cdot> v)" and notinC: "c \<notin> C"
- and Cv: "(C,v) \<in> foldSet f g z"
- hence AcC: "A = insert c C" and x': "x' = g c \<cdot> v" by auto
+ assume AcC: "A = insert c C" and x': "x' = g c \<cdot> v"
+ and notinC: "c \<notin> C" and Cv: "foldSet f g z C v"
from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
from insert_inj_onE [OF Beq notinB injh]
obtain hB mB where inj_onB: "inj_on hB {i. i < mB}"
@@ -604,15 +599,15 @@
using AbB AcC notinB notinC diff by(blast elim!:equalityE)+
have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
with AbB have "finite ?D" by simp
- then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g z"
+ then obtain d where Dfoldd: "foldSet f g z ?D d"
using finite_imp_foldSet by iprover
moreover have cinB: "c \<in> B" using B by auto
- ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
+ ultimately have "foldSet f g z B (g c \<cdot> d)"
by(rule Diff1_foldSet)
hence "g c \<cdot> d = u" by (rule IH [OF lessB Beq inj_onB Bu])
moreover have "g b \<cdot> d = v"
proof (rule IH[OF lessC Ceq inj_onC Cv])
- show "(C, g b \<cdot> d) \<in> foldSet f g z" using C notinB Dfoldd
+ show "foldSet f g z C (g b \<cdot> d)" using C notinB Dfoldd
by fastsimp
qed
ultimately show ?thesis using x x' by (auto simp: AC)
@@ -623,12 +618,12 @@
lemma (in ACf) foldSet_determ:
- "(A,x) : foldSet f g z ==> (A,y) : foldSet f g z ==> y = x"
+ "foldSet f g z A x ==> foldSet f g z A y ==> y = x"
apply (frule foldSet_imp_finite [THEN finite_imp_nat_seg_image_inj_on])
apply (blast intro: foldSet_determ_aux [rule_format])
done
-lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y"
+lemma (in ACf) fold_equality: "foldSet f g z A y ==> fold f g z A = y"
by (unfold fold_def) (blast intro: foldSet_determ)
text{* The base case for @{text fold}: *}
@@ -637,8 +632,8 @@
by (unfold fold_def) blast
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
- ((insert x A, v) : foldSet f g z) =
- (EX y. (A, y) : foldSet f g z & v = f (g x) y)"
+ (foldSet f g z (insert x A) v) =
+ (EX y. foldSet f g z A y & v = f (g x) y)"
apply auto
apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
apply (fastsimp dest: foldSet_imp_finite)
@@ -700,7 +695,7 @@
lemma (in ACf) fold_commute:
"finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
- apply (induct set: Finites)
+ apply (induct set: finite)
apply simp
apply (simp add: left_commute [of x])
done
@@ -708,7 +703,7 @@
lemma (in ACf) fold_nest_Un_Int:
"finite A ==> finite B
==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
- apply (induct set: Finites)
+ apply (induct set: finite)
apply simp
apply (simp add: fold_commute Int_insert_left insert_absorb)
done
@@ -730,7 +725,7 @@
"finite A ==> finite B ==>
fold f g e A \<cdot> fold f g e B =
fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
- apply (induct set: Finites, simp)
+ apply (induct set: finite, simp)
apply (simp add: AC insert_absorb Int_insert_left)
done
@@ -744,7 +739,7 @@
ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
\<Longrightarrow> fold f g e (UNION I A) =
fold f (%i. fold f g e (A i)) e I"
- apply (induct set: Finites, simp, atomize)
+ apply (induct set: finite, simp, atomize)
apply (subgoal_tac "ALL i:F. x \<noteq> i")
prefer 2 apply blast
apply (subgoal_tac "A x Int UNION F A = {}")
@@ -762,7 +757,7 @@
"finite A ==>
(!!x y. h (g x y) = f x (h y)) ==>
h (fold g j w A) = fold f j (h w) A"
- by (induct set: Finites) simp_all
+ by (induct set: finite) simp_all
lemma (in ACf) fold_cong:
"finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
@@ -954,7 +949,7 @@
lemma setsum_eq_0_iff [simp]:
"finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
- by (induct set: Finites) auto
+ by (induct set: finite) auto
lemma setsum_Un_nat: "finite A ==> finite B ==>
(setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
@@ -1064,7 +1059,7 @@
lemma setsum_negf:
"setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
proof (cases "finite A")
- case True thus ?thesis by (induct set: Finites) auto
+ case True thus ?thesis by (induct set: finite) auto
next
case False thus ?thesis by (simp add: setsum_def)
qed
@@ -1398,18 +1393,18 @@
lemma setprod_eq_1_iff [simp]:
"finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
- by (induct set: Finites) auto
+ by (induct set: finite) auto
lemma setprod_zero:
"finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
- apply (induct set: Finites, force, clarsimp)
+ apply (induct set: finite, force, clarsimp)
apply (erule disjE, auto)
done
lemma setprod_nonneg [rule_format]:
"(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
apply (case_tac "finite A")
- apply (induct set: Finites, force, clarsimp)
+ apply (induct set: finite, force, clarsimp)
apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
apply (rule mult_mono, assumption+)
apply (auto simp add: setprod_def)
@@ -1418,7 +1413,7 @@
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
--> 0 < setprod f A"
apply (case_tac "finite A")
- apply (induct set: Finites, force, clarsimp)
+ apply (induct set: finite, force, clarsimp)
apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
apply (rule mult_strict_mono, assumption+)
apply (auto simp add: setprod_def)
@@ -1546,7 +1541,7 @@
by (simp add: card_def setsum_mono2)
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
- apply (induct set: Finites, simp, clarify)
+ apply (induct set: finite, simp, clarify)
apply (subgoal_tac "finite A & A - {x} <= F")
prefer 2 apply (blast intro: finite_subset, atomize)
apply (drule_tac x = "A - {x}" in spec)
@@ -1698,7 +1693,7 @@
done
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
- apply (induct set: Finites)
+ apply (induct set: finite)
apply simp
apply (simp add: le_SucI finite_imageI card_insert_if)
done
@@ -1763,7 +1758,7 @@
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 (induct set: finite)
apply (simp_all add: Pow_insert)
apply (subst card_Un_disjoint, blast)
apply (blast intro: finite_imageI, blast)
@@ -1783,7 +1778,7 @@
k dvd card (Union C)"
apply(frule finite_UnionD)
apply(rotate_tac -1)
- apply (induct set: Finites, simp_all, clarify)
+ apply (induct set: finite, simp_all, clarify)
apply (subst card_Un_disjoint)
apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
done
@@ -1793,36 +1788,35 @@
text{* Does not require start value. *}
-consts
- fold1Set :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
-
-inductive "fold1Set f"
-intros
+inductive2
+ fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
+ for f :: "'a => 'a => 'a"
+where
fold1Set_insertI [intro]:
- "\<lbrakk> (A,x) \<in> foldSet f id a; a \<notin> A \<rbrakk> \<Longrightarrow> (insert a A, x) \<in> fold1Set f"
+ "\<lbrakk> foldSet f id a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
constdefs
fold1 :: "('a => 'a => 'a) => 'a set => 'a"
- "fold1 f A == THE x. (A, x) : fold1Set f"
+ "fold1 f A == THE x. fold1Set f A x"
lemma fold1Set_nonempty:
- "(A, x) : fold1Set f \<Longrightarrow> A \<noteq> {}"
+ "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
by(erule fold1Set.cases, simp_all)
-inductive_cases empty_fold1SetE [elim!]: "({}, x) : fold1Set f"
-
-inductive_cases insert_fold1SetE [elim!]: "(insert a X, x) : fold1Set f"
-
-
-lemma fold1Set_sing [iff]: "(({a},b) : fold1Set f) = (a = b)"
+inductive_cases2 empty_fold1SetE [elim!]: "fold1Set f {} x"
+
+inductive_cases2 insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
+
+
+lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
by (blast intro: foldSet.intros elim: foldSet.cases)
lemma fold1_singleton[simp]: "fold1 f {a} = a"
by (unfold fold1_def) blast
lemma finite_nonempty_imp_fold1Set:
- "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : fold1Set f"
+ "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
apply (induct A rule: finite_induct)
apply (auto dest: finite_imp_foldSet [of _ f id])
done
@@ -1830,26 +1824,26 @@
text{*First, some lemmas about @{term foldSet}.*}
lemma (in ACf) foldSet_insert_swap:
-assumes fold: "(A,y) \<in> foldSet f id b"
-shows "b \<notin> A \<Longrightarrow> (insert b A, z \<cdot> y) \<in> foldSet f id z"
+assumes fold: "foldSet f id b A y"
+shows "b \<notin> A \<Longrightarrow> foldSet f id z (insert b A) (z \<cdot> y)"
using fold
proof (induct rule: foldSet.induct)
case emptyI thus ?case by (force simp add: fold_insert_aux commute)
next
- case (insertI A x y)
- have "(insert x (insert b A), x \<cdot> (z \<cdot> y)) \<in> foldSet f (\<lambda>u. u) z"
+ case (insertI x A y)
+ have "foldSet f (\<lambda>u. u) z (insert x (insert b A)) (x \<cdot> (z \<cdot> y))"
using insertI by force --{*how does @{term id} get unfolded?*}
thus ?case by (simp add: insert_commute AC)
qed
lemma (in ACf) foldSet_permute_diff:
-assumes fold: "(A,x) \<in> foldSet f id b"
-shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> (insert b (A-{a}), x) \<in> foldSet f id a"
+assumes fold: "foldSet f id b A x"
+shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> foldSet f id a (insert b (A-{a})) x"
using fold
proof (induct rule: foldSet.induct)
case emptyI thus ?case by simp
next
- case (insertI A x y)
+ case (insertI x A y)
have "a = x \<or> a \<in> A" using insertI by simp
thus ?case
proof
@@ -1858,7 +1852,7 @@
by (simp add: id_def [symmetric], blast intro: foldSet_insert_swap)
next
assume ainA: "a \<in> A"
- hence "(insert x (insert b (A - {a})), x \<cdot> y) \<in> foldSet f id a"
+ hence "foldSet f id a (insert x (insert b (A - {a}))) (x \<cdot> y)"
using insertI by (force simp: id_def)
moreover
have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
@@ -1875,7 +1869,7 @@
apply (rule sym, clarify)
apply (case_tac "Aa=A")
apply (best intro: the_equality foldSet_determ)
-apply (subgoal_tac "(A,x) \<in> foldSet f id a")
+apply (subgoal_tac "foldSet f id a A x")
apply (best intro: the_equality foldSet_determ)
apply (subgoal_tac "insert aa (Aa - {a}) = A")
prefer 2 apply (blast elim: equalityE)
@@ -1943,18 +1937,18 @@
text{*Not actually used!!*}
lemma (in ACf) foldSet_permute:
- "[|(insert a A, x) \<in> foldSet f id b; a \<notin> A; b \<notin> A|]
- ==> (insert b A, x) \<in> foldSet f id a"
+ "[|foldSet f id b (insert a A) x; a \<notin> A; b \<notin> A|]
+ ==> foldSet f id a (insert b A) x"
apply (case_tac "a=b")
apply (auto dest: foldSet_permute_diff)
done
lemma (in ACf) fold1Set_determ:
- "(A, x) \<in> fold1Set f ==> (A, y) \<in> fold1Set f ==> y = x"
+ "fold1Set f A x ==> fold1Set f A y ==> y = x"
proof (clarify elim!: fold1Set.cases)
fix A x B y a b
- assume Ax: "(A, x) \<in> foldSet f id a"
- assume By: "(B, y) \<in> foldSet f id b"
+ assume Ax: "foldSet f id a A x"
+ assume By: "foldSet f id b B y"
assume anotA: "a \<notin> A"
assume bnotB: "b \<notin> B"
assume eq: "insert a A = insert b B"
@@ -1970,16 +1964,16 @@
and aB: "a \<in> B" and bA: "b \<in> A"
using eq anotA bnotB diff by (blast elim!:equalityE)+
with aB bnotB By
- have "(insert b ?D, y) \<in> foldSet f id a"
+ have "foldSet f id a (insert b ?D) y"
by (auto intro: foldSet_permute simp add: insert_absorb)
moreover
- have "(insert b ?D, x) \<in> foldSet f id a"
+ have "foldSet f id a (insert b ?D) x"
by (simp add: A [symmetric] Ax)
ultimately show ?thesis by (blast intro: foldSet_determ)
qed
qed
-lemma (in ACf) fold1Set_equality: "(A, y) : fold1Set f ==> fold1 f A = y"
+lemma (in ACf) fold1Set_equality: "fold1Set f A y ==> fold1 f A = y"
by (unfold fold1_def) (blast intro: fold1Set_determ)
declare