--- a/CONTRIBUTORS Sat Mar 23 17:11:06 2013 +0100
+++ b/CONTRIBUTORS Sat Mar 23 20:50:39 2013 +0100
@@ -7,6 +7,9 @@
--------------------------------------
* March 2013: Florian Haftmann, TUM
+ Reform of "big operators" on sets.
+
+* March 2013: Florian Haftmann, TUM
Algebraic locale hierarchy for orderings and (semi)lattices.
* Feb. 2013: Florian Haftmann, TUM
--- a/NEWS Sat Mar 23 17:11:06 2013 +0100
+++ b/NEWS Sat Mar 23 20:50:39 2013 +0100
@@ -33,6 +33,28 @@
*** HOL ***
+* Revised devices for recursive definitions over finite sets:
+ - Only one fundamental fold combinator on finite set remains:
+ Finite_Set.fold :: ('a => 'b => 'b) => 'b => 'a set => 'b
+ This is now identity on infinite sets.
+ - Locales (»mini packages«) for fundamental definitions with
+ Finite_Set.fold: folding, folding_idem.
+ - Locales comm_monoid_set, semilattice_order_set and
+ semilattice_neutr_order_set for big operators on sets.
+ See theory Big_Operators for canonical examples.
+ Note that foundational constants comm_monoid_set.F and
+ semilattice_set.F correspond to former combinators fold_image
+ and fold1 respectively. These are now gone. You may use
+ those foundational counstants as substitutes, but it is
+ preferable to interpret the above locales accordingly.
+ - Dropped class ab_semigroup_idem_mult (special case of lattice,
+ no longer needed in connection with Finite_Set.fold etc.)
+ - Fact renames:
+ card.union_inter ~> card_Un_Int [symmetric]
+ card.union_disjoint ~> card_Un_disjoint
+
+INCOMPATIBILITY.
+
* Locale hierarchy for abstract orderings and (semi)lattices.
* Discontinued theory src/HOL/Library/Eval_Witness.
--- a/src/Doc/Main/Main_Doc.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/Doc/Main/Main_Doc.thy Sat Mar 23 20:50:39 2013 +0100
@@ -406,7 +406,6 @@
@{const Finite_Set.finite} & @{term_type_only Finite_Set.finite "'a set\<Rightarrow>bool"}\\
@{const Finite_Set.card} & @{term_type_only Finite_Set.card "'a set => nat"}\\
@{const Finite_Set.fold} & @{term_type_only Finite_Set.fold "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"}\\
-@{const Finite_Set.fold_image} & @{typ "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"}\\
@{const Big_Operators.setsum} & @{term_type_only Big_Operators.setsum "('a => 'b) => 'a set => 'b::comm_monoid_add"}\\
@{const Big_Operators.setprod} & @{term_type_only Big_Operators.setprod "('a => 'b) => 'a set => 'b::comm_monoid_mult"}\\
\end{supertabular}
--- a/src/HOL/Algebra/poly/UnivPoly2.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Algebra/poly/UnivPoly2.thy Sat Mar 23 20:50:39 2013 +0100
@@ -687,7 +687,7 @@
proof (cases "finite A")
case True then show ?thesis by induct auto
next
- case False then show ?thesis by (simp add: setsum_def)
+ case False then show ?thesis by simp
qed
(* Instance of a more general result!!! *)
--- a/src/HOL/BNF/More_BNFs.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/BNF/More_BNFs.thy Sat Mar 23 20:50:39 2013 +0100
@@ -548,16 +548,16 @@
unfolding mcard_def by auto
lemma mcard_Plus[simp]: "mcard (M + N) = mcard M + mcard N"
-proof-
+proof -
have "setsum (count M) {a. 0 < count M a + count N a} =
setsum (count M) {a. a \<in># M}"
- apply(rule setsum_mono_zero_cong_right) by auto
+ apply (rule setsum_mono_zero_cong_right) by auto
moreover
have "setsum (count N) {a. 0 < count M a + count N a} =
setsum (count N) {a. a \<in># N}"
- apply(rule setsum_mono_zero_cong_right) by auto
+ apply (rule setsum_mono_zero_cong_right) by auto
ultimately show ?thesis
- unfolding mcard_def count_union[THEN ext] comm_monoid_add_class.setsum.F_fun_f by simp
+ unfolding mcard_def count_union [THEN ext] by (simp add: setsum.distrib)
qed
lemma setsum_gt_0_iff:
@@ -1207,7 +1207,7 @@
have "setsum L {aa. f aa = a \<and> 0 < L aa} = setsum L {aa. f aa = a \<and> 0 < K aa + L aa}"
apply(rule setsum_mono_zero_cong_left) using C by auto
ultimately show ?thesis
- unfolding mmap_def unfolding comm_monoid_add_class.setsum.F_fun_f by auto
+ unfolding mmap_def by (auto simp add: setsum.distrib)
qed
lemma multiset_map_Plus[simp]:
@@ -1265,10 +1265,10 @@
have "setsum (\<lambda> x. setsum (count M) (A x)) ?B = setsum (setsum (count M) o A) ?B"
unfolding comp_def ..
also have "... = (\<Sum>x\<in> A ` ?B. setsum (count M) x)"
- unfolding comm_monoid_add_class.setsum_reindex[OF i, symmetric] ..
+ unfolding setsum.reindex [OF i, symmetric] ..
also have "... = setsum (count M) (\<Union>x\<in>A ` {b. 0 < setsum (count M) (A b)}. x)"
(is "_ = setsum (count M) ?J")
- apply(rule comm_monoid_add_class.setsum_UN_disjoint[symmetric])
+ apply(rule setsum.UNION_disjoint[symmetric])
using 0 fin unfolding A_def by (auto intro!: finite_imageI)
also have "?J = {a. a \<in># M}" unfolding AB unfolding A_def by auto
finally have "setsum (\<lambda> x. setsum (count M) (A x)) ?B =
--- a/src/HOL/Big_Operators.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Big_Operators.thy Sat Mar 23 20:50:39 2013 +0100
@@ -6,7 +6,7 @@
header {* Big operators and finite (non-empty) sets *}
theory Big_Operators
-imports Finite_Set Metis
+imports Finite_Set Option Metis
begin
subsection {* Generic monoid operation over a set *}
@@ -14,46 +14,223 @@
no_notation times (infixl "*" 70)
no_notation Groups.one ("1")
-locale comm_monoid_big = comm_monoid +
- fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
- assumes F_eq: "F g A = (if finite A then fold_image (op *) g 1 A else 1)"
+locale comm_monoid_set = comm_monoid
+begin
-sublocale comm_monoid_big < folding_image proof
-qed (simp add: F_eq)
-
-context comm_monoid_big
-begin
+definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
+where
+ eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
lemma infinite [simp]:
"\<not> finite A \<Longrightarrow> F g A = 1"
- by (simp add: F_eq)
+ by (simp add: eq_fold)
+
+lemma empty [simp]:
+ "F g {} = 1"
+ by (simp add: eq_fold)
+
+lemma insert [simp]:
+ assumes "finite A" and "x \<notin> A"
+ shows "F g (insert x A) = g x * F g A"
+proof -
+ interpret comp_fun_commute f
+ by default (simp add: fun_eq_iff left_commute)
+ interpret comp_fun_commute "f \<circ> g"
+ by (rule comp_comp_fun_commute)
+ from assms show ?thesis by (simp add: eq_fold)
+qed
+
+lemma remove:
+ assumes "finite A" and "x \<in> A"
+ shows "F g A = g x * F g (A - {x})"
+proof -
+ from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
+ by (auto dest: mk_disjoint_insert)
+ moreover from `finite A` this have "finite B" by simp
+ ultimately show ?thesis by simp
+qed
+
+lemma insert_remove:
+ assumes "finite A"
+ shows "F g (insert x A) = g x * F g (A - {x})"
+ using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
+
+lemma neutral:
+ assumes "\<forall>x\<in>A. g x = 1"
+ shows "F g A = 1"
+proof (cases "finite A")
+ case True from `finite A` assms show ?thesis by (induct A) simp_all
+next
+ case False then show ?thesis by simp
+qed
-lemma F_cong:
- assumes "A = B" "\<And>x. x \<in> B \<Longrightarrow> h x = g x"
- shows "F h A = F g B"
-proof cases
- assume "finite A"
- with assms show ?thesis unfolding `A = B` by (simp cong: cong)
+lemma neutral_const [simp]:
+ "F (\<lambda>_. 1) A = 1"
+ by (simp add: neutral)
+
+lemma union_inter:
+ assumes "finite A" and "finite B"
+ shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
+ -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
+using assms proof (induct A)
+ case empty then show ?case by simp
next
- assume "\<not> finite A"
- then show ?thesis unfolding `A = B` by simp
+ case (insert x A) then show ?case
+ by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
+qed
+
+corollary union_inter_neutral:
+ assumes "finite A" and "finite B"
+ and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
+ shows "F g (A \<union> B) = F g A * F g B"
+ using assms by (simp add: union_inter [symmetric] neutral)
+
+corollary union_disjoint:
+ assumes "finite A" and "finite B"
+ assumes "A \<inter> B = {}"
+ shows "F g (A \<union> B) = F g A * F g B"
+ using assms by (simp add: union_inter_neutral)
+
+lemma subset_diff:
+ "B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> F g A = F g (A - B) * F g B"
+ by (metis Diff_partition union_disjoint Diff_disjoint finite_Un inf_commute sup_commute)
+
+lemma reindex:
+ assumes "inj_on h A"
+ shows "F g (h ` A) = F (g \<circ> h) A"
+proof (cases "finite A")
+ case True
+ interpret comp_fun_commute f
+ by default (simp add: fun_eq_iff left_commute)
+ interpret comp_fun_commute "f \<circ> g"
+ by (rule comp_comp_fun_commute)
+ from assms `finite A` show ?thesis by (simp add: eq_fold fold_image comp_assoc)
+next
+ case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
+ with False show ?thesis by simp
qed
-lemma strong_F_cong [cong]:
- "\<lbrakk> A = B; !!x. x:B =simp=> g x = h x \<rbrakk>
- \<Longrightarrow> F (%x. g x) A = F (%x. h x) B"
-by (rule F_cong) (simp_all add: simp_implies_def)
+lemma cong:
+ assumes "A = B"
+ assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
+ shows "F g A = F h B"
+proof (cases "finite A")
+ case True
+ then have "\<And>C. C \<subseteq> A \<longrightarrow> (\<forall>x\<in>C. g x = h x) \<longrightarrow> F g C = F h C"
+ proof induct
+ case empty then show ?case by simp
+ next
+ case (insert x F) then show ?case apply -
+ apply (simp add: subset_insert_iff, clarify)
+ apply (subgoal_tac "finite C")
+ prefer 2 apply (blast dest: finite_subset [rotated])
+ apply (subgoal_tac "C = insert x (C - {x})")
+ prefer 2 apply blast
+ apply (erule ssubst)
+ apply (simp add: Ball_def del: insert_Diff_single)
+ done
+ qed
+ with `A = B` g_h show ?thesis by simp
+next
+ case False
+ with `A = B` show ?thesis by simp
+qed
-lemma F_neutral[simp]: "F (%i. 1) A = 1"
-by (cases "finite A") (simp_all add: neutral)
+lemma strong_cong [cong]:
+ assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
+ shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
+ by (rule cong) (insert assms, simp_all add: simp_implies_def)
+
+lemma UNION_disjoint:
+ assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
+ and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
+ shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
+apply (insert assms)
+apply (induct rule: finite_induct)
+apply simp
+apply atomize
+apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
+ prefer 2 apply blast
+apply (subgoal_tac "A x Int UNION Fa A = {}")
+ prefer 2 apply blast
+apply (simp add: union_disjoint)
+done
+
+lemma Union_disjoint:
+ assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
+ shows "F g (Union C) = F (F g) C"
+proof cases
+ assume "finite C"
+ from UNION_disjoint [OF this assms]
+ show ?thesis
+ by (simp add: SUP_def)
+qed (auto dest: finite_UnionD intro: infinite)
-lemma F_neutral': "ALL a:A. g a = 1 \<Longrightarrow> F g A = 1"
-by simp
+lemma distrib:
+ "F (\<lambda>x. g x * h x) A = F g A * F h A"
+proof (cases "finite A")
+ case False then show ?thesis by simp
+next
+ case True then show ?thesis by (rule finite_induct) (simp_all add: assoc commute left_commute)
+qed
+
+lemma Sigma:
+ "finite A \<Longrightarrow> \<forall>x\<in>A. finite (B x) \<Longrightarrow> F (\<lambda>x. F (g x) (B x)) A = F (split g) (SIGMA x:A. B x)"
+apply (subst Sigma_def)
+apply (subst UNION_disjoint, assumption, simp)
+ apply blast
+apply (rule cong)
+apply rule
+apply (simp add: fun_eq_iff)
+apply (subst UNION_disjoint, simp, simp)
+ apply blast
+apply (simp add: comp_def)
+done
+
+lemma related:
+ assumes Re: "R 1 1"
+ and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
+ and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
+ shows "R (F h S) (F g S)"
+ using fS by (rule finite_subset_induct) (insert assms, auto)
-lemma F_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow> F g A = F g (A - B) * F g B"
-by (metis Diff_partition union_disjoint Diff_disjoint finite_Un inf_commute sup_commute)
+lemma eq_general:
+ assumes h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
+ and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
+ shows "F f1 S = F f2 S'"
+proof-
+ from h f12 have hS: "h ` S = S'" by blast
+ {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
+ from f12 h H have "x = y" by auto }
+ hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
+ from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
+ from hS have "F f2 S' = F f2 (h ` S)" by simp
+ also have "\<dots> = F (f2 o h) S" using reindex [OF hinj, of f2] .
+ also have "\<dots> = F f1 S " using th cong [of _ _ "f2 o h" f1]
+ by blast
+ finally show ?thesis ..
+qed
-lemma F_mono_neutral_cong_left:
+lemma eq_general_reverses:
+ assumes kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
+ and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
+ shows "F j S = F g T"
+ (* metis solves it, but not yet available here *)
+ apply (rule eq_general [of T S h g j])
+ apply (rule ballI)
+ apply (frule kh)
+ apply (rule ex1I[])
+ apply blast
+ apply clarsimp
+ apply (drule hk) apply simp
+ apply (rule sym)
+ apply (erule conjunct1[OF conjunct2[OF hk]])
+ apply (rule ballI)
+ apply (drule hk)
+ apply blast
+ done
+
+lemma mono_neutral_cong_left:
assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
proof-
@@ -62,25 +239,25 @@
from `finite T` `S \<subseteq> T` have f: "finite S" "finite (T - S)"
by (auto intro: finite_subset)
show ?thesis using assms(4)
- by (simp add: union_disjoint[OF f d, unfolded eq[symmetric]] F_neutral'[OF assms(3)])
+ by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
qed
-lemma F_mono_neutral_cong_right:
+lemma mono_neutral_cong_right:
"\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk>
\<Longrightarrow> F g T = F h S"
-by(auto intro!: F_mono_neutral_cong_left[symmetric])
+ by (auto intro!: mono_neutral_cong_left [symmetric])
-lemma F_mono_neutral_left:
+lemma mono_neutral_left:
"\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
-by(blast intro: F_mono_neutral_cong_left)
+ by (blast intro: mono_neutral_cong_left)
-lemma F_mono_neutral_right:
+lemma mono_neutral_right:
"\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
-by(blast intro!: F_mono_neutral_left[symmetric])
+ by (blast intro!: mono_neutral_left [symmetric])
-lemma F_delta:
+lemma delta:
assumes fS: "finite S"
- shows "F (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
+ shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
proof-
let ?f = "(\<lambda>k. if k=a then b k else 1)"
{ assume a: "a \<notin> S"
@@ -94,78 +271,71 @@
have dj: "?A \<inter> ?B = {}" by simp
from fS have fAB: "finite ?A" "finite ?B" by auto
have "F ?f S = F ?f ?A * F ?f ?B"
- using union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
+ using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
by simp
- then have ?thesis using a by simp }
+ then have ?thesis using a by simp }
ultimately show ?thesis by blast
qed
-lemma F_delta':
- assumes fS: "finite S" shows
- "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
-using F_delta[OF fS, of a b, symmetric] by (auto intro: F_cong)
-
-lemma F_fun_f: "F (%x. g x * h x) A = (F g A * F h A)"
-by (cases "finite A") (simp_all add: distrib)
-
-
-text {* for ad-hoc proofs for @{const fold_image} *}
-lemma comm_monoid_mult: "class.comm_monoid_mult (op *) 1"
-proof qed (auto intro: assoc commute)
-
-lemma F_Un_neutral:
- assumes fS: "finite S" and fT: "finite T"
- and I1: "\<forall>x \<in> S\<inter>T. g x = 1"
- shows "F g (S \<union> T) = F g S * F g T"
-proof -
- interpret comm_monoid_mult "op *" 1 by (fact comm_monoid_mult)
- show ?thesis
- using fS fT
- apply (simp add: F_eq)
- apply (rule fold_image_Un_one)
- using I1 by auto
-qed
+lemma delta':
+ assumes fS: "finite S"
+ shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
+ using delta [OF fS, of a b, symmetric] by (auto intro: cong)
lemma If_cases:
fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
assumes fA: "finite A"
shows "F (\<lambda>x. if P x then h x else g x) A =
- F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
-proof-
+ F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
+proof -
have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
"(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
by blast+
from fA
have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
let ?g = "\<lambda>x. if P x then h x else g x"
- from union_disjoint[OF f a(2), of ?g] a(1)
+ from union_disjoint [OF f a(2), of ?g] a(1)
show ?thesis
- by (subst (1 2) F_cong) simp_all
+ by (subst (1 2) cong) simp_all
qed
+lemma cartesian_product:
+ "F (\<lambda>x. F (g x) B) A = F (split g) (A <*> B)"
+apply (rule sym)
+apply (cases "finite A")
+ apply (cases "finite B")
+ apply (simp add: Sigma)
+ apply (cases "A={}", simp)
+ apply simp
+apply (auto intro: infinite dest: finite_cartesian_productD2)
+apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
+done
+
end
-text {* for ad-hoc proofs for @{const fold_image} *}
-
-lemma (in comm_monoid_add) comm_monoid_mult:
- "class.comm_monoid_mult (op +) 0"
-proof qed (auto intro: add_assoc add_commute)
-
notation times (infixl "*" 70)
notation Groups.one ("1")
subsection {* Generalized summation over a set *}
-definition (in comm_monoid_add) setsum :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
- "setsum f A = (if finite A then fold_image (op +) f 0 A else 0)"
+definition (in comm_monoid_add) setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
+where
+ "setsum = comm_monoid_set.F plus 0"
-sublocale comm_monoid_add < setsum!: comm_monoid_big "op +" 0 setsum proof
-qed (fact setsum_def)
+sublocale comm_monoid_add < setsum!: comm_monoid_set plus 0
+where
+ "setsum.F = setsum"
+proof -
+ show "comm_monoid_set plus 0" ..
+ then interpret setsum!: comm_monoid_set plus 0 .
+ show "setsum.F = setsum"
+ by (simp only: setsum_def)
+qed
abbreviation
- Setsum ("\<Sum>_" [1000] 999) where
- "\<Sum>A == setsum (%x. x) A"
+ Setsum ("\<Sum>_" [1000] 999) where
+ "\<Sum>A \<equiv> setsum (%x. x) A"
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
written @{text"\<Sum>x\<in>A. e"}. *}
@@ -211,48 +381,32 @@
in [(@{const_syntax setsum}, setsum_tr')] end
*}
-lemma setsum_empty:
- "setsum f {} = 0"
- by (fact setsum.empty)
+text {* TODO These are candidates for generalization *}
-lemma setsum_insert:
- "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
- by (fact setsum.insert)
-
-lemma setsum_infinite:
- "~ finite A ==> setsum f A = 0"
- by (fact setsum.infinite)
+context comm_monoid_add
+begin
-lemma (in comm_monoid_add) setsum_reindex:
- assumes "inj_on f B" shows "setsum h (f ` B) = setsum (h \<circ> f) B"
-proof -
- interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
- from assms show ?thesis by (auto simp add: setsum_def fold_image_reindex o_def dest!:finite_imageD)
-qed
-
-lemma setsum_reindex_id:
+lemma setsum_reindex_id:
"inj_on f B ==> setsum f B = setsum id (f ` B)"
-by (simp add: setsum_reindex)
+ by (simp add: setsum.reindex)
-lemma setsum_reindex_nonzero:
+lemma setsum_reindex_nonzero:
assumes fS: "finite S"
- and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
- shows "setsum h (f ` S) = setsum (h o f) S"
-using nz
-proof(induct rule: finite_induct[OF fS])
+ and nz: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
+ shows "setsum h (f ` S) = setsum (h \<circ> f) S"
+using nz proof (induct rule: finite_induct [OF fS])
case 1 thus ?case by simp
next
case (2 x F)
{ assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
then obtain y where y: "y \<in> F" "f x = f y" by auto
from "2.hyps" y have xy: "x \<noteq> y" by auto
-
- from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
+ from "2.prems" [of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
also have "\<dots> = setsum (h o f) (insert x F)"
unfolding setsum.insert[OF `finite F` `x\<notin>F`]
using h0
- apply (simp cong del:setsum.strong_F_cong)
+ apply (simp cong del: setsum.strong_cong)
apply (rule "2.hyps"(3))
apply (rule_tac y="y" in "2.prems")
apply simp_all
@@ -264,7 +418,7 @@
using fxF "2.hyps" by simp
also have "\<dots> = setsum (h o f) (insert x F)"
unfolding setsum.insert[OF `finite F` `x\<notin>F`]
- apply (simp cong del:setsum.strong_F_cong)
+ apply (simp cong del: setsum.strong_cong)
apply (rule cong [OF refl [of "op + (h (f x))"]])
apply (rule "2.hyps"(3))
apply (rule_tac y="y" in "2.prems")
@@ -274,59 +428,14 @@
ultimately show ?case by blast
qed
-lemma setsum_cong:
- "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
-by (fact setsum.F_cong)
-
-lemma strong_setsum_cong:
- "A = B ==> (!!x. x:B =simp=> f x = g x)
- ==> setsum (%x. f x) A = setsum (%x. g x) B"
-by (fact setsum.strong_F_cong)
-
-lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"
-by (auto intro: setsum_cong)
+lemma setsum_cong2:
+ "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
+ by (auto intro: setsum.cong)
lemma setsum_reindex_cong:
"[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
==> setsum h B = setsum g A"
-by (simp add: setsum_reindex)
-
-lemmas setsum_0 = setsum.F_neutral
-lemmas setsum_0' = setsum.F_neutral'
-
-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! *}
-by (fact setsum.union_inter)
-
-lemma setsum_Un_disjoint: "finite A ==> finite B
- ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
-by (fact setsum.union_disjoint)
-
-lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
- setsum f A = setsum f (A - B) + setsum f B"
-by(fact setsum.F_subset_diff)
-
-lemma setsum_mono_zero_left:
- "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
-by(fact setsum.F_mono_neutral_left)
-
-lemmas setsum_mono_zero_right = setsum.F_mono_neutral_right
-
-lemma setsum_mono_zero_cong_left:
- "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 0; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
- \<Longrightarrow> setsum f S = setsum g T"
-by(fact setsum.F_mono_neutral_cong_left)
-
-lemmas setsum_mono_zero_cong_right = setsum.F_mono_neutral_cong_right
-
-lemma setsum_delta: "finite S \<Longrightarrow>
- setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
-by(fact setsum.F_delta)
-
-lemma setsum_delta': "finite S \<Longrightarrow>
- setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
-by(fact setsum.F_delta')
+ by (simp add: setsum.reindex)
lemma setsum_restrict_set:
assumes fA: "finite A"
@@ -335,70 +444,20 @@
from fA have fab: "finite (A \<inter> B)" by auto
have aba: "A \<inter> B \<subseteq> A" by blast
let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
- from setsum_mono_zero_left[OF fA aba, of ?g]
+ from setsum.mono_neutral_left [OF fA aba, of ?g]
show ?thesis by simp
qed
-lemma setsum_cases:
- assumes fA: "finite A"
- shows "setsum (\<lambda>x. if P x then f x else g x) A =
- setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
- using setsum.If_cases[OF fA] .
-
-(*But we can't get rid of finite I. If infinite, although the rhs is 0,
- the lhs need not be, since UNION I A could still be finite.*)
-lemma (in comm_monoid_add) setsum_UN_disjoint:
- assumes "finite I" and "ALL i:I. finite (A i)"
- and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
- shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
-proof -
- interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
- from assms show ?thesis by (simp add: setsum_def fold_image_UN_disjoint)
-qed
-
-text{*No need to assume that @{term C} is finite. If infinite, the rhs is
-directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
lemma setsum_Union_disjoint:
assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
shows "setsum f (Union C) = setsum (setsum f) C"
-proof cases
- assume "finite C"
- from setsum_UN_disjoint[OF this assms]
- show ?thesis
- by (simp add: SUP_def)
-qed (force dest: finite_UnionD simp add: setsum_def)
-
-(*But we can't get rid of finite A. If infinite, although the lhs is 0,
- the rhs need not be, since SIGMA A B could still be finite.*)
-lemma (in comm_monoid_add) setsum_Sigma:
- assumes "finite A" and "ALL x:A. finite (B x)"
- shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
-proof -
- interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
- from assms show ?thesis by (simp add: setsum_def fold_image_Sigma split_def)
-qed
+ using assms by (fact setsum.Union_disjoint)
-text{*Here we can eliminate the finiteness assumptions, by cases.*}
-lemma setsum_cartesian_product:
- "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
-apply (cases "finite A")
- apply (cases "finite B")
- apply (simp add: setsum_Sigma)
- apply (cases "A={}", simp)
- apply (simp)
-apply (auto simp add: setsum_def
- dest: finite_cartesian_productD1 finite_cartesian_productD2)
-done
+lemma setsum_cartesian_product:
+ "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"
+ by (fact setsum.cartesian_product)
-lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
-by (fact setsum.F_fun_f)
-
-lemma setsum_Un_zero:
- "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
- setsum f (S \<union> T) = setsum f S + setsum f T"
-by(fact setsum.F_Un_neutral)
-
-lemma setsum_UNION_zero:
+lemma setsum_UNION_zero:
assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
@@ -412,36 +471,145 @@
from fTF have fUF: "finite (\<Union>F)" by auto
from "2.prems" TF fTF
show ?case
- by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f])
+ by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])
+qed
+
+text {* Commuting outer and inner summation *}
+
+lemma setsum_commute:
+ "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
+proof (simp add: setsum_cartesian_product)
+ have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
+ (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
+ (is "?s = _")
+ apply (simp add: setsum.reindex [where h = "%(i, j). (j, i)"] swap_inj_on)
+ apply (simp add: split_def)
+ done
+ also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
+ (is "_ = ?t")
+ apply (simp add: swap_product)
+ done
+ finally show "?s = ?t" .
+qed
+
+lemma setsum_Plus:
+ fixes A :: "'a set" and B :: "'b set"
+ assumes fin: "finite A" "finite B"
+ shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
+proof -
+ have "A <+> B = Inl ` A \<union> Inr ` B" by auto
+ moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
+ by auto
+ moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
+ moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
+ ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)
qed
+end
+
+text {* TODO These are legacy *}
+
+lemma setsum_empty:
+ "setsum f {} = 0"
+ by (fact setsum.empty)
+
+lemma setsum_insert:
+ "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
+ by (fact setsum.insert)
+
+lemma setsum_infinite:
+ "~ finite A ==> setsum f A = 0"
+ by (fact setsum.infinite)
+
+lemma setsum_reindex:
+ "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"
+ by (fact setsum.reindex)
+
+lemma setsum_cong:
+ "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
+ by (fact setsum.cong)
+
+lemma strong_setsum_cong:
+ "A = B ==> (!!x. x:B =simp=> f x = g x)
+ ==> setsum (%x. f x) A = setsum (%x. g x) B"
+ by (fact setsum.strong_cong)
+
+lemmas setsum_0 = setsum.neutral_const
+lemmas setsum_0' = setsum.neutral
+
+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! *}
+ by (fact setsum.union_inter)
+
+lemma setsum_Un_disjoint: "finite A ==> finite B
+ ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
+ by (fact setsum.union_disjoint)
+
+lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
+ setsum f A = setsum f (A - B) + setsum f B"
+ by (fact setsum.subset_diff)
+
+lemma setsum_mono_zero_left:
+ "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"
+ by (fact setsum.mono_neutral_left)
+
+lemmas setsum_mono_zero_right = setsum.mono_neutral_right
+
+lemma setsum_mono_zero_cong_left:
+ "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 0; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
+ \<Longrightarrow> setsum f S = setsum g T"
+ by (fact setsum.mono_neutral_cong_left)
+
+lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right
+
+lemma setsum_delta: "finite S \<Longrightarrow>
+ setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
+ by (fact setsum.delta)
+
+lemma setsum_delta': "finite S \<Longrightarrow>
+ setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"
+ by (fact setsum.delta')
+
+lemma setsum_cases:
+ assumes "finite A"
+ shows "setsum (\<lambda>x. if P x then f x else g x) A =
+ setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"
+ using assms by (fact setsum.If_cases)
+
+(*But we can't get rid of finite I. If infinite, although the rhs is 0,
+ the lhs need not be, since UNION I A could still be finite.*)
+lemma setsum_UN_disjoint:
+ assumes "finite I" and "ALL i:I. finite (A i)"
+ and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"
+ shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
+ using assms by (fact setsum.UNION_disjoint)
+
+(*But we can't get rid of finite A. If infinite, although the lhs is 0,
+ the rhs need not be, since SIGMA A B could still be finite.*)
+lemma setsum_Sigma:
+ assumes "finite A" and "ALL x:A. finite (B x)"
+ shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
+ using assms by (fact setsum.Sigma)
+
+lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
+ by (fact setsum.distrib)
+
+lemma setsum_Un_zero:
+ "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>
+ setsum f (S \<union> T) = setsum f S + setsum f T"
+ by (fact setsum.union_inter_neutral)
+
+lemma setsum_eq_general_reverses:
+ assumes fS: "finite S" and fT: "finite T"
+ and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
+ and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
+ shows "setsum f S = setsum g T"
+ using kh hk by (fact setsum.eq_general_reverses)
+
subsubsection {* Properties in more restricted classes of structures *}
-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, auto)
-done
-
-lemma setsum_eq_0_iff [simp]:
- "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
-by (induct set: finite) auto
-
-lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
- (setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
-apply(erule finite_induct)
-apply (auto simp add:add_is_1)
-done
-
-lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
-
-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)"
- -- {* For the natural numbers, we have subtraction. *}
-by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
-
lemma setsum_Un: "finite A ==> finite B ==>
(setsum f (A Un B) :: 'a :: ab_group_add) =
setsum f A + setsum f B - setsum f (A Int B)"
@@ -456,74 +624,11 @@
with assms show ?thesis by simp (subst setsum_Un_disjoint, auto)+
qed
-lemma (in comm_monoid_add) setsum_eq_general_reverses:
- assumes fS: "finite S" and fT: "finite T"
- and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
- and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
- shows "setsum f S = setsum g T"
-proof -
- interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
- show ?thesis
- apply (simp add: setsum_def fS fT)
- apply (rule fold_image_eq_general_inverses)
- apply (rule fS)
- apply (erule kh)
- apply (erule hk)
- done
-qed
-
-lemma setsum_diff1_nat: "(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, auto)
-done
-
lemma setsum_diff1: "finite A \<Longrightarrow>
(setsum f (A - {a}) :: ('a::ab_group_add)) =
(if a:A then setsum f A - f a else setsum f A)"
by (erule finite_induct) (auto simp add: insert_Diff_if)
-lemma setsum_diff1'[rule_format]:
- "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
-apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
-apply (auto simp add: insert_Diff_if add_ac)
-done
-
-lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
- shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
-unfolding setsum_diff1'[OF assms] by auto
-
-(* By Jeremy Siek: *)
-
-lemma setsum_diff_nat:
-assumes "finite B" and "B \<subseteq> A"
-shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
-using assms
-proof induct
- show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
-next
- fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
- and xFinA: "insert x F \<subseteq> A"
- and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
- from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
- from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
- by (simp add: setsum_diff1_nat)
- from xFinA have "F \<subseteq> A" by simp
- with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
- with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
- by simp
- from xnotinF have "A - insert x F = (A - F) - {x}" by auto
- with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
- by simp
- from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
- with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
- by simp
- thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
-qed
-
lemma setsum_diff:
assumes le: "finite A" "B \<subseteq> A"
shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
@@ -554,9 +659,7 @@
thus ?case using add_mono by fastforce
qed
next
- case False
- thus ?thesis
- by (simp add: setsum_def)
+ case False then show ?thesis by simp
qed
lemma setsum_strict_mono:
@@ -595,7 +698,7 @@
proof (cases "finite A")
case True thus ?thesis by (induct set: finite) auto
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
lemma setsum_subtractf:
@@ -604,7 +707,7 @@
proof (cases "finite A")
case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
lemma setsum_nonneg:
@@ -620,7 +723,7 @@
with insert show ?case by simp
qed
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
lemma setsum_nonpos:
@@ -636,7 +739,7 @@
with insert show ?case by simp
qed
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
lemma setsum_nonneg_leq_bound:
@@ -702,7 +805,7 @@
case (insert x A) thus ?case by (simp add: distrib_left)
qed
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
lemma setsum_left_distrib:
@@ -716,7 +819,7 @@
case (insert x A) thus ?case by (simp add: distrib_right)
qed
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
lemma setsum_divide_distrib:
@@ -730,7 +833,7 @@
case (insert x A) thus ?case by (simp add: add_divide_distrib)
qed
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
lemma setsum_abs[iff]:
@@ -746,7 +849,7 @@
thus ?case by (auto intro: abs_triangle_ineq order_trans)
qed
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
lemma setsum_abs_ge_zero[iff]:
@@ -761,7 +864,7 @@
case (insert x A) thus ?case by auto
qed
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
lemma abs_setsum_abs[simp]:
@@ -782,40 +885,18 @@
finally show ?case .
qed
next
- case False thus ?thesis by (simp add: setsum_def)
-qed
-
-lemma setsum_Plus:
- fixes A :: "'a set" and B :: "'b set"
- assumes fin: "finite A" "finite B"
- shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
-proof -
- have "A <+> B = Inl ` A \<union> Inr ` B" by auto
- moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
- by auto
- moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
- moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
- ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)
+ case False thus ?thesis by simp
qed
-
-text {* Commuting outer and inner summation *}
+lemma setsum_diff1'[rule_format]:
+ "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
+apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
+apply (auto simp add: insert_Diff_if add_ac)
+done
-lemma setsum_commute:
- "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
-proof (simp add: setsum_cartesian_product)
- have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
- (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
- (is "?s = _")
- apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
- apply (simp add: split_def)
- done
- also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
- (is "_ = ?t")
- apply (simp add: swap_product)
- done
- finally show "?s = ?t" .
-qed
+lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
+ shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
+unfolding setsum_diff1'[OF assms] by auto
lemma setsum_product:
fixes f :: "'a => ('b::semiring_0)"
@@ -829,7 +910,82 @@
by(auto simp: setsum_product setsum_cartesian_product
intro!: setsum_reindex_cong[symmetric])
-lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
+lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
+apply (case_tac "finite A")
+ prefer 2 apply simp
+apply (erule rev_mp)
+apply (erule finite_induct, auto)
+done
+
+lemma setsum_eq_0_iff [simp]:
+ "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
+ by (induct set: finite) auto
+
+lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
+ setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
+apply(erule finite_induct)
+apply (auto simp add:add_is_1)
+done
+
+lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
+
+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)"
+ -- {* For the natural numbers, we have subtraction. *}
+by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
+
+lemma setsum_diff1_nat: "(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
+apply (erule finite_induct)
+ apply (auto simp add: insert_Diff_if)
+apply (drule_tac a = a in mk_disjoint_insert, auto)
+done
+
+lemma setsum_diff_nat:
+assumes "finite B" and "B \<subseteq> A"
+shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
+using assms
+proof induct
+ show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
+next
+ fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
+ and xFinA: "insert x F \<subseteq> A"
+ and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
+ from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
+ from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
+ by (simp add: setsum_diff1_nat)
+ from xFinA have "F \<subseteq> A" by simp
+ with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
+ with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
+ by simp
+ from xnotinF have "A - insert x F = (A - F) - {x}" by auto
+ with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
+ by simp
+ from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
+ with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
+ by simp
+ thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
+qed
+
+
+subsubsection {* Cardinality as special case of @{const setsum} *}
+
+lemma card_eq_setsum:
+ "card A = setsum (\<lambda>x. 1) A"
+proof -
+ have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
+ by (simp add: fun_eq_iff)
+ then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
+ by (rule arg_cong)
+ then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
+ by (blast intro: fun_cong)
+ then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
+qed
+
+lemma setsum_constant [simp]:
+ "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
apply (cases "finite A")
apply (erule finite_induct)
apply (auto simp add: algebra_simps)
@@ -837,21 +993,14 @@
lemma setsum_bounded:
assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
- shows "setsum f A \<le> of_nat(card A) * K"
+ shows "setsum f A \<le> of_nat (card A) * K"
proof (cases "finite A")
case True
thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
next
- case False thus ?thesis by (simp add: setsum_def)
+ case False thus ?thesis by simp
qed
-
-subsubsection {* Cardinality as special case of @{const setsum} *}
-
-lemma card_eq_setsum:
- "card A = setsum (\<lambda>x. 1) A"
- by (simp only: card_def setsum_def)
-
lemma card_UN_disjoint:
assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
@@ -869,17 +1018,6 @@
apply (simp_all add: SUP_def id_def)
done
-text{*The image of a finite set can be expressed using @{term fold_image}.*}
-lemma image_eq_fold_image:
- "finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A"
-proof (induct rule: finite_induct)
- case empty then show ?case by simp
-next
- interpret ab_semigroup_mult "op Un"
- proof qed auto
- case insert
- then show ?case by simp
-qed
subsubsection {* Cardinality of products *}
@@ -904,15 +1042,23 @@
subsection {* Generalized product over a set *}
-definition (in comm_monoid_mult) setprod :: "('b \<Rightarrow> 'a) => 'b set => 'a" where
- "setprod f A = (if finite A then fold_image (op *) f 1 A else 1)"
+definition (in comm_monoid_mult) setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
+where
+ "setprod = comm_monoid_set.F times 1"
-sublocale comm_monoid_mult < setprod!: comm_monoid_big "op *" 1 setprod proof
-qed (fact setprod_def)
+sublocale comm_monoid_mult < setprod!: comm_monoid_set times 1
+where
+ "setprod.F = setprod"
+proof -
+ show "comm_monoid_set times 1" ..
+ then interpret setprod!: comm_monoid_set times 1 .
+ show "setprod.F = setprod"
+ by (simp only: setprod_def)
+qed
abbreviation
- Setprod ("\<Prod>_" [1000] 999) where
- "\<Prod>A == setprod (%x. x) A"
+ Setprod ("\<Prod>_" [1000] 999) where
+ "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
syntax
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3PROD _:_. _)" [0, 51, 10] 10)
@@ -939,6 +1085,55 @@
"PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
"\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
+text {* TODO These are candidates for generalization *}
+
+context comm_monoid_mult
+begin
+
+lemma setprod_reindex_id:
+ "inj_on f B ==> setprod f B = setprod id (f ` B)"
+ by (auto simp add: setprod.reindex)
+
+lemma setprod_reindex_cong:
+ "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
+ by (frule setprod.reindex, simp)
+
+lemma strong_setprod_reindex_cong:
+ assumes i: "inj_on f A"
+ and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
+ shows "setprod h B = setprod g A"
+proof-
+ have "setprod h B = setprod (h o f) A"
+ by (simp add: B setprod.reindex [OF i, of h])
+ then show ?thesis apply simp
+ apply (rule setprod.cong)
+ apply simp
+ by (simp add: eq)
+qed
+
+lemma setprod_Union_disjoint:
+ assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
+ shows "setprod f (Union C) = setprod (setprod f) C"
+ using assms by (fact setprod.Union_disjoint)
+
+text{*Here we can eliminate the finiteness assumptions, by cases.*}
+lemma setprod_cartesian_product:
+ "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
+ by (fact setprod.cartesian_product)
+
+lemma setprod_Un2:
+ assumes "finite (A \<union> B)"
+ shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
+proof -
+ have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
+ by auto
+ with assms show ?thesis by simp (subst setprod.union_disjoint, auto)+
+qed
+
+end
+
+text {* TODO These are legacy *}
+
lemma setprod_empty: "setprod f {} = 1"
by (fact setprod.empty)
@@ -950,126 +1145,91 @@
by (fact setprod.infinite)
lemma setprod_reindex:
- "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
-by(auto simp: setprod_def fold_image_reindex o_def dest!:finite_imageD)
-
-lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
-by (auto simp add: setprod_reindex)
+ "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
+ by (fact setprod.reindex)
lemma setprod_cong:
"A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
-by(fact setprod.F_cong)
+ by (fact setprod.cong)
lemma strong_setprod_cong:
"A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
-by(fact setprod.strong_F_cong)
-
-lemma setprod_reindex_cong: "inj_on f A ==>
- B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
-by (frule setprod_reindex, simp)
+ by (fact setprod.strong_cong)
-lemma strong_setprod_reindex_cong: assumes i: "inj_on f A"
- and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
- shows "setprod h B = setprod g A"
-proof-
- have "setprod h B = setprod (h o f) A"
- by (simp add: B setprod_reindex[OF i, of h])
- then show ?thesis apply simp
- apply (rule setprod_cong)
- apply simp
- by (simp add: eq)
-qed
+lemma setprod_Un_one:
+ "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
+ \<Longrightarrow> setprod f (S \<union> T) = setprod f S * setprod f T"
+ by (fact setprod.union_inter_neutral)
-lemma setprod_Un_one: "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>
- \<Longrightarrow> setprod f (S \<union> T) = setprod f S * setprod f T"
-by(fact setprod.F_Un_neutral)
-
-lemmas setprod_1 = setprod.F_neutral
-lemmas setprod_1' = setprod.F_neutral'
-
+lemmas setprod_1 = setprod.neutral_const
+lemmas setprod_1' = setprod.neutral
lemma setprod_Un_Int: "finite A ==> finite B
==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
-by (fact setprod.union_inter)
+ by (fact setprod.union_inter)
lemma setprod_Un_disjoint: "finite A ==> finite B
==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
-by (fact setprod.union_disjoint)
+ by (fact setprod.union_disjoint)
lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>
setprod f A = setprod f (A - B) * setprod f B"
-by(fact setprod.F_subset_diff)
+ by (fact setprod.subset_diff)
lemma setprod_mono_one_left:
"\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"
-by(fact setprod.F_mono_neutral_left)
+ by (fact setprod.mono_neutral_left)
-lemmas setprod_mono_one_right = setprod.F_mono_neutral_right
+lemmas setprod_mono_one_right = setprod.mono_neutral_right
lemma setprod_mono_one_cong_left:
"\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>
\<Longrightarrow> setprod f S = setprod g T"
-by(fact setprod.F_mono_neutral_cong_left)
+ by (fact setprod.mono_neutral_cong_left)
-lemmas setprod_mono_one_cong_right = setprod.F_mono_neutral_cong_right
+lemmas setprod_mono_one_cong_right = setprod.mono_neutral_cong_right
lemma setprod_delta: "finite S \<Longrightarrow>
setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
-by(fact setprod.F_delta)
+ by (fact setprod.delta)
lemma setprod_delta': "finite S \<Longrightarrow>
setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"
-by(fact setprod.F_delta')
+ by (fact setprod.delta')
lemma setprod_UN_disjoint:
"finite I ==> (ALL i:I. finite (A i)) ==>
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
- by (simp add: setprod_def fold_image_UN_disjoint)
-
-lemma setprod_Union_disjoint:
- assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
- shows "setprod f (Union C) = setprod (setprod f) C"
-proof cases
- assume "finite C"
- from setprod_UN_disjoint[OF this assms]
- show ?thesis
- by (simp add: SUP_def)
-qed (force dest: finite_UnionD simp add: setprod_def)
+ by (fact setprod.UNION_disjoint)
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
(\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
(\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
-by(simp add:setprod_def fold_image_Sigma split_def)
+ by (fact setprod.Sigma)
-text{*Here we can eliminate the finiteness assumptions, by cases.*}
-lemma setprod_cartesian_product:
- "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
-apply (cases "finite A")
- apply (cases "finite B")
- apply (simp add: setprod_Sigma)
- apply (cases "A={}", simp)
- apply (simp)
-apply (auto simp add: setprod_def
- dest: finite_cartesian_productD1 finite_cartesian_productD2)
-done
-
-lemma setprod_timesf: "setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
-by (fact setprod.F_fun_f)
+lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"
+ by (fact setprod.distrib)
subsubsection {* Properties in more restricted classes of structures *}
-lemma setprod_eq_1_iff [simp]:
- "finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
-by (induct set: finite) auto
-
lemma setprod_zero:
"finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
apply (induct set: finite, force, clarsimp)
apply (erule disjE, auto)
done
+lemma setprod_zero_iff[simp]: "finite A ==>
+ (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
+ (EX x: A. f x = 0)"
+by (erule finite_induct, auto simp:no_zero_divisors)
+
+lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
+ (setprod f (A Un B) :: 'a ::{field})
+ = setprod f A * setprod f B / setprod f (A Int B)"
+by (subst setprod_Un_Int [symmetric], auto)
+
lemma setprod_nonneg [rule_format]:
"(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
@@ -1078,33 +1238,6 @@
--> 0 < setprod f A"
by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
-lemma setprod_zero_iff[simp]: "finite A ==>
- (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
- (EX x: A. f x = 0)"
-by (erule finite_induct, auto simp:no_zero_divisors)
-
-lemma setprod_pos_nat:
- "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
-using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
-
-lemma setprod_pos_nat_iff[simp]:
- "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
-using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
-
-lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
- (setprod f (A Un B) :: 'a ::{field})
- = setprod f A * setprod f B / setprod f (A Int B)"
-by (subst setprod_Un_Int [symmetric], auto)
-
-lemma setprod_Un2:
- assumes "finite (A \<union> B)"
- shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod f (A \<inter> B)"
-proof -
- have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
- by auto
- with assms show ?thesis by simp (subst setprod_Un_disjoint, auto)+
-qed
-
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
(setprod f (A - {a}) :: 'a :: {field}) =
(if a:A then setprod f A / f a else setprod f A)"
@@ -1197,7 +1330,7 @@
lemma setprod_gen_delta:
assumes fS: "finite S"
- shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::{comm_monoid_mult}) * c^ (card S - 1) else c^ card S)"
+ shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
proof-
let ?f = "(\<lambda>k. if k=a then b k else c)"
{assume a: "a \<notin> S"
@@ -1222,150 +1355,431 @@
ultimately show ?thesis by blast
qed
+lemma setprod_eq_1_iff [simp]:
+ "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
+ by (induct set: finite) auto
-subsection {* Versions of @{const inf} and @{const sup} on non-empty sets *}
+lemma setprod_pos_nat:
+ "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
+using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
+
+lemma setprod_pos_nat_iff[simp]:
+ "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
+using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
+
+
+subsection {* Generic lattice operations over a set *}
no_notation times (infixl "*" 70)
no_notation Groups.one ("1")
-locale semilattice_big = semilattice +
- fixes F :: "'a set \<Rightarrow> 'a"
- assumes F_eq: "finite A \<Longrightarrow> F A = fold1 (op *) A"
+
+subsubsection {* Without neutral element *}
+
+locale semilattice_set = semilattice
+begin
+
+definition F :: "'a set \<Rightarrow> 'a"
+where
+ eq_fold': "F A = the (Finite_Set.fold (\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)) None A)"
+
+lemma eq_fold:
+ assumes "finite A"
+ shows "F (insert x A) = Finite_Set.fold f x A"
+proof (rule sym)
+ let ?f = "\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> f x z)"
+ interpret comp_fun_idem f
+ by default (simp_all add: fun_eq_iff left_commute)
+ interpret comp_fun_idem "?f"
+ by default (simp_all add: fun_eq_iff commute left_commute split: option.split)
+ from assms show "Finite_Set.fold f x A = F (insert x A)"
+ proof induct
+ case empty then show ?case by (simp add: eq_fold')
+ next
+ case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
+ qed
+qed
+
+lemma singleton [simp]:
+ "F {x} = x"
+ by (simp add: eq_fold)
+
+lemma insert_not_elem:
+ assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
+ shows "F (insert x A) = x * F A"
+proof -
+ interpret comp_fun_idem f
+ by default (simp_all add: fun_eq_iff left_commute)
+ from `A \<noteq> {}` obtain b where "b \<in> A" by blast
+ then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
+ with `finite A` and `x \<notin> A`
+ have "finite (insert x B)" and "b \<notin> insert x B" by auto
+ then have "F (insert b (insert x B)) = x * F (insert b B)"
+ by (simp add: eq_fold)
+ then show ?thesis by (simp add: * insert_commute)
+qed
+
+lemma subsumption:
+ assumes "finite A" and "x \<in> A"
+ shows "x * F A = F A"
+proof -
+ from assms have "A \<noteq> {}" by auto
+ with `finite A` show ?thesis using `x \<in> A`
+ by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
+qed
+
+lemma insert [simp]:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "F (insert x A) = x * F A"
+ using assms by (cases "x \<in> A") (simp_all add: insert_absorb subsumption insert_not_elem)
+
+lemma union:
+ assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
+ shows "F (A \<union> B) = F A * F B"
+ using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
+
+lemma remove:
+ assumes "finite A" and "x \<in> A"
+ shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
+proof -
+ from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
+ with assms show ?thesis by simp
+qed
+
+lemma insert_remove:
+ assumes "finite A"
+ shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
+ using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
+
+lemma subset:
+ assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
+ shows "F B * F A = F A"
+proof -
+ from assms have "A \<noteq> {}" and "finite B" by (auto dest: finite_subset)
+ with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
+qed
+
+lemma closed:
+ assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
+ shows "F A \<in> A"
+using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
+ case singleton then show ?case by simp
+next
+ case insert with elem show ?case by force
+qed
+
+lemma hom_commute:
+ assumes hom: "\<And>x y. h (x * y) = h x * h y"
+ and N: "finite N" "N \<noteq> {}"
+ shows "h (F N) = F (h ` N)"
+using N proof (induct rule: finite_ne_induct)
+ case singleton thus ?case by simp
+next
+ case (insert n N)
+ then have "h (F (insert n N)) = h (n * F N)" by simp
+ also have "\<dots> = h n * h (F N)" by (rule hom)
+ also have "h (F N) = F (h ` N)" by (rule insert)
+ also have "h n * \<dots> = F (insert (h n) (h ` N))"
+ using insert by simp
+ also have "insert (h n) (h ` N) = h ` insert n N" by simp
+ finally show ?case .
+qed
+
+end
+
+locale semilattice_order_set = semilattice_order + semilattice_set
+begin
+
+lemma bounded_iff:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
+ using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)
+
+lemma boundedI:
+ assumes "finite A"
+ assumes "A \<noteq> {}"
+ assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
+ shows "x \<preceq> F A"
+ using assms by (simp add: bounded_iff)
+
+lemma boundedE:
+ assumes "finite A" and "A \<noteq> {}" and "x \<preceq> F A"
+ obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
+ using assms by (simp add: bounded_iff)
-sublocale semilattice_big < folding_one_idem proof
-qed (simp_all add: F_eq)
+lemma coboundedI:
+ assumes "finite A"
+ and "a \<in> A"
+ shows "F A \<preceq> a"
+proof -
+ from assms have "A \<noteq> {}" by auto
+ from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
+ proof (induct rule: finite_ne_induct)
+ case singleton thus ?case by (simp add: refl)
+ next
+ case (insert x B)
+ from insert have "a = x \<or> a \<in> B" by simp
+ then show ?case using insert by (auto intro: coboundedI2)
+ qed
+qed
+
+lemma antimono:
+ assumes "A \<subseteq> B" and "A \<noteq> {}" and "finite B"
+ shows "F B \<preceq> F A"
+proof (cases "A = B")
+ case True then show ?thesis by (simp add: refl)
+next
+ case False
+ have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
+ then have "F B = F (A \<union> (B - A))" by simp
+ also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
+ also have "\<dots> \<preceq> F A" by simp
+ finally show ?thesis .
+qed
+
+end
+
+
+subsubsection {* With neutral element *}
+
+locale semilattice_neutr_set = semilattice_neutr
+begin
+
+definition F :: "'a set \<Rightarrow> 'a"
+where
+ eq_fold: "F A = Finite_Set.fold f 1 A"
+
+lemma infinite [simp]:
+ "\<not> finite A \<Longrightarrow> F A = 1"
+ by (simp add: eq_fold)
+
+lemma empty [simp]:
+ "F {} = 1"
+ by (simp add: eq_fold)
+
+lemma insert [simp]:
+ assumes "finite A"
+ shows "F (insert x A) = x * F A"
+proof -
+ interpret comp_fun_idem f
+ by default (simp_all add: fun_eq_iff left_commute)
+ from assms show ?thesis by (simp add: eq_fold)
+qed
+
+lemma subsumption:
+ assumes "finite A" and "x \<in> A"
+ shows "x * F A = F A"
+proof -
+ from assms have "A \<noteq> {}" by auto
+ with `finite A` show ?thesis using `x \<in> A`
+ by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
+qed
+
+lemma union:
+ assumes "finite A" and "finite B"
+ shows "F (A \<union> B) = F A * F B"
+ using assms by (induct A) (simp_all add: ac_simps)
+
+lemma remove:
+ assumes "finite A" and "x \<in> A"
+ shows "F A = x * F (A - {x})"
+proof -
+ from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
+ with assms show ?thesis by simp
+qed
+
+lemma insert_remove:
+ assumes "finite A"
+ shows "F (insert x A) = x * F (A - {x})"
+ using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
+
+lemma subset:
+ assumes "finite A" and "B \<subseteq> A"
+ shows "F B * F A = F A"
+proof -
+ from assms have "finite B" by (auto dest: finite_subset)
+ with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
+qed
+
+lemma closed:
+ assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
+ shows "F A \<in> A"
+using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
+ case singleton then show ?case by simp
+next
+ case insert with elem show ?case by force
+qed
+
+end
+
+locale semilattice_order_neutr_set = semilattice_neutr_order + semilattice_neutr_set
+begin
+
+lemma bounded_iff:
+ assumes "finite A"
+ shows "x \<preceq> F A \<longleftrightarrow> (\<forall>a\<in>A. x \<preceq> a)"
+ using assms by (induct A) (simp_all add: bounded_iff)
+
+lemma boundedI:
+ assumes "finite A"
+ assumes "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
+ shows "x \<preceq> F A"
+ using assms by (simp add: bounded_iff)
+
+lemma boundedE:
+ assumes "finite A" and "x \<preceq> F A"
+ obtains "\<And>a. a \<in> A \<Longrightarrow> x \<preceq> a"
+ using assms by (simp add: bounded_iff)
+
+lemma coboundedI:
+ assumes "finite A"
+ and "a \<in> A"
+ shows "F A \<preceq> a"
+proof -
+ from assms have "A \<noteq> {}" by auto
+ from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
+ proof (induct rule: finite_ne_induct)
+ case singleton thus ?case by (simp add: refl)
+ next
+ case (insert x B)
+ from insert have "a = x \<or> a \<in> B" by simp
+ then show ?case using insert by (auto intro: coboundedI2)
+ qed
+qed
+
+lemma antimono:
+ assumes "A \<subseteq> B" and "finite B"
+ shows "F B \<preceq> F A"
+proof (cases "A = B")
+ case True then show ?thesis by (simp add: refl)
+next
+ case False
+ have B: "B = A \<union> (B - A)" using `A \<subseteq> B` by blast
+ then have "F B = F (A \<union> (B - A))" by simp
+ also have "\<dots> = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
+ also have "\<dots> \<preceq> F A" by simp
+ finally show ?thesis .
+qed
+
+end
notation times (infixl "*" 70)
notation Groups.one ("1")
-context lattice
-begin
-definition Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900) where
- "Inf_fin = fold1 inf"
-
-definition Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900) where
- "Sup_fin = fold1 sup"
-
-end
+subsection {* Lattice operations on finite sets *}
-sublocale lattice < Inf_fin!: semilattice_big inf Inf_fin proof
-qed (simp add: Inf_fin_def)
-
-sublocale lattice < Sup_fin!: semilattice_big sup Sup_fin proof
-qed (simp add: Sup_fin_def)
+text {*
+ For historic reasons, there is the sublocale dependency from @{class distrib_lattice}
+ to @{class linorder}. This is badly designed: both should depend on a common abstract
+ distributive lattice rather than having this non-subclass dependecy between two
+ classes. But for the moment we have to live with it. This forces us to setup
+ this sublocale dependency simultaneously with the lattice operations on finite
+ sets, to avoid garbage.
+*}
-context semilattice_inf
-begin
+definition (in semilattice_inf) Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900)
+where
+ "Inf_fin = semilattice_set.F inf"
-lemma ab_semigroup_idem_mult_inf:
- "class.ab_semigroup_idem_mult inf"
-proof qed (rule inf_assoc inf_commute inf_idem)+
+definition (in semilattice_sup) Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900)
+where
+ "Sup_fin = semilattice_set.F sup"
-lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> Finite_Set.fold inf b (insert a A) = inf a (Finite_Set.fold inf b A)"
-by(rule comp_fun_idem.fold_insert_idem[OF ab_semigroup_idem_mult.comp_fun_idem[OF ab_semigroup_idem_mult_inf]])
+definition (in linorder) Min :: "'a set \<Rightarrow> 'a"
+where
+ "Min = semilattice_set.F min"
-lemma inf_le_fold_inf: "finite A \<Longrightarrow> ALL a:A. b \<le> a \<Longrightarrow> inf b c \<le> Finite_Set.fold inf c A"
-by (induct pred: finite) (auto intro: le_infI1)
+definition (in linorder) Max :: "'a set \<Rightarrow> 'a"
+where
+ "Max = semilattice_set.F max"
+
+text {* An aside: @{const min}/@{const max} on linear orders as special case of @{const inf}/@{const sup} *}
-lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> Finite_Set.fold inf b A \<le> inf a b"
-proof(induct arbitrary: a pred:finite)
- case empty thus ?case by simp
-next
- case (insert x A)
- show ?case
- proof cases
- assume "A = {}" thus ?thesis using insert by simp
- next
- assume "A \<noteq> {}" thus ?thesis using insert by (auto intro: le_infI2)
- qed
-qed
-
-lemma below_fold1_iff:
- assumes "finite A" "A \<noteq> {}"
- shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
+sublocale linorder < min_max!: distrib_lattice min less_eq less max
+where
+ "semilattice_inf.Inf_fin min = Min"
+ and "semilattice_sup.Sup_fin max = Max"
proof -
- interpret ab_semigroup_idem_mult inf
- by (rule ab_semigroup_idem_mult_inf)
- show ?thesis using assms by (induct rule: finite_ne_induct) simp_all
+ show "class.distrib_lattice min less_eq less max"
+ proof
+ fix x y z
+ show "max x (min y z) = min (max x y) (max x z)"
+ by (auto simp add: min_def max_def)
+ qed (auto simp add: min_def max_def not_le less_imp_le)
+ then interpret min_max!: distrib_lattice min less_eq less max .
+ show "semilattice_inf.Inf_fin min = Min"
+ by (simp only: min_max.Inf_fin_def Min_def)
+ show "semilattice_sup.Sup_fin max = Max"
+ by (simp only: min_max.Sup_fin_def Max_def)
qed
-lemma fold1_belowI:
- assumes "finite A"
- and "a \<in> A"
- shows "fold1 inf A \<le> a"
+lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{semilattice_inf, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
+ by (rule ext)+ (auto intro: antisym)
+
+lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{semilattice_sup, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
+ by (rule ext)+ (auto intro: antisym)
+
+lemmas le_maxI1 = min_max.sup_ge1
+lemmas le_maxI2 = min_max.sup_ge2
+
+lemmas min_ac = min_max.inf_assoc min_max.inf_commute
+ min_max.inf.left_commute
+
+lemmas max_ac = min_max.sup_assoc min_max.sup_commute
+ min_max.sup.left_commute
+
+
+text {* Lattice operations proper *}
+
+sublocale semilattice_inf < Inf_fin!: semilattice_order_set inf less_eq less
+where
+ "Inf_fin.F = Inf_fin"
proof -
- from assms have "A \<noteq> {}" by auto
- from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
- proof (induct rule: finite_ne_induct)
- case singleton thus ?case by simp
- next
- interpret ab_semigroup_idem_mult inf
- by (rule ab_semigroup_idem_mult_inf)
- case (insert x F)
- from insert(5) have "a = x \<or> a \<in> F" by simp
- thus ?case
- proof
- assume "a = x" thus ?thesis using insert
- by (simp add: mult_ac)
- next
- assume "a \<in> F"
- hence bel: "fold1 inf F \<le> a" by (rule insert)
- have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
- using insert by (simp add: mult_ac)
- also have "inf (fold1 inf F) a = fold1 inf F"
- using bel by (auto intro: antisym)
- also have "inf x \<dots> = fold1 inf (insert x F)"
- using insert by (simp add: mult_ac)
- finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
- moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
- ultimately show ?thesis by simp
- qed
- qed
+ show "semilattice_order_set inf less_eq less" ..
+ then interpret Inf_fin!: semilattice_order_set inf less_eq less.
+ show "Inf_fin.F = Inf_fin"
+ by (fact Inf_fin_def [symmetric])
qed
-end
-
-context semilattice_sup
-begin
-
-lemma ab_semigroup_idem_mult_sup: "class.ab_semigroup_idem_mult sup"
-by (rule semilattice_inf.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
+sublocale semilattice_sup < Sup_fin!: semilattice_order_set sup greater_eq greater
+where
+ "Sup_fin.F = Sup_fin"
+proof -
+ show "semilattice_order_set sup greater_eq greater" ..
+ then interpret Sup_fin!: semilattice_order_set sup greater_eq greater .
+ show "Sup_fin.F = Sup_fin"
+ by (fact Sup_fin_def [symmetric])
+qed
-lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> Finite_Set.fold sup b (insert a A) = sup a (Finite_Set.fold sup b A)"
-by(rule semilattice_inf.fold_inf_insert)(rule dual_semilattice)
+
+subsection {* Infimum and Supremum over non-empty sets *}
-lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> Finite_Set.fold sup c A \<le> sup b c"
-by(rule semilattice_inf.inf_le_fold_inf)(rule dual_semilattice)
-
-lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> Finite_Set.fold sup b A"
-by(rule semilattice_inf.fold_inf_le_inf)(rule dual_semilattice)
-
-end
+text {*
+ After this non-regular bootstrap, things continue canonically.
+*}
context lattice
begin
lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
-apply(unfold Sup_fin_def Inf_fin_def)
apply(subgoal_tac "EX a. a:A")
prefer 2 apply blast
apply(erule exE)
apply(rule order_trans)
-apply(erule (1) fold1_belowI)
-apply(erule (1) semilattice_inf.fold1_belowI [OF dual_semilattice])
+apply(erule (1) Inf_fin.coboundedI)
+apply(erule (1) Sup_fin.coboundedI)
done
lemma sup_Inf_absorb [simp]:
"finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
apply(subst sup_commute)
-apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)
+apply(simp add: sup_absorb2 Inf_fin.coboundedI)
done
lemma inf_Sup_absorb [simp]:
"finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
-by (simp add: Sup_fin_def inf_absorb1
- semilattice_inf.fold1_belowI [OF dual_semilattice])
+by (simp add: inf_absorb1 Sup_fin.coboundedI)
end
@@ -1376,27 +1790,19 @@
assumes "finite A"
and "A \<noteq> {}"
shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
-proof -
- interpret ab_semigroup_idem_mult inf
- by (rule ab_semigroup_idem_mult_inf)
- from assms show ?thesis
- by (simp add: Inf_fin_def image_def
- hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])
- (rule arg_cong [where f="fold1 inf"], blast)
-qed
+using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
+ (rule arg_cong [where f="Inf_fin"], blast)
lemma sup_Inf2_distrib:
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
shows "sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B) = \<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B}"
using A proof (induct rule: finite_ne_induct)
- case singleton thus ?case
+ case singleton then show ?case
by (simp add: sup_Inf1_distrib [OF B])
next
- interpret ab_semigroup_idem_mult inf
- by (rule ab_semigroup_idem_mult_inf)
case (insert x A)
have finB: "finite {sup x b |b. b \<in> B}"
- by(rule finite_surj[where f = "sup x", OF B(1)], auto)
+ by (rule finite_surj [where f = "sup x", OF B(1)], auto)
have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
proof -
have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
@@ -1412,7 +1818,7 @@
also have "\<dots> = \<Sqinter>\<^bsub>fin\<^esub>({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
(is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
using B insert
- by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
+ by (simp add: Inf_fin.union [OF finB _ finAB ne])
also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
by blast
finally show ?case .
@@ -1421,13 +1827,8 @@
lemma inf_Sup1_distrib:
assumes "finite A" and "A \<noteq> {}"
shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
-proof -
- interpret ab_semigroup_idem_mult sup
- by (rule ab_semigroup_idem_mult_sup)
- from assms show ?thesis
- by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])
- (rule arg_cong [where f="fold1 sup"], blast)
-qed
+using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
+ (rule arg_cong [where f="Sup_fin"], blast)
lemma inf_Sup2_distrib:
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
@@ -1446,8 +1847,6 @@
thus ?thesis by(simp add: insert(1) B(1))
qed
have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
- interpret ab_semigroup_idem_mult sup
- by (rule ab_semigroup_idem_mult_sup)
have "inf (\<Squnion>\<^bsub>fin\<^esub>(insert x A)) (\<Squnion>\<^bsub>fin\<^esub>B) = inf (sup x (\<Squnion>\<^bsub>fin\<^esub>A)) (\<Squnion>\<^bsub>fin\<^esub>B)"
using insert by simp
also have "\<dots> = sup (inf x (\<Squnion>\<^bsub>fin\<^esub>B)) (inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2)
@@ -1456,7 +1855,7 @@
also have "\<dots> = \<Squnion>\<^bsub>fin\<^esub>({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
(is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
using B insert
- by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
+ by (simp add: Sup_fin.union [OF finB _ finAB ne])
also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
by blast
finally show ?case .
@@ -1471,227 +1870,84 @@
assumes "finite A" and "A \<noteq> {}"
shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
proof -
- interpret ab_semigroup_idem_mult inf
- by (rule ab_semigroup_idem_mult_inf)
- from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
- moreover with `finite A` have "finite B" by simp
- ultimately show ?thesis
- by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])
+ from assms obtain b B where "A = insert b B" and "finite B" by auto
+ then show ?thesis
+ by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
qed
lemma Sup_fin_Sup:
assumes "finite A" and "A \<noteq> {}"
shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
proof -
- interpret ab_semigroup_idem_mult sup
- by (rule ab_semigroup_idem_mult_sup)
- from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
- moreover with `finite A` have "finite B" by simp
- ultimately show ?thesis
- by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric])
+ from assms obtain b B where "A = insert b B" and "finite B" by auto
+ then show ?thesis
+ by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
qed
end
-subsection {* Versions of @{const min} and @{const max} on non-empty sets *}
-
-definition (in linorder) Min :: "'a set \<Rightarrow> 'a" where
- "Min = fold1 min"
+subsection {* Minimum and Maximum over non-empty sets *}
-definition (in linorder) Max :: "'a set \<Rightarrow> 'a" where
- "Max = fold1 max"
-
-sublocale linorder < Min!: semilattice_big min Min proof
-qed (simp add: Min_def)
-
-sublocale linorder < Max!: semilattice_big max Max proof
-qed (simp add: Max_def)
+text {*
+ This case is already setup by the @{text min_max} sublocale dependency from above. But note
+ that this yields irregular prefixes, e.g.~@{text min_max.Inf_fin.insert} instead
+ of @{text Max.insert}.
+*}
context linorder
begin
-lemmas Min_singleton = Min.singleton
-lemmas Max_singleton = Max.singleton
-
-lemma Min_insert:
- assumes "finite A" and "A \<noteq> {}"
- shows "Min (insert x A) = min x (Min A)"
- using assms by simp
-
-lemma Max_insert:
- assumes "finite A" and "A \<noteq> {}"
- shows "Max (insert x A) = max x (Max A)"
- using assms by simp
-
-lemma Min_Un:
- assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
- shows "Min (A \<union> B) = min (Min A) (Min B)"
- using assms by (rule Min.union_idem)
-
-lemma Max_Un:
- assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
- shows "Max (A \<union> B) = max (Max A) (Max B)"
- using assms by (rule Max.union_idem)
-
-lemma hom_Min_commute:
- assumes "\<And>x y. h (min x y) = min (h x) (h y)"
- and "finite N" and "N \<noteq> {}"
- shows "h (Min N) = Min (h ` N)"
- using assms by (rule Min.hom_commute)
-
-lemma hom_Max_commute:
- assumes "\<And>x y. h (max x y) = max (h x) (h y)"
- and "finite N" and "N \<noteq> {}"
- shows "h (Max N) = Max (h ` N)"
- using assms by (rule Max.hom_commute)
-
-lemma ab_semigroup_idem_mult_min:
- "class.ab_semigroup_idem_mult min"
- proof qed (auto simp add: min_def)
-
-lemma ab_semigroup_idem_mult_max:
- "class.ab_semigroup_idem_mult max"
- proof qed (auto simp add: max_def)
-
-lemma max_lattice:
- "class.semilattice_inf max (op \<ge>) (op >)"
- by (fact min_max.dual_semilattice)
-
-lemma dual_max:
- "ord.max (op \<ge>) = min"
- by (auto simp add: ord.max_def min_def fun_eq_iff)
-
lemma dual_min:
- "ord.min (op \<ge>) = max"
+ "ord.min greater_eq = max"
by (auto simp add: ord.min_def max_def fun_eq_iff)
-lemma strict_below_fold1_iff:
- assumes "finite A" and "A \<noteq> {}"
- shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
+lemma dual_max:
+ "ord.max greater_eq = min"
+ by (auto simp add: ord.max_def min_def fun_eq_iff)
+
+lemma dual_Min:
+ "linorder.Min greater_eq = Max"
proof -
- interpret ab_semigroup_idem_mult min
- by (rule ab_semigroup_idem_mult_min)
- from assms show ?thesis
- by (induct rule: finite_ne_induct)
- (simp_all add: fold1_insert)
-qed
-
-lemma fold1_below_iff:
- assumes "finite A" and "A \<noteq> {}"
- shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
-proof -
- interpret ab_semigroup_idem_mult min
- by (rule ab_semigroup_idem_mult_min)
- from assms show ?thesis
- by (induct rule: finite_ne_induct)
- (simp_all add: fold1_insert min_le_iff_disj)
+ interpret dual!: linorder greater_eq greater by (fact dual_linorder)
+ show ?thesis by (simp add: dual.Min_def dual_min Max_def)
qed
-lemma fold1_strict_below_iff:
- assumes "finite A" and "A \<noteq> {}"
- shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
+lemma dual_Max:
+ "linorder.Max greater_eq = Min"
proof -
- interpret ab_semigroup_idem_mult min
- by (rule ab_semigroup_idem_mult_min)
- from assms show ?thesis
- by (induct rule: finite_ne_induct)
- (simp_all add: fold1_insert min_less_iff_disj)
+ interpret dual!: linorder greater_eq greater by (fact dual_linorder)
+ show ?thesis by (simp add: dual.Max_def dual_max Min_def)
qed
-lemma fold1_antimono:
- assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
- shows "fold1 min B \<le> fold1 min A"
-proof cases
- assume "A = B" thus ?thesis by simp
-next
- interpret ab_semigroup_idem_mult min
- by (rule ab_semigroup_idem_mult_min)
- assume neq: "A \<noteq> B"
- have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
- have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
- also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
- proof -
- have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
- moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`])
- moreover have "(B-A) \<noteq> {}" using assms neq by blast
- moreover have "A Int (B-A) = {}" using assms by blast
- ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
- qed
- also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
- finally show ?thesis .
-qed
+lemmas Min_singleton = min_max.Inf_fin.singleton
+lemmas Max_singleton = min_max.Sup_fin.singleton
+lemmas Min_insert = min_max.Inf_fin.insert
+lemmas Max_insert = min_max.Sup_fin.insert
+lemmas Min_Un = min_max.Inf_fin.union
+lemmas Max_Un = min_max.Sup_fin.union
+lemmas hom_Min_commute = min_max.Inf_fin.hom_commute
+lemmas hom_Max_commute = min_max.Sup_fin.hom_commute
lemma Min_in [simp]:
assumes "finite A" and "A \<noteq> {}"
shows "Min A \<in> A"
-proof -
- interpret ab_semigroup_idem_mult min
- by (rule ab_semigroup_idem_mult_min)
- from assms fold1_in show ?thesis by (fastforce simp: Min_def min_def)
-qed
+ using assms by (auto simp add: min_def min_max.Inf_fin.closed)
lemma Max_in [simp]:
assumes "finite A" and "A \<noteq> {}"
shows "Max A \<in> A"
-proof -
- interpret ab_semigroup_idem_mult max
- by (rule ab_semigroup_idem_mult_max)
- from assms fold1_in [of A] show ?thesis by (fastforce simp: Max_def max_def)
-qed
+ using assms by (auto simp add: max_def min_max.Sup_fin.closed)
lemma Min_le [simp]:
assumes "finite A" and "x \<in> A"
shows "Min A \<le> x"
- using assms by (simp add: Min_def min_max.fold1_belowI)
+ using assms by (fact min_max.Inf_fin.coboundedI)
lemma Max_ge [simp]:
assumes "finite A" and "x \<in> A"
shows "x \<le> Max A"
- by (simp add: Max_def semilattice_inf.fold1_belowI [OF max_lattice] assms)
-
-lemma Min_ge_iff [simp, no_atp]:
- assumes "finite A" and "A \<noteq> {}"
- shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
- using assms by (simp add: Min_def min_max.below_fold1_iff)
-
-lemma Max_le_iff [simp, no_atp]:
- assumes "finite A" and "A \<noteq> {}"
- shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
- by (simp add: Max_def semilattice_inf.below_fold1_iff [OF max_lattice] assms)
-
-lemma Min_gr_iff [simp, no_atp]:
- assumes "finite A" and "A \<noteq> {}"
- shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
- using assms by (simp add: Min_def strict_below_fold1_iff)
-
-lemma Max_less_iff [simp, no_atp]:
- assumes "finite A" and "A \<noteq> {}"
- shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
- by (simp add: Max_def linorder.dual_max [OF dual_linorder]
- linorder.strict_below_fold1_iff [OF dual_linorder] assms)
-
-lemma Min_le_iff [no_atp]:
- assumes "finite A" and "A \<noteq> {}"
- shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
- using assms by (simp add: Min_def fold1_below_iff)
-
-lemma Max_ge_iff [no_atp]:
- assumes "finite A" and "A \<noteq> {}"
- shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
- by (simp add: Max_def linorder.dual_max [OF dual_linorder]
- linorder.fold1_below_iff [OF dual_linorder] assms)
-
-lemma Min_less_iff [no_atp]:
- assumes "finite A" and "A \<noteq> {}"
- shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
- using assms by (simp add: Min_def fold1_strict_below_iff)
-
-lemma Max_gr_iff [no_atp]:
- assumes "finite A" and "A \<noteq> {}"
- shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
- by (simp add: Max_def linorder.dual_max [OF dual_linorder]
- linorder.fold1_strict_below_iff [OF dual_linorder] assms)
+ using assms by (fact min_max.Sup_fin.coboundedI)
lemma Min_eqI:
assumes "finite A"
@@ -1717,22 +1973,91 @@
from assms show "x \<le> Max A" by simp
qed
+lemma Min_ge_iff [simp, no_atp]:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
+ using assms by (fact min_max.Inf_fin.bounded_iff)
+
+lemma Max_le_iff [simp, no_atp]:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
+ using assms by (fact min_max.Sup_fin.bounded_iff)
+
+lemma Min_gr_iff [simp, no_atp]:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
+ using assms by (induct rule: finite_ne_induct) simp_all
+
+lemma Max_less_iff [simp, no_atp]:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
+ using assms by (induct rule: finite_ne_induct) simp_all
+
+lemma Min_le_iff [no_atp]:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
+ using assms by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)
+
+lemma Max_ge_iff [no_atp]:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
+ using assms by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)
+
+lemma Min_less_iff [no_atp]:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
+ using assms by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)
+
+lemma Max_gr_iff [no_atp]:
+ assumes "finite A" and "A \<noteq> {}"
+ shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
+ using assms by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)
+
lemma Min_antimono:
assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
shows "Min N \<le> Min M"
- using assms by (simp add: Min_def fold1_antimono)
+ using assms by (fact min_max.Inf_fin.antimono)
lemma Max_mono:
assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
shows "Max M \<le> Max N"
- by (simp add: Max_def linorder.dual_max [OF dual_linorder]
- linorder.fold1_antimono [OF dual_linorder] assms)
+ using assms by (fact min_max.Sup_fin.antimono)
+
+lemma mono_Min_commute:
+ assumes "mono f"
+ assumes "finite A" and "A \<noteq> {}"
+ shows "f (Min A) = Min (f ` A)"
+proof (rule linorder_class.Min_eqI [symmetric])
+ from `finite A` show "finite (f ` A)" by simp
+ from assms show "f (Min A) \<in> f ` A" by simp
+ fix x
+ assume "x \<in> f ` A"
+ then obtain y where "y \<in> A" and "x = f y" ..
+ with assms have "Min A \<le> y" by auto
+ with `mono f` have "f (Min A) \<le> f y" by (rule monoE)
+ with `x = f y` show "f (Min A) \<le> x" by simp
+qed
-lemma finite_linorder_max_induct[consumes 1, case_names empty insert]:
- assumes fin: "finite A"
- and empty: "P {}"
- and insert: "(!!b A. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))"
- shows "P A"
+lemma mono_Max_commute:
+ assumes "mono f"
+ assumes "finite A" and "A \<noteq> {}"
+ shows "f (Max A) = Max (f ` A)"
+proof (rule linorder_class.Max_eqI [symmetric])
+ from `finite A` show "finite (f ` A)" by simp
+ from assms show "f (Max A) \<in> f ` A" by simp
+ fix x
+ assume "x \<in> f ` A"
+ then obtain y where "y \<in> A" and "x = f y" ..
+ with assms have "y \<le> Max A" by auto
+ with `mono f` have "f y \<le> f (Max A)" by (rule monoE)
+ with `x = f y` show "x \<le> f (Max A)" by simp
+qed
+
+lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
+ assumes fin: "finite A"
+ and empty: "P {}"
+ and insert: "\<And>b A. finite A \<Longrightarrow> \<forall>a\<in>A. a < b \<Longrightarrow> P A \<Longrightarrow> P (insert b A)"
+ shows "P A"
using fin empty insert
proof (induct rule: finite_psubset_induct)
case (psubset A)
@@ -1751,16 +2076,16 @@
assume "A \<noteq> {}"
with `finite A` have "Max A : A" by auto
then have A: "?A = A" using insert_Diff_single insert_absorb by auto
- then have "P ?B" using `P {}` step IH[of ?B] by blast
+ then have "P ?B" using `P {}` step IH [of ?B] by blast
moreover
have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastforce
- ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastforce
+ ultimately show "P A" using A insert_Diff_single step [OF `finite ?B`] by fastforce
qed
qed
-lemma finite_linorder_min_induct[consumes 1, case_names empty insert]:
- "\<lbrakk>finite A; P {}; \<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"
-by(rule linorder.finite_linorder_max_induct[OF dual_linorder])
+lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
+ "\<lbrakk>finite A; P {}; \<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"
+ by (rule linorder.finite_linorder_max_induct [OF dual_linorder])
end
@@ -1799,29 +2124,14 @@
begin
lemma minus_Max_eq_Min [simp]:
- "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Max S) = Min (uminus ` S)"
+ "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Max S = Min (uminus ` S)"
by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)
lemma minus_Min_eq_Max [simp]:
- "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Min S) = Max (uminus ` S)"
+ "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - Min S = Max (uminus ` S)"
by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)
end
-lemma (in linorder) mono_Max_commute:
- assumes "mono f"
- assumes "finite A" and "A \<noteq> {}"
- shows "f (Max A) = Max (f ` A)"
-proof (rule linorder_class.Max_eqI [symmetric])
- from `finite A` show "finite (f ` A)" by simp
- from assms show "f (Max A) \<in> f ` A" by simp
- fix x
- assume "x \<in> f ` A"
- then obtain y where "y \<in> A" and "x = f y" ..
- with assms have "y \<le> Max A" by auto
- with `mono f` have "f y \<le> f (Max A)" by (rule monoE)
- with `x = f y` show "x \<le> f (Max A)" by simp
-qed (* FIXME augment also dual rule mono_Min_commute *)
-
end
--- a/src/HOL/Complete_Lattices.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Complete_Lattices.thy Sat Mar 23 20:50:39 2013 +0100
@@ -514,10 +514,10 @@
by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
lemma complete_linorder_inf_min: "inf = min"
- by (rule ext)+ (auto intro: antisym)
+ by (rule ext)+ (auto intro: antisym simp add: min_def)
lemma complete_linorder_sup_max: "sup = max"
- by (rule ext)+ (auto intro: antisym)
+ by (rule ext)+ (auto intro: antisym simp add: max_def)
lemma Inf_less_iff:
"\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
--- a/src/HOL/Finite_Set.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Finite_Set.thy Sat Mar 23 20:50:39 2013 +0100
@@ -564,9 +564,13 @@
assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
begin
-lemma fun_left_comm: "f x (f y z) = f y (f x z)"
+lemma fun_left_comm: "f y (f x z) = f x (f y z)"
using comp_fun_commute by (simp add: fun_eq_iff)
+lemma commute_left_comp:
+ "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
+ by (simp add: o_assoc comp_fun_commute)
+
end
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
@@ -578,7 +582,7 @@
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
- "fold f z A = (THE y. fold_graph f z A y)"
+ "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
text{*A tempting alternative for the definiens is
@{term "if finite A then THE y. fold_graph f z A y else e"}.
@@ -595,6 +599,11 @@
context comp_fun_commute
begin
+lemma fold_graph_finite:
+ assumes "fold_graph f z A y"
+ shows "finite A"
+ using assms by induct simp_all
+
lemma fold_graph_insertE_aux:
"fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
proof (induct set: fold_graph)
@@ -632,7 +641,7 @@
lemma fold_equality:
"fold_graph f z A y \<Longrightarrow> fold f z A = y"
-by (unfold fold_def) (blast intro: fold_graph_determ)
+ by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
lemma fold_graph_fold:
assumes "finite A"
@@ -642,13 +651,19 @@
moreover note fold_graph_determ
ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
- then show ?thesis by (unfold fold_def)
+ with assms show ?thesis by (simp add: fold_def)
qed
-text{* The base case for @{text fold}: *}
+text {* The base case for @{text fold}: *}
-lemma (in -) fold_empty [simp]: "fold f z {} = z"
-by (unfold fold_def) blast
+lemma (in -) fold_infinite [simp]:
+ assumes "\<not> finite A"
+ shows "fold f z A = z"
+ using assms by (auto simp add: fold_def)
+
+lemma (in -) fold_empty [simp]:
+ "fold f z {} = z"
+ by (auto simp add: fold_def)
text{* The various recursion equations for @{const fold}: *}
@@ -656,22 +671,27 @@
assumes "finite A" and "x \<notin> A"
shows "fold f z (insert x A) = f x (fold f z A)"
proof (rule fold_equality)
+ fix z
from `finite A` have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
- with `x \<notin> A`show "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
+ with `x \<notin> A` have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
+ then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
qed
-lemma fold_fun_comm:
+declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
+ -- {* No more proofs involve these. *}
+
+lemma fold_fun_left_comm:
"finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
proof (induct rule: finite_induct)
case empty then show ?case by simp
next
case (insert y A) then show ?case
- by (simp add: fun_left_comm[of x])
+ by (simp add: fun_left_comm [of x])
qed
lemma fold_insert2:
- "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
-by (simp add: fold_fun_comm)
+ "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
+ by (simp add: fold_fun_left_comm)
lemma fold_rec:
assumes "finite A" and "x \<in> A"
@@ -699,23 +719,23 @@
lemma fold_image:
assumes "finite A" and "inj_on g A"
- shows "fold f x (g ` A) = fold (f \<circ> g) x A"
+ shows "fold f z (g ` A) = fold (f \<circ> g) z A"
using assms
proof induction
case (insert a F)
interpret comp_fun_commute "\<lambda>x. f (g x)" by default (simp add: comp_fun_commute)
from insert show ?case by auto
-qed (simp)
+qed simp
end
lemma fold_cong:
assumes "comp_fun_commute f" "comp_fun_commute g"
assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
- and "A = B" and "s = t"
- shows "Finite_Set.fold f s A = Finite_Set.fold g t B"
+ and "s = t" and "A = B"
+ shows "fold f s A = fold g t B"
proof -
- have "Finite_Set.fold f s A = Finite_Set.fold g s A"
+ have "fold f s A = fold g s A"
using `finite A` cong proof (induct A)
case empty then show ?case by simp
next
@@ -728,10 +748,10 @@
qed
-text{* A simplified version for idempotent functions: *}
+text {* A simplified version for idempotent functions: *}
locale comp_fun_idem = comp_fun_commute +
- assumes comp_fun_idem: "f x o f x = f x"
+ assumes comp_fun_idem: "f x \<circ> f x = f x"
begin
lemma fun_left_idem: "f x (f x z) = f x z"
@@ -739,20 +759,20 @@
lemma fold_insert_idem:
assumes fin: "finite A"
- shows "fold f z (insert x A) = f x (fold f z A)"
+ shows "fold f z (insert x A) = f x (fold f z A)"
proof cases
assume "x \<in> A"
then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
- then show ?thesis using assms by (simp add:fun_left_idem)
+ then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
next
assume "x \<notin> A" then show ?thesis using assms by simp
qed
-declare fold_insert[simp del] fold_insert_idem[simp]
+declare fold_insert [simp del] fold_insert_idem [simp]
lemma fold_insert_idem2:
"finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
-by(simp add:fold_fun_comm)
+ by (simp add: fold_fun_left_comm)
end
@@ -810,6 +830,11 @@
subsubsection {* Expressing set operations via @{const fold} *}
+lemma comp_fun_commute_const:
+ "comp_fun_commute (\<lambda>_. f)"
+proof
+qed rule
+
lemma comp_fun_idem_insert:
"comp_fun_idem insert"
proof
@@ -847,7 +872,8 @@
then show ?thesis ..
qed
-lemma comp_fun_commute_filter_fold: "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
+lemma comp_fun_commute_filter_fold:
+ "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
proof -
interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
show ?thesis by default (auto simp: fun_eq_iff)
@@ -916,13 +942,13 @@
lemma comp_fun_commute_product_fold:
assumes "finite B"
- shows "comp_fun_commute (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B)"
+ shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
by default (auto simp: fold_union_pair[symmetric] assms)
lemma product_fold:
assumes "finite A"
assumes "finite B"
- shows "A \<times> B = fold (\<lambda>x A. fold (\<lambda>y. Set.insert (x, y)) A B) {} A"
+ shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
using assms unfolding Sigma_def
by (induct A)
(simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
@@ -933,20 +959,20 @@
lemma inf_Inf_fold_inf:
assumes "finite A"
- shows "inf B (Inf A) = fold inf B A"
+ shows "inf (Inf A) B = fold inf B A"
proof -
interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
- from `finite A` show ?thesis by (induct A arbitrary: B)
- (simp_all add: inf_commute fold_fun_comm)
+ from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
+ (simp_all add: inf_commute fun_eq_iff)
qed
lemma sup_Sup_fold_sup:
assumes "finite A"
- shows "sup B (Sup A) = fold sup B A"
+ shows "sup (Sup A) B = fold sup B A"
proof -
interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
- from `finite A` show ?thesis by (induct A arbitrary: B)
- (simp_all add: sup_commute fold_fun_comm)
+ from `finite A` fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
+ (simp_all add: sup_commute fun_eq_iff)
qed
lemma Inf_fold_inf:
@@ -994,503 +1020,42 @@
end
-subsection {* The derived combinator @{text fold_image} *}
-
-definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
- where "fold_image f g = fold (\<lambda>x y. f (g x) y)"
-
-lemma fold_image_empty[simp]: "fold_image f g z {} = z"
- by (simp add:fold_image_def)
-
-context ab_semigroup_mult
-begin
-
-lemma fold_image_insert[simp]:
- assumes "finite A" and "a \<notin> A"
- shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
-proof -
- interpret comp_fun_commute "%x y. (g x) * y"
- by default (simp add: fun_eq_iff mult_ac)
- from assms show ?thesis by (simp add: fold_image_def)
-qed
-
-lemma fold_image_reindex:
- assumes "finite A"
- shows "inj_on h A \<Longrightarrow> fold_image times g z (h ` A) = fold_image times (g \<circ> h) z A"
- using assms by induct auto
-
-lemma fold_image_cong:
- assumes "finite A" and g_h: "\<And>x. x\<in>A \<Longrightarrow> g x = h x"
- shows "fold_image times g z A = fold_image times h z A"
-proof -
- from `finite A`
- have "\<And>C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C"
- proof (induct arbitrary: C)
- case empty then show ?case by simp
- next
- case (insert x F) then show ?case apply -
- apply (simp add: subset_insert_iff, clarify)
- apply (subgoal_tac "finite C")
- prefer 2 apply (blast dest: finite_subset [rotated])
- apply (subgoal_tac "C = insert x (C - {x})")
- prefer 2 apply blast
- apply (erule ssubst)
- apply (simp add: Ball_def del: insert_Diff_single)
- done
- qed
- with g_h show ?thesis by simp
-qed
-
-end
-
-context comm_monoid_mult
-begin
-
-lemma fold_image_1:
- "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
- apply (induct rule: finite_induct)
- apply simp by auto
-
-lemma fold_image_Un_Int:
- "finite A ==> finite B ==>
- fold_image times g 1 A * fold_image times g 1 B =
- fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
- apply (induct rule: finite_induct)
-by (induct set: finite)
- (auto simp add: mult_ac insert_absorb Int_insert_left)
-
-lemma fold_image_Un_one:
- assumes fS: "finite S" and fT: "finite T"
- and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
- shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
-proof-
- have "fold_image op * f 1 (S \<inter> T) = 1"
- apply (rule fold_image_1)
- using fS fT I0 by auto
- with fold_image_Un_Int[OF fS fT] show ?thesis by simp
-qed
-
-corollary fold_Un_disjoint:
- "finite A ==> finite B ==> A Int B = {} ==>
- fold_image times g 1 (A Un B) =
- fold_image times g 1 A * fold_image times g 1 B"
-by (simp add: fold_image_Un_Int)
-
-lemma fold_image_UN_disjoint:
- "\<lbrakk> finite I; ALL i:I. finite (A i);
- ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
- \<Longrightarrow> fold_image times g 1 (UNION I A) =
- fold_image times (%i. fold_image times g 1 (A i)) 1 I"
-apply (induct rule: finite_induct)
-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: fold_Un_disjoint)
-done
-
-lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
- fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
- fold_image times (split g) 1 (SIGMA x:A. B x)"
-apply (subst Sigma_def)
-apply (subst fold_image_UN_disjoint, assumption, simp)
- apply blast
-apply (erule fold_image_cong)
-apply (subst fold_image_UN_disjoint, simp, simp)
- apply blast
-apply simp
-done
-
-lemma fold_image_distrib: "finite A \<Longrightarrow>
- fold_image times (%x. g x * h x) 1 A =
- fold_image times g 1 A * fold_image times h 1 A"
-by (erule finite_induct) (simp_all add: mult_ac)
-
-lemma fold_image_related:
- assumes Re: "R e e"
- and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
- and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
- shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
- using fS by (rule finite_subset_induct) (insert assms, auto)
-
-lemma fold_image_eq_general:
- assumes fS: "finite S"
- and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
- and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
- shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
-proof-
- from h f12 have hS: "h ` S = S'" by auto
- {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
- from f12 h H have "x = y" by auto }
- hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
- from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
- from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
- also have "\<dots> = fold_image (op *) (f2 o h) e S"
- using fold_image_reindex[OF fS hinj, of f2 e] .
- also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
- by blast
- finally show ?thesis ..
-qed
-
-lemma fold_image_eq_general_inverses:
- assumes fS: "finite S"
- and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
- and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
- shows "fold_image (op *) f e S = fold_image (op *) g e T"
- (* metis solves it, but not yet available here *)
- apply (rule fold_image_eq_general[OF fS, of T h g f e])
- apply (rule ballI)
- apply (frule kh)
- apply (rule ex1I[])
- apply blast
- apply clarsimp
- apply (drule hk) apply simp
- apply (rule sym)
- apply (erule conjunct1[OF conjunct2[OF hk]])
- apply (rule ballI)
- apply (drule hk)
- apply blast
- done
-
-end
-
-
-subsection {* A fold functional for non-empty sets *}
-
-text{* Does not require start value. *}
-
-inductive
- fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
- for f :: "'a => 'a => 'a"
-where
- fold1Set_insertI [intro]:
- "\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
-
-definition fold1 :: "('a => 'a => 'a) => 'a set => 'a" where
- "fold1 f A == THE x. fold1Set f A x"
-
-lemma fold1Set_nonempty:
- "fold1Set f A x \<Longrightarrow> A \<noteq> {}"
-by(erule fold1Set.cases, simp_all)
-
-inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
-
-inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
-
-
-lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
-by (blast elim: fold_graph.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. fold1Set f A x"
-apply (induct A rule: finite_induct)
-apply (auto dest: finite_imp_fold_graph [of _ f])
-done
-
-text{*First, some lemmas about @{const fold_graph}.*}
-
-context ab_semigroup_mult
-begin
-
-lemma comp_fun_commute: "comp_fun_commute (op *)"
- by default (simp add: fun_eq_iff mult_ac)
-
-lemma fold_graph_insert_swap:
-assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
-shows "fold_graph times z (insert b A) (z * y)"
-proof -
- interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
-from assms show ?thesis
-proof (induct rule: fold_graph.induct)
- case emptyI show ?case by (subst mult_commute [of z b], fast)
-next
- case (insertI x A y)
- have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
- using insertI by force --{*how does @{term id} get unfolded?*}
- thus ?case by (simp add: insert_commute mult_ac)
-qed
-qed
-
-lemma fold_graph_permute_diff:
-assumes fold: "fold_graph times b A x"
-shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
-using fold
-proof (induct rule: fold_graph.induct)
- case emptyI thus ?case by simp
-next
- case (insertI x A y)
- have "a = x \<or> a \<in> A" using insertI by simp
- thus ?case
- proof
- assume "a = x"
- with insertI show ?thesis
- by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
- next
- assume ainA: "a \<in> A"
- hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
- using insertI by force
- moreover
- have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
- using ainA insertI by blast
- ultimately show ?thesis by simp
- qed
-qed
-
-lemma fold1_eq_fold:
-assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
-proof -
- interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
- from assms show ?thesis
-apply (simp add: fold1_def fold_def)
-apply (rule the_equality)
-apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
-apply (rule sym, clarify)
-apply (case_tac "Aa=A")
- apply (best intro: fold_graph_determ)
-apply (subgoal_tac "fold_graph times a A x")
- apply (best intro: fold_graph_determ)
-apply (subgoal_tac "insert aa (Aa - {a}) = A")
- prefer 2 apply (blast elim: equalityE)
-apply (auto dest: fold_graph_permute_diff [where a=a])
-done
-qed
-
-lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
-apply safe
- apply simp
- apply (drule_tac x=x in spec)
- apply (drule_tac x="A-{x}" in spec, auto)
-done
-
-lemma fold1_insert:
- assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
- shows "fold1 times (insert x A) = x * fold1 times A"
-proof -
- interpret comp_fun_commute "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule comp_fun_commute)
- from nonempty obtain a A' where "A = insert a A' & a ~: A'"
- by (auto simp add: nonempty_iff)
- with A show ?thesis
- by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
-qed
-
-end
-
-class ab_semigroup_idem_mult = ab_semigroup_mult +
- assumes mult_idem: "x * x = x"
-
-sublocale ab_semigroup_idem_mult < times!: semilattice times proof
-qed (fact mult_idem)
-
-context ab_semigroup_idem_mult
-begin
-
-lemmas mult_left_idem = times.left_idem
-
-lemma comp_fun_idem: "comp_fun_idem (op *)"
- by default (simp_all add: fun_eq_iff mult_left_commute)
-
-lemma fold1_insert_idem [simp]:
- assumes nonempty: "A \<noteq> {}" and A: "finite A"
- shows "fold1 times (insert x A) = x * fold1 times A"
-proof -
- interpret comp_fun_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
- by (rule comp_fun_idem)
- from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
- by (auto simp add: nonempty_iff)
- show ?thesis
- proof cases
- assume a: "a = x"
- show ?thesis
- proof cases
- assume "A' = {}"
- with A' a show ?thesis by simp
- next
- assume "A' \<noteq> {}"
- with A A' a show ?thesis
- by (simp add: fold1_insert mult_assoc [symmetric])
- qed
- next
- assume "a \<noteq> x"
- with A A' show ?thesis
- by (simp add: insert_commute fold1_eq_fold)
- qed
-qed
-
-lemma hom_fold1_commute:
-assumes hom: "!!x y. h (x * y) = h x * h y"
-and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
-using N
-proof (induct rule: finite_ne_induct)
- case singleton thus ?case by simp
-next
- case (insert n N)
- then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
- also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
- also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
- also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
- using insert by(simp)
- also have "insert (h n) (h ` N) = h ` insert n N" by simp
- finally show ?case .
-qed
-
-lemma fold1_eq_fold_idem:
- assumes "finite A"
- shows "fold1 times (insert a A) = fold times a A"
-proof (cases "a \<in> A")
- case False
- with assms show ?thesis by (simp add: fold1_eq_fold)
-next
- interpret comp_fun_idem times by (fact comp_fun_idem)
- case True then obtain b B
- where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
- with assms have "finite B" by auto
- then have "fold times a (insert a B) = fold times (a * a) B"
- using `a \<notin> B` by (rule fold_insert2)
- then show ?thesis
- using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
-qed
-
-end
-
-
-text{* Now the recursion rules for definitions: *}
-
-lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
-by simp
-
-lemma (in ab_semigroup_mult) fold1_insert_def:
- "\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
-by (simp add:fold1_insert)
-
-lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
- "\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
-by simp
-
-subsubsection{* Determinacy for @{term fold1Set} *}
-
-(*Not actually used!!*)
-(*
-context ab_semigroup_mult
-begin
-
-lemma fold_graph_permute:
- "[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
- ==> fold_graph times id a (insert b A) x"
-apply (cases "a=b")
-apply (auto dest: fold_graph_permute_diff)
-done
-
-lemma fold1Set_determ:
- "fold1Set times A x ==> fold1Set times A y ==> y = x"
-proof (clarify elim!: fold1Set.cases)
- fix A x B y a b
- assume Ax: "fold_graph times id a A x"
- assume By: "fold_graph times id b B y"
- assume anotA: "a \<notin> A"
- assume bnotB: "b \<notin> B"
- assume eq: "insert a A = insert b B"
- show "y=x"
- proof cases
- assume same: "a=b"
- hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
- thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
- next
- assume diff: "a\<noteq>b"
- let ?D = "B - {a}"
- have B: "B = insert a ?D" and A: "A = insert b ?D"
- and aB: "a \<in> B" and bA: "b \<in> A"
- using eq anotA bnotB diff by (blast elim!:equalityE)+
- with aB bnotB By
- have "fold_graph times id a (insert b ?D) y"
- by (auto intro: fold_graph_permute simp add: insert_absorb)
- moreover
- have "fold_graph times id a (insert b ?D) x"
- by (simp add: A [symmetric] Ax)
- ultimately show ?thesis by (blast intro: fold_graph_determ)
- qed
-qed
-
-lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
- by (unfold fold1_def) (blast intro: fold1Set_determ)
-
-end
-*)
-
-declare
- empty_fold_graphE [rule del] fold_graph.intros [rule del]
- empty_fold1SetE [rule del] insert_fold1SetE [rule del]
- -- {* No more proofs involve these relations. *}
-
-subsubsection {* Lemmas about @{text fold1} *}
-
-context ab_semigroup_mult
-begin
-
-lemma fold1_Un:
-assumes A: "finite A" "A \<noteq> {}"
-shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
- fold1 times (A Un B) = fold1 times A * fold1 times B"
-using A by (induct rule: finite_ne_induct)
- (simp_all add: fold1_insert mult_assoc)
-
-lemma fold1_in:
- assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
- shows "fold1 times A \<in> A"
-using A
-proof (induct rule:finite_ne_induct)
- case singleton thus ?case by simp
-next
- case insert thus ?case using elem by (force simp add:fold1_insert)
-qed
-
-end
-
-lemma (in ab_semigroup_idem_mult) fold1_Un2:
-assumes A: "finite A" "A \<noteq> {}"
-shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
- fold1 times (A Un B) = fold1 times A * fold1 times B"
-using A
-proof(induct rule:finite_ne_induct)
- case singleton thus ?case by simp
-next
- case insert thus ?case by (simp add: mult_assoc)
-qed
-
-
subsection {* Locales as mini-packages for fold operations *}
subsubsection {* The natural case *}
locale folding =
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
- fixes F :: "'a set \<Rightarrow> 'b \<Rightarrow> 'b"
+ fixes z :: "'b"
assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
- assumes eq_fold: "finite A \<Longrightarrow> F A s = fold f s A"
begin
+definition F :: "'a set \<Rightarrow> 'b"
+where
+ eq_fold: "F A = fold f z A"
+
lemma empty [simp]:
- "F {} = id"
- by (simp add: eq_fold fun_eq_iff)
+ "F {} = z"
+ by (simp add: eq_fold)
+lemma infinite [simp]:
+ "\<not> finite A \<Longrightarrow> F A = z"
+ by (simp add: eq_fold)
+
lemma insert [simp]:
assumes "finite A" and "x \<notin> A"
- shows "F (insert x A) = F A \<circ> f x"
+ shows "F (insert x A) = f x (F A)"
proof -
interpret comp_fun_commute f
by default (insert comp_fun_commute, simp add: fun_eq_iff)
- from fold_insert2 assms
- have "\<And>s. fold f s (insert x A) = fold f (f x s) A" .
+ from fold_insert assms
+ have "fold f z (insert x A) = f x (fold f z A)" by simp
with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
qed
-
+
lemma remove:
assumes "finite A" and "x \<in> A"
- shows "F A = F (A - {x}) \<circ> f x"
+ shows "F A = f x (F (A - {x}))"
proof -
from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
by (auto dest: mk_disjoint_insert)
@@ -1500,524 +1065,69 @@
lemma insert_remove:
assumes "finite A"
- shows "F (insert x A) = F (A - {x}) \<circ> f x"
+ shows "F (insert x A) = f x (F (A - {x}))"
using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
-lemma commute_left_comp:
- "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
- by (simp add: o_assoc comp_fun_commute)
-
-lemma comp_fun_commute':
- assumes "finite A"
- shows "f x \<circ> F A = F A \<circ> f x"
- using assms by (induct A)
- (simp, simp del: o_apply add: o_assoc, simp del: o_apply add: comp_assoc comp_fun_commute)
-
-lemma commute_left_comp':
- assumes "finite A"
- shows "f x \<circ> (F A \<circ> g) = F A \<circ> (f x \<circ> g)"
- using assms by (simp add: o_assoc comp_fun_commute')
-
-lemma comp_fun_commute'':
- assumes "finite A" and "finite B"
- shows "F B \<circ> F A = F A \<circ> F B"
- using assms by (induct A)
- (simp_all add: o_assoc, simp add: comp_assoc comp_fun_commute')
-
-lemma commute_left_comp'':
- assumes "finite A" and "finite B"
- shows "F B \<circ> (F A \<circ> g) = F A \<circ> (F B \<circ> g)"
- using assms by (simp add: o_assoc comp_fun_commute'')
-
-lemmas comp_fun_commutes = comp_assoc comp_fun_commute commute_left_comp
- comp_fun_commute' commute_left_comp' comp_fun_commute'' commute_left_comp''
-
-lemma union_inter:
- assumes "finite A" and "finite B"
- shows "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B"
- using assms by (induct A)
- (simp_all del: o_apply add: insert_absorb Int_insert_left comp_fun_commutes,
- simp add: o_assoc)
-
-lemma union:
- assumes "finite A" and "finite B"
- and "A \<inter> B = {}"
- shows "F (A \<union> B) = F A \<circ> F B"
-proof -
- from union_inter `finite A` `finite B` have "F (A \<union> B) \<circ> F (A \<inter> B) = F A \<circ> F B" .
- with `A \<inter> B = {}` show ?thesis by simp
-qed
-
end
-subsubsection {* The natural case with idempotency *}
+subsubsection {* With idempotency *}
locale folding_idem = folding +
- assumes idem_comp: "f x \<circ> f x = f x"
+ assumes comp_fun_idem: "f x \<circ> f x = f x"
begin
-lemma idem_left_comp:
- "f x \<circ> (f x \<circ> g) = f x \<circ> g"
- by (simp add: o_assoc idem_comp)
-
-lemma in_comp_idem:
- assumes "finite A" and "x \<in> A"
- shows "F A \<circ> f x = F A"
-using assms by (induct A)
- (auto simp add: comp_fun_commutes idem_comp, simp add: commute_left_comp' [symmetric] comp_fun_commute')
-
-lemma subset_comp_idem:
- assumes "finite A" and "B \<subseteq> A"
- shows "F A \<circ> F B = F A"
-proof -
- from assms have "finite B" by (blast dest: finite_subset)
- then show ?thesis using `B \<subseteq> A` by (induct B)
- (simp_all add: o_assoc in_comp_idem `finite A`)
-qed
-
declare insert [simp del]
lemma insert_idem [simp]:
assumes "finite A"
- shows "F (insert x A) = F A \<circ> f x"
- using assms by (cases "x \<in> A") (simp_all add: insert in_comp_idem insert_absorb)
-
-lemma union_idem:
- assumes "finite A" and "finite B"
- shows "F (A \<union> B) = F A \<circ> F B"
+ shows "F (insert x A) = f x (F A)"
proof -
- from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
- then have "F (A \<union> B) \<circ> F (A \<inter> B) = F (A \<union> B)" by (rule subset_comp_idem)
- with assms show ?thesis by (simp add: union_inter)
+ interpret comp_fun_idem f
+ by default (insert comp_fun_commute comp_fun_idem, simp add: fun_eq_iff)
+ from fold_insert_idem assms
+ have "fold f z (insert x A) = f x (fold f z A)" by simp
+ with `finite A` show ?thesis by (simp add: eq_fold fun_eq_iff)
qed
end
-subsubsection {* The image case with fixed function *}
-
-no_notation times (infixl "*" 70)
-no_notation Groups.one ("1")
-
-locale folding_image_simple = comm_monoid +
- fixes g :: "('b \<Rightarrow> 'a)"
- fixes F :: "'b set \<Rightarrow> 'a"
- assumes eq_fold_g: "finite A \<Longrightarrow> F A = fold_image f g 1 A"
-begin
-
-lemma empty [simp]:
- "F {} = 1"
- by (simp add: eq_fold_g)
-
-lemma insert [simp]:
- assumes "finite A" and "x \<notin> A"
- shows "F (insert x A) = g x * F A"
-proof -
- interpret comp_fun_commute "%x y. (g x) * y"
- by default (simp add: ac_simps fun_eq_iff)
- from assms have "fold_image (op *) g 1 (insert x A) = g x * fold_image (op *) g 1 A"
- by (simp add: fold_image_def)
- with `finite A` show ?thesis by (simp add: eq_fold_g)
-qed
-
-lemma remove:
- assumes "finite A" and "x \<in> A"
- shows "F A = g x * F (A - {x})"
-proof -
- from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"
- by (auto dest: mk_disjoint_insert)
- moreover from `finite A` this have "finite B" by simp
- ultimately show ?thesis by simp
-qed
-
-lemma insert_remove:
- assumes "finite A"
- shows "F (insert x A) = g x * F (A - {x})"
- using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
-
-lemma neutral:
- assumes "finite A" and "\<forall>x\<in>A. g x = 1"
- shows "F A = 1"
- using assms by (induct A) simp_all
-
-lemma union_inter:
- assumes "finite A" and "finite B"
- shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
-using assms proof (induct A)
- case empty then show ?case by simp
-next
- case (insert x A) then show ?case
- by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
-qed
-
-corollary union_inter_neutral:
- assumes "finite A" and "finite B"
- and I0: "\<forall>x \<in> A\<inter>B. g x = 1"
- shows "F (A \<union> B) = F A * F B"
- using assms by (simp add: union_inter [symmetric] neutral)
-
-corollary union_disjoint:
- assumes "finite A" and "finite B"
- assumes "A \<inter> B = {}"
- shows "F (A \<union> B) = F A * F B"
- using assms by (simp add: union_inter_neutral)
-
-end
-
-
-subsubsection {* The image case with flexible function *}
-
-locale folding_image = comm_monoid +
- fixes F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
- assumes eq_fold: "\<And>g. finite A \<Longrightarrow> F g A = fold_image f g 1 A"
-
-sublocale folding_image < folding_image_simple "op *" 1 g "F g" proof
-qed (fact eq_fold)
-
-context folding_image
-begin
-
-lemma reindex: (* FIXME polymorhism *)
- assumes "finite A" and "inj_on h A"
- shows "F g (h ` A) = F (g \<circ> h) A"
- using assms by (induct A) auto
-
-lemma cong:
- assumes "finite A" and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
- shows "F g A = F h A"
-proof -
- from assms have "ALL C. C <= A --> (ALL x:C. g x = h x) --> F g C = F h C"
- apply - apply (erule finite_induct) apply simp
- apply (simp add: subset_insert_iff, clarify)
- apply (subgoal_tac "finite C")
- prefer 2 apply (blast dest: finite_subset [rotated])
- 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 del: insert_Diff_single)
- done
- with assms show ?thesis by simp
-qed
-
-lemma UNION_disjoint:
- assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
- and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
- shows "F g (UNION I A) = F (F g \<circ> A) I"
-apply (insert assms)
-apply (induct rule: finite_induct)
-apply simp
-apply atomize
-apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
- prefer 2 apply blast
-apply (subgoal_tac "A x Int UNION Fa A = {}")
- prefer 2 apply blast
-apply (simp add: union_disjoint)
-done
-
-lemma distrib:
- assumes "finite A"
- shows "F (\<lambda>x. g x * h x) A = F g A * F h A"
- using assms by (rule finite_induct) (simp_all add: assoc commute left_commute)
-
-lemma related:
- assumes Re: "R 1 1"
- and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
- and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
- shows "R (F h S) (F g S)"
- using fS by (rule finite_subset_induct) (insert assms, auto)
-
-lemma eq_general:
- assumes fS: "finite S"
- and h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
- and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
- shows "F f1 S = F f2 S'"
-proof-
- from h f12 have hS: "h ` S = S'" by blast
- {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
- from f12 h H have "x = y" by auto }
- hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
- from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
- from hS have "F f2 S' = F f2 (h ` S)" by simp
- also have "\<dots> = F (f2 o h) S" using reindex [OF fS hinj, of f2] .
- also have "\<dots> = F f1 S " using th cong [OF fS, of "f2 o h" f1]
- by blast
- finally show ?thesis ..
-qed
-
-lemma eq_general_inverses:
- assumes fS: "finite S"
- and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
- and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
- shows "F j S = F g T"
- (* metis solves it, but not yet available here *)
- apply (rule eq_general [OF fS, of T h g j])
- apply (rule ballI)
- apply (frule kh)
- apply (rule ex1I[])
- apply blast
- apply clarsimp
- apply (drule hk) apply simp
- apply (rule sym)
- apply (erule conjunct1[OF conjunct2[OF hk]])
- apply (rule ballI)
- apply (drule hk)
- apply blast
- done
-
-end
-
-
-subsubsection {* The image case with fixed function and idempotency *}
-
-locale folding_image_simple_idem = folding_image_simple +
- assumes idem: "x * x = x"
-
-sublocale folding_image_simple_idem < semilattice: semilattice proof
-qed (fact idem)
-
-context folding_image_simple_idem
-begin
-
-lemma in_idem:
- assumes "finite A" and "x \<in> A"
- shows "g x * F A = F A"
- using assms by (induct A) (auto simp add: left_commute)
-
-lemma subset_idem:
- assumes "finite A" and "B \<subseteq> A"
- shows "F B * F A = F A"
-proof -
- from assms have "finite B" by (blast dest: finite_subset)
- then show ?thesis using `B \<subseteq> A` by (induct B)
- (auto simp add: assoc in_idem `finite A`)
-qed
-
-declare insert [simp del]
-
-lemma insert_idem [simp]:
- assumes "finite A"
- shows "F (insert x A) = g x * F A"
- using assms by (cases "x \<in> A") (simp_all add: insert in_idem insert_absorb)
-
-lemma union_idem:
- assumes "finite A" and "finite B"
- shows "F (A \<union> B) = F A * F B"
-proof -
- from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
- then have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (rule subset_idem)
- with assms show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
-qed
-
-end
-
-
-subsubsection {* The image case with flexible function and idempotency *}
-
-locale folding_image_idem = folding_image +
- assumes idem: "x * x = x"
-
-sublocale folding_image_idem < folding_image_simple_idem "op *" 1 g "F g" proof
-qed (fact idem)
-
-
-subsubsection {* The neutral-less case *}
-
-locale folding_one = abel_semigroup +
- fixes F :: "'a set \<Rightarrow> 'a"
- assumes eq_fold: "finite A \<Longrightarrow> F A = fold1 f A"
-begin
-
-lemma singleton [simp]:
- "F {x} = x"
- by (simp add: eq_fold)
-
-lemma eq_fold':
- assumes "finite A" and "x \<notin> A"
- shows "F (insert x A) = fold (op *) x A"
-proof -
- interpret ab_semigroup_mult "op *" by default (simp_all add: ac_simps)
- from assms show ?thesis by (simp add: eq_fold fold1_eq_fold)
-qed
-
-lemma insert [simp]:
- assumes "finite A" and "x \<notin> A" and "A \<noteq> {}"
- shows "F (insert x A) = x * F A"
-proof -
- from `A \<noteq> {}` obtain b where "b \<in> A" by blast
- then obtain B where *: "A = insert b B" "b \<notin> B" by (blast dest: mk_disjoint_insert)
- with `finite A` have "finite B" by simp
- interpret fold: folding "op *" "\<lambda>a b. fold (op *) b a" proof
- qed (simp_all add: fun_eq_iff ac_simps)
- from `finite B` fold.comp_fun_commute' [of B x]
- have "op * x \<circ> (\<lambda>b. fold op * b B) = (\<lambda>b. fold op * b B) \<circ> op * x" by simp
- then have A: "x * fold op * b B = fold op * (b * x) B" by (simp add: fun_eq_iff commute)
- from `finite B` * fold.insert [of B b]
- have "(\<lambda>x. fold op * x (insert b B)) = (\<lambda>x. fold op * x B) \<circ> op * b" by simp
- then have B: "fold op * x (insert b B) = fold op * (b * x) B" by (simp add: fun_eq_iff)
- from A B assms * show ?thesis by (simp add: eq_fold' del: fold.insert)
-qed
-
-lemma remove:
- assumes "finite A" and "x \<in> A"
- shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
-proof -
- from assms obtain B where "A = insert x B" and "x \<notin> B" by (blast dest: mk_disjoint_insert)
- with assms show ?thesis by simp
-qed
-
-lemma insert_remove:
- assumes "finite A"
- shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
- using assms by (cases "x \<in> A") (simp_all add: insert_absorb remove)
-
-lemma union_disjoint:
- assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}" and "A \<inter> B = {}"
- shows "F (A \<union> B) = F A * F B"
- using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)
-
-lemma union_inter:
- assumes "finite A" and "finite B" and "A \<inter> B \<noteq> {}"
- shows "F (A \<union> B) * F (A \<inter> B) = F A * F B"
-proof -
- from assms have "A \<noteq> {}" and "B \<noteq> {}" by auto
- from `finite A` `A \<noteq> {}` `A \<inter> B \<noteq> {}` show ?thesis proof (induct A rule: finite_ne_induct)
- case (singleton x) then show ?case by (simp add: insert_absorb ac_simps)
- next
- case (insert x A) show ?case proof (cases "x \<in> B")
- case True then have "B \<noteq> {}" by auto
- with insert True `finite B` show ?thesis by (cases "A \<inter> B = {}")
- (simp_all add: insert_absorb ac_simps union_disjoint)
- next
- case False with insert have "F (A \<union> B) * F (A \<inter> B) = F A * F B" by simp
- moreover from False `finite B` insert have "finite (A \<union> B)" "x \<notin> A \<union> B" "A \<union> B \<noteq> {}"
- by auto
- ultimately show ?thesis using False `finite A` `x \<notin> A` `A \<noteq> {}` by (simp add: assoc)
- qed
- qed
-qed
-
-lemma closed:
- assumes "finite A" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x, y}"
- shows "F A \<in> A"
-using `finite A` `A \<noteq> {}` proof (induct rule: finite_ne_induct)
- case singleton then show ?case by simp
-next
- case insert with elem show ?case by force
-qed
-
-end
-
-
-subsubsection {* The neutral-less case with idempotency *}
-
-locale folding_one_idem = folding_one +
- assumes idem: "x * x = x"
-
-sublocale folding_one_idem < semilattice: semilattice proof
-qed (fact idem)
-
-context folding_one_idem
-begin
-
-lemma in_idem:
- assumes "finite A" and "x \<in> A"
- shows "x * F A = F A"
-proof -
- from assms have "A \<noteq> {}" by auto
- with `finite A` show ?thesis using `x \<in> A` by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
-qed
-
-lemma subset_idem:
- assumes "finite A" "B \<noteq> {}" and "B \<subseteq> A"
- shows "F B * F A = F A"
-proof -
- from assms have "finite B" by (blast dest: finite_subset)
- then show ?thesis using `B \<noteq> {}` `B \<subseteq> A` by (induct B rule: finite_ne_induct)
- (simp_all add: assoc in_idem `finite A`)
-qed
-
-lemma eq_fold_idem':
- assumes "finite A"
- shows "F (insert a A) = fold (op *) a A"
-proof -
- interpret ab_semigroup_idem_mult "op *" by default (simp_all add: ac_simps)
- from assms show ?thesis by (simp add: eq_fold fold1_eq_fold_idem)
-qed
-
-lemma insert_idem [simp]:
- assumes "finite A" and "A \<noteq> {}"
- shows "F (insert x A) = x * F A"
-proof (cases "x \<in> A")
- case False from `finite A` `x \<notin> A` `A \<noteq> {}` show ?thesis by (rule insert)
-next
- case True
- from `finite A` `A \<noteq> {}` show ?thesis by (simp add: in_idem insert_absorb True)
-qed
-
-lemma union_idem:
- assumes "finite A" "A \<noteq> {}" and "finite B" "B \<noteq> {}"
- shows "F (A \<union> B) = F A * F B"
-proof (cases "A \<inter> B = {}")
- case True with assms show ?thesis by (simp add: union_disjoint)
-next
- case False
- from assms have "finite (A \<union> B)" and "A \<inter> B \<subseteq> A \<union> B" by auto
- with False have "F (A \<inter> B) * F (A \<union> B) = F (A \<union> B)" by (auto intro: subset_idem)
- with assms False show ?thesis by (simp add: union_inter [of A B, symmetric] commute)
-qed
-
-lemma hom_commute:
- assumes hom: "\<And>x y. h (x * y) = h x * h y"
- and N: "finite N" "N \<noteq> {}" shows "h (F N) = F (h ` N)"
-using N proof (induct rule: finite_ne_induct)
- case singleton thus ?case by simp
-next
- case (insert n N)
- then have "h (F (insert n N)) = h (n * F N)" by simp
- also have "\<dots> = h n * h (F N)" by (rule hom)
- also have "h (F N) = F (h ` N)" by(rule insert)
- also have "h n * \<dots> = F (insert (h n) (h ` N))"
- using insert by(simp)
- also have "insert (h n) (h ` N) = h ` insert n N" by simp
- finally show ?case .
-qed
-
-end
-
-notation times (infixl "*" 70)
-notation Groups.one ("1")
-
-
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}"}.
-But now that we have @{text fold_image} things are easy:
+text {*
+ The traditional definition
+ @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
+ is ugly to work with.
+ But now that we have @{const fold} things are easy:
*}
definition card :: "'a set \<Rightarrow> nat" where
- "card A = (if finite A then fold_image (op +) (\<lambda>x. 1) 0 A else 0)"
+ "card = folding.F (\<lambda>_. Suc) 0"
-interpretation card: folding_image_simple "op +" 0 "\<lambda>x. 1" card proof
-qed (simp add: card_def)
+interpretation card!: folding "\<lambda>_. Suc" 0
+where
+ "card.F = card"
+proof -
+ show "folding (\<lambda>_. Suc)" by default rule
+ then interpret card!: folding "\<lambda>_. Suc" 0 .
+ show "card.F = card" by (simp only: card_def)
+qed
-lemma card_infinite [simp]:
+lemma card_infinite:
"\<not> finite A \<Longrightarrow> card A = 0"
- by (simp add: card_def)
+ by (fact card.infinite)
lemma card_empty:
"card {} = 0"
by (fact card.empty)
lemma card_insert_disjoint:
- "finite A ==> x \<notin> A ==> card (insert x A) = Suc (card A)"
- by simp
+ "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
+ by (fact card.insert)
lemma card_insert_if:
- "finite A ==> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
+ "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
by auto (simp add: card.insert_remove card.remove)
lemma card_ge_0_finite:
@@ -2040,29 +1150,30 @@
"0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
by (simp add: neq0_conv [symmetric] card_eq_0_iff)
-lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
+lemma card_Suc_Diff1:
+ "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
apply(simp del:insert_Diff_single)
done
lemma card_Diff_singleton:
- "finite A ==> x: A ==> card (A - {x}) = card A - 1"
-by (simp add: card_Suc_Diff1 [symmetric])
+ "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> 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)
+ "finite A \<Longrightarrow> card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
+ by (simp add: card_Diff_singleton)
lemma card_Diff_insert[simp]:
-assumes "finite A" and "a:A" and "a ~: B"
-shows "card(A - insert a B) = card(A - B) - 1"
+ assumes "finite A" and "a \<in> A" and "a \<notin> B"
+ shows "card (A - insert a B) = card (A - B) - 1"
proof -
have "A - insert a B = (A - B) - {a}" using assms by blast
- then show ?thesis using assms by(simp add:card_Diff_singleton)
+ then show ?thesis using assms by(simp add: card_Diff_singleton)
qed
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
-by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
+ by (fact card.insert_remove)
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
by (simp add: card_insert_if)
@@ -2105,13 +1216,21 @@
apply (blast dest: card_seteq)
done
-lemma card_Un_Int: "finite A ==> finite B
- ==> card A + card B = card (A Un B) + card (A Int B)"
- by (fact card.union_inter [symmetric])
+lemma card_Un_Int:
+ assumes "finite A" and "finite B"
+ shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
+using assms proof (induct A)
+ case empty then show ?case by simp
+next
+ case (insert x A) then show ?case
+ by (auto simp add: insert_absorb Int_insert_left)
+qed
-lemma card_Un_disjoint: "finite A ==> finite B
- ==> A Int B = {} ==> card (A Un B) = card A + card B"
- by (fact card.union_disjoint)
+lemma card_Un_disjoint:
+ assumes "finite A" and "finite B"
+ assumes "A \<inter> B = {}"
+ shows "card (A \<union> B) = card A + card B"
+using assms card_Un_Int [of A B] by simp
lemma card_Diff_subset:
assumes "finite B" and "B \<subseteq> A"
@@ -2241,7 +1360,7 @@
apply(rule iffI)
apply(erule card_eq_SucD)
apply(auto)
-apply(subst card_insert)
+apply(subst card.insert)
apply(auto intro:ccontr)
done
@@ -2439,25 +1558,26 @@
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
by(fastforce simp:surj_def dest!: endo_inj_surj)
-corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
+corollary infinite_UNIV_nat [iff]:
+ "\<not> finite (UNIV :: nat set)"
proof
- assume "finite(UNIV::nat set)"
- with finite_UNIV_inj_surj[of Suc]
+ assume "finite (UNIV :: nat set)"
+ with finite_UNIV_inj_surj [of Suc]
show False by simp (blast dest: Suc_neq_Zero surjD)
qed
(* Often leads to bogus ATP proofs because of reduced type information, hence no_atp *)
-lemma infinite_UNIV_char_0[no_atp]:
- "\<not> finite (UNIV::'a::semiring_char_0 set)"
+lemma infinite_UNIV_char_0 [no_atp]:
+ "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
proof
- assume "finite (UNIV::'a set)"
- with subset_UNIV have "finite (range of_nat::'a set)"
+ assume "finite (UNIV :: 'a set)"
+ with subset_UNIV have "finite (range of_nat :: 'a set)"
by (rule finite_subset)
- moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
+ moreover have "inj (of_nat :: nat \<Rightarrow> 'a)"
by (simp add: inj_on_def)
- ultimately have "finite (UNIV::nat set)"
+ ultimately have "finite (UNIV :: nat set)"
by (rule finite_imageD)
- then show "False"
+ then show False
by simp
qed
--- a/src/HOL/GCD.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/GCD.thy Sat Mar 23 20:50:39 2013 +0100
@@ -1462,6 +1462,10 @@
subsection {* The complete divisibility lattice *}
+lemma semilattice_neutr_set_lcm_one_nat:
+ "semilattice_neutr_set lcm (1::nat)"
+ by default simp_all
+
interpretation gcd_semilattice_nat: semilattice_inf gcd "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)"
proof
case goal3 thus ?case by(metis gcd_unique_nat)
@@ -1486,33 +1490,62 @@
begin
definition
- "Lcm (M::nat set) = (if finite M then Finite_Set.fold lcm 1 M else 0)"
+ "Lcm (M::nat set) = (if finite M then semilattice_neutr_set.F lcm 1 M else 0)"
+
+lemma Lcm_nat_infinite:
+ "\<not> finite M \<Longrightarrow> Lcm M = (0::nat)"
+ by (simp add: Lcm_nat_def)
+
+lemma Lcm_nat_empty:
+ "Lcm {} = (1::nat)"
+proof -
+ interpret semilattice_neutr_set lcm "1::nat"
+ by (fact semilattice_neutr_set_lcm_one_nat)
+ show ?thesis by (simp add: Lcm_nat_def)
+qed
+
+lemma Lcm_nat_insert:
+ "Lcm (insert n M) = lcm (n::nat) (Lcm M)"
+proof (cases "finite M")
+ interpret semilattice_neutr_set lcm "1::nat"
+ by (fact semilattice_neutr_set_lcm_one_nat)
+ case True
+ then show ?thesis by (simp add: Lcm_nat_def)
+next
+ case False then show ?thesis by (simp add: Lcm_nat_infinite)
+qed
definition
"Gcd (M::nat set) = Lcm {d. \<forall>m\<in>M. d dvd m}"
instance ..
+
end
lemma dvd_Lcm_nat [simp]:
- fixes M :: "nat set" assumes "m \<in> M" shows "m dvd Lcm M"
- using lcm_semilattice_nat.sup_le_fold_sup[OF _ assms, of 1]
- by (simp add: Lcm_nat_def)
+ fixes M :: "nat set"
+ assumes "m \<in> M"
+ shows "m dvd Lcm M"
+proof (cases "finite M")
+ case False then show ?thesis by (simp add: Lcm_nat_infinite)
+next
+ case True then show ?thesis using assms by (induct M) (auto simp add: Lcm_nat_insert)
+qed
lemma Lcm_dvd_nat [simp]:
- fixes M :: "nat set" assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
+ fixes M :: "nat set"
+ assumes "\<forall>m\<in>M. m dvd n"
+ shows "Lcm M dvd n"
proof (cases "n = 0")
assume "n \<noteq> 0"
hence "finite {d. d dvd n}" by (rule finite_divisors_nat)
moreover have "M \<subseteq> {d. d dvd n}" using assms by fast
ultimately have "finite M" by (rule rev_finite_subset)
- thus ?thesis
- using lcm_semilattice_nat.fold_sup_le_sup [OF _ assms, of 1]
- by (simp add: Lcm_nat_def)
+ then show ?thesis using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
qed simp
interpretation gcd_lcm_complete_lattice_nat:
- complete_lattice Gcd Lcm gcd "op dvd" "%m n::nat. m dvd n & ~ n dvd m" lcm 1 0
+ complete_lattice Gcd Lcm gcd "op dvd" "\<lambda>m n::nat. m dvd n \<and> \<not> n dvd m" lcm 1 0
proof
case goal1 show ?case by simp
next
--- a/src/HOL/HOLCF/Compact_Basis.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/HOLCF/Compact_Basis.thy Sat Mar 23 20:50:39 2013 +0100
@@ -96,20 +96,23 @@
definition
fold_pd ::
"('a compact_basis \<Rightarrow> 'b::type) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a pd_basis \<Rightarrow> 'b"
- where "fold_pd g f t = fold1 f (g ` Rep_pd_basis t)"
+ where "fold_pd g f t = semilattice_set.F f (g ` Rep_pd_basis t)"
lemma fold_pd_PDUnit:
- assumes "class.ab_semigroup_idem_mult f"
+ assumes "semilattice f"
shows "fold_pd g f (PDUnit x) = g x"
-unfolding fold_pd_def Rep_PDUnit by simp
+proof -
+ from assms interpret semilattice_set f by (rule semilattice_set.intro)
+ show ?thesis by (simp add: fold_pd_def Rep_PDUnit)
+qed
lemma fold_pd_PDPlus:
- assumes "class.ab_semigroup_idem_mult f"
+ assumes "semilattice f"
shows "fold_pd g f (PDPlus t u) = f (fold_pd g f t) (fold_pd g f u)"
proof -
- interpret ab_semigroup_idem_mult f by fact
- show ?thesis unfolding fold_pd_def Rep_PDPlus
- by (simp add: image_Un fold1_Un2)
+ from assms interpret semilattice_set f by (rule semilattice_set.intro)
+ show ?thesis by (simp add: image_Un fold_pd_def Rep_PDPlus union)
qed
end
+
--- a/src/HOL/HOLCF/ConvexPD.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/HOLCF/ConvexPD.thy Sat Mar 23 20:50:39 2013 +0100
@@ -316,7 +316,7 @@
(\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<natural> y\<cdot>f)"
lemma ACI_convex_bind:
- "class.ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<natural> y\<cdot>f)"
+ "semilattice (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<natural> y\<cdot>f)"
apply unfold_locales
apply (simp add: convex_plus_assoc)
apply (simp add: convex_plus_commute)
--- a/src/HOL/HOLCF/LowerPD.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/HOLCF/LowerPD.thy Sat Mar 23 20:50:39 2013 +0100
@@ -309,7 +309,7 @@
(\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<flat> y\<cdot>f)"
lemma ACI_lower_bind:
- "class.ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<flat> y\<cdot>f)"
+ "semilattice (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<flat> y\<cdot>f)"
apply unfold_locales
apply (simp add: lower_plus_assoc)
apply (simp add: lower_plus_commute)
--- a/src/HOL/HOLCF/UpperPD.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/HOLCF/UpperPD.thy Sat Mar 23 20:50:39 2013 +0100
@@ -302,7 +302,7 @@
(\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<sharp> y\<cdot>f)"
lemma ACI_upper_bind:
- "class.ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<sharp> y\<cdot>f)"
+ "semilattice (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<sharp> y\<cdot>f)"
apply unfold_locales
apply (simp add: upper_plus_assoc)
apply (simp add: upper_plus_commute)
--- a/src/HOL/Lattices.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Lattices.thy Sat Mar 23 20:50:39 2013 +0100
@@ -716,32 +716,6 @@
qed
-subsection {* @{const min}/@{const max} on linear orders as
- special case of @{const inf}/@{const sup} *}
-
-sublocale linorder < min_max!: distrib_lattice min less_eq less max
-proof
- fix x y z
- show "max x (min y z) = min (max x y) (max x z)"
- by (auto simp add: min_def max_def)
-qed (auto simp add: min_def max_def not_le less_imp_le)
-
-lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{semilattice_inf, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
- by (rule ext)+ (auto intro: antisym)
-
-lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{semilattice_sup, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
- by (rule ext)+ (auto intro: antisym)
-
-lemmas le_maxI1 = min_max.sup_ge1
-lemmas le_maxI2 = min_max.sup_ge2
-
-lemmas min_ac = min_max.inf_assoc min_max.inf_commute
- min_max.inf.left_commute
-
-lemmas max_ac = min_max.sup_assoc min_max.sup_commute
- min_max.sup.left_commute
-
-
subsection {* Lattice on @{typ bool} *}
instantiation bool :: boolean_algebra
--- a/src/HOL/Library/Finite_Lattice.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Library/Finite_Lattice.thy Sat Mar 23 20:50:39 2013 +0100
@@ -39,6 +39,30 @@
by (metis finite_UNIV inf_Sup_absorb inf_top_left iso_tuple_UNIV_I)
-- "Derived definition of @{const top}."
+lemma finite_lattice_complete_Inf_empty:
+ "Inf {} = (top :: 'a::finite_lattice_complete)"
+ by (simp add: Inf_def)
+
+lemma finite_lattice_complete_Sup_empty:
+ "Sup {} = (bot :: 'a::finite_lattice_complete)"
+ by (simp add: Sup_def)
+
+lemma finite_lattice_complete_Inf_insert:
+ fixes A :: "'a::finite_lattice_complete set"
+ shows "Inf (insert x A) = inf x (Inf A)"
+proof -
+ interpret comp_fun_idem "inf :: 'a \<Rightarrow> _" by (fact comp_fun_idem_inf)
+ show ?thesis by (simp add: Inf_def)
+qed
+
+lemma finite_lattice_complete_Sup_insert:
+ fixes A :: "'a::finite_lattice_complete set"
+ shows "Sup (insert x A) = sup x (Sup A)"
+proof -
+ interpret comp_fun_idem "sup :: 'a \<Rightarrow> _" by (fact comp_fun_idem_sup)
+ show ?thesis by (simp add: Sup_def)
+qed
+
text {* The definitional assumptions
on the operators @{const Inf} and @{const Sup}
of class @{class finite_lattice_complete}
@@ -47,19 +71,19 @@
lemma finite_lattice_complete_Inf_lower:
"(x::'a::finite_lattice_complete) \<in> A \<Longrightarrow> Inf A \<le> x"
-unfolding Inf_def by (metis finite_code le_inf_iff fold_inf_le_inf)
+ using finite [of A] by (induct A) (auto simp add: finite_lattice_complete_Inf_insert intro: le_infI2)
lemma finite_lattice_complete_Inf_greatest:
"\<forall>x::'a::finite_lattice_complete \<in> A. z \<le> x \<Longrightarrow> z \<le> Inf A"
-unfolding Inf_def by (metis finite_code inf_le_fold_inf inf_top_right)
+ using finite [of A] by (induct A) (auto simp add: finite_lattice_complete_Inf_empty finite_lattice_complete_Inf_insert)
lemma finite_lattice_complete_Sup_upper:
"(x::'a::finite_lattice_complete) \<in> A \<Longrightarrow> Sup A \<ge> x"
-unfolding Sup_def by (metis finite_code le_sup_iff sup_le_fold_sup)
+ using finite [of A] by (induct A) (auto simp add: finite_lattice_complete_Sup_insert intro: le_supI2)
lemma finite_lattice_complete_Sup_least:
"\<forall>x::'a::finite_lattice_complete \<in> A. z \<ge> x \<Longrightarrow> z \<ge> Sup A"
-unfolding Sup_def by (metis finite_code fold_sup_le_sup sup_bot_right)
+ using finite [of A] by (induct A) (auto simp add: finite_lattice_complete_Sup_empty finite_lattice_complete_Sup_insert)
instance finite_lattice_complete \<subseteq> complete_lattice
proof
--- a/src/HOL/Library/Formal_Power_Series.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Library/Formal_Power_Series.thy Sat Mar 23 20:50:39 2013 +0100
@@ -1022,7 +1022,7 @@
also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m - i))" by (simp add: fps_mult_nth)
also have "\<dots> = 0" apply (rule setsum_0')
apply auto
- apply (case_tac "aa = m")
+ apply (case_tac "x = m")
using a0
apply simp
apply (rule H[rule_format])
@@ -2270,10 +2270,10 @@
unfolding fps_mult_nth
apply (rule setsum_0')
apply (clarsimp simp add: not_le)
- apply (case_tac "aaa < aa")
+ apply (case_tac "x < aa")
apply (rule startsby_zero_power_prefix[OF c0, rule_format])
apply simp
- apply (subgoal_tac "n - aaa < ba")
+ apply (subgoal_tac "n - x < ba")
apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format])
apply simp
apply arith
--- a/src/HOL/Library/Function_Algebras.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Library/Function_Algebras.thy Sat Mar 23 20:50:39 2013 +0100
@@ -97,9 +97,6 @@
instance "fun" :: (type, ab_semigroup_mult) ab_semigroup_mult
by default (simp add: fun_eq_iff mult.commute)
-instance "fun" :: (type, ab_semigroup_idem_mult) ab_semigroup_idem_mult
- by default (simp add: fun_eq_iff)
-
instance "fun" :: (type, monoid_mult) monoid_mult
by default (simp_all add: fun_eq_iff)
--- a/src/HOL/Library/Nat_Bijection.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Library/Nat_Bijection.thy Sat Mar 23 20:50:39 2013 +0100
@@ -377,7 +377,7 @@
by (metis finite_set_decode set_decode_inverse)
thus ?thesis using assms
apply auto
- apply (simp add: set_encode_def nat_add_commute setsum.F_subset_diff)
+ apply (simp add: set_encode_def nat_add_commute setsum.subset_diff)
done
qed
thus ?thesis
@@ -385,3 +385,4 @@
qed
end
+
--- a/src/HOL/Library/Permutations.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Library/Permutations.thy Sat Mar 23 20:50:39 2013 +0100
@@ -216,36 +216,36 @@
(* Permutations of index set for iterated operations. *)
(* ------------------------------------------------------------------------- *)
-lemma (in ab_semigroup_mult) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"
- shows "fold_image times f z S = fold_image times (f o p) z S"
- using fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z]
- unfolding permutes_image[OF pS] .
-lemma (in ab_semigroup_add) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"
- shows "fold_image plus f z S = fold_image plus (f o p) z S"
-proof-
- interpret ab_semigroup_mult plus apply unfold_locales apply (simp add: add_assoc)
- apply (simp add: add_commute) done
- from fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z]
- show ?thesis
- unfolding permutes_image[OF pS] .
+lemma (in comm_monoid_set) permute:
+ assumes "p permutes S"
+ shows "F g S = F (g o p) S"
+proof -
+ from `p permutes S` have "inj p" by (rule permutes_inj)
+ then have "inj_on p S" by (auto intro: subset_inj_on)
+ then have "F g (p ` S) = F (g o p) S" by (rule reindex)
+ moreover from `p permutes S` have "p ` S = S" by (rule permutes_image)
+ ultimately show ?thesis by simp
qed
-lemma setsum_permute: assumes pS: "p permutes S"
+lemma setsum_permute:
+ assumes "p permutes S"
shows "setsum f S = setsum (f o p) S"
- unfolding setsum_def using fold_image_permute[of S p f 0] pS by clarsimp
+ using assms by (fact setsum.permute)
-lemma setsum_permute_natseg:assumes pS: "p permutes {m .. n}"
+lemma setsum_permute_natseg:
+ assumes pS: "p permutes {m .. n}"
shows "setsum f {m .. n} = setsum (f o p) {m .. n}"
- using setsum_permute[OF pS, of f ] pS by blast
+ using setsum_permute [OF pS, of f ] pS by blast
-lemma setprod_permute: assumes pS: "p permutes S"
+lemma setprod_permute:
+ assumes "p permutes S"
shows "setprod f S = setprod (f o p) S"
- unfolding setprod_def
- using ab_semigroup_mult_class.fold_image_permute[of S p f 1] pS by clarsimp
+ using assms by (fact setprod.permute)
-lemma setprod_permute_natseg:assumes pS: "p permutes {m .. n}"
+lemma setprod_permute_natseg:
+ assumes pS: "p permutes {m .. n}"
shows "setprod f {m .. n} = setprod (f o p) {m .. n}"
- using setprod_permute[OF pS, of f ] pS by blast
+ using setprod_permute [OF pS, of f ] pS by blast
(* ------------------------------------------------------------------------- *)
(* Various combinations of transpositions with 2, 1 and 0 common elements. *)
@@ -835,7 +835,6 @@
by (simp add: o_def)
with bc have "b = c \<and> p = q" by blast
}
-
then show "inj_on ?f (insert a S \<times> ?P)"
unfolding inj_on_def
apply clarify by metis
@@ -843,3 +842,4 @@
qed
end
+
--- a/src/HOL/Library/RBT_Set.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Library/RBT_Set.thy Sat Mar 23 20:50:39 2013 +0100
@@ -316,11 +316,10 @@
assumes "is_rbt t"
shows "rbt_min t = rbt_min_opt t"
proof -
- interpret ab_semigroup_idem_mult "(min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" using ab_semigroup_idem_mult_min
- unfolding class.ab_semigroup_idem_mult_def by blast
- show ?thesis
- by (simp add: Min_eqI rbt_min_opt_is_min rbt_min_opt_in_set assms Min_def[symmetric]
- non_empty_rbt_keys fold1_set_fold[symmetric] rbt_min_def rbt_fold1_keys_def)
+ from assms have "hd (RBT_Impl.keys t) # tl (RBT_Impl.keys t) = RBT_Impl.keys t" by (cases t) simp_all
+ with assms show ?thesis
+ by (simp add: rbt_min_def rbt_fold1_keys_def rbt_min_opt_is_min
+ min_max.Inf_fin.set_eq_fold [symmetric] Min_eqI rbt_min_opt_in_set)
qed
(* maximum *)
@@ -337,12 +336,7 @@
fixes xs :: "('a :: linorder) list"
assumes "xs \<noteq> []"
shows "List.fold max (tl xs) (hd xs) = List.fold max (tl (rev xs)) (hd (rev xs))"
-proof -
- interpret ab_semigroup_idem_mult "(max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" using ab_semigroup_idem_mult_max
- unfolding class.ab_semigroup_idem_mult_def by blast
- show ?thesis
- using assms by (auto simp add: fold1_set_fold[symmetric])
-qed
+ using assms by (simp add: min_max.Sup_fin.set_eq_fold [symmetric])
lemma rbt_max_simps:
assumes "is_rbt (Branch c lt k v RBT_Impl.Empty)"
@@ -416,11 +410,10 @@
assumes "is_rbt t"
shows "rbt_max t = rbt_max_opt t"
proof -
- interpret ab_semigroup_idem_mult "(max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" using ab_semigroup_idem_mult_max
- unfolding class.ab_semigroup_idem_mult_def by blast
- show ?thesis
- by (simp add: Max_eqI rbt_max_opt_is_max rbt_max_opt_in_set assms Max_def[symmetric]
- non_empty_rbt_keys fold1_set_fold[symmetric] rbt_max_def rbt_fold1_keys_def)
+ from assms have "hd (RBT_Impl.keys t) # tl (RBT_Impl.keys t) = RBT_Impl.keys t" by (cases t) simp_all
+ with assms show ?thesis
+ by (simp add: rbt_max_def rbt_fold1_keys_def rbt_max_opt_is_max
+ min_max.Sup_fin.set_eq_fold [symmetric] Max_eqI rbt_max_opt_in_set)
qed
@@ -434,13 +427,13 @@
by transfer (simp add: rbt_fold1_keys_def)
lemma finite_fold1_fold1_keys:
- assumes "class.ab_semigroup_mult f"
- assumes "\<not> (is_empty t)"
- shows "Finite_Set.fold1 f (Set t) = fold1_keys f t"
+ assumes "semilattice f"
+ assumes "\<not> is_empty t"
+ shows "semilattice_set.F f (Set t) = fold1_keys f t"
proof -
- interpret ab_semigroup_mult f by fact
+ from `semilattice f` interpret semilattice_set f by (rule semilattice_set.intro)
show ?thesis using assms
- by (auto simp: fold1_keys_def_alt set_keys fold_def_alt fold1_distinct_set_fold non_empty_keys)
+ by (auto simp: fold1_keys_def_alt set_keys fold_def_alt non_empty_keys set_eq_fold [symmetric])
qed
(* minimum *)
@@ -658,14 +651,14 @@
lemma card_Set [code]:
"card (Set t) = fold_keys (\<lambda>_ n. n + 1) t 0"
-by (auto simp add: card_def fold_image_def intro!: finite_fold_fold_keys) (default, simp)
+ by (auto simp add: card.eq_fold intro: finite_fold_fold_keys comp_fun_commute_const)
lemma setsum_Set [code]:
"setsum f (Set xs) = fold_keys (plus o f) xs 0"
proof -
have "comp_fun_commute (\<lambda>x. op + (f x))" by default (auto simp: add_ac)
then show ?thesis
- by (auto simp add: setsum_def fold_image_def finite_fold_fold_keys o_def)
+ by (auto simp add: setsum.eq_fold finite_fold_fold_keys o_def)
qed
definition not_a_singleton_tree where [code del]: "not_a_singleton_tree x y = x y"
@@ -743,11 +736,10 @@
lemma Min_fin_set_fold [code]:
"Min (Set t) = (if is_empty t then not_non_empty_tree Min (Set t) else r_min_opt t)"
proof -
- have *:"(class.ab_semigroup_mult (min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a))" using ab_semigroup_idem_mult_min
- unfolding class.ab_semigroup_idem_mult_def by blast
+ have *: "semilattice (min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" ..
+ with finite_fold1_fold1_keys [OF *, folded Min_def]
show ?thesis
- by (auto simp add: Min_def not_non_empty_tree_def finite_fold1_fold1_keys[OF *] r_min_alt_def
- r_min_eq_r_min_opt[symmetric])
+ by (simp add: not_non_empty_tree_def r_min_alt_def r_min_eq_r_min_opt [symmetric])
qed
lemma Inf_fin_set_fold [code]:
@@ -781,11 +773,10 @@
lemma Max_fin_set_fold [code]:
"Max (Set t) = (if is_empty t then not_non_empty_tree Max (Set t) else r_max_opt t)"
proof -
- have *:"(class.ab_semigroup_mult (max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a))" using ab_semigroup_idem_mult_max
- unfolding class.ab_semigroup_idem_mult_def by blast
+ have *: "semilattice (max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" ..
+ with finite_fold1_fold1_keys [OF *, folded Max_def]
show ?thesis
- by (auto simp add: Max_def not_non_empty_tree_def finite_fold1_fold1_keys[OF *] r_max_alt_def
- r_max_eq_r_max_opt[symmetric])
+ by (simp add: not_non_empty_tree_def r_max_alt_def r_max_eq_r_max_opt [symmetric])
qed
lemma Sup_fin_set_fold [code]:
--- a/src/HOL/Library/Saturated.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Library/Saturated.thy Sat Mar 23 20:50:39 2013 +0100
@@ -207,47 +207,65 @@
end
-instantiation sat :: (len) complete_lattice
+instantiation sat :: (len) "{Inf, Sup}"
begin
definition
- "Inf (A :: 'a sat set) = Finite_Set.fold min top A"
+ "Inf = (semilattice_neutr_set.F min top :: 'a sat set \<Rightarrow> 'a sat)"
definition
- "Sup (A :: 'a sat set) = Finite_Set.fold max bot A"
+ "Sup = (semilattice_neutr_set.F max bot :: 'a sat set \<Rightarrow> 'a sat)"
+
+instance ..
+
+end
-instance proof
+interpretation Inf_sat!: semilattice_neutr_set min "top :: 'a::len sat"
+where
+ "semilattice_neutr_set.F min (top :: 'a sat) = Inf"
+proof -
+ show "semilattice_neutr_set min (top :: 'a sat)" by default (simp add: min_def)
+ show "semilattice_neutr_set.F min (top :: 'a sat) = Inf" by (simp add: Inf_sat_def)
+qed
+
+interpretation Sup_sat!: semilattice_neutr_set max "bot :: 'a::len sat"
+where
+ "semilattice_neutr_set.F max (bot :: 'a sat) = Sup"
+proof -
+ show "semilattice_neutr_set max (bot :: 'a sat)" by default (simp add: max_def bot.extremum_unique)
+ show "semilattice_neutr_set.F max (bot :: 'a sat) = Sup" by (simp add: Sup_sat_def)
+qed
+
+instance sat :: (len) complete_lattice
+proof
fix x :: "'a sat"
fix A :: "'a sat set"
note finite
moreover assume "x \<in> A"
- ultimately have "Finite_Set.fold min top A \<le> min x top" by (rule min_max.fold_inf_le_inf)
- then show "Inf A \<le> x" by (simp add: Inf_sat_def)
+ ultimately show "Inf A \<le> x"
+ by (induct A) (auto intro: min_max.le_infI2)
next
fix z :: "'a sat"
fix A :: "'a sat set"
note finite
moreover assume z: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
- ultimately have "min z top \<le> Finite_Set.fold min top A" by (blast intro: min_max.inf_le_fold_inf)
- then show "z \<le> Inf A" by (simp add: Inf_sat_def min_def)
+ ultimately show "z \<le> Inf A" by (induct A) simp_all
next
fix x :: "'a sat"
fix A :: "'a sat set"
note finite
moreover assume "x \<in> A"
- ultimately have "max x bot \<le> Finite_Set.fold max bot A" by (rule min_max.sup_le_fold_sup)
- then show "x \<le> Sup A" by (simp add: Sup_sat_def)
+ ultimately show "x \<le> Sup A"
+ by (induct A) (auto intro: min_max.le_supI2)
next
fix z :: "'a sat"
fix A :: "'a sat set"
note finite
moreover assume z: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
- ultimately have "Finite_Set.fold max bot A \<le> max z bot" by (blast intro: min_max.fold_sup_le_sup)
- then show "Sup A \<le> z" by (simp add: Sup_sat_def max_def bot_unique)
+ ultimately show "Sup A \<le> z" by (induct A) auto
qed
-end
-
hide_const (open) sat_of_nat
end
+
--- a/src/HOL/List.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/List.thy Sat Mar 23 20:50:39 2013 +0100
@@ -2734,51 +2734,11 @@
lemma (in comp_fun_commute) fold_set_fold_remdups:
"Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
- by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
-
-lemma (in ab_semigroup_mult) fold1_distinct_set_fold:
- assumes "xs \<noteq> []"
- assumes d: "distinct xs"
- shows "Finite_Set.fold1 times (set xs) = List.fold times (tl xs) (hd xs)"
-proof -
- interpret comp_fun_commute times by (fact comp_fun_commute)
- from assms obtain y ys where xs: "xs = y # ys"
- by (cases xs) auto
- then have *: "y \<notin> set ys" using assms by simp
- from xs d have **: "remdups ys = ys" by safe (induct ys, auto)
- show ?thesis
- proof (cases "set ys = {}")
- case True with xs show ?thesis by simp
- next
- case False
- then have "fold1 times (Set.insert y (set ys)) = Finite_Set.fold times y (set ys)"
- by (simp_all add: fold1_eq_fold *)
- with xs show ?thesis
- by (simp add: fold_set_fold_remdups **)
- qed
-qed
+ by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_left_comm insert_absorb)
lemma (in comp_fun_idem) fold_set_fold:
"Finite_Set.fold f y (set xs) = fold f xs y"
- by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
-
-lemma (in ab_semigroup_idem_mult) fold1_set_fold:
- assumes "xs \<noteq> []"
- shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
-proof -
- interpret comp_fun_idem times by (fact comp_fun_idem)
- from assms obtain y ys where xs: "xs = y # ys"
- by (cases xs) auto
- show ?thesis
- proof (cases "set ys = {}")
- case True with xs show ?thesis by simp
- next
- case False
- then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
- by (simp only: finite_set fold1_eq_fold_idem)
- with xs show ?thesis by (simp add: fold_set_fold mult_commute)
- qed
-qed
+ by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_left_comm)
lemma union_set_fold [code]:
"set xs \<union> A = fold Set.insert xs A"
@@ -2813,49 +2773,18 @@
"A \<inter> List.coset xs = fold Set.remove xs A"
by (simp add: Diff_eq [symmetric] minus_set_fold)
-lemma (in lattice) Inf_fin_set_fold:
- "Inf_fin (set (x # xs)) = fold inf xs x"
+lemma (in semilattice_set) set_eq_fold:
+ "F (set (x # xs)) = fold f xs x"
proof -
- interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
- by (fact ab_semigroup_idem_mult_inf)
- show ?thesis
- by (simp add: Inf_fin_def fold1_set_fold del: set.simps)
-qed
-
-declare Inf_fin_set_fold [code]
-
-lemma (in lattice) Sup_fin_set_fold:
- "Sup_fin (set (x # xs)) = fold sup xs x"
-proof -
- interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
- by (fact ab_semigroup_idem_mult_sup)
- show ?thesis
- by (simp add: Sup_fin_def fold1_set_fold del: set.simps)
+ interpret comp_fun_idem f
+ by default (simp_all add: fun_eq_iff left_commute)
+ show ?thesis by (simp add: eq_fold fold_set_fold)
qed
-declare Sup_fin_set_fold [code]
-
-lemma (in linorder) Min_fin_set_fold:
- "Min (set (x # xs)) = fold min xs x"
-proof -
- interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
- by (fact ab_semigroup_idem_mult_min)
- show ?thesis
- by (simp add: Min_def fold1_set_fold del: set.simps)
-qed
-
-declare Min_fin_set_fold [code]
-
-lemma (in linorder) Max_fin_set_fold:
- "Max (set (x # xs)) = fold max xs x"
-proof -
- interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
- by (fact ab_semigroup_idem_mult_max)
- show ?thesis
- by (simp add: Max_def fold1_set_fold del: set.simps)
-qed
-
-declare Max_fin_set_fold [code]
+declare Inf_fin.set_eq_fold [code]
+declare Sup_fin.set_eq_fold [code]
+declare min_max.Inf_fin.set_eq_fold [code]
+declare min_max.Sup_fin.set_eq_fold [code]
lemma (in complete_lattice) Inf_set_fold:
"Inf (set xs) = fold inf xs top"
@@ -4915,24 +4844,36 @@
sets to lists but one should convert in the other direction (via
@{const set}). *}
+subsubsection {* @{text sorted_list_of_set} *}
+
+text{* This function maps (finite) linearly ordered sets to sorted
+lists. Warning: in most cases it is not a good idea to convert from
+sets to lists but one should convert in the other direction (via
+@{const set}). *}
+
+definition (in linorder) sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
+ "sorted_list_of_set = folding.F insort []"
+
+sublocale linorder < sorted_list_of_set!: folding insort Nil
+where
+ "folding.F insort [] = sorted_list_of_set"
+proof -
+ interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
+ show "folding insort" by default (fact comp_fun_commute)
+ show "folding.F insort [] = sorted_list_of_set" by (simp only: sorted_list_of_set_def)
+qed
+
context linorder
begin
-definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
-"sorted_list_of_set = Finite_Set.fold insort []"
-
-lemma sorted_list_of_set_empty [simp]:
+lemma sorted_list_of_set_empty:
"sorted_list_of_set {} = []"
- by (simp add: sorted_list_of_set_def)
+ by (fact sorted_list_of_set.empty)
lemma sorted_list_of_set_insert [simp]:
assumes "finite A"
shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
-proof -
- interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
- from assms show ?thesis
- by (simp add: sorted_list_of_set_def fold_insert_remove)
-qed
+ using assms by (fact sorted_list_of_set.insert_remove)
lemma sorted_list_of_set [simp]:
"finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A)
@@ -4943,7 +4884,7 @@
"sorted_list_of_set (set xs) = sort (remdups xs)"
proof -
interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
- show ?thesis by (simp add: sorted_list_of_set_def sort_conv_fold fold_set_fold_remdups)
+ show ?thesis by (simp add: sorted_list_of_set.eq_fold sort_conv_fold fold_set_fold_remdups)
qed
lemma sorted_list_of_set_remove:
--- a/src/HOL/MacLaurin.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/MacLaurin.thy Sat Mar 23 20:50:39 2013 +0100
@@ -428,7 +428,7 @@
apply (simp (no_asm))
apply (simp (no_asm) add: sin_expansion_lemma)
apply (force intro!: DERIV_intros)
-apply (subst (asm) setsum_0', clarify, case_tac "a", simp, simp)
+apply (subst (asm) setsum_0', clarify, case_tac "x", simp, simp)
apply (cases n, simp, simp)
apply (rule ccontr, simp)
apply (drule_tac x = x in spec, simp)
--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Sat Mar 23 20:50:39 2013 +0100
@@ -188,9 +188,6 @@
instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
by default (vector mult_commute)
-instance vec :: (ab_semigroup_idem_mult, finite) ab_semigroup_idem_mult
- by default (vector mult_idem)
-
instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
by default vector
--- a/src/HOL/Multivariate_Analysis/Determinants.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Determinants.thy Sat Mar 23 20:50:39 2013 +0100
@@ -103,18 +103,7 @@
lemma setprod_permute:
assumes p: "p permutes S"
shows "setprod f S = setprod (f o p) S"
-proof-
- {assume "\<not> finite S" hence ?thesis by simp}
- moreover
- {assume fS: "finite S"
- then have ?thesis
- apply (simp add: setprod_def cong del:strong_setprod_cong)
- apply (rule ab_semigroup_mult.fold_image_permute)
- apply (auto simp add: p)
- apply unfold_locales
- done}
- ultimately show ?thesis by blast
-qed
+ using assms by (fact setprod.permute)
lemma setproduct_permute_nat_interval: "p permutes {m::nat .. n} ==> setprod f {m..n} = setprod (f o p) {m..n}"
by (blast intro!: setprod_permute)
--- a/src/HOL/Multivariate_Analysis/Integration.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Sat Mar 23 20:50:39 2013 +0100
@@ -2744,12 +2744,15 @@
subsection {* Additivity of content. *}
-lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
-proof- have *:"setsum f s = setsum f (support op + f s)"
- apply(rule setsum_mono_zero_right)
+lemma setsum_iterate:
+ assumes "finite s" shows "setsum f s = iterate op + s f"
+proof -
+ have *: "setsum f s = setsum f (support op + f s)"
+ apply (rule setsum_mono_zero_right)
unfolding support_def neutral_monoid using assms by auto
- thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
- unfolding neutral_monoid . qed
+ then show ?thesis unfolding * iterate_def fold'_def setsum.eq_fold
+ unfolding neutral_monoid by (simp add: comp_def)
+qed
lemma additive_content_division: assumes "d division_of {a..b}"
shows "setsum content d = content({a..b})"
--- a/src/HOL/Number_Theory/UniqueFactorization.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Number_Theory/UniqueFactorization.thy Sat Mar 23 20:50:39 2013 +0100
@@ -36,36 +36,71 @@
"ALL i :# M. P i"?
*)
+no_notation times (infixl "*" 70)
+no_notation Groups.one ("1")
+
+locale comm_monoid_mset = comm_monoid
+begin
+
+definition F :: "'a multiset \<Rightarrow> 'a"
+where
+ eq_fold: "F M = Multiset.fold f 1 M"
+
+lemma empty [simp]:
+ "F {#} = 1"
+ by (simp add: eq_fold)
+
+lemma singleton [simp]:
+ "F {#x#} = x"
+proof -
+ interpret comp_fun_commute
+ by default (simp add: fun_eq_iff left_commute)
+ show ?thesis by (simp add: eq_fold)
+qed
+
+lemma union [simp]:
+ "F (M + N) = F M * F N"
+proof -
+ interpret comp_fun_commute f
+ by default (simp add: fun_eq_iff left_commute)
+ show ?thesis by (induct N) (simp_all add: left_commute eq_fold)
+qed
+
+end
+
+notation times (infixl "*" 70)
+notation Groups.one ("1")
+
+definition (in comm_monoid_mult) msetprod :: "'a multiset \<Rightarrow> 'a"
+where
+ "msetprod = comm_monoid_mset.F times 1"
+
+sublocale comm_monoid_mult < msetprod!: comm_monoid_mset times 1
+where
+ "comm_monoid_mset.F times 1 = msetprod"
+proof -
+ show "comm_monoid_mset times 1" ..
+ from msetprod_def show "comm_monoid_mset.F times 1 = msetprod" by rule
+qed
+
context comm_monoid_mult
begin
-definition msetprod :: "'a multiset \<Rightarrow> 'a"
-where
- "msetprod M = Multiset.fold times 1 M"
+lemma msetprod_empty:
+ "msetprod {#} = 1"
+ by (fact msetprod.empty)
-lemma msetprod_empty [simp]:
- "msetprod {#} = 1"
- by (simp add: msetprod_def)
-
-lemma msetprod_singleton [simp]:
+lemma msetprod_singleton:
"msetprod {#x#} = x"
-proof -
- interpret comp_fun_commute times
- by (fact comp_fun_commute)
- show ?thesis by (simp add: msetprod_def)
-qed
+ by (fact msetprod.singleton)
-lemma msetprod_Un [simp]:
+lemma msetprod_Un:
"msetprod (A + B) = msetprod A * msetprod B"
-proof -
- interpret comp_fun_commute times
- by (fact comp_fun_commute)
- show ?thesis by (induct B) (simp_all add: msetprod_def mult_ac)
-qed
+ by (fact msetprod.union)
lemma msetprod_multiplicity:
"msetprod M = setprod (\<lambda>x. x ^ count M x) (set_of M)"
- by (simp add: msetprod_def setprod_def Multiset.fold_def fold_image_def funpow_times_power)
+ by (simp add: Multiset.fold_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)
abbreviation msetprod_image :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b multiset \<Rightarrow> 'a"
where
@@ -111,8 +146,7 @@
by arith
moreover {
assume "n = 1"
- then have "(ALL p : set_of {#}. prime p) & n = (PROD i :# {#}. i)"
- by (auto simp add: msetprod_def)
+ then have "(ALL p : set_of {#}. prime p) & n = (PROD i :# {#}. i)" by auto
} moreover {
assume "n > 1" and "prime n"
then have "(ALL p : set_of {# n #}. prime p) & n = (PROD i :# {# n #}. i)"
--- a/src/HOL/Old_Number_Theory/Finite2.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Old_Number_Theory/Finite2.thy Sat Mar 23 20:50:39 2013 +0100
@@ -23,7 +23,7 @@
assume "finite S"
thus ?thesis using a by induct (simp_all add: zcong_zadd)
next
- assume "infinite S" thus ?thesis by(simp add:setsum_def)
+ assume "infinite S" thus ?thesis by simp
qed
lemma setprod_same_function_zcong:
@@ -33,7 +33,7 @@
assume "finite S"
thus ?thesis using a by induct (simp_all add: zcong_zmult)
next
- assume "infinite S" thus ?thesis by(simp add:setprod_def)
+ assume "infinite S" thus ?thesis by simp
qed
lemma setsum_const: "finite X ==> setsum (%x. (c :: int)) X = c * int(card X)"
--- a/src/HOL/Old_Number_Theory/Pocklington.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Old_Number_Theory/Pocklington.thy Sat Mar 23 20:50:39 2013 +0100
@@ -566,7 +566,7 @@
[x1 = x2] (mod n) \<and> [y1 = y2] (mod n) \<longrightarrow> [x1 * y1 = x2 * y2] (mod n)"
by blast
have th4:"\<forall>x\<in>S. [a x mod n = a x] (mod n)" by (simp add: modeq_def)
- from fold_image_related[where h="(\<lambda>m. a(m) mod n)" and g=a, OF th1 th3 fS, OF th4] show ?thesis unfolding setprod_def by (simp add: fS)
+ from setprod.related [where h="(\<lambda>m. a(m) mod n)" and g=a, OF th1 th3 fS, OF th4] show ?thesis by (simp add: fS)
qed
lemma nproduct_cmul:
@@ -577,7 +577,7 @@
lemma coprime_nproduct:
assumes fS: "finite S" and Sn: "\<forall>x\<in>S. coprime n (a x)"
shows "coprime n (setprod a S)"
- using fS unfolding setprod_def by (rule finite_subset_induct)
+ using fS by (rule finite_subset_induct)
(insert Sn, auto simp add: coprime_mul)
lemma fermat_little: assumes an: "coprime a n"
@@ -607,12 +607,8 @@
hence hS: "?h ` ?S = ?S"by (auto simp add: image_iff)
have "a\<noteq>0" using an n1 nz apply- apply (rule ccontr) by simp
hence inj: "inj_on (op * a) ?S" unfolding inj_on_def by simp
-
- have eq0: "fold_image op * (?h \<circ> op * a) 1 {m. coprime m n \<and> m < n} =
- fold_image op * (\<lambda>m. m) 1 {m. coprime m n \<and> m < n}"
- proof (rule fold_image_eq_general[where h="?h o (op * a)"])
- show "finite ?S" using fS .
- next
+ have eq0: "setprod (?h \<circ> op * a) {m. coprime m n \<and> m < n} = setprod (\<lambda>m. m) {m. coprime m n \<and> m < n}"
+ proof (rule setprod.eq_general [where h="?h o (op * a)"])
{fix y assume yS: "y \<in> ?S" hence y: "coprime y n" "y < n" by simp_all
from cong_solve_unique[OF an nz, of y]
obtain x where x:"x < n" "[a * x = y] (mod n)" "\<forall>z. z < n \<and> [a * z = y] (mod n) \<longrightarrow> z=x" by blast
--- a/src/HOL/Probability/Fin_Map.thy Sat Mar 23 17:11:06 2013 +0100
+++ b/src/HOL/Probability/Fin_Map.thy Sat Mar 23 20:50:39 2013 +0100
@@ -406,7 +406,7 @@
next
fix P Q::"'a \<Rightarrow>\<^isub>F 'b"
have Max_eq_iff: "\<And>A m. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> (Max A = m) = (m \<in> A \<and> (\<forall>a\<in>A. a \<le> m))"
- by (metis Max.in_idem Max_in max_def min_max.sup.commute order_refl)
+ by (auto intro: Max_in Max_eqI)
show "dist P Q = 0 \<longleftrightarrow> P = Q"
by (auto simp: finmap_eq_iff dist_finmap_def Max_ge_iff finite_proj_diag Max_eq_iff
intro!: Max_eqI image_eqI[where x=undefined])