--- a/src/HOL/BNF/More_BNFs.thy Tue Jul 16 10:18:25 2013 +0200
+++ b/src/HOL/BNF/More_BNFs.thy Tue Jul 16 15:59:55 2013 +0200
@@ -231,7 +231,7 @@
lemmas [simp] = fset.map_comp' fset.map_id' fset.set_map'
lemma fset_rel_fset: "set_rel \<chi> (fset A1) (fset A2) = fset_rel \<chi> A1 A2"
- unfolding fset_rel_def set_rel_def by auto
+ unfolding fset_rel_def set_rel_def by auto
(* Countable sets *)
@@ -263,16 +263,15 @@
by (erule finite_Collect_subsets)
lemma rcset_to_rcset: "countable A \<Longrightarrow> rcset (the_inv rcset A) = A"
-apply (rule f_the_inv_into_f)
-unfolding inj_on_def Rep_cset_inject using rcset_surj by auto
+ apply (rule f_the_inv_into_f[unfolded inj_on_def image_iff])
+ apply transfer' apply simp
+ apply transfer' apply simp
+ done
lemma Collect_Int_Times:
"{(x, y). R x y} \<inter> A \<times> B = {(x, y). R x y \<and> x \<in> A \<and> y \<in> B}"
by auto
-lemma rcset_map': "rcset (cIm f x) = f ` rcset x"
-unfolding cIm_def[abs_def] by simp
-
definition cset_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a cset \<Rightarrow> 'b cset \<Rightarrow> bool" where
"cset_rel R a b \<longleftrightarrow>
(\<forall>t \<in> rcset a. \<exists>u \<in> rcset b. R t u) \<and>
@@ -280,68 +279,55 @@
lemma cset_rel_aux:
"(\<forall>t \<in> rcset a. \<exists>u \<in> rcset b. R t u) \<and> (\<forall>t \<in> rcset b. \<exists>u \<in> rcset a. R u t) \<longleftrightarrow>
- ((Grp {x. rcset x \<subseteq> {(a, b). R a b}} (cIm fst))\<inverse>\<inverse> OO
- Grp {x. rcset x \<subseteq> {(a, b). R a b}} (cIm snd)) a b" (is "?L = ?R")
+ ((Grp {x. rcset x \<subseteq> {(a, b). R a b}} (cimage fst))\<inverse>\<inverse> OO
+ Grp {x. rcset x \<subseteq> {(a, b). R a b}} (cimage snd)) a b" (is "?L = ?R")
proof
assume ?L
def R' \<equiv> "the_inv rcset (Collect (split R) \<inter> (rcset a \<times> rcset b))"
(is "the_inv rcset ?L'")
- have "countable ?L'" by auto
+ have L: "countable ?L'" by auto
hence *: "rcset R' = ?L'" unfolding R'_def using fset_to_fset by (intro rcset_to_rcset)
- show ?R unfolding Grp_def relcompp.simps conversep.simps
+ thus ?R unfolding Grp_def relcompp.simps conversep.simps
proof (intro CollectI prod_caseI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
- have "rcset a = fst ` ({(x, y). R x y} \<inter> rcset a \<times> rcset b)" (is "_ = ?A")
- using conjunct1[OF `?L`] unfolding image_def by (auto simp add: Collect_Int_Times)
- hence "a = acset ?A" by (metis acset_rcset)
- thus "a = cIm fst R'" unfolding cIm_def * by auto
- have "rcset b = snd ` ({(x, y). R x y} \<inter> rcset a \<times> rcset b)" (is "_ = ?B")
- using conjunct2[OF `?L`] unfolding image_def by (auto simp add: Collect_Int_Times)
- hence "b = acset ?B" by (metis acset_rcset)
- thus "b = cIm snd R'" unfolding cIm_def * by auto
- qed (auto simp add: *)
+ from * `?L` show "a = cimage fst R'" by transfer (auto simp: image_def Collect_Int_Times)
+ next
+ from * `?L` show "b = cimage snd R'" by transfer (auto simp: image_def Collect_Int_Times)
+ qed simp_all
next
assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
- apply (simp add: subset_eq Ball_def)
- apply (rule conjI)
- apply (clarsimp, metis (lifting, no_types) rcset_map' image_iff surjective_pairing)
- apply (clarsimp)
- by (metis Domain.intros Range.simps rcset_map' fst_eq_Domain snd_eq_Range)
+ by transfer force
qed
-bnf cIm [rcset] "\<lambda>_::'a cset. natLeq" ["cEmp"] cset_rel
+bnf cimage [rcset] "\<lambda>_::'a cset. natLeq" ["cempty"] cset_rel
proof -
- show "cIm id = id" unfolding cIm_def[abs_def] id_def by auto
+ show "cimage id = id" by transfer' simp
next
- fix f g show "cIm (g \<circ> f) = cIm g \<circ> cIm f"
- unfolding cIm_def[abs_def] apply(rule ext) unfolding comp_def by auto
+ fix f g show "cimage (g \<circ> f) = cimage g \<circ> cimage f" by transfer' fastforce
next
fix C f g assume eq: "\<And>a. a \<in> rcset C \<Longrightarrow> f a = g a"
- thus "cIm f C = cIm g C"
- unfolding cIm_def[abs_def] unfolding image_def by auto
+ thus "cimage f C = cimage g C" by transfer force
next
- fix f show "rcset \<circ> cIm f = op ` f \<circ> rcset" unfolding cIm_def[abs_def] by auto
+ fix f show "rcset \<circ> cimage f = op ` f \<circ> rcset" by transfer' fastforce
next
show "card_order natLeq" by (rule natLeq_card_order)
next
show "cinfinite natLeq" by (rule natLeq_cinfinite)
next
- fix C show "|rcset C| \<le>o natLeq" using rcset unfolding countable_card_le_natLeq .
+ fix C show "|rcset C| \<le>o natLeq" by transfer (unfold countable_card_le_natLeq)
next
fix A B1 B2 f1 f2 p1 p2
assume wp: "wpull A B1 B2 f1 f2 p1 p2"
show "wpull {x. rcset x \<subseteq> A} {x. rcset x \<subseteq> B1} {x. rcset x \<subseteq> B2}
- (cIm f1) (cIm f2) (cIm p1) (cIm p2)"
+ (cimage f1) (cimage f2) (cimage p1) (cimage p2)"
unfolding wpull_def proof safe
fix y1 y2
assume Y1: "rcset y1 \<subseteq> B1" and Y2: "rcset y2 \<subseteq> B2"
- assume "cIm f1 y1 = cIm f2 y2"
- hence EQ: "f1 ` (rcset y1) = f2 ` (rcset y2)"
- unfolding cIm_def by auto
+ assume "cimage f1 y1 = cimage f2 y2"
+ hence EQ: "f1 ` (rcset y1) = f2 ` (rcset y2)" by transfer
with Y1 Y2 obtain X where X: "X \<subseteq> A"
and Y1: "p1 ` X = rcset y1" and Y2: "p2 ` X = rcset y2"
- using wpull_image[OF wp] unfolding wpull_def Pow_def
- unfolding Bex_def mem_Collect_eq apply -
- apply(erule allE[of _ "rcset y1"], erule allE[of _ "rcset y2"]) by auto
+ using wpull_image[OF wp] unfolding wpull_def Pow_def Bex_def mem_Collect_eq
+ by (auto elim!: allE[of _ "rcset y1"] allE[of _ "rcset y2"])
have "\<forall> y1' \<in> rcset y1. \<exists> x. x \<in> X \<and> y1' = p1 x" using Y1 by auto
then obtain q1 where q1: "\<forall> y1' \<in> rcset y1. q1 y1' \<in> X \<and> y1' = p1 (q1 y1')" by metis
have "\<forall> y2' \<in> rcset y2. \<exists> x. x \<in> X \<and> y2' = p2 x" using Y2 by auto
@@ -350,16 +336,17 @@
have X': "X' \<subseteq> A" and Y1: "p1 ` X' = rcset y1" and Y2: "p2 ` X' = rcset y2"
using X Y1 Y2 q1 q2 unfolding X'_def by fast+
have fX': "countable X'" unfolding X'_def by simp
- then obtain x where X'eq: "X' = rcset x" by (metis rcset_acset)
- show "\<exists>x\<in>{x. rcset x \<subseteq> A}. cIm p1 x = y1 \<and> cIm p2 x = y2"
- apply(intro bexI[of _ "x"]) using X' Y1 Y2 unfolding X'eq cIm_def by auto
+ then obtain x where X'eq: "X' = rcset x" by transfer blast
+ show "\<exists>x\<in>{x. rcset x \<subseteq> A}. cimage p1 x = y1 \<and> cimage p2 x = y2"
+ using X' Y1 Y2 unfolding X'eq by (intro bexI[of _ "x"]) (transfer, auto)
qed
next
fix R
show "cset_rel R =
- (Grp {x. rcset x \<subseteq> Collect (split R)} (cIm fst))\<inverse>\<inverse> OO Grp {x. rcset x \<subseteq> Collect (split R)} (cIm snd)"
+ (Grp {x. rcset x \<subseteq> Collect (split R)} (cimage fst))\<inverse>\<inverse> OO
+ Grp {x. rcset x \<subseteq> Collect (split R)} (cimage snd)"
unfolding cset_rel_def[abs_def] cset_rel_aux by simp
-qed (unfold cEmp_def, auto)
+qed (transfer, simp)
(* Multisets *)
@@ -375,108 +362,76 @@
finally show ?thesis .
qed
-(* *)
-definition mmap :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> nat) \<Rightarrow> 'b \<Rightarrow> nat" where
-"mmap h f b = setsum f {a. h a = b \<and> f a > 0}"
-
-lemma mmap_id: "mmap id = id"
-proof (rule ext)+
- fix f a show "mmap id f a = id f a"
- proof(cases "f a = 0")
- case False
- hence 1: "{aa. aa = a \<and> 0 < f aa} = {a}" by auto
- show ?thesis by (simp add: mmap_def id_apply 1)
- qed(unfold mmap_def, auto)
-qed
-
-lemma inj_on_setsum_inv:
-assumes f: "f \<in> multiset"
-and 1: "(0::nat) < setsum f {a. h a = b' \<and> 0 < f a}" (is "0 < setsum f ?A'")
-and 2: "{a. h a = b \<and> 0 < f a} = {a. h a = b' \<and> 0 < f a}" (is "?A = ?A'")
-shows "b = b'"
-proof-
- have "finite ?A'" using f unfolding multiset_def by auto
- hence "?A' \<noteq> {}" using 1 by (auto simp add: setsum_gt_0_iff)
- thus ?thesis using 2 by auto
-qed
-
-lemma mmap_comp:
-fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
-assumes f: "f \<in> multiset"
-shows "mmap (h2 o h1) f = (mmap h2 o mmap h1) f"
-unfolding mmap_def[abs_def] comp_def proof(rule ext)+
- fix c :: 'c
- let ?A = "{a. h2 (h1 a) = c \<and> 0 < f a}"
- let ?As = "\<lambda> b. {a. h1 a = b \<and> 0 < f a}"
- let ?B = "{b. h2 b = c \<and> 0 < setsum f (?As b)}"
- have 0: "{?As b | b. b \<in> ?B} = ?As ` ?B" by auto
- have "\<And> b. finite (?As b)" using f unfolding multiset_def by simp
- hence "?B = {b. h2 b = c \<and> ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
- hence A: "?A = \<Union> {?As b | b. b \<in> ?B}" by auto
- have "setsum f ?A = setsum (setsum f) {?As b | b. b \<in> ?B}"
- unfolding A apply(rule setsum_Union_disjoint)
- using f unfolding multiset_def by auto
- also have "... = setsum (setsum f) (?As ` ?B)" unfolding 0 ..
- also have "... = setsum (setsum f o ?As) ?B" apply(rule setsum_reindex)
- unfolding inj_on_def apply auto using inj_on_setsum_inv[OF f, of h1] by blast
- also have "... = setsum (\<lambda> b. setsum f (?As b)) ?B" unfolding comp_def ..
- finally show "setsum f ?A = setsum (\<lambda> b. setsum f (?As b)) ?B" .
-qed
-
-lemma mmap_comp1:
-fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
-assumes "f \<in> multiset"
-shows "mmap (\<lambda> a. h2 (h1 a)) f = mmap h2 (mmap h1 f)"
-using mmap_comp[OF assms] unfolding comp_def by auto
-
-lemma mmap:
-assumes "f \<in> multiset"
-shows "mmap h f \<in> multiset"
-using assms unfolding mmap_def[abs_def] multiset_def proof safe
+lift_definition mmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" is
+ "\<lambda>h f b. setsum f {a. h a = b \<and> f a > 0} :: nat"
+unfolding multiset_def proof safe
+ fix h :: "'a \<Rightarrow> 'b" and f :: "'a \<Rightarrow> nat"
assume fin: "finite {a. 0 < f a}" (is "finite ?A")
show "finite {b. 0 < setsum f {a. h a = b \<and> 0 < f a}}"
(is "finite {b. 0 < setsum f (?As b)}")
proof- let ?B = "{b. 0 < setsum f (?As b)}"
- have "\<And> b. finite (?As b)" using assms unfolding multiset_def by simp
+ have "\<And> b. finite (?As b)" using fin by simp
hence B: "?B = {b. ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
hence "?B \<subseteq> h ` ?A" by auto
thus ?thesis using finite_surj[OF fin] by auto
qed
qed
-lemma mmap_cong:
-assumes "\<And>a. a \<in># M \<Longrightarrow> f a = g a"
-shows "mmap f (count M) = mmap g (count M)"
-using assms unfolding mmap_def[abs_def] by (intro ext, intro setsum_cong) auto
-
-abbreviation supp where "supp f \<equiv> {a. f a > 0}"
-
-lemma mmap_image_comp:
-assumes f: "f \<in> multiset"
-shows "(supp o mmap h) f = (image h o supp) f"
-unfolding mmap_def[abs_def] comp_def proof-
- have "\<And> b. finite {a. h a = b \<and> 0 < f a}" (is "\<And> b. finite (?As b)")
- using f unfolding multiset_def by auto
- thus "{b. 0 < setsum f (?As b)} = h ` {a. 0 < f a}"
- by (auto simp add: setsum_gt_0_iff)
+lemma mmap_id: "mmap id = id"
+proof (intro ext multiset_eqI)
+ fix f a show "count (mmap id f) a = count (id f) a"
+ proof (cases "count f a = 0")
+ case False
+ hence 1: "{aa. aa = a \<and> aa \<in># f} = {a}" by auto
+ thus ?thesis by transfer auto
+ qed (transfer, simp)
qed
-lemma mmap_image:
-assumes f: "f \<in> multiset"
-shows "supp (mmap h f) = h ` (supp f)"
-using mmap_image_comp[OF assms] unfolding comp_def .
+lemma inj_on_setsum_inv:
+assumes 1: "(0::nat) < setsum (count f) {a. h a = b' \<and> a \<in># f}" (is "0 < setsum (count f) ?A'")
+and 2: "{a. h a = b \<and> a \<in># f} = {a. h a = b' \<and> a \<in># f}" (is "?A = ?A'")
+shows "b = b'"
+using assms by (auto simp add: setsum_gt_0_iff)
-lemma set_of_Abs_multiset:
-assumes f: "f \<in> multiset"
-shows "set_of (Abs_multiset f) = supp f"
-using assms unfolding set_of_def by (auto simp: Abs_multiset_inverse)
+lemma mmap_comp:
+fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
+shows "mmap (h2 o h1) = mmap h2 o mmap h1"
+proof (intro ext multiset_eqI)
+ fix f :: "'a multiset" fix c :: 'c
+ let ?A = "{a. h2 (h1 a) = c \<and> a \<in># f}"
+ let ?As = "\<lambda> b. {a. h1 a = b \<and> a \<in># f}"
+ let ?B = "{b. h2 b = c \<and> 0 < setsum (count f) (?As b)}"
+ have 0: "{?As b | b. b \<in> ?B} = ?As ` ?B" by auto
+ have "\<And> b. finite (?As b)" by transfer (simp add: multiset_def)
+ hence "?B = {b. h2 b = c \<and> ?As b \<noteq> {}}" by (auto simp add: setsum_gt_0_iff)
+ hence A: "?A = \<Union> {?As b | b. b \<in> ?B}" by auto
+ have "setsum (count f) ?A = setsum (setsum (count f)) {?As b | b. b \<in> ?B}"
+ unfolding A by transfer (intro setsum_Union_disjoint, auto simp: multiset_def)
+ also have "... = setsum (setsum (count f)) (?As ` ?B)" unfolding 0 ..
+ also have "... = setsum (setsum (count f) o ?As) ?B"
+ by(intro setsum_reindex) (auto simp add: setsum_gt_0_iff inj_on_def)
+ also have "... = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" unfolding comp_def ..
+ finally have "setsum (count f) ?A = setsum (\<lambda> b. setsum (count f) (?As b)) ?B" .
+ thus "count (mmap (h2 \<circ> h1) f) c = count ((mmap h2 \<circ> mmap h1) f) c"
+ by transfer (unfold o_apply, blast)
+qed
-lemma supp_count:
-"supp (count M) = set_of M"
-using assms unfolding set_of_def by auto
+lemma mmap_cong:
+assumes "\<And>a. a \<in># M \<Longrightarrow> f a = g a"
+shows "mmap f M = mmap g M"
+using assms by transfer (auto intro!: setsum_cong)
+
+lemma set_of_transfer[transfer_rule]: "(pcr_multiset op = ===> op =) (\<lambda>f. {a. 0 < f a}) set_of"
+ unfolding set_of_def pcr_multiset_def cr_multiset_def fun_rel_def by auto
+
+lemma set_of_mmap: "set_of o mmap h = image h o set_of"
+proof (rule ext, unfold o_apply)
+ fix M show "set_of (mmap h M) = h ` set_of M"
+ by transfer (auto simp add: multiset_def setsum_gt_0_iff)
+qed
lemma multiset_of_surj:
-"multiset_of ` {as. set as \<subseteq> A} = {M. set_of M \<subseteq> A}"
+ "multiset_of ` {as. set as \<subseteq> A} = {M. set_of M \<subseteq> A}"
proof safe
fix M assume M: "set_of M \<subseteq> A"
obtain as where eq: "M = multiset_of as" using surj_multiset_of unfolding surj_def by auto
@@ -666,18 +621,11 @@
qed
qed
-lemma supp_vimage_mmap:
-assumes "M \<in> multiset"
-shows "supp M \<subseteq> f -` (supp (mmap f M))"
-using assms by (auto simp: mmap_image)
+lemma supp_vimage_mmap: "set_of M \<subseteq> f -` (set_of (mmap f M))"
+ by transfer (auto simp: multiset_def setsum_gt_0_iff)
-lemma mmap_ge_0:
-assumes "M \<in> multiset"
-shows "0 < mmap f M b \<longleftrightarrow> (\<exists>a. 0 < M a \<and> f a = b)"
-proof-
- have f: "finite {a. f a = b \<and> 0 < M a}" using assms unfolding multiset_def by auto
- show ?thesis unfolding mmap_def setsum_gt_0_iff[OF f] by auto
-qed
+lemma mmap_ge_0: "b \<in># mmap f M \<longleftrightarrow> (\<exists>a. a \<in># M \<and> f a = b)"
+ by transfer (auto simp: multiset_def setsum_gt_0_iff)
lemma finite_twosets:
assumes "finite B1" and "finite B2"
@@ -687,74 +635,71 @@
show ?thesis unfolding A using finite_cartesian_product[OF assms] by auto
qed
-lemma wp_mmap:
+lemma wpull_mmap:
fixes A :: "'a set" and B1 :: "'b1 set" and B2 :: "'b2 set"
assumes wp: "wpull A B1 B2 f1 f2 p1 p2"
shows
-"wpull {M. M \<in> multiset \<and> supp M \<subseteq> A}
- {N1. N1 \<in> multiset \<and> supp N1 \<subseteq> B1} {N2. N2 \<in> multiset \<and> supp N2 \<subseteq> B2}
+"wpull {M. set_of M \<subseteq> A}
+ {N1. set_of N1 \<subseteq> B1} {N2. set_of N2 \<subseteq> B2}
(mmap f1) (mmap f2) (mmap p1) (mmap p2)"
unfolding wpull_def proof (safe, unfold Bex_def mem_Collect_eq)
- fix N1 :: "'b1 \<Rightarrow> nat" and N2 :: "'b2 \<Rightarrow> nat"
+ fix N1 :: "'b1 multiset" and N2 :: "'b2 multiset"
assume mmap': "mmap f1 N1 = mmap f2 N2"
- and N1[simp]: "N1 \<in> multiset" "supp N1 \<subseteq> B1"
- and N2[simp]: "N2 \<in> multiset" "supp N2 \<subseteq> B2"
- have mN1[simp]: "mmap f1 N1 \<in> multiset" using N1 by (auto simp: mmap)
- have mN2[simp]: "mmap f2 N2 \<in> multiset" using N2 by (auto simp: mmap)
+ and N1[simp]: "set_of N1 \<subseteq> B1"
+ and N2[simp]: "set_of N2 \<subseteq> B2"
def P \<equiv> "mmap f1 N1"
have P1: "P = mmap f1 N1" and P2: "P = mmap f2 N2" unfolding P_def using mmap' by auto
note P = P1 P2
- have P_mult[simp]: "P \<in> multiset" unfolding P_def using N1 by auto
- have fin_N1[simp]: "finite (supp N1)" using N1(1) unfolding multiset_def by auto
- have fin_N2[simp]: "finite (supp N2)" using N2(1) unfolding multiset_def by auto
- have fin_P[simp]: "finite (supp P)" using P_mult unfolding multiset_def by auto
+ have fin_N1[simp]: "finite (set_of N1)"
+ and fin_N2[simp]: "finite (set_of N2)"
+ and fin_P[simp]: "finite (set_of P)" by auto
(* *)
- def set1 \<equiv> "\<lambda> c. {b1 \<in> supp N1. f1 b1 = c}"
+ def set1 \<equiv> "\<lambda> c. {b1 \<in> set_of N1. f1 b1 = c}"
have set1[simp]: "\<And> c b1. b1 \<in> set1 c \<Longrightarrow> f1 b1 = c" unfolding set1_def by auto
- have fin_set1: "\<And> c. c \<in> supp P \<Longrightarrow> finite (set1 c)"
- using N1(1) unfolding set1_def multiset_def by auto
- have set1_NE: "\<And> c. c \<in> supp P \<Longrightarrow> set1 c \<noteq> {}"
- unfolding set1_def P1 mmap_ge_0[OF N1(1)] by auto
- have supp_N1_set1: "supp N1 = (\<Union> c \<in> supp P. set1 c)"
- using supp_vimage_mmap[OF N1(1), of f1] unfolding set1_def P1 by auto
- hence set1_inclN1: "\<And>c. c \<in> supp P \<Longrightarrow> set1 c \<subseteq> supp N1" by auto
- hence set1_incl: "\<And> c. c \<in> supp P \<Longrightarrow> set1 c \<subseteq> B1" using N1(2) by blast
+ have fin_set1: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (set1 c)"
+ using N1(1) unfolding set1_def multiset_def by auto
+ have set1_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<noteq> {}"
+ unfolding set1_def set_of_def P mmap_ge_0 by auto
+ have supp_N1_set1: "set_of N1 = (\<Union> c \<in> set_of P. set1 c)"
+ using supp_vimage_mmap[of N1 f1] unfolding set1_def P1 by auto
+ hence set1_inclN1: "\<And>c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> set_of N1" by auto
+ hence set1_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set1 c \<subseteq> B1" using N1 by blast
have set1_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set1 c \<inter> set1 c' = {}"
- unfolding set1_def by auto
- have setsum_set1: "\<And> c. setsum N1 (set1 c) = P c"
- unfolding P1 set1_def mmap_def apply(rule setsum_cong) by auto
+ unfolding set1_def by auto
+ have setsum_set1: "\<And> c. setsum (count N1) (set1 c) = count P c"
+ unfolding P1 set1_def by transfer (auto intro: setsum_cong)
(* *)
- def set2 \<equiv> "\<lambda> c. {b2 \<in> supp N2. f2 b2 = c}"
+ def set2 \<equiv> "\<lambda> c. {b2 \<in> set_of N2. f2 b2 = c}"
have set2[simp]: "\<And> c b2. b2 \<in> set2 c \<Longrightarrow> f2 b2 = c" unfolding set2_def by auto
- have fin_set2: "\<And> c. c \<in> supp P \<Longrightarrow> finite (set2 c)"
+ have fin_set2: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (set2 c)"
using N2(1) unfolding set2_def multiset_def by auto
- have set2_NE: "\<And> c. c \<in> supp P \<Longrightarrow> set2 c \<noteq> {}"
- unfolding set2_def P2 mmap_ge_0[OF N2(1)] by auto
- have supp_N2_set2: "supp N2 = (\<Union> c \<in> supp P. set2 c)"
- using supp_vimage_mmap[OF N2(1), of f2] unfolding set2_def P2 by auto
- hence set2_inclN2: "\<And>c. c \<in> supp P \<Longrightarrow> set2 c \<subseteq> supp N2" by auto
- hence set2_incl: "\<And> c. c \<in> supp P \<Longrightarrow> set2 c \<subseteq> B2" using N2(2) by blast
+ have set2_NE: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<noteq> {}"
+ unfolding set2_def P2 mmap_ge_0 set_of_def by auto
+ have supp_N2_set2: "set_of N2 = (\<Union> c \<in> set_of P. set2 c)"
+ using supp_vimage_mmap[of N2 f2] unfolding set2_def P2 by auto
+ hence set2_inclN2: "\<And>c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> set_of N2" by auto
+ hence set2_incl: "\<And> c. c \<in> set_of P \<Longrightarrow> set2 c \<subseteq> B2" using N2 by blast
have set2_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set2 c \<inter> set2 c' = {}"
- unfolding set2_def by auto
- have setsum_set2: "\<And> c. setsum N2 (set2 c) = P c"
- unfolding P2 set2_def mmap_def apply(rule setsum_cong) by auto
+ unfolding set2_def by auto
+ have setsum_set2: "\<And> c. setsum (count N2) (set2 c) = count P c"
+ unfolding P2 set2_def by transfer (auto intro: setsum_cong)
(* *)
- have ss: "\<And> c. c \<in> supp P \<Longrightarrow> setsum N1 (set1 c) = setsum N2 (set2 c)"
- unfolding setsum_set1 setsum_set2 ..
- have "\<forall> c \<in> supp P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
+ have ss: "\<And> c. c \<in> set_of P \<Longrightarrow> setsum (count N1) (set1 c) = setsum (count N2) (set2 c)"
+ unfolding setsum_set1 setsum_set2 ..
+ have "\<forall> c \<in> set_of P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
\<exists> a \<in> A. p1 a = fst b1b2 \<and> p2 a = snd b1b2"
- using wp set1_incl set2_incl unfolding wpull_def Ball_def mem_Collect_eq
- by simp (metis set1 set2 set_rev_mp)
+ using wp set1_incl set2_incl unfolding wpull_def Ball_def mem_Collect_eq
+ by simp (metis set1 set2 set_rev_mp)
then obtain uu where uu:
- "\<forall> c \<in> supp P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
+ "\<forall> c \<in> set_of P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
uu c b1b2 \<in> A \<and> p1 (uu c b1b2) = fst b1b2 \<and> p2 (uu c b1b2) = snd b1b2" by metis
def u \<equiv> "\<lambda> c b1 b2. uu c (b1,b2)"
have u[simp]:
- "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> A"
- "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p1 (u c b1 b2) = b1"
- "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p2 (u c b1 b2) = b2"
- using uu unfolding u_def by auto
- {fix c assume c: "c \<in> supp P"
+ "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> A"
+ "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p1 (u c b1 b2) = b1"
+ "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p2 (u c b1 b2) = b2"
+ using uu unfolding u_def by auto
+ {fix c assume c: "c \<in> set_of P"
have "inj2 (u c) (set1 c) (set2 c)" unfolding inj2_def proof clarify
fix b1 b1' b2 b2'
assume "{b1, b1'} \<subseteq> set1 c" "{b2, b2'} \<subseteq> set2 c" and 0: "u c b1 b2 = u c b1' b2'"
@@ -765,10 +710,10 @@
qed
} note inj = this
def sset \<equiv> "\<lambda> c. {u c b1 b2 | b1 b2. b1 \<in> set1 c \<and> b2 \<in> set2 c}"
- have fin_sset[simp]: "\<And> c. c \<in> supp P \<Longrightarrow> finite (sset c)" unfolding sset_def
- using fin_set1 fin_set2 finite_twosets by blast
- have sset_A: "\<And> c. c \<in> supp P \<Longrightarrow> sset c \<subseteq> A" unfolding sset_def by auto
- {fix c a assume c: "c \<in> supp P" and ac: "a \<in> sset c"
+ have fin_sset[simp]: "\<And> c. c \<in> set_of P \<Longrightarrow> finite (sset c)" unfolding sset_def
+ using fin_set1 fin_set2 finite_twosets by blast
+ have sset_A: "\<And> c. c \<in> set_of P \<Longrightarrow> sset c \<subseteq> A" unfolding sset_def by auto
+ {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
then obtain b1 b2 where b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
and a: "a = u c b1 b2" unfolding sset_def by auto
have "p1 a \<in> set1 c" and p2a: "p2 a \<in> set2 c"
@@ -776,260 +721,198 @@
hence "u c (p1 a) (p2 a) = a" unfolding a using b1 b2 inj[OF c]
unfolding inj2_def by (metis c u(2) u(3))
} note u_p12[simp] = this
- {fix c a assume c: "c \<in> supp P" and ac: "a \<in> sset c"
+ {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
hence "p1 a \<in> set1 c" unfolding sset_def by auto
}note p1[simp] = this
- {fix c a assume c: "c \<in> supp P" and ac: "a \<in> sset c"
+ {fix c a assume c: "c \<in> set_of P" and ac: "a \<in> sset c"
hence "p2 a \<in> set2 c" unfolding sset_def by auto
}note p2[simp] = this
(* *)
- {fix c assume c: "c \<in> supp P"
- hence "\<exists> M. (\<forall> b1 \<in> set1 c. setsum (\<lambda> b2. M (u c b1 b2)) (set2 c) = N1 b1) \<and>
- (\<forall> b2 \<in> set2 c. setsum (\<lambda> b1. M (u c b1 b2)) (set1 c) = N2 b2)"
+ {fix c assume c: "c \<in> set_of P"
+ hence "\<exists> M. (\<forall> b1 \<in> set1 c. setsum (\<lambda> b2. M (u c b1 b2)) (set2 c) = count N1 b1) \<and>
+ (\<forall> b2 \<in> set2 c. setsum (\<lambda> b1. M (u c b1 b2)) (set1 c) = count N2 b2)"
unfolding sset_def
using matrix_setsum_finite[OF set1_NE[OF c] fin_set1[OF c]
set2_NE[OF c] fin_set2[OF c] inj[OF c] ss[OF c]] by auto
}
then obtain Ms where
- ss1: "\<And> c b1. \<lbrakk>c \<in> supp P; b1 \<in> set1 c\<rbrakk> \<Longrightarrow>
- setsum (\<lambda> b2. Ms c (u c b1 b2)) (set2 c) = N1 b1" and
- ss2: "\<And> c b2. \<lbrakk>c \<in> supp P; b2 \<in> set2 c\<rbrakk> \<Longrightarrow>
- setsum (\<lambda> b1. Ms c (u c b1 b2)) (set1 c) = N2 b2"
+ ss1: "\<And> c b1. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c\<rbrakk> \<Longrightarrow>
+ setsum (\<lambda> b2. Ms c (u c b1 b2)) (set2 c) = count N1 b1" and
+ ss2: "\<And> c b2. \<lbrakk>c \<in> set_of P; b2 \<in> set2 c\<rbrakk> \<Longrightarrow>
+ setsum (\<lambda> b1. Ms c (u c b1 b2)) (set1 c) = count N2 b2"
by metis
- def SET \<equiv> "\<Union> c \<in> supp P. sset c"
+ def SET \<equiv> "\<Union> c \<in> set_of P. sset c"
have fin_SET[simp]: "finite SET" unfolding SET_def apply(rule finite_UN_I) by auto
- have SET_A: "SET \<subseteq> A" unfolding SET_def using sset_A by auto
- have u_SET[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> SET"
- unfolding SET_def sset_def by blast
- {fix c a assume c: "c \<in> supp P" and a: "a \<in> SET" and p1a: "p1 a \<in> set1 c"
- then obtain c' where c': "c' \<in> supp P" and ac': "a \<in> sset c'"
- unfolding SET_def by auto
+ have SET_A: "SET \<subseteq> A" unfolding SET_def using sset_A by blast
+ have u_SET[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> SET"
+ unfolding SET_def sset_def by blast
+ {fix c a assume c: "c \<in> set_of P" and a: "a \<in> SET" and p1a: "p1 a \<in> set1 c"
+ then obtain c' where c': "c' \<in> set_of P" and ac': "a \<in> sset c'"
+ unfolding SET_def by auto
hence "p1 a \<in> set1 c'" unfolding sset_def by auto
hence eq: "c = c'" using p1a c c' set1_disj by auto
hence "a \<in> sset c" using ac' by simp
} note p1_rev = this
- {fix c a assume c: "c \<in> supp P" and a: "a \<in> SET" and p2a: "p2 a \<in> set2 c"
- then obtain c' where c': "c' \<in> supp P" and ac': "a \<in> sset c'"
+ {fix c a assume c: "c \<in> set_of P" and a: "a \<in> SET" and p2a: "p2 a \<in> set2 c"
+ then obtain c' where c': "c' \<in> set_of P" and ac': "a \<in> sset c'"
unfolding SET_def by auto
hence "p2 a \<in> set2 c'" unfolding sset_def by auto
hence eq: "c = c'" using p2a c c' set2_disj by auto
hence "a \<in> sset c" using ac' by simp
} note p2_rev = this
(* *)
- have "\<forall> a \<in> SET. \<exists> c \<in> supp P. a \<in> sset c" unfolding SET_def by auto
- then obtain h where h: "\<forall> a \<in> SET. h a \<in> supp P \<and> a \<in> sset (h a)" by metis
- have h_u[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
+ have "\<forall> a \<in> SET. \<exists> c \<in> set_of P. a \<in> sset c" unfolding SET_def by auto
+ then obtain h where h: "\<forall> a \<in> SET. h a \<in> set_of P \<and> a \<in> sset (h a)" by metis
+ have h_u[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
\<Longrightarrow> h (u c b1 b2) = c"
by (metis h p2 set2 u(3) u_SET)
- have h_u1: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
+ have h_u1: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
\<Longrightarrow> h (u c b1 b2) = f1 b1"
using h unfolding sset_def by auto
- have h_u2: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
+ have h_u2: "\<And> c b1 b2. \<lbrakk>c \<in> set_of P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
\<Longrightarrow> h (u c b1 b2) = f2 b2"
using h unfolding sset_def by auto
- def M \<equiv> "\<lambda> a. if a \<in> SET \<and> p1 a \<in> supp N1 \<and> p2 a \<in> supp N2 then Ms (h a) a else 0"
- have sM: "supp M \<subseteq> SET" "supp M \<subseteq> p1 -` (supp N1)" "supp M \<subseteq> p2 -` (supp N2)"
- unfolding M_def by auto
- show "\<exists>M. (M \<in> multiset \<and> supp M \<subseteq> A) \<and> mmap p1 M = N1 \<and> mmap p2 M = N2"
+ def M \<equiv>
+ "Abs_multiset (\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0)"
+ have "(\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0) \<in> multiset"
+ unfolding multiset_def by auto
+ hence [transfer_rule]: "pcr_multiset op = (\<lambda> a. if a \<in> SET \<and> p1 a \<in> set_of N1 \<and> p2 a \<in> set_of N2 then Ms (h a) a else 0) M"
+ unfolding M_def pcr_multiset_def cr_multiset_def by (auto simp: Abs_multiset_inverse)
+ have sM: "set_of M \<subseteq> SET" "set_of M \<subseteq> p1 -` (set_of N1)" "set_of M \<subseteq> p2 -` set_of N2"
+ by (transfer, auto split: split_if_asm)+
+ show "\<exists>M. set_of M \<subseteq> A \<and> mmap p1 M = N1 \<and> mmap p2 M = N2"
proof(rule exI[of _ M], safe)
- show "M \<in> multiset"
- unfolding multiset_def using finite_subset[OF sM(1) fin_SET] by simp
- next
- fix a assume "0 < M a"
- thus "a \<in> A" unfolding M_def using SET_A by (cases "a \<in> SET") auto
+ fix a assume *: "a \<in> set_of M"
+ from SET_A show "a \<in> A"
+ proof (cases "a \<in> SET")
+ case False thus ?thesis using * by transfer' auto
+ qed blast
next
show "mmap p1 M = N1"
- unfolding mmap_def[abs_def] proof(rule ext)
+ proof(intro multiset_eqI)
fix b1
- let ?K = "{a. p1 a = b1 \<and> 0 < M a}"
- show "setsum M ?K = N1 b1"
- proof(cases "b1 \<in> supp N1")
+ let ?K = "{a. p1 a = b1 \<and> a \<in># M}"
+ have "setsum (count M) ?K = count N1 b1"
+ proof(cases "b1 \<in> set_of N1")
case False
hence "?K = {}" using sM(2) by auto
thus ?thesis using False by auto
next
case True
def c \<equiv> "f1 b1"
- have c: "c \<in> supp P" and b1: "b1 \<in> set1 c"
- unfolding set1_def c_def P1 using True by (auto simp: mmap_image)
- have "setsum M ?K = setsum M {a. p1 a = b1 \<and> a \<in> SET}"
- apply(rule setsum_mono_zero_cong_left) unfolding M_def by auto
- also have "... = setsum M ((\<lambda> b2. u c b1 b2) ` (set2 c))"
- apply(rule setsum_cong) using c b1 proof safe
- fix a assume p1a: "p1 a \<in> set1 c" and "0 < P c" and "a \<in> SET"
+ have c: "c \<in> set_of P" and b1: "b1 \<in> set1 c"
+ unfolding set1_def c_def P1 using True by (auto simp: o_eq_dest[OF set_of_mmap])
+ with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p1 a = b1 \<and> a \<in> SET}"
+ by transfer (force intro: setsum_mono_zero_cong_left split: split_if_asm)
+ also have "... = setsum (count M) ((\<lambda> b2. u c b1 b2) ` (set2 c))"
+ apply(rule setsum_cong) using c b1 proof safe
+ fix a assume p1a: "p1 a \<in> set1 c" and "c \<in> set_of P" and "a \<in> SET"
hence ac: "a \<in> sset c" using p1_rev by auto
hence "a = u c (p1 a) (p2 a)" using c by auto
moreover have "p2 a \<in> set2 c" using ac c by auto
ultimately show "a \<in> u c (p1 a) ` set2 c" by auto
- next
- fix b2 assume b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
- hence "u c b1 b2 \<in> SET" using c by auto
qed auto
- also have "... = setsum (\<lambda> b2. M (u c b1 b2)) (set2 c)"
- unfolding comp_def[symmetric] apply(rule setsum_reindex)
- using inj unfolding inj_on_def inj2_def using b1 c u(3) by blast
- also have "... = N1 b1" unfolding ss1[OF c b1, symmetric]
- apply(rule setsum_cong[OF refl]) unfolding M_def
+ also have "... = setsum (\<lambda> b2. count M (u c b1 b2)) (set2 c)"
+ unfolding comp_def[symmetric] apply(rule setsum_reindex)
+ using inj unfolding inj_on_def inj2_def using b1 c u(3) by blast
+ also have "... = count N1 b1" unfolding ss1[OF c b1, symmetric]
+ apply(rule setsum_cong[OF refl]) apply (transfer fixing: Ms u c b1 set2)
using True h_u[OF c b1] set2_def u(2,3)[OF c b1] u_SET[OF c b1] by fastforce
finally show ?thesis .
qed
+ thus "count (mmap p1 M) b1 = count N1 b1" by transfer
qed
next
+next
show "mmap p2 M = N2"
- unfolding mmap_def[abs_def] proof(rule ext)
+ proof(intro multiset_eqI)
fix b2
- let ?K = "{a. p2 a = b2 \<and> 0 < M a}"
- show "setsum M ?K = N2 b2"
- proof(cases "b2 \<in> supp N2")
+ let ?K = "{a. p2 a = b2 \<and> a \<in># M}"
+ have "setsum (count M) ?K = count N2 b2"
+ proof(cases "b2 \<in> set_of N2")
case False
hence "?K = {}" using sM(3) by auto
thus ?thesis using False by auto
next
case True
def c \<equiv> "f2 b2"
- have c: "c \<in> supp P" and b2: "b2 \<in> set2 c"
- unfolding set2_def c_def P2 using True by (auto simp: mmap_image)
- have "setsum M ?K = setsum M {a. p2 a = b2 \<and> a \<in> SET}"
- apply(rule setsum_mono_zero_cong_left) unfolding M_def by auto
- also have "... = setsum M ((\<lambda> b1. u c b1 b2) ` (set1 c))"
- apply(rule setsum_cong) using c b2 proof safe
- fix a assume p2a: "p2 a \<in> set2 c" and "0 < P c" and "a \<in> SET"
+ have c: "c \<in> set_of P" and b2: "b2 \<in> set2 c"
+ unfolding set2_def c_def P2 using True by (auto simp: o_eq_dest[OF set_of_mmap])
+ with sM(1) have "setsum (count M) ?K = setsum (count M) {a. p2 a = b2 \<and> a \<in> SET}"
+ by transfer (force intro: setsum_mono_zero_cong_left split: split_if_asm)
+ also have "... = setsum (count M) ((\<lambda> b1. u c b1 b2) ` (set1 c))"
+ apply(rule setsum_cong) using c b2 proof safe
+ fix a assume p2a: "p2 a \<in> set2 c" and "c \<in> set_of P" and "a \<in> SET"
hence ac: "a \<in> sset c" using p2_rev by auto
hence "a = u c (p1 a) (p2 a)" using c by auto
moreover have "p1 a \<in> set1 c" using ac c by auto
- ultimately show "a \<in> (\<lambda>b1. u c b1 (p2 a)) ` set1 c" by auto
- next
- fix b2 assume b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
- hence "u c b1 b2 \<in> SET" using c by auto
+ ultimately show "a \<in> (\<lambda>x. u c x (p2 a)) ` set1 c" by auto
qed auto
- also have "... = setsum (M o (\<lambda> b1. u c b1 b2)) (set1 c)"
- apply(rule setsum_reindex)
- using inj unfolding inj_on_def inj2_def using b2 c u(2) by blast
- also have "... = setsum (\<lambda> b1. M (u c b1 b2)) (set1 c)"
- unfolding comp_def[symmetric] by simp
- also have "... = N2 b2" unfolding ss2[OF c b2, symmetric]
- apply(rule setsum_cong[OF refl]) unfolding M_def set2_def
- using True h_u1[OF c _ b2] u(2,3)[OF c _ b2] u_SET[OF c _ b2]
- unfolding set1_def by fastforce
+ also have "... = setsum (count M o (\<lambda> b1. u c b1 b2)) (set1 c)"
+ apply(rule setsum_reindex)
+ using inj unfolding inj_on_def inj2_def using b2 c u(2) by blast
+ also have "... = setsum (\<lambda> b1. count M (u c b1 b2)) (set1 c)" by simp
+ also have "... = count N2 b2" unfolding ss2[OF c b2, symmetric] o_def
+ apply(rule setsum_cong[OF refl]) apply (transfer fixing: Ms u c b2 set1)
+ using True h_u1[OF c _ b2] u(2,3)[OF c _ b2] u_SET[OF c _ b2] set1_def by fastforce
finally show ?thesis .
qed
+ thus "count (mmap p2 M) b2 = count N2 b2" by transfer
qed
qed
qed
-definition multiset_map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
-"multiset_map h = Abs_multiset \<circ> mmap h \<circ> count"
+lemma set_of_bd: "|set_of x| \<le>o natLeq"
+ by transfer
+ (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
-bnf multiset_map [set_of] "\<lambda>_::'a multiset. natLeq" ["{#}"]
-unfolding multiset_map_def
-proof -
- show "Abs_multiset \<circ> mmap id \<circ> count = id" unfolding mmap_id by (auto simp: count_inverse)
-next
- fix f g
- show "Abs_multiset \<circ> mmap (g \<circ> f) \<circ> count =
- Abs_multiset \<circ> mmap g \<circ> count \<circ> (Abs_multiset \<circ> mmap f \<circ> count)"
- unfolding comp_def apply(rule ext)
- by (auto simp: Abs_multiset_inverse count mmap_comp1 mmap)
-next
- fix M f g assume eq: "\<And>a. a \<in> set_of M \<Longrightarrow> f a = g a"
- thus "(Abs_multiset \<circ> mmap f \<circ> count) M = (Abs_multiset \<circ> mmap g \<circ> count) M" apply auto
- unfolding cIm_def[abs_def] image_def
- by (auto intro!: mmap_cong simp: Abs_multiset_inject count mmap)
-next
- fix f show "set_of \<circ> (Abs_multiset \<circ> mmap f \<circ> count) = op ` f \<circ> set_of"
- by (auto simp: count mmap mmap_image set_of_Abs_multiset supp_count)
-next
- show "card_order natLeq" by (rule natLeq_card_order)
-next
- show "cinfinite natLeq" by (rule natLeq_cinfinite)
-next
- fix M show "|set_of M| \<le>o natLeq"
- apply(rule ordLess_imp_ordLeq)
- unfolding finite_iff_ordLess_natLeq[symmetric] using finite_set_of .
-next
- fix A B1 B2 f1 f2 p1 p2
- let ?map = "\<lambda> f. Abs_multiset \<circ> mmap f \<circ> count"
- assume wp: "wpull A B1 B2 f1 f2 p1 p2"
- show "wpull {x. set_of x \<subseteq> A} {x. set_of x \<subseteq> B1} {x. set_of x \<subseteq> B2}
- (?map f1) (?map f2) (?map p1) (?map p2)"
- unfolding wpull_def proof safe
- fix y1 y2
- assume y1: "set_of y1 \<subseteq> B1" and y2: "set_of y2 \<subseteq> B2"
- and m: "?map f1 y1 = ?map f2 y2"
- def N1 \<equiv> "count y1" def N2 \<equiv> "count y2"
- have "N1 \<in> multiset \<and> supp N1 \<subseteq> B1" and "N2 \<in> multiset \<and> supp N2 \<subseteq> B2"
- and "mmap f1 N1 = mmap f2 N2"
- using y1 y2 m unfolding N1_def N2_def
- by (auto simp: Abs_multiset_inject count mmap)
- then obtain M where M: "M \<in> multiset \<and> supp M \<subseteq> A"
- and N1: "mmap p1 M = N1" and N2: "mmap p2 M = N2"
- using wp_mmap[OF wp] unfolding wpull_def by auto
- def x \<equiv> "Abs_multiset M"
- show "\<exists>x\<in>{x. set_of x \<subseteq> A}. ?map p1 x = y1 \<and> ?map p2 x = y2"
- apply(intro bexI[of _ x]) using M N1 N2 unfolding N1_def N2_def x_def
- by (auto simp: count_inverse Abs_multiset_inverse)
- qed
-qed (unfold set_of_empty, auto)
+bnf mmap [set_of] "\<lambda>_::'a multiset. natLeq" ["{#}"]
+by (auto simp add: mmap_id mmap_comp set_of_mmap natLeq_card_order natLeq_cinfinite set_of_bd
+ intro: mmap_cong wpull_mmap)
inductive multiset_rel' where
Zero: "multiset_rel' R {#} {#}"
|
Plus: "\<lbrakk>R a b; multiset_rel' R M N\<rbrakk> \<Longrightarrow> multiset_rel' R (M + {#a#}) (N + {#b#})"
-lemma multiset_map_Zero_iff[simp]: "multiset_map f M = {#} \<longleftrightarrow> M = {#}"
+lemma multiset_map_Zero_iff[simp]: "mmap f M = {#} \<longleftrightarrow> M = {#}"
by (metis image_is_empty multiset.set_map' set_of_eq_empty_iff)
-lemma multiset_map_Zero[simp]: "multiset_map f {#} = {#}" by simp
+lemma multiset_map_Zero[simp]: "mmap f {#} = {#}" by simp
lemma multiset_rel_Zero: "multiset_rel R {#} {#}"
unfolding multiset_rel_def Grp_def by auto
declare multiset.count[simp]
-declare mmap[simp]
declare Abs_multiset_inverse[simp]
declare multiset.count_inverse[simp]
declare union_preserves_multiset[simp]
-lemma mmap_Plus[simp]:
-assumes "K \<in> multiset" and "L \<in> multiset"
-shows "mmap f (\<lambda>a. K a + L a) a = mmap f K a + mmap f L a"
-proof-
- have "{aa. f aa = a \<and> (0 < K aa \<or> 0 < L aa)} \<subseteq>
- {aa. 0 < K aa} \<union> {aa. 0 < L aa}" (is "?C \<subseteq> ?A \<union> ?B") by auto
- moreover have "finite (?A \<union> ?B)" apply(rule finite_UnI)
- using assms unfolding multiset_def by auto
- ultimately have C: "finite ?C" using finite_subset by blast
- have "setsum K {aa. f aa = a \<and> 0 < K aa} = setsum K {aa. f aa = a \<and> 0 < K aa + L aa}"
- apply(rule setsum_mono_zero_cong_left) using C by auto
+
+lemma multiset_map_Plus[simp]: "mmap f (M1 + M2) = mmap f M1 + mmap f M2"
+proof (intro multiset_eqI, transfer fixing: f)
+ fix x :: 'a and M1 M2 :: "'b \<Rightarrow> nat"
+ assume "M1 \<in> multiset" "M2 \<in> multiset"
moreover
- 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 by (auto simp add: setsum.distrib)
+ hence "setsum M1 {a. f a = x \<and> 0 < M1 a} = setsum M1 {a. f a = x \<and> 0 < M1 a + M2 a}"
+ "setsum M2 {a. f a = x \<and> 0 < M2 a} = setsum M2 {a. f a = x \<and> 0 < M1 a + M2 a}"
+ by (auto simp: multiset_def intro!: setsum_mono_zero_cong_left)
+ ultimately show "(\<Sum>a | f a = x \<and> 0 < M1 a + M2 a. M1 a + M2 a) =
+ setsum M1 {a. f a = x \<and> 0 < M1 a} +
+ setsum M2 {a. f a = x \<and> 0 < M2 a}"
+ by (auto simp: setsum.distrib[symmetric])
qed
-lemma multiset_map_Plus[simp]:
-"multiset_map f (M1 + M2) = multiset_map f M1 + multiset_map f M2"
-unfolding multiset_map_def
-apply(subst multiset.count_inject[symmetric])
-unfolding plus_multiset.rep_eq comp_def by auto
-
-lemma multiset_map_singl[simp]: "multiset_map f {#a#} = {#f a#}"
-proof-
- have 0: "\<And> b. card {aa. a = aa \<and> (a = aa \<longrightarrow> f aa = b)} =
- (if b = f a then 1 else 0)" by auto
- thus ?thesis
- unfolding multiset_map_def comp_def mmap_def[abs_def] map_fun_def
- by (simp, simp add: single_def)
-qed
+lemma multiset_map_singl[simp]: "mmap f {#a#} = {#f a#}"
+ by transfer auto
lemma multiset_rel_Plus:
assumes ab: "R a b" and MN: "multiset_rel R M N"
shows "multiset_rel R (M + {#a#}) (N + {#b#})"
proof-
{fix y assume "R a b" and "set_of y \<subseteq> {(x, y). R x y}"
- hence "\<exists>ya. multiset_map fst y + {#a#} = multiset_map fst ya \<and>
- multiset_map snd y + {#b#} = multiset_map snd ya \<and>
+ hence "\<exists>ya. mmap fst y + {#a#} = mmap fst ya \<and>
+ mmap snd y + {#b#} = mmap snd ya \<and>
set_of ya \<subseteq> {(x, y). R x y}"
apply(intro exI[of _ "y + {#(a,b)#}"]) by auto
}
@@ -1043,7 +926,7 @@
apply(induct rule: multiset_rel'.induct)
using multiset_rel_Zero multiset_rel_Plus by auto
-lemma mcard_multiset_map[simp]: "mcard (multiset_map f M) = mcard M"
+lemma mcard_mmap[simp]: "mcard (mmap f M) = mcard M"
proof -
def A \<equiv> "\<lambda> b. {a. f a = b \<and> a \<in># M}"
let ?B = "{b. 0 < setsum (count M) (A b)}"
@@ -1067,11 +950,11 @@
also have "... = setsum (count M) (\<Union>x\<in>A ` {b. 0 < setsum (count M) (A b)}. x)"
(is "_ = setsum (count M) ?J")
apply(rule setsum.UNION_disjoint[symmetric])
- using 0 fin unfolding A_def by (auto intro!: finite_imageI)
+ using 0 fin unfolding A_def by auto
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 =
setsum (count M) {a. a \<in># M}" .
- then show ?thesis by (simp add: A_def mcard_unfold_setsum multiset_map_def set_of_def mmap_def)
+ then show ?thesis unfolding mcard_unfold_setsum A_def by transfer
qed
lemma multiset_rel_mcard:
@@ -1109,25 +992,25 @@
qed
lemma msed_map_invL:
-assumes "multiset_map f (M + {#a#}) = N"
-shows "\<exists> N1. N = N1 + {#f a#} \<and> multiset_map f M = N1"
+assumes "mmap f (M + {#a#}) = N"
+shows "\<exists> N1. N = N1 + {#f a#} \<and> mmap f M = N1"
proof-
have "f a \<in># N"
using assms multiset.set_map'[of f "M + {#a#}"] by auto
then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
- have "multiset_map f M = N1" using assms unfolding N by simp
+ have "mmap f M = N1" using assms unfolding N by simp
thus ?thesis using N by blast
qed
lemma msed_map_invR:
-assumes "multiset_map f M = N + {#b#}"
-shows "\<exists> M1 a. M = M1 + {#a#} \<and> f a = b \<and> multiset_map f M1 = N"
+assumes "mmap f M = N + {#b#}"
+shows "\<exists> M1 a. M = M1 + {#a#} \<and> f a = b \<and> mmap f M1 = N"
proof-
obtain a where a: "a \<in># M" and fa: "f a = b"
using multiset.set_map'[of f M] unfolding assms
by (metis image_iff mem_set_of_iff union_single_eq_member)
then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
- have "multiset_map f M1 = N" using assms unfolding M fa[symmetric] by simp
+ have "mmap f M1 = N" using assms unfolding M fa[symmetric] by simp
thus ?thesis using M fa by blast
qed
@@ -1135,13 +1018,13 @@
assumes "multiset_rel R (M + {#a#}) N"
shows "\<exists> N1 b. N = N1 + {#b#} \<and> R a b \<and> multiset_rel R M N1"
proof-
- obtain K where KM: "multiset_map fst K = M + {#a#}"
- and KN: "multiset_map snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
+ obtain K where KM: "mmap fst K = M + {#a#}"
+ and KN: "mmap snd K = N" and sK: "set_of K \<subseteq> {(a, b). R a b}"
using assms
unfolding multiset_rel_def Grp_def by auto
obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
- and K1M: "multiset_map fst K1 = M" using msed_map_invR[OF KM] by auto
- obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "multiset_map snd K1 = N1"
+ and K1M: "mmap fst K1 = M" using msed_map_invR[OF KM] by auto
+ obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "mmap snd K1 = N1"
using msed_map_invL[OF KN[unfolded K]] by auto
have Rab: "R a (snd ab)" using sK a unfolding K by auto
have "multiset_rel R M N1" using sK K1M K1N1
@@ -1153,13 +1036,13 @@
assumes "multiset_rel R M (N + {#b#})"
shows "\<exists> M1 a. M = M1 + {#a#} \<and> R a b \<and> multiset_rel R M1 N"
proof-
- obtain K where KN: "multiset_map snd K = N + {#b#}"
- and KM: "multiset_map fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
+ obtain K where KN: "mmap snd K = N + {#b#}"
+ and KM: "mmap fst K = M" and sK: "set_of K \<subseteq> {(a, b). R a b}"
using assms
unfolding multiset_rel_def Grp_def by auto
obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
- and K1N: "multiset_map snd K1 = N" using msed_map_invR[OF KN] by auto
- obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "multiset_map fst K1 = M1"
+ and K1N: "mmap snd K1 = N" using msed_map_invR[OF KN] by auto
+ obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "mmap fst K1 = M1"
using msed_map_invL[OF KM[unfolded K]] by auto
have Rab: "R (fst ab) b" using sK b unfolding K by auto
have "multiset_rel R M1 N" using sK K1N K1M1