--- a/src/HOL/Library/Multiset.thy Mon Aug 06 13:12:06 2001 +0200
+++ b/src/HOL/Library/Multiset.thy Mon Aug 06 13:43:24 2001 +0200
@@ -16,7 +16,7 @@
typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
proof
- show "(\\<lambda>x. 0::nat) \\<in> ?multiset" by simp
+ show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
qed
lemmas multiset_typedef [simp] =
@@ -25,23 +25,23 @@
constdefs
Mempty :: "'a multiset" ("{#}")
- "{#} == Abs_multiset (\\<lambda>a. 0)"
+ "{#} == Abs_multiset (\<lambda>a. 0)"
single :: "'a => 'a multiset" ("{#_#}")
- "{#a#} == Abs_multiset (\\<lambda>b. if b = a then 1 else 0)"
+ "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1' else 0)"
count :: "'a multiset => 'a => nat"
"count == Rep_multiset"
MCollect :: "'a multiset => ('a => bool) => 'a multiset"
- "MCollect M P == Abs_multiset (\\<lambda>x. if P x then Rep_multiset M x else 0)"
+ "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
syntax
"_Melem" :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50)
"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset" ("(1{# _ : _./ _#})")
translations
"a :# M" == "0 < count M a"
- "{#x:M. P#}" == "MCollect M (\\<lambda>x. P)"
+ "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
constdefs
set_of :: "'a multiset => 'a set"
@@ -52,8 +52,8 @@
instance multiset :: ("term") zero ..
defs (overloaded)
- union_def: "M + N == Abs_multiset (\\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
- diff_def: "M - N == Abs_multiset (\\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
+ union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
+ diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
Zero_def [simp]: "0 == {#}"
size_def: "size M == setsum (count M) (set_of M)"
@@ -62,16 +62,16 @@
\medskip Preservation of the representing set @{term multiset}.
*}
-lemma const0_in_multiset [simp]: "(\\<lambda>a. 0) \\<in> multiset"
+lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
apply (simp add: multiset_def)
done
-lemma only1_in_multiset [simp]: "(\\<lambda>b. if b = a then 1 else 0) \\<in> multiset"
+lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1' else 0) \<in> multiset"
apply (simp add: multiset_def)
done
lemma union_preserves_multiset [simp]:
- "M \\<in> multiset ==> N \\<in> multiset ==> (\\<lambda>a. M a + N a) \\<in> multiset"
+ "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
apply (unfold multiset_def)
apply simp
apply (drule finite_UnI)
@@ -80,7 +80,7 @@
done
lemma diff_preserves_multiset [simp]:
- "M \\<in> multiset ==> (\\<lambda>a. M a - N a) \\<in> multiset"
+ "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
apply (unfold multiset_def)
apply simp
apply (rule finite_subset)
@@ -94,7 +94,7 @@
subsubsection {* Union *}
-theorem union_empty [simp]: "M + {#} = M \\<and> {#} + M = M"
+theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
apply (simp add: union_def Mempty_def)
done
@@ -124,7 +124,7 @@
subsubsection {* Difference *}
-theorem diff_empty [simp]: "M - {#} = M \\<and> {#} - M = {#}"
+theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
apply (simp add: Mempty_def diff_def)
done
@@ -139,7 +139,7 @@
apply (simp add: count_def Mempty_def)
done
-theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
+theorem count_single [simp]: "count {#b#} a = (if b = a then 1' else 0)"
apply (simp add: count_def single_def)
done
@@ -162,7 +162,7 @@
apply (simp add: set_of_def)
done
-theorem set_of_union [simp]: "set_of (M + N) = set_of M \\<union> set_of N"
+theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
apply (auto simp add: set_of_def)
done
@@ -170,7 +170,7 @@
apply (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
done
-theorem mem_set_of_iff [simp]: "(x \\<in> set_of M) = (x :# M)"
+theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
apply (auto simp add: set_of_def)
done
@@ -191,7 +191,7 @@
done
theorem setsum_count_Int:
- "finite A ==> setsum (count N) (A \\<inter> set_of N) = setsum (count N) A"
+ "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
apply (erule finite_induct)
apply simp
apply (simp add: Int_insert_left set_of_def)
@@ -199,7 +199,7 @@
theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
apply (unfold size_def)
- apply (subgoal_tac "count (M + N) = (\\<lambda>a. count M a + count N a)")
+ apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
prefer 2
apply (rule ext)
apply simp
@@ -214,7 +214,7 @@
apply (simp add: set_of_def count_def expand_fun_eq)
done
-theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \\<exists>a. a :# M"
+theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
apply (unfold size_def)
apply (drule setsum_SucD)
apply auto
@@ -223,11 +223,11 @@
subsubsection {* Equality of multisets *}
-theorem multiset_eq_conv_count_eq: "(M = N) = (\\<forall>a. count M a = count N a)"
+theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
apply (simp add: count_def expand_fun_eq)
done
-theorem single_not_empty [simp]: "{#a#} \\<noteq> {#} \\<and> {#} \\<noteq> {#a#}"
+theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
apply (simp add: single_def Mempty_def expand_fun_eq)
done
@@ -235,11 +235,11 @@
apply (auto simp add: single_def expand_fun_eq)
done
-theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \\<and> N = {#})"
+theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
apply (auto simp add: union_def Mempty_def expand_fun_eq)
done
-theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \\<and> N = {#})"
+theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
apply (auto simp add: union_def Mempty_def expand_fun_eq)
done
@@ -252,7 +252,7 @@
done
theorem union_is_single:
- "(M + N = {#a#}) = (M = {#a#} \\<and> N={#} \\<or> M = {#} \\<and> N = {#a#})"
+ "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
apply (unfold Mempty_def single_def union_def)
apply (simp add: add_is_1 expand_fun_eq)
apply blast
@@ -260,16 +260,16 @@
theorem single_is_union:
"({#a#} = M + N) =
- ({#a#} = M \\<and> N = {#} \\<or> M = {#} \\<and> {#a#} = N)"
+ ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
apply (unfold Mempty_def single_def union_def)
- apply (simp add: add_is_1 expand_fun_eq)
+ apply (simp add: add_is_1 one_is_add expand_fun_eq)
apply (blast dest: sym)
done
theorem add_eq_conv_diff:
"(M + {#a#} = N + {#b#}) =
- (M = N \\<and> a = b \\<or>
- M = N - {#a#} + {#b#} \\<and> N = M - {#b#} + {#a#})"
+ (M = N \<and> a = b \<or>
+ M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
apply (unfold single_def union_def diff_def)
apply (simp (no_asm) add: expand_fun_eq)
apply (rule conjI)
@@ -291,7 +291,7 @@
(*
val prems = Goal
"[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
-by (res_inst_tac [("a","F"),("f","\\<lambda>A. if finite A then card A else 0")]
+by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
measure_induct 1);
by (Clarify_tac 1);
by (resolve_tac prems 1);
@@ -320,7 +320,7 @@
lemma setsum_decr:
"finite F ==> 0 < f a ==>
- setsum (f (a := f a - 1)) F = (if a \\<in> F then setsum f F - 1 else setsum f F)"
+ setsum (f (a := f a - 1')) F = (if a \<in> F then setsum f F - 1 else setsum f F)"
apply (erule finite_induct)
apply auto
apply (drule_tac a = a in mk_disjoint_insert)
@@ -328,8 +328,8 @@
done
lemma rep_multiset_induct_aux:
- "P (\\<lambda>a. 0) ==> (!!f b. f \\<in> multiset ==> P f ==> P (f (b := f b + 1)))
- ==> \\<forall>f. f \\<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
+ "P (\<lambda>a. 0) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1')))
+ ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
proof -
case antecedent
note prems = this [unfolded multiset_def]
@@ -338,7 +338,7 @@
apply (induct_tac n)
apply simp
apply clarify
- apply (subgoal_tac "f = (\\<lambda>a.0)")
+ apply (subgoal_tac "f = (\<lambda>a.0)")
apply simp
apply (rule prems)
apply (rule ext)
@@ -347,14 +347,14 @@
apply (frule setsum_SucD)
apply clarify
apply (rename_tac a)
- apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
+ apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1')) x}")
prefer 2
apply (rule finite_subset)
prefer 2
apply assumption
apply simp
apply blast
- apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
+ apply (subgoal_tac "f = (f (a := f a - 1'))(a := (f (a := f a - 1')) a + 1')")
prefer 2
apply (rule ext)
apply (simp (no_asm_simp))
@@ -363,10 +363,10 @@
apply (erule allE, erule impE, erule_tac [2] mp)
apply blast
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply)
- apply (subgoal_tac "{x. x \\<noteq> a --> 0 < f x} = {x. 0 < f x}")
+ apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
prefer 2
apply blast
- apply (subgoal_tac "{x. x \\<noteq> a \\<and> 0 < f x} = {x. 0 < f x} - {a}")
+ apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
prefer 2
apply blast
apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)
@@ -374,8 +374,8 @@
qed
theorem rep_multiset_induct:
- "f \\<in> multiset ==> P (\\<lambda>a. 0) ==>
- (!!f b. f \\<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
+ "f \<in> multiset ==> P (\<lambda>a. 0) ==>
+ (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1'))) ==> P f"
apply (insert rep_multiset_induct_aux)
apply blast
done
@@ -390,7 +390,7 @@
apply (rule Rep_multiset_inverse [THEN subst])
apply (rule Rep_multiset [THEN rep_multiset_induct])
apply (rule prem1)
- apply (subgoal_tac "f (b := f b + 1) = (\\<lambda>a. f a + (if a = b then 1 else 0))")
+ apply (subgoal_tac "f (b := f b + 1') = (\<lambda>a. f a + (if a = b then 1' else 0))")
prefer 2
apply (simp add: expand_fun_eq)
apply (erule ssubst)
@@ -401,7 +401,7 @@
lemma MCollect_preserves_multiset:
- "M \\<in> multiset ==> (\\<lambda>x. if P x then M x else 0) \\<in> multiset"
+ "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
apply (simp add: multiset_def)
apply (rule finite_subset)
apply auto
@@ -413,11 +413,11 @@
apply (simp add: MCollect_preserves_multiset)
done
-theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \\<inter> {x. P x}"
+theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
apply (auto simp add: set_of_def)
done
-theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \\<not> P x #}"
+theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
apply (subst multiset_eq_conv_count_eq)
apply auto
done
@@ -427,7 +427,7 @@
theorem add_eq_conv_ex:
"(M + {#a#} = N + {#b#}) =
- (M = N \\<and> a = b \\<or> (\\<exists>K. M = K + {#b#} \\<and> N = K + {#a#}))"
+ (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
apply (auto simp add: add_eq_conv_diff)
done
@@ -437,41 +437,41 @@
subsubsection {* Well-foundedness *}
constdefs
- mult1 :: "('a \\<times> 'a) set => ('a multiset \\<times> 'a multiset) set"
+ mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
"mult1 r ==
- {(N, M). \\<exists>a M0 K. M = M0 + {#a#} \\<and> N = M0 + K \\<and>
- (\\<forall>b. b :# K --> (b, a) \\<in> r)}"
+ {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
+ (\<forall>b. b :# K --> (b, a) \<in> r)}"
- mult :: "('a \\<times> 'a) set => ('a multiset \\<times> 'a multiset) set"
+ mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
"mult r == (mult1 r)\<^sup>+"
-lemma not_less_empty [iff]: "(M, {#}) \\<notin> mult1 r"
+lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
by (simp add: mult1_def)
-lemma less_add: "(N, M0 + {#a#}) \\<in> mult1 r ==>
- (\\<exists>M. (M, M0) \\<in> mult1 r \\<and> N = M + {#a#}) \\<or>
- (\\<exists>K. (\\<forall>b. b :# K --> (b, a) \\<in> r) \\<and> N = M0 + K)"
- (concl is "?case1 (mult1 r) \\<or> ?case2")
+lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
+ (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
+ (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
+ (concl is "?case1 (mult1 r) \<or> ?case2")
proof (unfold mult1_def)
- let ?r = "\\<lambda>K a. \\<forall>b. b :# K --> (b, a) \\<in> r"
- let ?R = "\\<lambda>N M. \\<exists>a M0 K. M = M0 + {#a#} \\<and> N = M0 + K \\<and> ?r K a"
+ let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
+ let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
let ?case1 = "?case1 {(N, M). ?R N M}"
- assume "(N, M0 + {#a#}) \\<in> {(N, M). ?R N M}"
- hence "\\<exists>a' M0' K.
- M0 + {#a#} = M0' + {#a'#} \\<and> N = M0' + K \\<and> ?r K a'" by simp
- thus "?case1 \\<or> ?case2"
+ assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
+ hence "\<exists>a' M0' K.
+ M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
+ thus "?case1 \<or> ?case2"
proof (elim exE conjE)
fix a' M0' K
assume N: "N = M0' + K" and r: "?r K a'"
assume "M0 + {#a#} = M0' + {#a'#}"
- hence "M0 = M0' \\<and> a = a' \\<or>
- (\\<exists>K'. M0 = K' + {#a'#} \\<and> M0' = K' + {#a#})"
+ hence "M0 = M0' \<and> a = a' \<or>
+ (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
by (simp only: add_eq_conv_ex)
thus ?thesis
proof (elim disjE conjE exE)
assume "M0 = M0'" "a = a'"
- with N r have "?r K a \\<and> N = M0 + K" by simp
+ with N r have "?r K a \<and> N = M0 + K" by simp
hence ?case2 .. thus ?thesis ..
next
fix K'
@@ -485,78 +485,78 @@
qed
qed
-lemma all_accessible: "wf r ==> \\<forall>M. M \\<in> acc (mult1 r)"
+lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
proof
let ?R = "mult1 r"
let ?W = "acc ?R"
{
fix M M0 a
- assume M0: "M0 \\<in> ?W"
- and wf_hyp: "\\<forall>b. (b, a) \\<in> r --> (\\<forall>M \\<in> ?W. M + {#b#} \\<in> ?W)"
- and acc_hyp: "\\<forall>M. (M, M0) \\<in> ?R --> M + {#a#} \\<in> ?W"
- have "M0 + {#a#} \\<in> ?W"
+ assume M0: "M0 \<in> ?W"
+ and wf_hyp: "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
+ and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
+ have "M0 + {#a#} \<in> ?W"
proof (rule accI [of "M0 + {#a#}"])
fix N
- assume "(N, M0 + {#a#}) \\<in> ?R"
- hence "((\\<exists>M. (M, M0) \\<in> ?R \\<and> N = M + {#a#}) \\<or>
- (\\<exists>K. (\\<forall>b. b :# K --> (b, a) \\<in> r) \\<and> N = M0 + K))"
+ assume "(N, M0 + {#a#}) \<in> ?R"
+ hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
+ (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
by (rule less_add)
- thus "N \\<in> ?W"
+ thus "N \<in> ?W"
proof (elim exE disjE conjE)
- fix M assume "(M, M0) \\<in> ?R" and N: "N = M + {#a#}"
- from acc_hyp have "(M, M0) \\<in> ?R --> M + {#a#} \\<in> ?W" ..
- hence "M + {#a#} \\<in> ?W" ..
- thus "N \\<in> ?W" by (simp only: N)
+ fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
+ from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
+ hence "M + {#a#} \<in> ?W" ..
+ thus "N \<in> ?W" by (simp only: N)
next
fix K
assume N: "N = M0 + K"
- assume "\\<forall>b. b :# K --> (b, a) \\<in> r"
- have "?this --> M0 + K \\<in> ?W" (is "?P K")
+ assume "\<forall>b. b :# K --> (b, a) \<in> r"
+ have "?this --> M0 + K \<in> ?W" (is "?P K")
proof (induct K)
- from M0 have "M0 + {#} \\<in> ?W" by simp
+ from M0 have "M0 + {#} \<in> ?W" by simp
thus "?P {#}" ..
fix K x assume hyp: "?P K"
show "?P (K + {#x#})"
proof
- assume a: "\\<forall>b. b :# (K + {#x#}) --> (b, a) \\<in> r"
- hence "(x, a) \\<in> r" by simp
- with wf_hyp have b: "\\<forall>M \\<in> ?W. M + {#x#} \\<in> ?W" by blast
+ assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
+ hence "(x, a) \<in> r" by simp
+ with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
- from a hyp have "M0 + K \\<in> ?W" by simp
- with b have "(M0 + K) + {#x#} \\<in> ?W" ..
- thus "M0 + (K + {#x#}) \\<in> ?W" by (simp only: union_assoc)
+ from a hyp have "M0 + K \<in> ?W" by simp
+ with b have "(M0 + K) + {#x#} \<in> ?W" ..
+ thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
qed
qed
- hence "M0 + K \\<in> ?W" ..
- thus "N \\<in> ?W" by (simp only: N)
+ hence "M0 + K \<in> ?W" ..
+ thus "N \<in> ?W" by (simp only: N)
qed
qed
} note tedious_reasoning = this
assume wf: "wf r"
fix M
- show "M \\<in> ?W"
+ show "M \<in> ?W"
proof (induct M)
- show "{#} \\<in> ?W"
+ show "{#} \<in> ?W"
proof (rule accI)
- fix b assume "(b, {#}) \\<in> ?R"
- with not_less_empty show "b \\<in> ?W" by contradiction
+ fix b assume "(b, {#}) \<in> ?R"
+ with not_less_empty show "b \<in> ?W" by contradiction
qed
- fix M a assume "M \\<in> ?W"
- from wf have "\\<forall>M \\<in> ?W. M + {#a#} \\<in> ?W"
+ fix M a assume "M \<in> ?W"
+ from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
proof induct
fix a
- assume "\\<forall>b. (b, a) \\<in> r --> (\\<forall>M \\<in> ?W. M + {#b#} \\<in> ?W)"
- show "\\<forall>M \\<in> ?W. M + {#a#} \\<in> ?W"
+ assume "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
+ show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
proof
- fix M assume "M \\<in> ?W"
- thus "M + {#a#} \\<in> ?W"
+ fix M assume "M \<in> ?W"
+ thus "M + {#a#} \<in> ?W"
by (rule acc_induct) (rule tedious_reasoning)
qed
qed
- thus "M + {#a#} \\<in> ?W" ..
+ thus "M + {#a#} \<in> ?W" ..
qed
qed
@@ -578,9 +578,9 @@
text {* One direction. *}
lemma mult_implies_one_step:
- "trans r ==> (M, N) \\<in> mult r ==>
- \\<exists>I J K. N = I + J \\<and> M = I + K \\<and> J \\<noteq> {#} \\<and>
- (\\<forall>k \\<in> set_of K. \\<exists>j \\<in> set_of J. (k, j) \\<in> r)"
+ "trans r ==> (M, N) \<in> mult r ==>
+ \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
+ (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
apply (unfold mult_def mult1_def set_of_def)
apply (erule converse_trancl_induct)
apply clarify
@@ -592,7 +592,7 @@
apply (simp (no_asm))
apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
apply (simp (no_asm_simp) add: union_assoc [symmetric])
- apply (drule_tac f = "\\<lambda>M. M - {#a#}" in arg_cong)
+ apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
apply (simp add: diff_union_single_conv)
apply (simp (no_asm_use) add: trans_def)
apply blast
@@ -603,7 +603,7 @@
apply (rule conjI)
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
apply (rule conjI)
- apply (drule_tac f = "\\<lambda>M. M - {#a#}" in arg_cong)
+ apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
apply simp
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
apply (simp (no_asm_use) add: trans_def)
@@ -617,7 +617,7 @@
apply (simp add: multiset_eq_conv_count_eq)
done
-lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \\<exists>a N. M = N + {#a#}"
+lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
apply (erule size_eq_Suc_imp_elem [THEN exE])
apply (drule elem_imp_eq_diff_union)
apply auto
@@ -625,8 +625,8 @@
lemma one_step_implies_mult_aux:
"trans r ==>
- \\<forall>I J K. (size J = n \\<and> J \\<noteq> {#} \\<and> (\\<forall>k \\<in> set_of K. \\<exists>j \\<in> set_of J. (k, j) \\<in> r))
- --> (I + K, I + J) \\<in> mult r"
+ \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
+ --> (I + K, I + J) \<in> mult r"
apply (induct_tac n)
apply auto
apply (frule size_eq_Suc_imp_eq_union)
@@ -640,15 +640,15 @@
apply (rule r_into_trancl)
apply (simp add: mult1_def set_of_def)
apply blast
- txt {* Now we know @{term "J' \\<noteq> {#}"}. *}
- apply (cut_tac M = K and P = "\\<lambda>x. (x, a) \\<in> r" in multiset_partition)
- apply (erule_tac P = "\\<forall>k \\<in> set_of K. ?P k" in rev_mp)
+ txt {* Now we know @{term "J' \<noteq> {#}"}. *}
+ apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
+ apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
apply (erule ssubst)
apply (simp add: Ball_def)
apply auto
apply (subgoal_tac
- "((I + {# x : K. (x, a) \\<in> r #}) + {# x : K. (x, a) \\<notin> r #},
- (I + {# x : K. (x, a) \\<in> r #}) + J') \\<in> mult r")
+ "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
+ (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
prefer 2
apply force
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
@@ -661,8 +661,8 @@
done
theorem one_step_implies_mult:
- "trans r ==> J \\<noteq> {#} ==> \\<forall>k \\<in> set_of K. \\<exists>j \\<in> set_of J. (k, j) \\<in> r
- ==> (I + K, I + J) \\<in> mult r"
+ "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
+ ==> (I + K, I + J) \<in> mult r"
apply (insert one_step_implies_mult_aux)
apply blast
done
@@ -673,8 +673,8 @@
instance multiset :: ("term") ord ..
defs (overloaded)
- less_multiset_def: "M' < M == (M', M) \\<in> mult {(x', x). x' < x}"
- le_multiset_def: "M' <= M == M' = M \\<or> M' < (M::'a multiset)"
+ less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
+ le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
apply (unfold trans_def)
@@ -686,12 +686,12 @@
*}
lemma mult_irrefl_aux:
- "finite A ==> (\\<forall>x \\<in> A. \\<exists>y \\<in> A. x < (y::'a::order)) --> A = {}"
+ "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
apply (erule finite_induct)
apply (auto intro: order_less_trans)
done
-theorem mult_less_not_refl: "\\<not> M < (M::'a::order multiset)"
+theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
apply (unfold less_multiset_def)
apply auto
apply (drule trans_base_order [THEN mult_implies_one_step])
@@ -715,7 +715,7 @@
text {* Asymmetry. *}
-theorem mult_less_not_sym: "M < N ==> \\<not> N < (M::'a::order multiset)"
+theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
apply auto
apply (rule mult_less_not_refl [THEN notE])
apply (erule mult_less_trans)
@@ -723,7 +723,7 @@
done
theorem mult_less_asym:
- "M < N ==> (\\<not> P ==> N < (M::'a::order multiset)) ==> P"
+ "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
apply (insert mult_less_not_sym)
apply blast
done
@@ -749,7 +749,7 @@
apply (blast intro: mult_less_trans)
done
-theorem mult_less_le: "M < N = (M <= N \\<and> M \\<noteq> (N::'a::order multiset))"
+theorem mult_less_le: "M < N = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
apply (unfold le_multiset_def)
apply auto
done
@@ -770,7 +770,7 @@
subsubsection {* Monotonicity of multiset union *}
theorem mult1_union:
- "(B, D) \\<in> mult1 r ==> trans r ==> (C + B, C + D) \\<in> mult1 r"
+ "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
apply (unfold mult1_def)
apply auto
apply (rule_tac x = a in exI)
@@ -806,7 +806,7 @@
apply (unfold le_multiset_def less_multiset_def)
apply (case_tac "M = {#}")
prefer 2
- apply (subgoal_tac "({#} + {#}, {#} + M) \\<in> mult (Collect (split op <))")
+ apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
prefer 2
apply (rule one_step_implies_mult)
apply (simp only: trans_def)