author paulson Thu Dec 21 10:16:07 2000 +0100 (2000-12-21) changeset 10714 07f75bf77a33 parent 10713 69c9fc1d11f2 child 10715 c838477b5c18
re-orientation of 0=... (no idea why the backslashes have changed too)
```     1.1 --- a/src/HOL/Library/Multiset.thy	Thu Dec 21 10:11:10 2000 +0100
1.2 +++ b/src/HOL/Library/Multiset.thy	Thu Dec 21 10:16:07 2000 +0100
1.3 @@ -16,7 +16,7 @@
1.4
1.5  typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
1.6  proof
1.7 -  show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
1.8 +  show "(\\<lambda>x. 0::nat) \\<in> ?multiset" by simp
1.9  qed
1.10
1.11  lemmas multiset_typedef [simp] =
1.12 @@ -25,23 +25,23 @@
1.13
1.14  constdefs
1.15    Mempty :: "'a multiset"    ("{#}")
1.16 -  "{#} == Abs_multiset (\<lambda>a. 0)"
1.17 +  "{#} == Abs_multiset (\\<lambda>a. 0)"
1.18
1.19    single :: "'a => 'a multiset"    ("{#_#}")
1.20 -  "{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
1.21 +  "{#a#} == Abs_multiset (\\<lambda>b. if b = a then 1 else 0)"
1.22
1.23    count :: "'a multiset => 'a => nat"
1.24    "count == Rep_multiset"
1.25
1.26    MCollect :: "'a multiset => ('a => bool) => 'a multiset"
1.27 -  "MCollect M P == Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
1.28 +  "MCollect M P == Abs_multiset (\\<lambda>x. if P x then Rep_multiset M x else 0)"
1.29
1.30  syntax
1.31    "_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
1.32    "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
1.33  translations
1.34    "a :# M" == "0 < count M a"
1.35 -  "{#x:M. P#}" == "MCollect M (\<lambda>x. P)"
1.36 +  "{#x:M. P#}" == "MCollect M (\\<lambda>x. P)"
1.37
1.38  constdefs
1.39    set_of :: "'a multiset => 'a set"
1.40 @@ -52,8 +52,8 @@
1.41  instance multiset :: ("term") zero ..
1.42
1.44 -  union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
1.45 -  diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
1.46 +  union_def: "M + N == Abs_multiset (\\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
1.47 +  diff_def: "M - N == Abs_multiset (\\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
1.48    Zero_def [simp]: "0 == {#}"
1.49    size_def: "size M == setsum (count M) (set_of M)"
1.50
1.51 @@ -62,16 +62,16 @@
1.52   \medskip Preservation of the representing set @{term multiset}.
1.53  *}
1.54
1.55 -lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset"
1.56 +lemma const0_in_multiset [simp]: "(\\<lambda>a. 0) \\<in> multiset"
1.58    done
1.59
1.60 -lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
1.61 +lemma only1_in_multiset [simp]: "(\\<lambda>b. if b = a then 1 else 0) \\<in> multiset"
1.63    done
1.64
1.65  lemma union_preserves_multiset [simp]:
1.66 -    "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
1.67 +    "M \\<in> multiset ==> N \\<in> multiset ==> (\\<lambda>a. M a + N a) \\<in> multiset"
1.68    apply (unfold multiset_def)
1.69    apply simp
1.70    apply (drule finite_UnI)
1.71 @@ -80,7 +80,7 @@
1.72    done
1.73
1.74  lemma diff_preserves_multiset [simp]:
1.75 -    "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
1.76 +    "M \\<in> multiset ==> (\\<lambda>a. M a - N a) \\<in> multiset"
1.77    apply (unfold multiset_def)
1.78    apply simp
1.79    apply (rule finite_subset)
1.80 @@ -94,7 +94,7 @@
1.81
1.82  subsubsection {* Union *}
1.83
1.84 -theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
1.85 +theorem union_empty [simp]: "M + {#} = M \\<and> {#} + M = M"
1.86    apply (simp add: union_def Mempty_def)
1.87    done
1.88
1.89 @@ -124,7 +124,7 @@
1.90
1.91  subsubsection {* Difference *}
1.92
1.93 -theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
1.94 +theorem diff_empty [simp]: "M - {#} = M \\<and> {#} - M = {#}"
1.95    apply (simp add: Mempty_def diff_def)
1.96    done
1.97
1.98 @@ -162,7 +162,7 @@
1.100    done
1.101
1.102 -theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
1.103 +theorem set_of_union [simp]: "set_of (M + N) = set_of M \\<union> set_of N"
1.104    apply (auto simp add: set_of_def)
1.105    done
1.106
1.107 @@ -170,7 +170,7 @@
1.108    apply (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
1.109    done
1.110
1.111 -theorem mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
1.112 +theorem mem_set_of_iff [simp]: "(x \\<in> set_of M) = (x :# M)"
1.113    apply (auto simp add: set_of_def)
1.114    done
1.115
1.116 @@ -191,7 +191,7 @@
1.117    done
1.118
1.119  theorem setsum_count_Int:
1.120 -    "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
1.121 +    "finite A ==> setsum (count N) (A \\<inter> set_of N) = setsum (count N) A"
1.122    apply (erule finite_induct)
1.123     apply simp
1.124    apply (simp add: Int_insert_left set_of_def)
1.125 @@ -199,7 +199,7 @@
1.126
1.127  theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
1.128    apply (unfold size_def)
1.129 -  apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
1.130 +  apply (subgoal_tac "count (M + N) = (\\<lambda>a. count M a + count N a)")
1.131     prefer 2
1.132     apply (rule ext)
1.133     apply simp
1.134 @@ -214,7 +214,7 @@
1.135    apply (simp add: set_of_def count_def expand_fun_eq)
1.136    done
1.137
1.138 -theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
1.139 +theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \\<exists>a. a :# M"
1.140    apply (unfold size_def)
1.141    apply (drule setsum_SucD)
1.142    apply auto
1.143 @@ -223,11 +223,11 @@
1.144
1.145  subsubsection {* Equality of multisets *}
1.146
1.147 -theorem multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
1.148 +theorem multiset_eq_conv_count_eq: "(M = N) = (\\<forall>a. count M a = count N a)"
1.149    apply (simp add: count_def expand_fun_eq)
1.150    done
1.151
1.152 -theorem single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
1.153 +theorem single_not_empty [simp]: "{#a#} \\<noteq> {#} \\<and> {#} \\<noteq> {#a#}"
1.154    apply (simp add: single_def Mempty_def expand_fun_eq)
1.155    done
1.156
1.157 @@ -235,11 +235,11 @@
1.158    apply (auto simp add: single_def expand_fun_eq)
1.159    done
1.160
1.161 -theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
1.162 +theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \\<and> N = {#})"
1.163    apply (auto simp add: union_def Mempty_def expand_fun_eq)
1.164    done
1.165
1.166 -theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
1.167 +theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \\<and> N = {#})"
1.168    apply (auto simp add: union_def Mempty_def expand_fun_eq)
1.169    done
1.170
1.171 @@ -252,7 +252,7 @@
1.172    done
1.173
1.174  theorem union_is_single:
1.175 -    "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
1.176 +    "(M + N = {#a#}) = (M = {#a#} \\<and> N={#} \\<or> M = {#} \\<and> N = {#a#})"
1.177    apply (unfold Mempty_def single_def union_def)
1.179    apply blast
1.180 @@ -260,16 +260,16 @@
1.181
1.182  theorem single_is_union:
1.183    "({#a#} = M + N) =
1.184 -    ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
1.185 +    ({#a#} = M \\<and> N = {#} \\<or> M = {#} \\<and> {#a#} = N)"
1.186    apply (unfold Mempty_def single_def union_def)
1.189    apply (blast dest: sym)
1.190    done
1.191
1.193    "(M + {#a#} = N + {#b#}) =
1.194 -    (M = N \<and> a = b \<or>
1.195 -      M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
1.196 +    (M = N \\<and> a = b \\<or>
1.197 +      M = N - {#a#} + {#b#} \\<and> N = M - {#b#} + {#a#})"
1.198    apply (unfold single_def union_def diff_def)
1.199    apply (simp (no_asm) add: expand_fun_eq)
1.200    apply (rule conjI)
1.201 @@ -291,7 +291,7 @@
1.202  (*
1.203  val prems = Goal
1.204   "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
1.205 -by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")]
1.206 +by (res_inst_tac [("a","F"),("f","\\<lambda>A. if finite A then card A else 0")]
1.207       measure_induct 1);
1.208  by (Clarify_tac 1);
1.209  by (resolve_tac prems 1);
1.210 @@ -320,7 +320,7 @@
1.211
1.212  lemma setsum_decr:
1.213    "finite F ==> 0 < f a ==>
1.214 -    setsum (f (a := f a - 1)) F = (if a \<in> F then setsum f F - 1 else setsum f F)"
1.215 +    setsum (f (a := f a - 1)) F = (if a \\<in> F then setsum f F - 1 else setsum f F)"
1.216    apply (erule finite_induct)
1.217     apply auto
1.218    apply (drule_tac a = a in mk_disjoint_insert)
1.219 @@ -328,8 +328,8 @@
1.220    done
1.221
1.222  lemma rep_multiset_induct_aux:
1.223 -  "P (\<lambda>a. 0) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1)))
1.224 -    ==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
1.225 +  "P (\\<lambda>a. 0) ==> (!!f b. f \\<in> multiset ==> P f ==> P (f (b := f b + 1)))
1.226 +    ==> \\<forall>f. f \\<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
1.227  proof -
1.228    case antecedent
1.229    note prems = this [unfolded multiset_def]
1.230 @@ -338,7 +338,7 @@
1.231      apply (induct_tac n)
1.232       apply simp
1.233       apply clarify
1.234 -     apply (subgoal_tac "f = (\<lambda>a.0)")
1.235 +     apply (subgoal_tac "f = (\\<lambda>a.0)")
1.236        apply simp
1.237        apply (rule prems)
1.238       apply (rule ext)
1.239 @@ -363,10 +363,10 @@
1.240      apply (erule allE, erule impE, erule_tac [2] mp)
1.241       apply blast
1.242      apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply)
1.243 -    apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}")
1.244 +    apply (subgoal_tac "{x. x \\<noteq> a --> 0 < f x} = {x. 0 < f x}")
1.245       prefer 2
1.246       apply blast
1.247 -    apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}")
1.248 +    apply (subgoal_tac "{x. x \\<noteq> a \\<and> 0 < f x} = {x. 0 < f x} - {a}")
1.249       prefer 2
1.250       apply blast
1.252 @@ -374,8 +374,8 @@
1.253  qed
1.254
1.255  theorem rep_multiset_induct:
1.256 -  "f \<in> multiset ==> P (\<lambda>a. 0) ==>
1.257 -    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
1.258 +  "f \\<in> multiset ==> P (\\<lambda>a. 0) ==>
1.259 +    (!!f b. f \\<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
1.260    apply (insert rep_multiset_induct_aux)
1.261    apply blast
1.262    done
1.263 @@ -390,7 +390,7 @@
1.264      apply (rule Rep_multiset_inverse [THEN subst])
1.265      apply (rule Rep_multiset [THEN rep_multiset_induct])
1.266       apply (rule prem1)
1.267 -    apply (subgoal_tac "f (b := f b + 1) = (\<lambda>a. f a + (if a = b then 1 else 0))")
1.268 +    apply (subgoal_tac "f (b := f b + 1) = (\\<lambda>a. f a + (if a = b then 1 else 0))")
1.269       prefer 2
1.271      apply (erule ssubst)
1.272 @@ -401,7 +401,7 @@
1.273
1.274
1.275  lemma MCollect_preserves_multiset:
1.276 -    "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
1.277 +    "M \\<in> multiset ==> (\\<lambda>x. if P x then M x else 0) \\<in> multiset"
1.279    apply (rule finite_subset)
1.280     apply auto
1.281 @@ -413,11 +413,11 @@
1.283    done
1.284
1.285 -theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
1.286 +theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \\<inter> {x. P x}"
1.287    apply (auto simp add: set_of_def)
1.288    done
1.289
1.290 -theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}"
1.291 +theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \\<not> P x #}"
1.292    apply (subst multiset_eq_conv_count_eq)
1.293    apply auto
1.294    done
1.295 @@ -427,7 +427,7 @@
1.296
1.298    "(M + {#a#} = N + {#b#}) =
1.299 -    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
1.300 +    (M = N \\<and> a = b \\<or> (\\<exists>K. M = K + {#b#} \\<and> N = K + {#a#}))"
1.302    done
1.303
1.304 @@ -437,41 +437,41 @@
1.305  subsubsection {* Well-foundedness *}
1.306
1.307  constdefs
1.308 -  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
1.309 +  mult1 :: "('a \\<times> 'a) set => ('a multiset \\<times> 'a multiset) set"
1.310    "mult1 r ==
1.311 -    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
1.312 -      (\<forall>b. b :# K --> (b, a) \<in> r)}"
1.313 +    {(N, M). \\<exists>a M0 K. M = M0 + {#a#} \\<and> N = M0 + K \\<and>
1.314 +      (\\<forall>b. b :# K --> (b, a) \\<in> r)}"
1.315
1.316 -  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set"
1.317 +  mult :: "('a \\<times> 'a) set => ('a multiset \\<times> 'a multiset) set"
1.318    "mult r == (mult1 r)\<^sup>+"
1.319
1.320 -lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
1.321 +lemma not_less_empty [iff]: "(M, {#}) \\<notin> mult1 r"
1.323
1.324 -lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
1.325 -    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
1.326 -    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
1.327 -  (concl is "?case1 (mult1 r) \<or> ?case2")
1.328 +lemma less_add: "(N, M0 + {#a#}) \\<in> mult1 r ==>
1.329 +    (\\<exists>M. (M, M0) \\<in> mult1 r \\<and> N = M + {#a#}) \\<or>
1.330 +    (\\<exists>K. (\\<forall>b. b :# K --> (b, a) \\<in> r) \\<and> N = M0 + K)"
1.331 +  (concl is "?case1 (mult1 r) \\<or> ?case2")
1.332  proof (unfold mult1_def)
1.333 -  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
1.334 -  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
1.335 +  let ?r = "\\<lambda>K a. \\<forall>b. b :# K --> (b, a) \\<in> r"
1.336 +  let ?R = "\\<lambda>N M. \\<exists>a M0 K. M = M0 + {#a#} \\<and> N = M0 + K \\<and> ?r K a"
1.337    let ?case1 = "?case1 {(N, M). ?R N M}"
1.338
1.339 -  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
1.340 -  hence "\<exists>a' M0' K.
1.341 -      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
1.342 -  thus "?case1 \<or> ?case2"
1.343 +  assume "(N, M0 + {#a#}) \\<in> {(N, M). ?R N M}"
1.344 +  hence "\\<exists>a' M0' K.
1.345 +      M0 + {#a#} = M0' + {#a'#} \\<and> N = M0' + K \\<and> ?r K a'" by simp
1.346 +  thus "?case1 \\<or> ?case2"
1.347    proof (elim exE conjE)
1.348      fix a' M0' K
1.349      assume N: "N = M0' + K" and r: "?r K a'"
1.350      assume "M0 + {#a#} = M0' + {#a'#}"
1.351 -    hence "M0 = M0' \<and> a = a' \<or>
1.352 -        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
1.353 +    hence "M0 = M0' \\<and> a = a' \\<or>
1.354 +        (\\<exists>K'. M0 = K' + {#a'#} \\<and> M0' = K' + {#a#})"
1.356      thus ?thesis
1.357      proof (elim disjE conjE exE)
1.358        assume "M0 = M0'" "a = a'"
1.359 -      with N r have "?r K a \<and> N = M0 + K" by simp
1.360 +      with N r have "?r K a \\<and> N = M0 + K" by simp
1.361        hence ?case2 .. thus ?thesis ..
1.362      next
1.363        fix K'
1.364 @@ -485,78 +485,78 @@
1.365    qed
1.366  qed
1.367
1.368 -lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
1.369 +lemma all_accessible: "wf r ==> \\<forall>M. M \\<in> acc (mult1 r)"
1.370  proof
1.371    let ?R = "mult1 r"
1.372    let ?W = "acc ?R"
1.373    {
1.374      fix M M0 a
1.375 -    assume M0: "M0 \<in> ?W"
1.376 -      and wf_hyp: "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1.377 -      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
1.378 -    have "M0 + {#a#} \<in> ?W"
1.379 +    assume M0: "M0 \\<in> ?W"
1.380 +      and wf_hyp: "\\<forall>b. (b, a) \\<in> r --> (\\<forall>M \\<in> ?W. M + {#b#} \\<in> ?W)"
1.381 +      and acc_hyp: "\\<forall>M. (M, M0) \\<in> ?R --> M + {#a#} \\<in> ?W"
1.382 +    have "M0 + {#a#} \\<in> ?W"
1.383      proof (rule accI [of "M0 + {#a#}"])
1.384        fix N
1.385 -      assume "(N, M0 + {#a#}) \<in> ?R"
1.386 -      hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
1.387 -          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
1.388 +      assume "(N, M0 + {#a#}) \\<in> ?R"
1.389 +      hence "((\\<exists>M. (M, M0) \\<in> ?R \\<and> N = M + {#a#}) \\<or>
1.390 +          (\\<exists>K. (\\<forall>b. b :# K --> (b, a) \\<in> r) \\<and> N = M0 + K))"
1.392 -      thus "N \<in> ?W"
1.393 +      thus "N \\<in> ?W"
1.394        proof (elim exE disjE conjE)
1.395 -        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
1.396 -        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
1.397 -        hence "M + {#a#} \<in> ?W" ..
1.398 -        thus "N \<in> ?W" by (simp only: N)
1.399 +        fix M assume "(M, M0) \\<in> ?R" and N: "N = M + {#a#}"
1.400 +        from acc_hyp have "(M, M0) \\<in> ?R --> M + {#a#} \\<in> ?W" ..
1.401 +        hence "M + {#a#} \\<in> ?W" ..
1.402 +        thus "N \\<in> ?W" by (simp only: N)
1.403        next
1.404          fix K
1.405          assume N: "N = M0 + K"
1.406 -        assume "\<forall>b. b :# K --> (b, a) \<in> r"
1.407 -        have "?this --> M0 + K \<in> ?W" (is "?P K")
1.408 +        assume "\\<forall>b. b :# K --> (b, a) \\<in> r"
1.409 +        have "?this --> M0 + K \\<in> ?W" (is "?P K")
1.410          proof (induct K)
1.411 -          from M0 have "M0 + {#} \<in> ?W" by simp
1.412 +          from M0 have "M0 + {#} \\<in> ?W" by simp
1.413            thus "?P {#}" ..
1.414
1.415            fix K x assume hyp: "?P K"
1.416            show "?P (K + {#x#})"
1.417            proof
1.418 -            assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r"
1.419 -            hence "(x, a) \<in> r" by simp
1.420 -            with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
1.421 +            assume a: "\\<forall>b. b :# (K + {#x#}) --> (b, a) \\<in> r"
1.422 +            hence "(x, a) \\<in> r" by simp
1.423 +            with wf_hyp have b: "\\<forall>M \\<in> ?W. M + {#x#} \\<in> ?W" by blast
1.424
1.425 -            from a hyp have "M0 + K \<in> ?W" by simp
1.426 -            with b have "(M0 + K) + {#x#} \<in> ?W" ..
1.427 -            thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
1.428 +            from a hyp have "M0 + K \\<in> ?W" by simp
1.429 +            with b have "(M0 + K) + {#x#} \\<in> ?W" ..
1.430 +            thus "M0 + (K + {#x#}) \\<in> ?W" by (simp only: union_assoc)
1.431            qed
1.432          qed
1.433 -        hence "M0 + K \<in> ?W" ..
1.434 -        thus "N \<in> ?W" by (simp only: N)
1.435 +        hence "M0 + K \\<in> ?W" ..
1.436 +        thus "N \\<in> ?W" by (simp only: N)
1.437        qed
1.438      qed
1.439    } note tedious_reasoning = this
1.440
1.441    assume wf: "wf r"
1.442    fix M
1.443 -  show "M \<in> ?W"
1.444 +  show "M \\<in> ?W"
1.445    proof (induct M)
1.446 -    show "{#} \<in> ?W"
1.447 +    show "{#} \\<in> ?W"
1.448      proof (rule accI)
1.449 -      fix b assume "(b, {#}) \<in> ?R"
1.450 -      with not_less_empty show "b \<in> ?W" by contradiction
1.451 +      fix b assume "(b, {#}) \\<in> ?R"
1.452 +      with not_less_empty show "b \\<in> ?W" by contradiction
1.453      qed
1.454
1.455 -    fix M a assume "M \<in> ?W"
1.456 -    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1.457 +    fix M a assume "M \\<in> ?W"
1.458 +    from wf have "\\<forall>M \\<in> ?W. M + {#a#} \\<in> ?W"
1.459      proof induct
1.460        fix a
1.461 -      assume "\<forall>b. (b, a) \<in> r --> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
1.462 -      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
1.463 +      assume "\\<forall>b. (b, a) \\<in> r --> (\\<forall>M \\<in> ?W. M + {#b#} \\<in> ?W)"
1.464 +      show "\\<forall>M \\<in> ?W. M + {#a#} \\<in> ?W"
1.465        proof
1.466 -        fix M assume "M \<in> ?W"
1.467 -        thus "M + {#a#} \<in> ?W"
1.468 +        fix M assume "M \\<in> ?W"
1.469 +        thus "M + {#a#} \\<in> ?W"
1.470            by (rule acc_induct) (rule tedious_reasoning)
1.471        qed
1.472      qed
1.473 -    thus "M + {#a#} \<in> ?W" ..
1.474 +    thus "M + {#a#} \\<in> ?W" ..
1.475    qed
1.476  qed
1.477
1.478 @@ -578,9 +578,9 @@
1.479  text {* One direction. *}
1.480
1.481  lemma mult_implies_one_step:
1.482 -  "trans r ==> (M, N) \<in> mult r ==>
1.483 -    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
1.484 -    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
1.485 +  "trans r ==> (M, N) \\<in> mult r ==>
1.486 +    \\<exists>I J K. N = I + J \\<and> M = I + K \\<and> J \\<noteq> {#} \\<and>
1.487 +    (\\<forall>k \\<in> set_of K. \\<exists>j \\<in> set_of J. (k, j) \\<in> r)"
1.488    apply (unfold mult_def mult1_def set_of_def)
1.489    apply (erule converse_trancl_induct)
1.490    apply clarify
1.491 @@ -592,7 +592,7 @@
1.492     apply (simp (no_asm))
1.493     apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
1.494     apply (simp (no_asm_simp) add: union_assoc [symmetric])
1.495 -   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
1.496 +   apply (drule_tac f = "\\<lambda>M. M - {#a#}" in arg_cong)
1.498     apply (simp (no_asm_use) add: trans_def)
1.499     apply blast
1.500 @@ -603,7 +603,7 @@
1.501     apply (rule conjI)
1.502      apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
1.503     apply (rule conjI)
1.504 -    apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
1.505 +    apply (drule_tac f = "\\<lambda>M. M - {#a#}" in arg_cong)
1.506      apply simp
1.507      apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
1.508     apply (simp (no_asm_use) add: trans_def)
1.509 @@ -617,7 +617,7 @@
1.511    done
1.512
1.513 -lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
1.514 +lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \\<exists>a N. M = N + {#a#}"
1.515    apply (erule size_eq_Suc_imp_elem [THEN exE])
1.516    apply (drule elem_imp_eq_diff_union)
1.517    apply auto
1.518 @@ -625,8 +625,8 @@
1.519
1.520  lemma one_step_implies_mult_aux:
1.521    "trans r ==>
1.522 -    \<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))
1.523 -      --> (I + K, I + J) \<in> mult r"
1.524 +    \\<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))
1.525 +      --> (I + K, I + J) \\<in> mult r"
1.526    apply (induct_tac n)
1.527     apply auto
1.528    apply (frule size_eq_Suc_imp_eq_union)
1.529 @@ -640,15 +640,15 @@
1.530     apply (rule r_into_trancl)
1.531     apply (simp add: mult1_def set_of_def)
1.532     apply blast
1.533 -  txt {* Now we know @{term "J' \<noteq> {#}"}. *}
1.534 -  apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
1.535 -  apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
1.536 +  txt {* Now we know @{term "J' \\<noteq> {#}"}. *}
1.537 +  apply (cut_tac M = K and P = "\\<lambda>x. (x, a) \\<in> r" in multiset_partition)
1.538 +  apply (erule_tac P = "\\<forall>k \\<in> set_of K. ?P k" in rev_mp)
1.539    apply (erule ssubst)
1.541    apply auto
1.542    apply (subgoal_tac
1.543 -    "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
1.544 -      (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
1.545 +    "((I + {# x : K. (x, a) \\<in> r #}) + {# x : K. (x, a) \\<notin> r #},
1.546 +      (I + {# x : K. (x, a) \\<in> r #}) + J') \\<in> mult r")
1.547     prefer 2
1.548     apply force
1.549    apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
1.550 @@ -661,8 +661,8 @@
1.551    done
1.552
1.553  theorem one_step_implies_mult:
1.554 -  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
1.555 -    ==> (I + K, I + J) \<in> mult r"
1.556 +  "trans r ==> J \\<noteq> {#} ==> \\<forall>k \\<in> set_of K. \\<exists>j \\<in> set_of J. (k, j) \\<in> r
1.557 +    ==> (I + K, I + J) \\<in> mult r"
1.558    apply (insert one_step_implies_mult_aux)
1.559    apply blast
1.560    done
1.561 @@ -673,8 +673,8 @@
1.562  instance multiset :: ("term") ord ..
1.563
1.565 -  less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
1.566 -  le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
1.567 +  less_multiset_def: "M' < M == (M', M) \\<in> mult {(x', x). x' < x}"
1.568 +  le_multiset_def: "M' <= M == M' = M \\<or> M' < (M::'a multiset)"
1.569
1.570  lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
1.571    apply (unfold trans_def)
1.572 @@ -686,12 +686,12 @@
1.573  *}
1.574
1.575  lemma mult_irrefl_aux:
1.576 -    "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}"
1.577 +    "finite A ==> (\\<forall>x \\<in> A. \\<exists>y \\<in> A. x < (y::'a::order)) --> A = {}"
1.578    apply (erule finite_induct)
1.579     apply (auto intro: order_less_trans)
1.580    done
1.581
1.582 -theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
1.583 +theorem mult_less_not_refl: "\\<not> M < (M::'a::order multiset)"
1.584    apply (unfold less_multiset_def)
1.585    apply auto
1.586    apply (drule trans_base_order [THEN mult_implies_one_step])
1.587 @@ -715,7 +715,7 @@
1.588
1.589  text {* Asymmetry. *}
1.590
1.591 -theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
1.592 +theorem mult_less_not_sym: "M < N ==> \\<not> N < (M::'a::order multiset)"
1.593    apply auto
1.594    apply (rule mult_less_not_refl [THEN notE])
1.595    apply (erule mult_less_trans)
1.596 @@ -723,7 +723,7 @@
1.597    done
1.598
1.599  theorem mult_less_asym:
1.600 -    "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
1.601 +    "M < N ==> (\\<not> P ==> N < (M::'a::order multiset)) ==> P"
1.602    apply (insert mult_less_not_sym)
1.603    apply blast
1.604    done
1.605 @@ -749,7 +749,7 @@
1.606    apply (blast intro: mult_less_trans)
1.607    done
1.608
1.609 -theorem mult_less_le: "M < N = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
1.610 +theorem mult_less_le: "M < N = (M <= N \\<and> M \\<noteq> (N::'a::order multiset))"
1.611    apply (unfold le_multiset_def)
1.612    apply auto
1.613    done
1.614 @@ -770,7 +770,7 @@
1.615  subsubsection {* Monotonicity of multiset union *}
1.616
1.617  theorem mult1_union:
1.618 -    "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
1.619 +    "(B, D) \\<in> mult1 r ==> trans r ==> (C + B, C + D) \\<in> mult1 r"
1.620    apply (unfold mult1_def)
1.621    apply auto
1.622    apply (rule_tac x = a in exI)
1.623 @@ -806,7 +806,7 @@
1.624    apply (unfold le_multiset_def less_multiset_def)
1.625    apply (case_tac "M = {#}")
1.626     prefer 2
1.627 -   apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
1.628 +   apply (subgoal_tac "({#} + {#}, {#} + M) \\<in> mult (Collect (split op <))")
1.629      prefer 2
1.630      apply (rule one_step_implies_mult)
1.631        apply (simp only: trans_def)
```