src/HOL/Library/Multiset.thy
 changeset 10277 081c8641aa11 parent 10249 e4d13d8a9011 child 10313 51e830bb7abe
```     1.1 --- a/src/HOL/Library/Multiset.thy	Thu Oct 19 21:22:05 2000 +0200
1.2 +++ b/src/HOL/Library/Multiset.thy	Thu Oct 19 21:22:44 2000 +0200
1.3 @@ -16,12 +16,12 @@
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> {f. finite {x. 0 < f x}}"
1.8 -    by simp
1.9 +  show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
1.10  qed
1.11
1.12  lemmas multiset_typedef [simp] =
1.13 -  Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
1.14 +    Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
1.15 +  and [simp] = Rep_multiset_inject [symmetric]
1.16
1.17  constdefs
1.18    Mempty :: "'a multiset"    ("{#}")
1.19 @@ -89,33 +89,6 @@
1.20    apply auto
1.21    done
1.22
1.23 -text {*
1.24 - \medskip Injectivity of @{term Rep_multiset}.
1.25 -*}  (* FIXME typedef package (!?) *)
1.26 -
1.27 -lemma multiset_eq_conv_Rep_eq [simp]:
1.28 -    "(M = N) = (Rep_multiset M = Rep_multiset N)"
1.29 -  apply (rule iffI)
1.30 -   apply simp
1.31 -  apply (drule_tac f = Abs_multiset in arg_cong)
1.32 -  apply simp
1.33 -  done
1.34 -
1.35 -(* FIXME
1.36 -Goal
1.37 - "[| f : multiset; g : multiset |] ==> \
1.38 -\ (Abs_multiset f = Abs_multiset g) = (!x. f x = g x)";
1.39 -by (rtac iffI 1);
1.40 - by (dres_inst_tac [("f","Rep_multiset")] arg_cong 1);
1.41 - by (Asm_full_simp_tac 1);
1.42 -by (subgoal_tac "f = g" 1);
1.43 - by (Asm_simp_tac 1);
1.44 -by (rtac ext 1);
1.45 -by (Blast_tac 1);
1.46 -qed "Abs_multiset_eq";
1.48 -*)
1.49 -
1.50
1.51  subsection {* Algebraic properties of multisets *}
1.52
1.53 @@ -141,6 +114,13 @@
1.54
1.55  theorems union_ac = union_assoc union_commute union_lcomm
1.56
1.57 +instance multiset :: ("term") plus_ac0
1.58 +  apply intro_classes
1.59 +    apply (rule union_commute)
1.60 +   apply (rule union_assoc)
1.61 +  apply simp
1.62 +  done
1.63 +
1.64
1.65  subsubsection {* Difference *}
1.66
1.67 @@ -442,7 +422,7 @@
1.68    apply auto
1.69    done
1.70
1.71 -declare multiset_eq_conv_Rep_eq [simp del]
1.72 +declare Rep_multiset_inject [symmetric, simp del]
1.73  declare multiset_typedef [simp del]
1.74
1.76 @@ -466,8 +446,7 @@
1.77    "mult r == (mult1 r)^+"
1.78
1.79  lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
1.80 -  apply (simp add: mult1_def)
1.81 -  done
1.82 +  by (simp add: mult1_def)
1.83
1.84  lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
1.85      (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
1.86 @@ -629,7 +608,7 @@
1.87      apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
1.88     apply (simp (no_asm_use) add: trans_def)
1.89     apply blast
1.90 -  apply (subgoal_tac "a :# (M0 +{#a#})")
1.91 +  apply (subgoal_tac "a :# (M0 + {#a#})")
1.92     apply simp
1.93    apply (simp (no_asm))
1.94    done
1.95 @@ -775,6 +754,18 @@
1.96    apply auto
1.97    done
1.98
1.99 +text {* Partial order. *}
1.100 +
1.101 +instance multiset :: (order) order
1.102 +  apply intro_classes
1.103 +     apply (rule mult_le_refl)
1.104 +    apply (erule mult_le_trans)
1.105 +    apply assumption
1.106 +   apply (erule mult_le_antisym)
1.107 +   apply assumption
1.108 +  apply (rule mult_less_le)
1.109 +  done
1.110 +
1.111
1.112  subsubsection {* Monotonicity of multiset union *}
1.113
1.114 @@ -834,21 +825,4 @@
1.115    apply (subst union_commute, rule union_upper1)
1.116    done
1.117
1.118 -instance multiset :: (order) order
1.119 -  apply intro_classes
1.120 -     apply (rule mult_le_refl)
1.121 -    apply (erule mult_le_trans)
1.122 -    apply assumption
1.123 -   apply (erule mult_le_antisym)
1.124 -   apply assumption
1.125 -  apply (rule mult_less_le)
1.126 -  done
1.127 -
1.128 -instance multiset :: ("term") plus_ac0
1.129 -  apply intro_classes
1.130 -    apply (rule union_commute)
1.131 -   apply (rule union_assoc)
1.132 -  apply simp
1.133 -  done
1.134 -
1.135  end
```