src/HOL/Library/RBT_Set.thy
changeset 53745 788730ab7da4
parent 51540 eea5c4ca4a0e
child 53955 436649a2ed62
     1.1 --- a/src/HOL/Library/RBT_Set.thy	Fri Sep 20 00:08:42 2013 +0200
     1.2 +++ b/src/HOL/Library/RBT_Set.thy	Fri Sep 20 10:09:16 2013 +0200
     1.3 @@ -628,20 +628,16 @@
     1.4    "A \<le> Coset t \<longleftrightarrow> (\<forall>y\<in>Set t. y \<notin> A)"
     1.5  by auto
     1.6  
     1.7 -definition non_empty_trees where [code del]: "non_empty_trees t1 t2 \<longleftrightarrow> Coset t1 \<le> Set t2"
     1.8 -
     1.9 -code_abort non_empty_trees
    1.10 -
    1.11  lemma subset_Coset_empty_Set_empty [code]:
    1.12    "Coset t1 \<le> Set t2 \<longleftrightarrow> (case (impl_of t1, impl_of t2) of 
    1.13      (rbt.Empty, rbt.Empty) => False |
    1.14 -    (_, _) => non_empty_trees t1 t2)"
    1.15 +    (_, _) => Code.abort (STR ''non_empty_trees'') (\<lambda>_. Coset t1 \<le> Set t2))"
    1.16  proof -
    1.17    have *: "\<And>t. impl_of t = rbt.Empty \<Longrightarrow> t = RBT rbt.Empty"
    1.18      by (subst(asm) RBT_inverse[symmetric]) (auto simp: impl_of_inject)
    1.19    have **: "Lifting.invariant is_rbt rbt.Empty rbt.Empty" unfolding Lifting.invariant_def by simp
    1.20    show ?thesis  
    1.21 -    by (auto simp: Set_def lookup.abs_eq[OF **] dest!: * split: rbt.split simp: non_empty_trees_def)
    1.22 +    by (auto simp: Set_def lookup.abs_eq[OF **] dest!: * split: rbt.split)
    1.23  qed
    1.24  
    1.25  text {* A frequent case – avoid intermediate sets *}
    1.26 @@ -661,15 +657,11 @@
    1.27      by (auto simp add: setsum.eq_fold finite_fold_fold_keys o_def)
    1.28  qed
    1.29  
    1.30 -definition not_a_singleton_tree  where [code del]: "not_a_singleton_tree x y = x y"
    1.31 -
    1.32 -code_abort not_a_singleton_tree
    1.33 -
    1.34  lemma the_elem_set [code]:
    1.35    fixes t :: "('a :: linorder, unit) rbt"
    1.36    shows "the_elem (Set t) = (case impl_of t of 
    1.37      (Branch RBT_Impl.B RBT_Impl.Empty x () RBT_Impl.Empty) \<Rightarrow> x
    1.38 -    | _ \<Rightarrow> not_a_singleton_tree the_elem (Set t))"
    1.39 +    | _ \<Rightarrow> Code.abort (STR ''not_a_singleton_tree'') (\<lambda>_. the_elem (Set t)))"
    1.40  proof -
    1.41    {
    1.42      fix x :: "'a :: linorder"
    1.43 @@ -681,7 +673,7 @@
    1.44        by (subst(asm) RBT_inverse[symmetric, OF *])
    1.45          (auto simp: Set_def the_elem_def lookup.abs_eq[OF **] impl_of_inject)
    1.46    }
    1.47 -  then show ?thesis unfolding not_a_singleton_tree_def
    1.48 +  then show ?thesis
    1.49      by(auto split: rbt.split unit.split color.split)
    1.50  qed
    1.51  
    1.52 @@ -729,17 +721,16 @@
    1.53    "wf (Set t) = acyclic (Set t)"
    1.54  by (simp add: wf_iff_acyclic_if_finite)
    1.55  
    1.56 -definition not_non_empty_tree  where [code del]: "not_non_empty_tree x y = x y"
    1.57 -
    1.58 -code_abort not_non_empty_tree
    1.59 -
    1.60  lemma Min_fin_set_fold [code]:
    1.61 -  "Min (Set t) = (if is_empty t then not_non_empty_tree Min (Set t) else r_min_opt t)"
    1.62 +  "Min (Set t) = 
    1.63 +  (if is_empty t
    1.64 +   then Code.abort (STR ''not_non_empty_tree'') (\<lambda>_. Min (Set t))
    1.65 +   else r_min_opt t)"
    1.66  proof -
    1.67    have *: "semilattice (min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" ..
    1.68    with finite_fold1_fold1_keys [OF *, folded Min_def]
    1.69    show ?thesis
    1.70 -    by (simp add: not_non_empty_tree_def r_min_alt_def r_min_eq_r_min_opt [symmetric])  
    1.71 +    by (simp add: r_min_alt_def r_min_eq_r_min_opt [symmetric])  
    1.72  qed
    1.73  
    1.74  lemma Inf_fin_set_fold [code]:
    1.75 @@ -771,12 +762,15 @@
    1.76  qed
    1.77  
    1.78  lemma Max_fin_set_fold [code]:
    1.79 -  "Max (Set t) = (if is_empty t then not_non_empty_tree Max (Set t) else r_max_opt t)"
    1.80 +  "Max (Set t) = 
    1.81 +  (if is_empty t
    1.82 +   then Code.abort (STR ''not_non_empty_tree'') (\<lambda>_. Max (Set t))
    1.83 +   else r_max_opt t)"
    1.84  proof -
    1.85    have *: "semilattice (max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" ..
    1.86    with finite_fold1_fold1_keys [OF *, folded Max_def]
    1.87    show ?thesis
    1.88 -    by (simp add: not_non_empty_tree_def r_max_alt_def r_max_eq_r_max_opt [symmetric])  
    1.89 +    by (simp add: r_max_alt_def r_max_eq_r_max_opt [symmetric])  
    1.90  qed
    1.91  
    1.92  lemma Sup_fin_set_fold [code]: