1.1 --- a/src/HOL/Library/Product_ord.thy	Wed May 23 13:33:35 2012 +0200
     1.2 +++ b/src/HOL/Library/Product_ord.thy	Wed May 23 14:17:32 2012 +0200
     1.3 @@ -22,30 +22,30 @@
     1.4  end
     1.5  
     1.6  lemma [code]:
     1.7 -  "(x1\<Colon>'a\<Colon>{ord, equal}, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 \<le> y2"
     1.8 -  "(x1\<Colon>'a\<Colon>{ord, equal}, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 < y2"
     1.9 +  "(x1::'a::{ord, equal}, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 \<le> y2"
    1.10 +  "(x1::'a::{ord, equal}, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 < y2"
    1.11    unfolding prod_le_def prod_less_def by simp_all
    1.12  
    1.13 -instance prod :: (preorder, preorder) preorder proof
    1.14 -qed (auto simp: prod_le_def prod_less_def less_le_not_le intro: order_trans)
    1.15 +instance prod :: (preorder, preorder) preorder
    1.16 +  by default (auto simp: prod_le_def prod_less_def less_le_not_le intro: order_trans)
    1.17  
    1.18 -instance prod :: (order, order) order proof
    1.19 -qed (auto simp add: prod_le_def)
    1.20 +instance prod :: (order, order) order
    1.21 +  by default (auto simp add: prod_le_def)
    1.22  
    1.23 -instance prod :: (linorder, linorder) linorder proof
    1.24 -qed (auto simp: prod_le_def)
    1.25 +instance prod :: (linorder, linorder) linorder
    1.26 +  by default (auto simp: prod_le_def)
    1.27  
    1.28  instantiation prod :: (linorder, linorder) distrib_lattice
    1.29  begin
    1.30  
    1.31  definition
    1.32 -  inf_prod_def: "(inf \<Colon> 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = min"
    1.33 +  inf_prod_def: "(inf :: 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = min"
    1.34  
    1.35  definition
    1.36 -  sup_prod_def: "(sup \<Colon> 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = max"
    1.37 +  sup_prod_def: "(sup :: 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = max"
    1.38  
    1.39 -instance proof
    1.40 -qed (auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
    1.41 +instance
    1.42 +  by default (auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
    1.43  
    1.44  end
    1.45  
    1.46 @@ -55,8 +55,8 @@
    1.47  definition
    1.48    bot_prod_def: "bot = (bot, bot)"
    1.49  
    1.50 -instance proof
    1.51 -qed (auto simp add: bot_prod_def prod_le_def)
    1.52 +instance
    1.53 +  by default (auto simp add: bot_prod_def prod_le_def)
    1.54  
    1.55  end
    1.56  
    1.57 @@ -66,8 +66,8 @@
    1.58  definition
    1.59    top_prod_def: "top = (top, top)"
    1.60  
    1.61 -instance proof
    1.62 -qed (auto simp add: top_prod_def prod_le_def)
    1.63 +instance
    1.64 +  by default (auto simp add: top_prod_def prod_le_def)
    1.65  
    1.66  end
    1.67  
    1.68 @@ -86,11 +86,29 @@
    1.69    proof (induct z)
    1.70      case (Pair a b)
    1.71      show "P (a, b)"
    1.72 -      apply (induct a arbitrary: b rule: less_induct)
    1.73 -      apply (rule less_induct [where 'a='b])
    1.74 -      apply (rule P)
    1.75 -      apply (auto simp add: prod_less_eq)
    1.76 -      done
    1.77 +    proof (induct a arbitrary: b rule: less_induct)
    1.78 +      case (less a\<^isub>1) note a\<^isub>1 = this
    1.79 +      show "P (a\<^isub>1, b)"
    1.80 +      proof (induct b rule: less_induct)
    1.81 +        case (less b\<^isub>1) note b\<^isub>1 = this
    1.82 +        show "P (a\<^isub>1, b\<^isub>1)"
    1.83 +        proof (rule P)
    1.84 +          fix p assume p: "p < (a\<^isub>1, b\<^isub>1)"
    1.85 +          show "P p"
    1.86 +          proof (cases "fst p < a\<^isub>1")
    1.87 +            case True
    1.88 +            then have "P (fst p, snd p)" by (rule a\<^isub>1)
    1.89 +            then show ?thesis by simp
    1.90 +          next
    1.91 +            case False
    1.92 +            with p have 1: "a\<^isub>1 = fst p" and 2: "snd p < b\<^isub>1"
    1.93 +              by (simp_all add: prod_less_eq)
    1.94 +            from 2 have "P (a\<^isub>1, snd p)" by (rule b\<^isub>1)
    1.95 +            with 1 show ?thesis by simp
    1.96 +          qed
    1.97 +        qed
    1.98 +      qed
    1.99 +    qed
   1.100    qed
   1.101  qed
   1.102