src/HOL/Library/List_lexord.thy

--- a/src/HOL/Library/List_lexord.thy	Wed Aug 31 15:46:36 2005 +0200
+++ b/src/HOL/Library/List_lexord.thy	Wed Aug 31 15:46:37 2005 +0200
@@ -3,51 +3,54 @@
     Author:     Norbert Voelker
 *)
 
-header {* Instantiation of order classes for lexord on lists *}
+header {* Lexicographic order on lists *}
 
 theory List_lexord
 imports Main
 begin
 
 instance list :: (ord) ord ..
-defs(overloaded)
-  list_le_def:  "(xs::('a::ord) list) \<le> ys \<equiv> (xs < ys \<or> xs = ys)" 
-  list_less_def: "(xs::('a::ord) list) < ys \<equiv> (xs,ys) \<in> lexord {(u,v). u < v}"
+defs (overloaded)
+  list_le_def:  "(xs::('a::ord) list) \<le> ys \<equiv> (xs < ys \<or> xs = ys)"
+  list_less_def: "(xs::('a::ord) list) < ys \<equiv> (xs, ys) \<in> lexord {(u,v). u < v}"
 
 lemmas list_ord_defs = list_less_def list_le_def
 
-instance list::(order)order
+instance list :: (order) order
   apply (intro_classes, unfold list_ord_defs)
-  apply (rule disjI2, safe)
-  apply (blast intro: lexord_trans transI order_less_trans)
+     apply (rule disjI2, safe)
+    apply (blast intro: lexord_trans transI order_less_trans)
+   apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE])
+    apply simp
+   apply (blast intro: lexord_trans transI order_less_trans)
   apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE])
   apply simp
-  apply (blast intro: lexord_trans transI order_less_trans)
-  apply (rule_tac r1 = "{(a::'a,b). a < b}" in lexord_irreflexive [THEN notE])
-  apply simp
-  by assumption
+  apply assumption
+  done
 
 instance list::(linorder)linorder
   apply (intro_classes, unfold list_le_def list_less_def, safe)
-  apply (cut_tac x="x" and y="y" and  r = "{(a,b). a < b}"  in lexord_linear)
-  by (force, simp)
+  apply (cut_tac x = x and y = y and  r = "{(a,b). a < b}"  in lexord_linear)
+   apply force
+  apply simp
+  done
 
-lemma not_less_Nil[simp]: "~(x < [])";
-  by (unfold list_less_def, simp);
+lemma not_less_Nil[simp]: "~(x < [])"
+  by (unfold list_less_def) simp
 
-lemma Nil_less_Cons[simp]: "[] < a # x";
-  by (unfold list_less_def, simp);
+lemma Nil_less_Cons[simp]: "[] < a # x"
+  by (unfold list_less_def) simp
 
-lemma Cons_less_Cons[simp]: "(a # x < b # y) = (a < b | a = b & x < y)";
-  by (unfold list_less_def, simp);
+lemma Cons_less_Cons[simp]: "(a # x < b # y) = (a < b | a = b & x < y)"
+  by (unfold list_less_def) simp
 
-lemma le_Nil[simp]: "(x <= [])   = (x = [])";
-  by (unfold list_ord_defs, case_tac x, auto);
+lemma le_Nil[simp]: "(x <= []) = (x = [])"
+  by (unfold list_ord_defs, cases x) auto
 
-lemma Nil_le_Cons[simp]: "([] <= x)";
-  by (unfold list_ord_defs, case_tac x, auto);
+lemma Nil_le_Cons [simp]: "([] <= x)"
+  by (unfold list_ord_defs, cases x) auto
 
-lemma Cons_le_Cons[simp]: "(a # x <= b # y) = (a < b | a = b & x <= y)";
-  by (unfold list_ord_defs, auto);
+lemma Cons_le_Cons[simp]: "(a # x <= b # y) = (a < b | a = b & x <= y)"
+  by (unfold list_ord_defs) auto
 
-end
\ No newline at end of file
+end