some derived properties;

--- a/src/HOL/Library/Ring_and_Field.thy	Wed Dec 06 13:59:42 2000 +0100
+++ b/src/HOL/Library/Ring_and_Field.thy	Wed Dec 06 17:03:26 2000 +0100
@@ -15,14 +15,14 @@
 axclass ring < zero, plus, minus, times, number
   add_assoc: "(a + b) + c = a + (b + c)"
   add_commute: "a + b = b + a"
-  left_zero: "0 + a = a"
-  right_minus: "a + - a = 0"
-  diff_minus: "a - b = a + (-b)"
+  left_zero [simp]: "0 + a = a"
+  left_minus [simp]: "- a + a = 0"
+  diff_minus [simp]: "a - b = a + (-b)"
   zero_number: "0 = #0"
 
   mult_assoc: "(a * b) * c = a * (b * c)"
   mult_commute: "a * b = b * a"
-  left_one: "#1 * a = a"
+  left_one [simp]: "#1 * a = a"
 
   left_distrib: "(a + b) * c = a * c + b * c"
 
@@ -32,20 +32,106 @@
   abs_if: "\<bar>a\<bar> = (if a < 0 then -a else a)"
 
 axclass field < ring, inverse
-  left_inverse: "a \<noteq> 0 ==> inverse a * a = #1"
-  divides_inverse: "b \<noteq> 0 ==> a / b = a * inverse b"
+  left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = #1"
+  divides_inverse [simp]: "b \<noteq> 0 ==> a / b = a * inverse b" (* FIXME unconditional ?? *)
 
 axclass ordered_field < ordered_ring, field
 
-subsection {* simplification rules *}
+
+
+subsection {* Derived rules *}
+
+subsubsection {* Derived rules for addition *}
+
+lemma right_zero [simp]: "a + 0 = (a::'a::ring)" 
+proof -
+  have "a + 0 = 0 + a" by (simp only: add_commute)
+  also have "\<dots> = a" by simp
+  finally show ?thesis .
+qed
+
+lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::ring))"
+proof -
+  have "a + (b + c) = (a + b) + c" by (simp only: add_assoc)
+  also have "\<dots> = (b + a) + c" by (simp only: add_commute)
+  finally show ?thesis by (simp only: add_assoc) 
+qed
 
-lemma left_minus [simp]: "- (a::'a::field) + a = 0"
-  by (simp only: add_commute right_minus)
+theorems ring_add_ac = add_assoc add_commute add_left_commute
+
+lemma right_minus [simp]: "a + -(a::'a::ring) = 0" 
+proof -
+  have "a + -a = -a + a" by (simp add: ring_add_ac)
+  also have "\<dots> = 0" by simp
+  finally show ?thesis .
+qed
+
+lemma right_minus_eq [simp]: "(a - b = 0) = (a = (b::'a::ring))"
+proof 
+  have "a = a - b + b" by (simp add: ring_add_ac)
+  also assume "a - b = 0"
+  finally show "a = b" by simp
+qed simp
+
+lemma diff_self [simp]: "a - (a::'a::ring) = 0"
+  by simp
+
+
+
+subsubsection {* Derived rules for multiplication *}
 
-lemma diff_self [simp]: "a - (a::'a::field) = 0"
-  by (simp only: diff_minus right_minus)
+lemma right_one [simp]: "a = a * (#1\<Colon>'a::field)"
+proof -
+  have "a = #1 * a" by simp
+  also have "\<dots> = a * #1" by (simp add: mult_commute)
+  finally show ?thesis .
+qed
+
+lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::ring))"
+proof -
+  have "a * (b * c) = (a * b) * c" by (simp only: mult_assoc)
+  also have "\<dots> = (b * a) * c" by (simp only: mult_commute)
+  finally show ?thesis by (simp only: mult_assoc)
+qed
+
+theorems ring_mult_ac = mult_assoc mult_commute mult_left_commute
+
+lemma right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow>  a * inverse (a::'a::field) = #1" 
+proof -
+  have "a * inverse a = inverse a * a" by (simp add: ring_mult_ac)
+  also assume "a \<noteq> 0"
+  hence "inverse a * a = #1" by simp
+  finally show ?thesis .
+qed
 
-subsection {* The ordered ring of integers *}
+lemma right_inverse_eq [simp]: "b \<noteq> 0 \<Longrightarrow> (a / b = #1) = (a = (b::'a::field))"
+proof 
+  assume "b \<noteq> 0"
+  hence "a = (a / b) * b" by (simp add: ring_mult_ac)
+  also assume "a / b = #1"
+  finally show "a = b" by simp
+qed simp
+
+lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / (a::'a::field) = #1"
+  by simp
+
+
+
+subsubsection {* Distribution rules *}
+
+lemma right_distrib: "a * (b + c) = a * b + a * (c::'a::ring)"
+proof -
+  have "a * (b + c) = (b + c) * a" by (simp add: ring_mult_ac)
+  also have "\<dots> = b * a + c * a" by (simp only: left_distrib)
+  also have "\<dots> = a * b + a * c" by (simp add: ring_mult_ac)
+  finally show "?thesis" .
+qed
+
+theorems ring_distrib = right_distrib left_distrib
+
+
+
+subsection {* Example: The ordered ring of integers *}
 
 instance int :: ordered_ring
 proof
@@ -53,7 +139,7 @@
   show "(i + j) + k = i + (j + k)" by simp
   show "i + j = j + i" by simp
   show "0 + i = i" by simp
-  show "i + - i = 0" by simp
+  show "- i + i = 0" by simp
   show "i - j = i + (-j)" by simp
   show "(i * j) * k = i * (j * k)" by simp
   show "i * j = j * i" by simp