src/HOL/Ring_and_Field.thy

+(*  Title:   HOL/Ring_and_Field.thy
+    ID:      $Id$
+    Author:  Gertrud Bauer and Markus Wenzel, TU Muenchen
+    License: GPL (GNU GENERAL PUBLIC LICENSE)
+*)
+
+header {*
+  \title{Ring and field structures}
+  \author{Gertrud Bauer and Markus Wenzel}
+*}
+
+theory Ring_and_Field = Inductive:
+
+text{*Lemmas and extension to semirings by L. C. Paulson*}
+
+subsection {* Abstract algebraic structures *}
+
+axclass semiring \<subseteq> zero, one, plus, times
+  add_assoc: "(a + b) + c = a + (b + c)"
+  add_commute: "a + b = b + a"
+  left_zero [simp]: "0 + a = a"
+
+  mult_assoc: "(a * b) * c = a * (b * c)"
+  mult_commute: "a * b = b * a"
+  left_one [simp]: "1 * a = a"
+
+  left_distrib: "(a + b) * c = a * c + b * c"
+  zero_neq_one [simp]: "0 \<noteq> 1"
+
+axclass ring \<subseteq> semiring, minus
+  left_minus [simp]: "- a + a = 0"
+  diff_minus: "a - b = a + (-b)"
+
+axclass ordered_semiring \<subseteq> semiring, linorder
+  add_left_mono: "a \<le> b ==> c + a \<le> c + b"
+  mult_strict_left_mono: "a < b ==> 0 < c ==> c * a < c * b"
+
+axclass ordered_ring \<subseteq> ordered_semiring, ring
+  abs_if: "\<bar>a\<bar> = (if a < 0 then -a else a)"
+
+axclass field \<subseteq> ring, inverse
+  left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"
+  divide_inverse:      "b \<noteq> 0 ==> a / b = a * inverse b"
+
+axclass ordered_field \<subseteq> ordered_ring, field
+
+axclass division_by_zero \<subseteq> zero, inverse
+  inverse_zero: "inverse 0 = 0"
+  divide_zero: "a / 0 = 0"
+
+
+subsection {* Derived rules for addition *}
+
+lemma right_zero [simp]: "a + 0 = (a::'a::semiring)"
+proof -
+  have "a + 0 = 0 + a" by (simp only: add_commute)
+  also have "... = a" by simp
+  finally show ?thesis .
+qed
+
+lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::semiring))"
+  by (rule mk_left_commute [of "op +", OF add_assoc add_commute])
+
+theorems 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: add_ac)
+  also have "... = 0" by simp
+  finally show ?thesis .
+qed
+
+lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::ring))"
+proof
+  have "a = a - b + b" by (simp add: diff_minus add_ac)
+  also assume "a - b = 0"
+  finally show "a = b" by simp
+next
+  assume "a = b"
+  thus "a - b = 0" by (simp add: diff_minus)
+qed
+
+lemma diff_self [simp]: "a - (a::'a::ring) = 0"
+  by (simp add: diff_minus)
+
+lemma add_left_cancel [simp]:
+     "(a + b = a + c) = (b = (c::'a::ring))"
+proof
+  assume eq: "a + b = a + c"
+  then have "(-a + a) + b = (-a + a) + c"
+    by (simp only: eq add_assoc)
+  then show "b = c" by simp
+next
+  assume eq: "b = c"
+  then show "a + b = a + c" by simp
+qed
+
+lemma add_right_cancel [simp]:
+     "(b + a = c + a) = (b = (c::'a::ring))"
+  by (simp add: add_commute)
+
+lemma minus_minus [simp]: "- (- (a::'a::ring)) = a"
+  proof (rule add_left_cancel [of "-a", THEN iffD1])
+    show "(-a + -(-a) = -a + a)"
+    by simp
+  qed
+
+lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::ring)"
+apply (rule right_minus_eq [THEN iffD1, symmetric])
+apply (simp add: diff_minus add_commute) 
+done
+
+lemma minus_zero [simp]: "- 0 = (0::'a::ring)"
+by (simp add: equals_zero_I)
+
+lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ring))" 
+  proof 
+    assume "- a = - b"
+    then have "- (- a) = - (- b)"
+      by simp
+    then show "a=b"
+      by simp
+  next
+    assume "a=b"
+    then show "-a = -b"
+      by simp
+  qed
+
+lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ring))"
+by (subst neg_equal_iff_equal [symmetric], simp)
+
+lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ring))"
+by (subst neg_equal_iff_equal [symmetric], simp)
+
+
+subsection {* Derived rules for multiplication *}
+
+lemma right_one [simp]: "a = a * (1::'a::semiring)"
+proof -
+  have "a = 1 * a" by simp
+  also have "... = a * 1" by (simp add: mult_commute)
+  finally show ?thesis .
+qed
+
+lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::semiring))"
+  by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute])
+
+theorems mult_ac = mult_assoc mult_commute mult_left_commute
+
+lemma right_inverse [simp]: "a \<noteq> 0 ==>  a * inverse (a::'a::field) = 1"
+proof -
+  have "a * inverse a = inverse a * a" by (simp add: mult_ac)
+  also assume "a \<noteq> 0"
+  hence "inverse a * a = 1" by simp
+  finally show ?thesis .
+qed
+
+lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
+proof
+  assume neq: "b \<noteq> 0"
+  {
+    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
+    also assume "a / b = 1"
+    finally show "a = b" by simp
+  next
+    assume "a = b"
+    with neq show "a / b = 1" by (simp add: divide_inverse)
+  }
+qed
+
+lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
+  by (simp add: divide_inverse)
+
+lemma mult_left_zero [simp]: "0 * a = (0::'a::ring)"
+proof -
+  have "0*a + 0*a = 0*a + 0"
+    by (simp add: left_distrib [symmetric])
+  then show ?thesis by (simp only: add_left_cancel)
+qed
+
+lemma mult_right_zero [simp]: "a * 0 = (0::'a::ring)"
+  by (simp add: mult_commute)
+
+
+subsection {* Distribution rules *}
+
+lemma right_distrib: "a * (b + c) = a * b + a * (c::'a::semiring)"
+proof -
+  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
+  also have "... = b * a + c * a" by (simp only: left_distrib)
+  also have "... = a * b + a * c" by (simp add: mult_ac)
+  finally show ?thesis .
+qed
+
+theorems ring_distrib = right_distrib left_distrib
+
+lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ring)"
+apply (rule equals_zero_I)
+apply (simp add: add_ac) 
+done
+
+lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
+apply (rule equals_zero_I)
+apply (simp add: left_distrib [symmetric]) 
+done
+
+lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
+apply (rule equals_zero_I)
+apply (simp add: right_distrib [symmetric]) 
+done
+
+lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
+by (simp add: right_distrib diff_minus 
+              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
+
+
+subsection {* Ordering rules *}
+
+lemma add_right_mono: "a \<le> (b::'a::ordered_semiring) ==> a + c \<le> b + c"
+by (simp add: add_commute [of _ c] add_left_mono)
+
+lemma le_imp_neg_le:
+   assumes "a \<le> (b::'a::ordered_ring)" shows "-b \<le> -a"
+  proof -
+  have "-a+a \<le> -a+b"
+    by (rule add_left_mono) 
+  then have "0 \<le> -a+b"
+    by simp
+  then have "0 + (-b) \<le> (-a + b) + (-b)"
+    by (rule add_right_mono) 
+  then show ?thesis
+    by (simp add: add_assoc)
+  qed
+
+lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::ordered_ring))"
+  proof 
+    assume "- b \<le> - a"
+    then have "- (- a) \<le> - (- b)"
+      by (rule le_imp_neg_le)
+    then show "a\<le>b"
+      by simp
+  next
+    assume "a\<le>b"
+    then show "-b \<le> -a"
+      by (rule le_imp_neg_le)
+  qed
+
+lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::ordered_ring))"
+by (subst neg_le_iff_le [symmetric], simp)
+
+lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::ordered_ring))"
+by (subst neg_le_iff_le [symmetric], simp)
+
+lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::ordered_ring))"
+by (force simp add: order_less_le) 
+
+lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::ordered_ring))"
+by (subst neg_less_iff_less [symmetric], simp)
+
+lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::ordered_ring))"
+by (subst neg_less_iff_less [symmetric], simp)
+
+lemma mult_strict_right_mono:
+     "[|a < b; 0 < c|] ==> a * c < b * (c::'a::ordered_semiring)"
+by (simp add: mult_commute [of _ c] mult_strict_left_mono)
+
+lemma mult_left_mono:
+     "[|a \<le> b; 0 < c|] ==> c * a \<le> c * (b::'a::ordered_semiring)"
+by (force simp add: mult_strict_left_mono order_le_less) 
+
+lemma mult_right_mono:
+     "[|a \<le> b; 0 < c|] ==> a*c \<le> b * (c::'a::ordered_semiring)"
+by (force simp add: mult_strict_right_mono order_le_less) 
+
+lemma mult_strict_left_mono_neg:
+     "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring)"
+apply (drule mult_strict_left_mono [of _ _ "-c"])
+apply (simp_all add: minus_mult_left [symmetric]) 
+done
+
+lemma mult_strict_right_mono_neg:
+     "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring)"
+apply (drule mult_strict_right_mono [of _ _ "-c"])
+apply (simp_all add: minus_mult_right [symmetric]) 
+done
+
+lemma mult_left_mono_neg:
+     "[|b \<le> a; c < 0|] ==> c * a \<le> c * (b::'a::ordered_ring)"
+by (force simp add: mult_strict_left_mono_neg order_le_less) 
+
+lemma mult_right_mono_neg:
+     "[|b \<le> a; c < 0|] ==> a * c \<le> b * (c::'a::ordered_ring)"
+by (force simp add: mult_strict_right_mono_neg order_le_less) 
+
+
+subsection{* Products of Signs *}
+
+lemma mult_pos: "[| (0::'a::ordered_ring) < a; 0 < b |] ==> 0 < a*b"
+by (drule mult_strict_left_mono [of 0 b], auto)
+
+lemma mult_pos_neg: "[| (0::'a::ordered_ring) < a; b < 0 |] ==> a*b < 0"
+by (drule mult_strict_left_mono [of b 0], auto)
+
+lemma mult_neg: "[| a < (0::'a::ordered_ring); b < 0 |] ==> 0 < a*b"
+by (drule mult_strict_right_mono_neg, auto)
+
+lemma zero_less_mult_pos: "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_ring)"
+apply (case_tac "b\<le>0") 
+ apply (auto simp add: order_le_less linorder_not_less)
+apply (drule_tac mult_pos_neg [of a b]) 
+ apply (auto dest: order_less_not_sym)
+done
+
+lemma zero_less_mult_iff:
+     "((0::'a::ordered_ring) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
+apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
+apply (blast dest: zero_less_mult_pos) 
+apply (simp add: mult_commute [of a b]) 
+apply (blast dest: zero_less_mult_pos) 
+done
+
+
+lemma mult_eq_0_iff [iff]: "(a*b = (0::'a::ordered_ring)) = (a = 0 | b = 0)"
+apply (case_tac "a < 0")
+apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
+apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
+done
+
+lemma zero_le_mult_iff:
+     "((0::'a::ordered_ring) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
+by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
+                   zero_less_mult_iff)
+
+lemma mult_less_0_iff:
+     "(a*b < (0::'a::ordered_ring)) = (0 < a & b < 0 | a < 0 & 0 < b)"
+apply (insert zero_less_mult_iff [of "-a" b]) 
+apply (force simp add: minus_mult_left[symmetric]) 
+done
+
+lemma mult_le_0_iff:
+     "(a*b \<le> (0::'a::ordered_ring)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
+apply (insert zero_le_mult_iff [of "-a" b]) 
+apply (force simp add: minus_mult_left[symmetric]) 
+done
+
+lemma zero_le_square: "(0::'a::ordered_ring) \<le> a*a"
+by (simp add: zero_le_mult_iff linorder_linear) 
+
+lemma zero_less_one: "(0::'a::ordered_ring) < 1"
+apply (insert zero_le_square [of 1]) 
+apply (simp add: order_less_le) 
+done
+
+
+subsection {* Absolute Value *}
+
+text{*But is it really better than just rewriting with @{text abs_if}?*}
+lemma abs_split:
+     "P(abs(a::'a::ordered_ring)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
+by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
+
+lemma abs_zero [simp]: "abs 0 = (0::'a::ordered_ring)"
+by (simp add: abs_if)
+
+lemma abs_mult: "abs (x * y) = abs x * abs (y::'a::ordered_ring)" 
+apply (case_tac "x=0 | y=0", force) 
+apply (auto elim: order_less_asym
+            simp add: abs_if mult_less_0_iff linorder_neq_iff
+                  minus_mult_left [symmetric] minus_mult_right [symmetric])  
+done
+
+lemma abs_eq_0 [iff]: "(abs x = 0) = (x = (0::'a::ordered_ring))"
+by (simp add: abs_if)
+
+lemma zero_less_abs_iff [iff]: "(0 < abs x) = (x ~= (0::'a::ordered_ring))"
+by (simp add: abs_if linorder_neq_iff)
+
+
+subsection {* Fields *}
+
+
+end