(*  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