(*  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