(*  Title:     HOL/Algebra/abstract/Ring2.thy

     Author:    Clemens Ballarin

 The algebraic hierarchy of rings as axiomatic classes.

 *)

 header {* The algebraic hierarchy of rings as type classes *}

 theory Ring2

 imports Main

 begin

 subsection {* Ring axioms *}

 class ring = zero + one + plus + minus + uminus + times + inverse + power + Ring_and_Field.dvd +

   assumes a_assoc:      "(a + b) + c = a + (b + c)"

   and l_zero:           "0 + a = a"

   and l_neg:            "(-a) + a = 0"

   and a_comm:           "a + b = b + a"

   assumes m_assoc:      "(a * b) * c = a * (b * c)"

   and l_one:            "1 * a = a"

   assumes l_distr:      "(a + b) * c = a * c + b * c"

   assumes m_comm:       "a * b = b * a"

   assumes minus_def:    "a - b = a + (-b)"

   and inverse_def:      "inverse a = (if a dvd 1 then THE x. a*x = 1 else 0)"

   and divide_def:       "a / b = a * inverse b"

   and power_0 [simp]:   "a ^ 0 = 1"

   and power_Suc [simp]: "a ^ Suc n = a ^ n * a"

 begin

 definition assoc :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "assoc" 50) where

   assoc_def: "a assoc b \<longleftrightarrow> a dvd b & b dvd a"

 definition irred :: "'a \<Rightarrow> bool" where

   irred_def: "irred a \<longleftrightarrow> a ~= 0 & ~ a dvd 1

                           & (ALL d. d dvd a --> d dvd 1 | a dvd d)"

 definition prime :: "'a \<Rightarrow> bool" where

   prime_def: "prime p \<longleftrightarrow> p ~= 0 & ~ p dvd 1

                           & (ALL a b. p dvd (a*b) --> p dvd a | p dvd b)"

 end

 subsection {* Integral domains *}

 class "domain" = ring +

   assumes one_not_zero: "1 ~= 0"

   and integral: "a * b = 0 ==> a = 0 | b = 0"

 subsection {* Factorial domains *}

 class factorial = "domain" +

 (*

   Proper definition using divisor chain condition currently not supported.

   factorial_divisor:    "wf {(a, b). a dvd b & ~ (b dvd a)}"

 *)

   (*assumes factorial_divisor: "True"*)

   assumes factorial_prime: "irred a ==> prime a"

 subsection {* Euclidean domains *}

 (*

 axclass

   euclidean < "domain"

   euclidean_ax:  "b ~= 0 ==> Ex (% (q, r, e_size::('a::ringS)=>nat).

                    a = b * q + r & e_size r < e_size b)"

   Nothing has been proved about Euclidean domains, yet.

   Design question:

     Fix quo, rem and e_size as constants that are axiomatised with

     euclidean_ax?

     - advantage: more pragmatic and easier to use

     - disadvantage: for every type, one definition of quo and rem will

         be fixed, users may want to use differing ones;

         also, it seems not possible to prove that fields are euclidean

         domains, because that would require generic (type-independent)

         definitions of quo and rem.

 *)

 subsection {* Fields *}

 class field = ring +

   assumes field_one_not_zero: "1 ~= 0"

                 (* Avoid a common superclass as the first thing we will

                    prove about fields is that they are domains. *)

   and field_ax: "a ~= 0 ==> a dvd 1"

 section {* Basic facts *}

 subsection {* Normaliser for rings *}

 (* derived rewrite rules *)

 lemma a_lcomm: "(a::'a::ring)+(b+c) = b+(a+c)"

   apply (rule a_comm [THEN trans])

   apply (rule a_assoc [THEN trans])

   apply (rule a_comm [THEN arg_cong])

   done

 lemma r_zero: "(a::'a::ring) + 0 = a"

   apply (rule a_comm [THEN trans])

   apply (rule l_zero)

   done

 lemma r_neg: "(a::'a::ring) + (-a) = 0"

   apply (rule a_comm [THEN trans])

   apply (rule l_neg)

   done

 lemma r_neg2: "(a::'a::ring) + (-a + b) = b"

   apply (rule a_assoc [symmetric, THEN trans])

   apply (simp add: r_neg l_zero)

   done

 lemma r_neg1: "-(a::'a::ring) + (a + b) = b"

   apply (rule a_assoc [symmetric, THEN trans])

   apply (simp add: l_neg l_zero)

   done

 (* auxiliary *)

 lemma a_lcancel: "!! a::'a::ring. a + b = a + c ==> b = c"

   apply (rule box_equals)

   prefer 2

   apply (rule l_zero)

   prefer 2

   apply (rule l_zero)

   apply (rule_tac a1 = a in l_neg [THEN subst])

   apply (simp add: a_assoc)

   done

 lemma minus_add: "-((a::'a::ring) + b) = (-a) + (-b)"

   apply (rule_tac a = "a + b" in a_lcancel)

   apply (simp add: r_neg l_neg l_zero a_assoc a_comm a_lcomm)

   done

 lemma minus_minus: "-(-(a::'a::ring)) = a"

   apply (rule a_lcancel)

   apply (rule r_neg [THEN trans])

   apply (rule l_neg [symmetric])

   done

 lemma minus0: "- 0 = (0::'a::ring)"

   apply (rule a_lcancel)

   apply (rule r_neg [THEN trans])

   apply (rule l_zero [symmetric])

   done

 (* derived rules for multiplication *)

 lemma m_lcomm: "(a::'a::ring)*(b*c) = b*(a*c)"

   apply (rule m_comm [THEN trans])

   apply (rule m_assoc [THEN trans])

   apply (rule m_comm [THEN arg_cong])

   done

 lemma r_one: "(a::'a::ring) * 1 = a"

   apply (rule m_comm [THEN trans])

   apply (rule l_one)

   done

 lemma r_distr: "(a::'a::ring) * (b + c) = a * b + a * c"

   apply (rule m_comm [THEN trans])

   apply (rule l_distr [THEN trans])

   apply (simp add: m_comm)

   done

 (* the following proof is from Jacobson, Basic Algebra I, pp. 88-89 *)

 lemma l_null: "0 * (a::'a::ring) = 0"

   apply (rule a_lcancel)

   apply (rule l_distr [symmetric, THEN trans])

   apply (simp add: r_zero)

   done

 lemma r_null: "(a::'a::ring) * 0 = 0"

   apply (rule m_comm [THEN trans])

   apply (rule l_null)

   done

 lemma l_minus: "(-(a::'a::ring)) * b = - (a * b)"

   apply (rule a_lcancel)

   apply (rule r_neg [symmetric, THEN [2] trans])

   apply (rule l_distr [symmetric, THEN trans])

   apply (simp add: l_null r_neg)

   done

 lemma r_minus: "(a::'a::ring) * (-b) = - (a * b)"

   apply (rule a_lcancel)

   apply (rule r_neg [symmetric, THEN [2] trans])

   apply (rule r_distr [symmetric, THEN trans])

   apply (simp add: r_null r_neg)

   done

 (*** Term order for commutative rings ***)

 ML {*

 fun ring_ord (Const (a, _)) =

     find_index (fn a' => a = a')

       [@{const_name HOL.zero}, @{const_name HOL.plus}, @{const_name HOL.uminus},

         @{const_name HOL.minus}, @{const_name HOL.one}, @{const_name HOL.times}]

   | ring_ord _ = ~1;

 fun termless_ring (a, b) = (TermOrd.term_lpo ring_ord (a, b) = LESS);

 val ring_ss = HOL_basic_ss settermless termless_ring addsimps

   [thm "a_assoc", thm "l_zero", thm "l_neg", thm "a_comm", thm "m_assoc",

    thm "l_one", thm "l_distr", thm "m_comm", thm "minus_def",

    thm "r_zero", thm "r_neg", thm "r_neg2", thm "r_neg1", thm "minus_add",

    thm "minus_minus", thm "minus0", thm "a_lcomm", thm "m_lcomm", (*thm "r_one",*)

    thm "r_distr", thm "l_null", thm "r_null", thm "l_minus", thm "r_minus"];

 *}   (* Note: r_one is not necessary in ring_ss *)

 method_setup ring =

   {* Method.no_args (Method.SIMPLE_METHOD' (full_simp_tac ring_ss)) *}

   {* computes distributive normal form in rings *}

 subsection {* Rings and the summation operator *}

 (* Basic facts --- move to HOL!!! *)

 (* needed because natsum_cong (below) disables atMost_0 *)

 lemma natsum_0 [simp]: "setsum f {..(0::nat)} = (f 0::'a::comm_monoid_add)"

 by simp

 (*

 lemma natsum_Suc [simp]:

   "setsum f {..Suc n} = (f (Suc n) + setsum f {..n}::'a::comm_monoid_add)"

 by (simp add: atMost_Suc)

 *)

 lemma natsum_Suc2:

   "setsum f {..Suc n} = (f 0::'a::comm_monoid_add) + (setsum (%i. f (Suc i)) {..n})"

 proof (induct n)

   case 0 show ?case by simp

 next

   case Suc thus ?case by (simp add: add_assoc) 

 qed

 lemma natsum_cong [cong]:

   "!!k. [| j = k; !!i::nat. i <= k ==> f i = (g i::'a::comm_monoid_add) |] ==>

         setsum f {..j} = setsum g {..k}"

 by (induct j) auto

 lemma natsum_zero [simp]: "setsum (%i. 0) {..n::nat} = (0::'a::comm_monoid_add)"

 by (induct n) simp_all

 lemma natsum_add [simp]:

   "!!f::nat=>'a::comm_monoid_add.

    setsum (%i. f i + g i) {..n::nat} = setsum f {..n} + setsum g {..n}"

 by (induct n) (simp_all add: add_ac)

 (* Facts specific to rings *)

 subclass (in ring) comm_monoid_add

 proof

   fix x y z

   show "x + y = y + x" by (rule a_comm)

   show "(x + y) + z = x + (y + z)" by (rule a_assoc)

   show "0 + x = x" by (rule l_zero)

 qed

 ML {*

   local

     val lhss = 

         ["t + u::'a::ring",

 	 "t - u::'a::ring",

 	 "t * u::'a::ring",

 	 "- t::'a::ring"];

     fun proc ss t = 

       let val rew = Goal.prove (Simplifier.the_context ss) [] []

             (HOLogic.mk_Trueprop

               (HOLogic.mk_eq (t, Var (("x", Term.maxidx_of_term t + 1), fastype_of t))))

                 (fn _ => simp_tac (Simplifier.inherit_context ss ring_ss) 1)

             |> mk_meta_eq;

           val (t', u) = Logic.dest_equals (Thm.prop_of rew);

       in if t' aconv u 

         then NONE

         else SOME rew 

     end;

   in

     val ring_simproc = Simplifier.simproc (the_context ()) "ring" lhss (K proc);

   end;

 *}

 ML {* Addsimprocs [ring_simproc] *}

 lemma natsum_ldistr:

   "!!a::'a::ring. setsum f {..n::nat} * a = setsum (%i. f i * a) {..n}"

 by (induct n) simp_all

 lemma natsum_rdistr:

   "!!a::'a::ring. a * setsum f {..n::nat} = setsum (%i. a * f i) {..n}"

 by (induct n) simp_all

 subsection {* Integral Domains *}

 declare one_not_zero [simp]

 lemma zero_not_one [simp]:

   "0 ~= (1::'a::domain)" 

 by (rule not_sym) simp

 lemma integral_iff: (* not by default a simp rule! *)

   "(a * b = (0::'a::domain)) = (a = 0 | b = 0)"

 proof

   assume "a * b = 0" then show "a = 0 | b = 0" by (simp add: integral)

 next

   assume "a = 0 | b = 0" then show "a * b = 0" by auto

 qed

 (*

 lemma "(a::'a::ring) - (a - b) = b" apply simp

  simproc seems to fail on this example (fixed with new term order)

 *)

 (*

 lemma bug: "(b::'a::ring) - (b - a) = a" by simp

    simproc for rings cannot prove "(a::'a::ring) - (a - b) = b" 

 *)

 lemma m_lcancel:

   assumes prem: "(a::'a::domain) ~= 0" shows conc: "(a * b = a * c) = (b = c)"

 proof

   assume eq: "a * b = a * c"

   then have "a * (b - c) = 0" by simp

   then have "a = 0 | (b - c) = 0" by (simp only: integral_iff)

   with prem have "b - c = 0" by auto 

   then have "b = b - (b - c)" by simp 

   also have "b - (b - c) = c" by simp

   finally show "b = c" .

 next

   assume "b = c" then show "a * b = a * c" by simp

 qed

 lemma m_rcancel:

   "(a::'a::domain) ~= 0 ==> (b * a = c * a) = (b = c)"

 by (simp add: m_lcancel)

 declare power_Suc [simp]

 lemma power_one [simp]:

   "1 ^ n = (1::'a::ring)" by (induct n) simp_all

 lemma power_zero [simp]:

   "n \<noteq> 0 \<Longrightarrow> 0 ^ n = (0::'a::ring)" by (induct n) simp_all

 lemma power_mult [simp]:

   "(a::'a::ring) ^ m * a ^ n = a ^ (m + n)"

   by (induct m) simp_all

 section "Divisibility"

 lemma dvd_zero_right [simp]:

   "(a::'a::ring) dvd 0"

 proof

   show "0 = a * 0" by simp

 qed

 lemma dvd_zero_left:

   "0 dvd (a::'a::ring) \<Longrightarrow> a = 0" unfolding dvd_def by simp

 lemma dvd_refl_ring [simp]:

   "(a::'a::ring) dvd a"

 proof

   show "a = a * 1" by simp

 qed

 lemma dvd_trans_ring:

   fixes a b c :: "'a::ring"

   assumes a_dvd_b: "a dvd b"

   and b_dvd_c: "b dvd c"

   shows "a dvd c"

 proof -

   from a_dvd_b obtain l where "b = a * l" using dvd_def by blast

   moreover from b_dvd_c obtain j where "c = b * j" using dvd_def by blast

   ultimately have "c = a * (l * j)" by simp

   then have "\<exists>k. c = a * k" ..

   then show ?thesis using dvd_def by blast

 qed

 lemma unit_mult: 

   "!!a::'a::ring. [| a dvd 1; b dvd 1 |] ==> a * b dvd 1"

   apply (unfold dvd_def)

   apply clarify

   apply (rule_tac x = "k * ka" in exI)

   apply simp

   done

 lemma unit_power: "!!a::'a::ring. a dvd 1 ==> a^n dvd 1"

   apply (induct_tac n)

    apply simp

   apply (simp add: unit_mult)

   done

 lemma dvd_add_right [simp]:

   "!! a::'a::ring. [| a dvd b; a dvd c |] ==> a dvd b + c"

   apply (unfold dvd_def)

   apply clarify

   apply (rule_tac x = "k + ka" in exI)

   apply (simp add: r_distr)

   done

 lemma dvd_uminus_right [simp]:

   "!! a::'a::ring. a dvd b ==> a dvd -b"

   apply (unfold dvd_def)

   apply clarify

   apply (rule_tac x = "-k" in exI)

   apply (simp add: r_minus)

   done

 lemma dvd_l_mult_right [simp]:

   "!! a::'a::ring. a dvd b ==> a dvd c*b"

   apply (unfold dvd_def)

   apply clarify

   apply (rule_tac x = "c * k" in exI)

   apply simp

   done

 lemma dvd_r_mult_right [simp]:

   "!! a::'a::ring. a dvd b ==> a dvd b*c"

   apply (unfold dvd_def)

   apply clarify

   apply (rule_tac x = "k * c" in exI)

   apply simp

   done

 (* Inverse of multiplication *)

 section "inverse"

 lemma inverse_unique: "!! a::'a::ring. [| a * x = 1; a * y = 1 |] ==> x = y"

   apply (rule_tac a = "(a*y) * x" and b = "y * (a*x)" in box_equals)

     apply (simp (no_asm))

   apply auto

   done

 lemma r_inverse_ring: "!! a::'a::ring. a dvd 1 ==> a * inverse a = 1"

   apply (unfold inverse_def dvd_def)

   apply (tactic {* asm_full_simp_tac (@{simpset} delsimprocs [ring_simproc]) 1 *})

   apply clarify

   apply (rule theI)

    apply assumption

   apply (rule inverse_unique)

    apply assumption

   apply assumption

   done

 lemma l_inverse_ring: "!! a::'a::ring. a dvd 1 ==> inverse a * a = 1"

   by (simp add: r_inverse_ring)

 (* Fields *)

 section "Fields"

 lemma field_unit [simp]: "!! a::'a::field. (a dvd 1) = (a ~= 0)"

   by (auto dest: field_ax dvd_zero_left simp add: field_one_not_zero)

 lemma r_inverse [simp]: "!! a::'a::field. a ~= 0 ==> a * inverse a = 1"

   by (simp add: r_inverse_ring)

 lemma l_inverse [simp]: "!! a::'a::field. a ~= 0 ==> inverse a * a= 1"

   by (simp add: l_inverse_ring)

 (* fields are integral domains *)

 lemma field_integral: "!! a::'a::field. a * b = 0 ==> a = 0 | b = 0"

   apply (tactic "step_tac @{claset} 1")

   apply (rule_tac a = " (a*b) * inverse b" in box_equals)

     apply (rule_tac [3] refl)

    prefer 2

    apply (simp (no_asm))

    apply auto

   done

 (* fields are factorial domains *)

 lemma field_fact_prime: "!! a::'a::field. irred a ==> prime a"

   unfolding prime_def irred_def by (blast intro: field_ax)

 end