src/HOL/Algebra/abstract/Ring2.thy
 author wenzelm Mon Sep 06 19:13:10 2010 +0200 (2010-09-06) changeset 39159 0dec18004e75 parent 38715 6513ea67d95d child 42768 4db4a8b164c1 permissions -rw-r--r--
more antiquotations;
```     1 (*  Title:     HOL/Algebra/abstract/Ring2.thy
```
```     2     Author:    Clemens Ballarin
```
```     3
```
```     4 The algebraic hierarchy of rings as axiomatic classes.
```
```     5 *)
```
```     6
```
```     7 header {* The algebraic hierarchy of rings as type classes *}
```
```     8
```
```     9 theory Ring2
```
```    10 imports Main
```
```    11 begin
```
```    12
```
```    13 subsection {* Ring axioms *}
```
```    14
```
```    15 class ring = zero + one + plus + minus + uminus + times + inverse + power + dvd +
```
```    16   assumes a_assoc:      "(a + b) + c = a + (b + c)"
```
```    17   and l_zero:           "0 + a = a"
```
```    18   and l_neg:            "(-a) + a = 0"
```
```    19   and a_comm:           "a + b = b + a"
```
```    20
```
```    21   assumes m_assoc:      "(a * b) * c = a * (b * c)"
```
```    22   and l_one:            "1 * a = a"
```
```    23
```
```    24   assumes l_distr:      "(a + b) * c = a * c + b * c"
```
```    25
```
```    26   assumes m_comm:       "a * b = b * a"
```
```    27
```
```    28   assumes minus_def:    "a - b = a + (-b)"
```
```    29   and inverse_def:      "inverse a = (if a dvd 1 then THE x. a*x = 1 else 0)"
```
```    30   and divide_def:       "a / b = a * inverse b"
```
```    31 begin
```
```    32
```
```    33 definition
```
```    34   assoc :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "assoc" 50)
```
```    35   where "a assoc b \<longleftrightarrow> a dvd b & b dvd a"
```
```    36
```
```    37 definition
```
```    38   irred :: "'a \<Rightarrow> bool" where
```
```    39   "irred a \<longleftrightarrow> a ~= 0 & ~ a dvd 1 & (ALL d. d dvd a --> d dvd 1 | a dvd d)"
```
```    40
```
```    41 definition
```
```    42   prime :: "'a \<Rightarrow> bool" where
```
```    43   "prime p \<longleftrightarrow> p ~= 0 & ~ p dvd 1 & (ALL a b. p dvd (a*b) --> p dvd a | p dvd b)"
```
```    44
```
```    45 end
```
```    46
```
```    47
```
```    48 subsection {* Integral domains *}
```
```    49
```
```    50 class "domain" = ring +
```
```    51   assumes one_not_zero: "1 ~= 0"
```
```    52   and integral: "a * b = 0 ==> a = 0 | b = 0"
```
```    53
```
```    54 subsection {* Factorial domains *}
```
```    55
```
```    56 class factorial = "domain" +
```
```    57 (*
```
```    58   Proper definition using divisor chain condition currently not supported.
```
```    59   factorial_divisor:    "wf {(a, b). a dvd b & ~ (b dvd a)}"
```
```    60 *)
```
```    61   (*assumes factorial_divisor: "True"*)
```
```    62   assumes factorial_prime: "irred a ==> prime a"
```
```    63
```
```    64
```
```    65 subsection {* Euclidean domains *}
```
```    66
```
```    67 (*
```
```    68 class euclidean = "domain" +
```
```    69   assumes euclidean_ax: "b ~= 0 ==> Ex (% (q, r, e_size::('a::ringS)=>nat).
```
```    70                    a = b * q + r & e_size r < e_size b)"
```
```    71
```
```    72   Nothing has been proved about Euclidean domains, yet.
```
```    73   Design question:
```
```    74     Fix quo, rem and e_size as constants that are axiomatised with
```
```    75     euclidean_ax?
```
```    76     - advantage: more pragmatic and easier to use
```
```    77     - disadvantage: for every type, one definition of quo and rem will
```
```    78         be fixed, users may want to use differing ones;
```
```    79         also, it seems not possible to prove that fields are euclidean
```
```    80         domains, because that would require generic (type-independent)
```
```    81         definitions of quo and rem.
```
```    82 *)
```
```    83
```
```    84 subsection {* Fields *}
```
```    85
```
```    86 class field = ring +
```
```    87   assumes field_one_not_zero: "1 ~= 0"
```
```    88                 (* Avoid a common superclass as the first thing we will
```
```    89                    prove about fields is that they are domains. *)
```
```    90   and field_ax: "a ~= 0 ==> a dvd 1"
```
```    91
```
```    92
```
```    93 section {* Basic facts *}
```
```    94
```
```    95 subsection {* Normaliser for rings *}
```
```    96
```
```    97 (* derived rewrite rules *)
```
```    98
```
```    99 lemma a_lcomm: "(a::'a::ring)+(b+c) = b+(a+c)"
```
```   100   apply (rule a_comm [THEN trans])
```
```   101   apply (rule a_assoc [THEN trans])
```
```   102   apply (rule a_comm [THEN arg_cong])
```
```   103   done
```
```   104
```
```   105 lemma r_zero: "(a::'a::ring) + 0 = a"
```
```   106   apply (rule a_comm [THEN trans])
```
```   107   apply (rule l_zero)
```
```   108   done
```
```   109
```
```   110 lemma r_neg: "(a::'a::ring) + (-a) = 0"
```
```   111   apply (rule a_comm [THEN trans])
```
```   112   apply (rule l_neg)
```
```   113   done
```
```   114
```
```   115 lemma r_neg2: "(a::'a::ring) + (-a + b) = b"
```
```   116   apply (rule a_assoc [symmetric, THEN trans])
```
```   117   apply (simp add: r_neg l_zero)
```
```   118   done
```
```   119
```
```   120 lemma r_neg1: "-(a::'a::ring) + (a + b) = b"
```
```   121   apply (rule a_assoc [symmetric, THEN trans])
```
```   122   apply (simp add: l_neg l_zero)
```
```   123   done
```
```   124
```
```   125
```
```   126 (* auxiliary *)
```
```   127
```
```   128 lemma a_lcancel: "!! a::'a::ring. a + b = a + c ==> b = c"
```
```   129   apply (rule box_equals)
```
```   130   prefer 2
```
```   131   apply (rule l_zero)
```
```   132   prefer 2
```
```   133   apply (rule l_zero)
```
```   134   apply (rule_tac a1 = a in l_neg [THEN subst])
```
```   135   apply (simp add: a_assoc)
```
```   136   done
```
```   137
```
```   138 lemma minus_add: "-((a::'a::ring) + b) = (-a) + (-b)"
```
```   139   apply (rule_tac a = "a + b" in a_lcancel)
```
```   140   apply (simp add: r_neg l_neg l_zero a_assoc a_comm a_lcomm)
```
```   141   done
```
```   142
```
```   143 lemma minus_minus: "-(-(a::'a::ring)) = a"
```
```   144   apply (rule a_lcancel)
```
```   145   apply (rule r_neg [THEN trans])
```
```   146   apply (rule l_neg [symmetric])
```
```   147   done
```
```   148
```
```   149 lemma minus0: "- 0 = (0::'a::ring)"
```
```   150   apply (rule a_lcancel)
```
```   151   apply (rule r_neg [THEN trans])
```
```   152   apply (rule l_zero [symmetric])
```
```   153   done
```
```   154
```
```   155
```
```   156 (* derived rules for multiplication *)
```
```   157
```
```   158 lemma m_lcomm: "(a::'a::ring)*(b*c) = b*(a*c)"
```
```   159   apply (rule m_comm [THEN trans])
```
```   160   apply (rule m_assoc [THEN trans])
```
```   161   apply (rule m_comm [THEN arg_cong])
```
```   162   done
```
```   163
```
```   164 lemma r_one: "(a::'a::ring) * 1 = a"
```
```   165   apply (rule m_comm [THEN trans])
```
```   166   apply (rule l_one)
```
```   167   done
```
```   168
```
```   169 lemma r_distr: "(a::'a::ring) * (b + c) = a * b + a * c"
```
```   170   apply (rule m_comm [THEN trans])
```
```   171   apply (rule l_distr [THEN trans])
```
```   172   apply (simp add: m_comm)
```
```   173   done
```
```   174
```
```   175
```
```   176 (* the following proof is from Jacobson, Basic Algebra I, pp. 88-89 *)
```
```   177 lemma l_null: "0 * (a::'a::ring) = 0"
```
```   178   apply (rule a_lcancel)
```
```   179   apply (rule l_distr [symmetric, THEN trans])
```
```   180   apply (simp add: r_zero)
```
```   181   done
```
```   182
```
```   183 lemma r_null: "(a::'a::ring) * 0 = 0"
```
```   184   apply (rule m_comm [THEN trans])
```
```   185   apply (rule l_null)
```
```   186   done
```
```   187
```
```   188 lemma l_minus: "(-(a::'a::ring)) * b = - (a * b)"
```
```   189   apply (rule a_lcancel)
```
```   190   apply (rule r_neg [symmetric, THEN [2] trans])
```
```   191   apply (rule l_distr [symmetric, THEN trans])
```
```   192   apply (simp add: l_null r_neg)
```
```   193   done
```
```   194
```
```   195 lemma r_minus: "(a::'a::ring) * (-b) = - (a * b)"
```
```   196   apply (rule a_lcancel)
```
```   197   apply (rule r_neg [symmetric, THEN [2] trans])
```
```   198   apply (rule r_distr [symmetric, THEN trans])
```
```   199   apply (simp add: r_null r_neg)
```
```   200   done
```
```   201
```
```   202 (*** Term order for commutative rings ***)
```
```   203
```
```   204 ML {*
```
```   205 fun ring_ord (Const (a, _)) =
```
```   206     find_index (fn a' => a = a')
```
```   207       [@{const_name Groups.zero}, @{const_name Groups.plus}, @{const_name Groups.uminus},
```
```   208         @{const_name Groups.minus}, @{const_name Groups.one}, @{const_name Groups.times}]
```
```   209   | ring_ord _ = ~1;
```
```   210
```
```   211 fun termless_ring (a, b) = (Term_Ord.term_lpo ring_ord (a, b) = LESS);
```
```   212
```
```   213 val ring_ss = HOL_basic_ss settermless termless_ring addsimps
```
```   214   [@{thm a_assoc}, @{thm l_zero}, @{thm l_neg}, @{thm a_comm}, @{thm m_assoc},
```
```   215    @{thm l_one}, @{thm l_distr}, @{thm m_comm}, @{thm minus_def},
```
```   216    @{thm r_zero}, @{thm r_neg}, @{thm r_neg2}, @{thm r_neg1}, @{thm minus_add},
```
```   217    @{thm minus_minus}, @{thm minus0}, @{thm a_lcomm}, @{thm m_lcomm}, (*@{thm r_one},*)
```
```   218    @{thm r_distr}, @{thm l_null}, @{thm r_null}, @{thm l_minus}, @{thm r_minus}];
```
```   219 *}   (* Note: r_one is not necessary in ring_ss *)
```
```   220
```
```   221 method_setup ring =
```
```   222   {* Scan.succeed (K (SIMPLE_METHOD' (full_simp_tac ring_ss))) *}
```
```   223   {* computes distributive normal form in rings *}
```
```   224
```
```   225
```
```   226 subsection {* Rings and the summation operator *}
```
```   227
```
```   228 (* Basic facts --- move to HOL!!! *)
```
```   229
```
```   230 (* needed because natsum_cong (below) disables atMost_0 *)
```
```   231 lemma natsum_0 [simp]: "setsum f {..(0::nat)} = (f 0::'a::comm_monoid_add)"
```
```   232 by simp
```
```   233 (*
```
```   234 lemma natsum_Suc [simp]:
```
```   235   "setsum f {..Suc n} = (f (Suc n) + setsum f {..n}::'a::comm_monoid_add)"
```
```   236 by (simp add: atMost_Suc)
```
```   237 *)
```
```   238 lemma natsum_Suc2:
```
```   239   "setsum f {..Suc n} = (f 0::'a::comm_monoid_add) + (setsum (%i. f (Suc i)) {..n})"
```
```   240 proof (induct n)
```
```   241   case 0 show ?case by simp
```
```   242 next
```
```   243   case Suc thus ?case by (simp add: add_assoc)
```
```   244 qed
```
```   245
```
```   246 lemma natsum_cong [cong]:
```
```   247   "!!k. [| j = k; !!i::nat. i <= k ==> f i = (g i::'a::comm_monoid_add) |] ==>
```
```   248         setsum f {..j} = setsum g {..k}"
```
```   249 by (induct j) auto
```
```   250
```
```   251 lemma natsum_zero [simp]: "setsum (%i. 0) {..n::nat} = (0::'a::comm_monoid_add)"
```
```   252 by (induct n) simp_all
```
```   253
```
```   254 lemma natsum_add [simp]:
```
```   255   "!!f::nat=>'a::comm_monoid_add.
```
```   256    setsum (%i. f i + g i) {..n::nat} = setsum f {..n} + setsum g {..n}"
```
```   257 by (induct n) (simp_all add: add_ac)
```
```   258
```
```   259 (* Facts specific to rings *)
```
```   260
```
```   261 subclass (in ring) comm_monoid_add
```
```   262 proof
```
```   263   fix x y z
```
```   264   show "x + y = y + x" by (rule a_comm)
```
```   265   show "(x + y) + z = x + (y + z)" by (rule a_assoc)
```
```   266   show "0 + x = x" by (rule l_zero)
```
```   267 qed
```
```   268
```
```   269 ML {*
```
```   270   local
```
```   271     val lhss =
```
```   272         ["t + u::'a::ring",
```
```   273          "t - u::'a::ring",
```
```   274          "t * u::'a::ring",
```
```   275          "- t::'a::ring"];
```
```   276     fun proc ss t =
```
```   277       let val rew = Goal.prove (Simplifier.the_context ss) [] []
```
```   278             (HOLogic.mk_Trueprop
```
```   279               (HOLogic.mk_eq (t, Var (("x", Term.maxidx_of_term t + 1), fastype_of t))))
```
```   280                 (fn _ => simp_tac (Simplifier.inherit_context ss ring_ss) 1)
```
```   281             |> mk_meta_eq;
```
```   282           val (t', u) = Logic.dest_equals (Thm.prop_of rew);
```
```   283       in if t' aconv u
```
```   284         then NONE
```
```   285         else SOME rew
```
```   286     end;
```
```   287   in
```
```   288     val ring_simproc = Simplifier.simproc_global @{theory} "ring" lhss (K proc);
```
```   289   end;
```
```   290 *}
```
```   291
```
```   292 ML {* Addsimprocs [ring_simproc] *}
```
```   293
```
```   294 lemma natsum_ldistr:
```
```   295   "!!a::'a::ring. setsum f {..n::nat} * a = setsum (%i. f i * a) {..n}"
```
```   296 by (induct n) simp_all
```
```   297
```
```   298 lemma natsum_rdistr:
```
```   299   "!!a::'a::ring. a * setsum f {..n::nat} = setsum (%i. a * f i) {..n}"
```
```   300 by (induct n) simp_all
```
```   301
```
```   302 subsection {* Integral Domains *}
```
```   303
```
```   304 declare one_not_zero [simp]
```
```   305
```
```   306 lemma zero_not_one [simp]:
```
```   307   "0 ~= (1::'a::domain)"
```
```   308 by (rule not_sym) simp
```
```   309
```
```   310 lemma integral_iff: (* not by default a simp rule! *)
```
```   311   "(a * b = (0::'a::domain)) = (a = 0 | b = 0)"
```
```   312 proof
```
```   313   assume "a * b = 0" then show "a = 0 | b = 0" by (simp add: integral)
```
```   314 next
```
```   315   assume "a = 0 | b = 0" then show "a * b = 0" by auto
```
```   316 qed
```
```   317
```
```   318 (*
```
```   319 lemma "(a::'a::ring) - (a - b) = b" apply simp
```
```   320  simproc seems to fail on this example (fixed with new term order)
```
```   321 *)
```
```   322 (*
```
```   323 lemma bug: "(b::'a::ring) - (b - a) = a" by simp
```
```   324    simproc for rings cannot prove "(a::'a::ring) - (a - b) = b"
```
```   325 *)
```
```   326 lemma m_lcancel:
```
```   327   assumes prem: "(a::'a::domain) ~= 0" shows conc: "(a * b = a * c) = (b = c)"
```
```   328 proof
```
```   329   assume eq: "a * b = a * c"
```
```   330   then have "a * (b - c) = 0" by simp
```
```   331   then have "a = 0 | (b - c) = 0" by (simp only: integral_iff)
```
```   332   with prem have "b - c = 0" by auto
```
```   333   then have "b = b - (b - c)" by simp
```
```   334   also have "b - (b - c) = c" by simp
```
```   335   finally show "b = c" .
```
```   336 next
```
```   337   assume "b = c" then show "a * b = a * c" by simp
```
```   338 qed
```
```   339
```
```   340 lemma m_rcancel:
```
```   341   "(a::'a::domain) ~= 0 ==> (b * a = c * a) = (b = c)"
```
```   342 by (simp add: m_lcancel)
```
```   343
```
```   344 declare power_Suc [simp]
```
```   345
```
```   346 lemma power_one [simp]:
```
```   347   "1 ^ n = (1::'a::ring)" by (induct n) simp_all
```
```   348
```
```   349 lemma power_zero [simp]:
```
```   350   "n \<noteq> 0 \<Longrightarrow> 0 ^ n = (0::'a::ring)" by (induct n) simp_all
```
```   351
```
```   352 lemma power_mult [simp]:
```
```   353   "(a::'a::ring) ^ m * a ^ n = a ^ (m + n)"
```
```   354   by (induct m) simp_all
```
```   355
```
```   356
```
```   357 section "Divisibility"
```
```   358
```
```   359 lemma dvd_zero_right [simp]:
```
```   360   "(a::'a::ring) dvd 0"
```
```   361 proof
```
```   362   show "0 = a * 0" by simp
```
```   363 qed
```
```   364
```
```   365 lemma dvd_zero_left:
```
```   366   "0 dvd (a::'a::ring) \<Longrightarrow> a = 0" unfolding dvd_def by simp
```
```   367
```
```   368 lemma dvd_refl_ring [simp]:
```
```   369   "(a::'a::ring) dvd a"
```
```   370 proof
```
```   371   show "a = a * 1" by simp
```
```   372 qed
```
```   373
```
```   374 lemma dvd_trans_ring:
```
```   375   fixes a b c :: "'a::ring"
```
```   376   assumes a_dvd_b: "a dvd b"
```
```   377   and b_dvd_c: "b dvd c"
```
```   378   shows "a dvd c"
```
```   379 proof -
```
```   380   from a_dvd_b obtain l where "b = a * l" using dvd_def by blast
```
```   381   moreover from b_dvd_c obtain j where "c = b * j" using dvd_def by blast
```
```   382   ultimately have "c = a * (l * j)" by simp
```
```   383   then have "\<exists>k. c = a * k" ..
```
```   384   then show ?thesis using dvd_def by blast
```
```   385 qed
```
```   386
```
```   387
```
```   388 lemma unit_mult:
```
```   389   "!!a::'a::ring. [| a dvd 1; b dvd 1 |] ==> a * b dvd 1"
```
```   390   apply (unfold dvd_def)
```
```   391   apply clarify
```
```   392   apply (rule_tac x = "k * ka" in exI)
```
```   393   apply simp
```
```   394   done
```
```   395
```
```   396 lemma unit_power: "!!a::'a::ring. a dvd 1 ==> a^n dvd 1"
```
```   397   apply (induct_tac n)
```
```   398    apply simp
```
```   399   apply (simp add: unit_mult)
```
```   400   done
```
```   401
```
```   402 lemma dvd_add_right [simp]:
```
```   403   "!! a::'a::ring. [| a dvd b; a dvd c |] ==> a dvd b + c"
```
```   404   apply (unfold dvd_def)
```
```   405   apply clarify
```
```   406   apply (rule_tac x = "k + ka" in exI)
```
```   407   apply (simp add: r_distr)
```
```   408   done
```
```   409
```
```   410 lemma dvd_uminus_right [simp]:
```
```   411   "!! a::'a::ring. a dvd b ==> a dvd -b"
```
```   412   apply (unfold dvd_def)
```
```   413   apply clarify
```
```   414   apply (rule_tac x = "-k" in exI)
```
```   415   apply (simp add: r_minus)
```
```   416   done
```
```   417
```
```   418 lemma dvd_l_mult_right [simp]:
```
```   419   "!! a::'a::ring. a dvd b ==> a dvd c*b"
```
```   420   apply (unfold dvd_def)
```
```   421   apply clarify
```
```   422   apply (rule_tac x = "c * k" in exI)
```
```   423   apply simp
```
```   424   done
```
```   425
```
```   426 lemma dvd_r_mult_right [simp]:
```
```   427   "!! a::'a::ring. a dvd b ==> a dvd b*c"
```
```   428   apply (unfold dvd_def)
```
```   429   apply clarify
```
```   430   apply (rule_tac x = "k * c" in exI)
```
```   431   apply simp
```
```   432   done
```
```   433
```
```   434
```
```   435 (* Inverse of multiplication *)
```
```   436
```
```   437 section "inverse"
```
```   438
```
```   439 lemma inverse_unique: "!! a::'a::ring. [| a * x = 1; a * y = 1 |] ==> x = y"
```
```   440   apply (rule_tac a = "(a*y) * x" and b = "y * (a*x)" in box_equals)
```
```   441     apply (simp (no_asm))
```
```   442   apply auto
```
```   443   done
```
```   444
```
```   445 lemma r_inverse_ring: "!! a::'a::ring. a dvd 1 ==> a * inverse a = 1"
```
```   446   apply (unfold inverse_def dvd_def)
```
```   447   apply (tactic {* asm_full_simp_tac (@{simpset} delsimprocs [ring_simproc]) 1 *})
```
```   448   apply clarify
```
```   449   apply (rule theI)
```
```   450    apply assumption
```
```   451   apply (rule inverse_unique)
```
```   452    apply assumption
```
```   453   apply assumption
```
```   454   done
```
```   455
```
```   456 lemma l_inverse_ring: "!! a::'a::ring. a dvd 1 ==> inverse a * a = 1"
```
```   457   by (simp add: r_inverse_ring)
```
```   458
```
```   459
```
```   460 (* Fields *)
```
```   461
```
```   462 section "Fields"
```
```   463
```
```   464 lemma field_unit [simp]: "!! a::'a::field. (a dvd 1) = (a ~= 0)"
```
```   465   by (auto dest: field_ax dvd_zero_left simp add: field_one_not_zero)
```
```   466
```
```   467 lemma r_inverse [simp]: "!! a::'a::field. a ~= 0 ==> a * inverse a = 1"
```
```   468   by (simp add: r_inverse_ring)
```
```   469
```
```   470 lemma l_inverse [simp]: "!! a::'a::field. a ~= 0 ==> inverse a * a= 1"
```
```   471   by (simp add: l_inverse_ring)
```
```   472
```
```   473
```
```   474 (* fields are integral domains *)
```
```   475
```
```   476 lemma field_integral: "!! a::'a::field. a * b = 0 ==> a = 0 | b = 0"
```
```   477   apply (tactic "step_tac @{claset} 1")
```
```   478   apply (rule_tac a = " (a*b) * inverse b" in box_equals)
```
```   479     apply (rule_tac [3] refl)
```
```   480    prefer 2
```
```   481    apply (simp (no_asm))
```
```   482    apply auto
```
```   483   done
```
```   484
```
```   485
```
```   486 (* fields are factorial domains *)
```
```   487
```
```   488 lemma field_fact_prime: "!! a::'a::field. irred a ==> prime a"
```
```   489   unfolding prime_def irred_def by (blast intro: field_ax)
```
```   490
```
```   491 end
```