src/HOL/Algebra/abstract/Ring2.thy
 author haftmann Wed Jan 02 15:14:02 2008 +0100 (2008-01-02) changeset 25762 c03e9d04b3e4 parent 24993 92dfacb32053 child 26342 0f65fa163304 permissions -rw-r--r--
splitted class uminus from class minus
```     1 (*
```
```     2   Title:     The algebraic hierarchy of rings as axiomatic classes
```
```     3   Id:        \$Id\$
```
```     4   Author:    Clemens Ballarin, started 9 December 1996
```
```     5   Copyright: Clemens Ballarin
```
```     6 *)
```
```     7
```
```     8 header {* The algebraic hierarchy of rings as axiomatic classes *}
```
```     9
```
```    10 theory Ring2 imports Main
```
```    11 begin
```
```    12
```
```    13 section {* Constants *}
```
```    14
```
```    15 text {* Most constants are already declared by HOL. *}
```
```    16
```
```    17 consts
```
```    18   assoc         :: "['a::Divides.div, 'a] => bool"              (infixl "assoc" 50)
```
```    19   irred         :: "'a::{zero, one, Divides.div} => bool"
```
```    20   prime         :: "'a::{zero, one, Divides.div} => bool"
```
```    21
```
```    22 section {* Axioms *}
```
```    23
```
```    24 subsection {* Ring axioms *}
```
```    25
```
```    26 axclass ring < zero, one, plus, minus, uminus, times, inverse, power, Divides.div
```
```    27
```
```    28   a_assoc:      "(a + b) + c = a + (b + c)"
```
```    29   l_zero:       "0 + a = a"
```
```    30   l_neg:        "(-a) + a = 0"
```
```    31   a_comm:       "a + b = b + a"
```
```    32
```
```    33   m_assoc:      "(a * b) * c = a * (b * c)"
```
```    34   l_one:        "1 * a = a"
```
```    35
```
```    36   l_distr:      "(a + b) * c = a * c + b * c"
```
```    37
```
```    38   m_comm:       "a * b = b * a"
```
```    39
```
```    40   -- {* Definition of derived operations *}
```
```    41
```
```    42   minus_def:    "a - b = a + (-b)"
```
```    43   inverse_def:  "inverse a = (if a dvd 1 then THE x. a*x = 1 else 0)"
```
```    44   divide_def:   "a / b = a * inverse b"
```
```    45   power_def:    "a ^ n = nat_rec 1 (%u b. b * a) n"
```
```    46
```
```    47 defs
```
```    48   assoc_def:    "a assoc b == a dvd b & b dvd a"
```
```    49   irred_def:    "irred a == a ~= 0 & ~ a dvd 1
```
```    50                           & (ALL d. d dvd a --> d dvd 1 | a dvd d)"
```
```    51   prime_def:    "prime p == p ~= 0 & ~ p dvd 1
```
```    52                           & (ALL a b. p dvd (a*b) --> p dvd a | p dvd b)"
```
```    53
```
```    54 subsection {* Integral domains *}
```
```    55
```
```    56 axclass
```
```    57   "domain" < ring
```
```    58
```
```    59   one_not_zero: "1 ~= 0"
```
```    60   integral:     "a * b = 0 ==> a = 0 | b = 0"
```
```    61
```
```    62 subsection {* Factorial domains *}
```
```    63
```
```    64 axclass
```
```    65   factorial < "domain"
```
```    66
```
```    67 (*
```
```    68   Proper definition using divisor chain condition currently not supported.
```
```    69   factorial_divisor:    "wf {(a, b). a dvd b & ~ (b dvd a)}"
```
```    70 *)
```
```    71   factorial_divisor:	"True"
```
```    72   factorial_prime:      "irred a ==> prime a"
```
```    73
```
```    74 subsection {* Euclidean domains *}
```
```    75
```
```    76 (*
```
```    77 axclass
```
```    78   euclidean < "domain"
```
```    79
```
```    80   euclidean_ax:  "b ~= 0 ==> Ex (% (q, r, e_size::('a::ringS)=>nat).
```
```    81                    a = b * q + r & e_size r < e_size b)"
```
```    82
```
```    83   Nothing has been proved about Euclidean domains, yet.
```
```    84   Design question:
```
```    85     Fix quo, rem and e_size as constants that are axiomatised with
```
```    86     euclidean_ax?
```
```    87     - advantage: more pragmatic and easier to use
```
```    88     - disadvantage: for every type, one definition of quo and rem will
```
```    89         be fixed, users may want to use differing ones;
```
```    90         also, it seems not possible to prove that fields are euclidean
```
```    91         domains, because that would require generic (type-independent)
```
```    92         definitions of quo and rem.
```
```    93 *)
```
```    94
```
```    95 subsection {* Fields *}
```
```    96
```
```    97 axclass
```
```    98   field < ring
```
```    99
```
```   100   field_one_not_zero:    "1 ~= 0"
```
```   101                 (* Avoid a common superclass as the first thing we will
```
```   102                    prove about fields is that they are domains. *)
```
```   103   field_ax:        "a ~= 0 ==> a dvd 1"
```
```   104
```
```   105
```
```   106 section {* Basic facts *}
```
```   107
```
```   108 subsection {* Normaliser for rings *}
```
```   109
```
```   110 (* derived rewrite rules *)
```
```   111
```
```   112 lemma a_lcomm: "(a::'a::ring)+(b+c) = b+(a+c)"
```
```   113   apply (rule a_comm [THEN trans])
```
```   114   apply (rule a_assoc [THEN trans])
```
```   115   apply (rule a_comm [THEN arg_cong])
```
```   116   done
```
```   117
```
```   118 lemma r_zero: "(a::'a::ring) + 0 = a"
```
```   119   apply (rule a_comm [THEN trans])
```
```   120   apply (rule l_zero)
```
```   121   done
```
```   122
```
```   123 lemma r_neg: "(a::'a::ring) + (-a) = 0"
```
```   124   apply (rule a_comm [THEN trans])
```
```   125   apply (rule l_neg)
```
```   126   done
```
```   127
```
```   128 lemma r_neg2: "(a::'a::ring) + (-a + b) = b"
```
```   129   apply (rule a_assoc [symmetric, THEN trans])
```
```   130   apply (simp add: r_neg l_zero)
```
```   131   done
```
```   132
```
```   133 lemma r_neg1: "-(a::'a::ring) + (a + b) = b"
```
```   134   apply (rule a_assoc [symmetric, THEN trans])
```
```   135   apply (simp add: l_neg l_zero)
```
```   136   done
```
```   137
```
```   138
```
```   139 (* auxiliary *)
```
```   140
```
```   141 lemma a_lcancel: "!! a::'a::ring. a + b = a + c ==> b = c"
```
```   142   apply (rule box_equals)
```
```   143   prefer 2
```
```   144   apply (rule l_zero)
```
```   145   prefer 2
```
```   146   apply (rule l_zero)
```
```   147   apply (rule_tac a1 = a in l_neg [THEN subst])
```
```   148   apply (simp add: a_assoc)
```
```   149   done
```
```   150
```
```   151 lemma minus_add: "-((a::'a::ring) + b) = (-a) + (-b)"
```
```   152   apply (rule_tac a = "a + b" in a_lcancel)
```
```   153   apply (simp add: r_neg l_neg l_zero a_assoc a_comm a_lcomm)
```
```   154   done
```
```   155
```
```   156 lemma minus_minus: "-(-(a::'a::ring)) = a"
```
```   157   apply (rule a_lcancel)
```
```   158   apply (rule r_neg [THEN trans])
```
```   159   apply (rule l_neg [symmetric])
```
```   160   done
```
```   161
```
```   162 lemma minus0: "- 0 = (0::'a::ring)"
```
```   163   apply (rule a_lcancel)
```
```   164   apply (rule r_neg [THEN trans])
```
```   165   apply (rule l_zero [symmetric])
```
```   166   done
```
```   167
```
```   168
```
```   169 (* derived rules for multiplication *)
```
```   170
```
```   171 lemma m_lcomm: "(a::'a::ring)*(b*c) = b*(a*c)"
```
```   172   apply (rule m_comm [THEN trans])
```
```   173   apply (rule m_assoc [THEN trans])
```
```   174   apply (rule m_comm [THEN arg_cong])
```
```   175   done
```
```   176
```
```   177 lemma r_one: "(a::'a::ring) * 1 = a"
```
```   178   apply (rule m_comm [THEN trans])
```
```   179   apply (rule l_one)
```
```   180   done
```
```   181
```
```   182 lemma r_distr: "(a::'a::ring) * (b + c) = a * b + a * c"
```
```   183   apply (rule m_comm [THEN trans])
```
```   184   apply (rule l_distr [THEN trans])
```
```   185   apply (simp add: m_comm)
```
```   186   done
```
```   187
```
```   188
```
```   189 (* the following proof is from Jacobson, Basic Algebra I, pp. 88-89 *)
```
```   190 lemma l_null: "0 * (a::'a::ring) = 0"
```
```   191   apply (rule a_lcancel)
```
```   192   apply (rule l_distr [symmetric, THEN trans])
```
```   193   apply (simp add: r_zero)
```
```   194   done
```
```   195
```
```   196 lemma r_null: "(a::'a::ring) * 0 = 0"
```
```   197   apply (rule m_comm [THEN trans])
```
```   198   apply (rule l_null)
```
```   199   done
```
```   200
```
```   201 lemma l_minus: "(-(a::'a::ring)) * b = - (a * b)"
```
```   202   apply (rule a_lcancel)
```
```   203   apply (rule r_neg [symmetric, THEN [2] trans])
```
```   204   apply (rule l_distr [symmetric, THEN trans])
```
```   205   apply (simp add: l_null r_neg)
```
```   206   done
```
```   207
```
```   208 lemma r_minus: "(a::'a::ring) * (-b) = - (a * b)"
```
```   209   apply (rule a_lcancel)
```
```   210   apply (rule r_neg [symmetric, THEN [2] trans])
```
```   211   apply (rule r_distr [symmetric, THEN trans])
```
```   212   apply (simp add: r_null r_neg)
```
```   213   done
```
```   214
```
```   215 (*** Term order for commutative rings ***)
```
```   216
```
```   217 ML {*
```
```   218 fun ring_ord (Const (a, _)) =
```
```   219     find_index (fn a' => a = a')
```
```   220       [@{const_name HOL.zero}, @{const_name HOL.plus}, @{const_name HOL.uminus},
```
```   221         @{const_name HOL.minus}, @{const_name HOL.one}, @{const_name HOL.times}]
```
```   222   | ring_ord _ = ~1;
```
```   223
```
```   224 fun termless_ring (a, b) = (Term.term_lpo ring_ord (a, b) = LESS);
```
```   225
```
```   226 val ring_ss = HOL_basic_ss settermless termless_ring addsimps
```
```   227   [thm "a_assoc", thm "l_zero", thm "l_neg", thm "a_comm", thm "m_assoc",
```
```   228    thm "l_one", thm "l_distr", thm "m_comm", thm "minus_def",
```
```   229    thm "r_zero", thm "r_neg", thm "r_neg2", thm "r_neg1", thm "minus_add",
```
```   230    thm "minus_minus", thm "minus0", thm "a_lcomm", thm "m_lcomm", (*thm "r_one",*)
```
```   231    thm "r_distr", thm "l_null", thm "r_null", thm "l_minus", thm "r_minus"];
```
```   232 *}   (* Note: r_one is not necessary in ring_ss *)
```
```   233
```
```   234 method_setup ring =
```
```   235   {* Method.no_args (Method.SIMPLE_METHOD' (full_simp_tac ring_ss)) *}
```
```   236   {* computes distributive normal form in rings *}
```
```   237
```
```   238 lemmas ring_simps =
```
```   239   l_zero r_zero l_neg r_neg minus_minus minus0
```
```   240   l_one r_one l_null r_null l_minus r_minus
```
```   241
```
```   242
```
```   243 subsection {* Rings and the summation operator *}
```
```   244
```
```   245 (* Basic facts --- move to HOL!!! *)
```
```   246
```
```   247 (* needed because natsum_cong (below) disables atMost_0 *)
```
```   248 lemma natsum_0 [simp]: "setsum f {..(0::nat)} = (f 0::'a::comm_monoid_add)"
```
```   249 by simp
```
```   250 (*
```
```   251 lemma natsum_Suc [simp]:
```
```   252   "setsum f {..Suc n} = (f (Suc n) + setsum f {..n}::'a::comm_monoid_add)"
```
```   253 by (simp add: atMost_Suc)
```
```   254 *)
```
```   255 lemma natsum_Suc2:
```
```   256   "setsum f {..Suc n} = (f 0::'a::comm_monoid_add) + (setsum (%i. f (Suc i)) {..n})"
```
```   257 proof (induct n)
```
```   258   case 0 show ?case by simp
```
```   259 next
```
```   260   case Suc thus ?case by (simp add: add_assoc)
```
```   261 qed
```
```   262
```
```   263 lemma natsum_cong [cong]:
```
```   264   "!!k. [| j = k; !!i::nat. i <= k ==> f i = (g i::'a::comm_monoid_add) |] ==>
```
```   265         setsum f {..j} = setsum g {..k}"
```
```   266 by (induct j) auto
```
```   267
```
```   268 lemma natsum_zero [simp]: "setsum (%i. 0) {..n::nat} = (0::'a::comm_monoid_add)"
```
```   269 by (induct n) simp_all
```
```   270
```
```   271 lemma natsum_add [simp]:
```
```   272   "!!f::nat=>'a::comm_monoid_add.
```
```   273    setsum (%i. f i + g i) {..n::nat} = setsum f {..n} + setsum g {..n}"
```
```   274 by (induct n) (simp_all add: add_ac)
```
```   275
```
```   276 (* Facts specific to rings *)
```
```   277
```
```   278 instance ring < comm_monoid_add
```
```   279 proof
```
```   280   fix x y z
```
```   281   show "(x::'a::ring) + y = y + x" by (rule a_comm)
```
```   282   show "((x::'a::ring) + y) + z = x + (y + z)" by (rule a_assoc)
```
```   283   show "0 + (x::'a::ring) = x" by (rule l_zero)
```
```   284 qed
```
```   285
```
```   286 ML {*
```
```   287   local
```
```   288     val lhss =
```
```   289         ["t + u::'a::ring",
```
```   290 	 "t - u::'a::ring",
```
```   291 	 "t * u::'a::ring",
```
```   292 	 "- t::'a::ring"];
```
```   293     fun proc ss t =
```
```   294       let val rew = Goal.prove (Simplifier.the_context ss) [] []
```
```   295             (HOLogic.mk_Trueprop
```
```   296               (HOLogic.mk_eq (t, Var (("x", Term.maxidx_of_term t + 1), fastype_of t))))
```
```   297                 (fn _ => simp_tac (Simplifier.inherit_context ss ring_ss) 1)
```
```   298             |> mk_meta_eq;
```
```   299           val (t', u) = Logic.dest_equals (Thm.prop_of rew);
```
```   300       in if t' aconv u
```
```   301         then NONE
```
```   302         else SOME rew
```
```   303     end;
```
```   304   in
```
```   305     val ring_simproc = Simplifier.simproc (the_context ()) "ring" lhss (K proc);
```
```   306   end;
```
```   307 *}
```
```   308
```
```   309 ML_setup {* Addsimprocs [ring_simproc] *}
```
```   310
```
```   311 lemma natsum_ldistr:
```
```   312   "!!a::'a::ring. setsum f {..n::nat} * a = setsum (%i. f i * a) {..n}"
```
```   313 by (induct n) simp_all
```
```   314
```
```   315 lemma natsum_rdistr:
```
```   316   "!!a::'a::ring. a * setsum f {..n::nat} = setsum (%i. a * f i) {..n}"
```
```   317 by (induct n) simp_all
```
```   318
```
```   319 subsection {* Integral Domains *}
```
```   320
```
```   321 declare one_not_zero [simp]
```
```   322
```
```   323 lemma zero_not_one [simp]:
```
```   324   "0 ~= (1::'a::domain)"
```
```   325 by (rule not_sym) simp
```
```   326
```
```   327 lemma integral_iff: (* not by default a simp rule! *)
```
```   328   "(a * b = (0::'a::domain)) = (a = 0 | b = 0)"
```
```   329 proof
```
```   330   assume "a * b = 0" then show "a = 0 | b = 0" by (simp add: integral)
```
```   331 next
```
```   332   assume "a = 0 | b = 0" then show "a * b = 0" by auto
```
```   333 qed
```
```   334
```
```   335 (*
```
```   336 lemma "(a::'a::ring) - (a - b) = b" apply simp
```
```   337  simproc seems to fail on this example (fixed with new term order)
```
```   338 *)
```
```   339 (*
```
```   340 lemma bug: "(b::'a::ring) - (b - a) = a" by simp
```
```   341    simproc for rings cannot prove "(a::'a::ring) - (a - b) = b"
```
```   342 *)
```
```   343 lemma m_lcancel:
```
```   344   assumes prem: "(a::'a::domain) ~= 0" shows conc: "(a * b = a * c) = (b = c)"
```
```   345 proof
```
```   346   assume eq: "a * b = a * c"
```
```   347   then have "a * (b - c) = 0" by simp
```
```   348   then have "a = 0 | (b - c) = 0" by (simp only: integral_iff)
```
```   349   with prem have "b - c = 0" by auto
```
```   350   then have "b = b - (b - c)" by simp
```
```   351   also have "b - (b - c) = c" by simp
```
```   352   finally show "b = c" .
```
```   353 next
```
```   354   assume "b = c" then show "a * b = a * c" by simp
```
```   355 qed
```
```   356
```
```   357 lemma m_rcancel:
```
```   358   "(a::'a::domain) ~= 0 ==> (b * a = c * a) = (b = c)"
```
```   359 by (simp add: m_lcancel)
```
```   360
```
```   361 lemma power_0 [simp]:
```
```   362   "(a::'a::ring) ^ 0 = 1" unfolding power_def by simp
```
```   363
```
```   364 lemma power_Suc [simp]:
```
```   365   "(a::'a::ring) ^ Suc n = a ^ n * a" unfolding power_def by simp
```
```   366
```
```   367 lemma power_one [simp]:
```
```   368   "1 ^ n = (1::'a::ring)" by (induct n) simp_all
```
```   369
```
```   370 lemma power_zero [simp]:
```
```   371   "n \<noteq> 0 \<Longrightarrow> 0 ^ n = (0::'a::ring)" by (induct n) simp_all
```
```   372
```
```   373 lemma power_mult [simp]:
```
```   374   "(a::'a::ring) ^ m * a ^ n = a ^ (m + n)"
```
```   375   by (induct m) simp_all
```
```   376
```
```   377
```
```   378 section "Divisibility"
```
```   379
```
```   380 lemma dvd_zero_right [simp]:
```
```   381   "(a::'a::ring) dvd 0"
```
```   382 proof
```
```   383   show "0 = a * 0" by simp
```
```   384 qed
```
```   385
```
```   386 lemma dvd_zero_left:
```
```   387   "0 dvd (a::'a::ring) \<Longrightarrow> a = 0" unfolding dvd_def by simp
```
```   388
```
```   389 lemma dvd_refl_ring [simp]:
```
```   390   "(a::'a::ring) dvd a"
```
```   391 proof
```
```   392   show "a = a * 1" by simp
```
```   393 qed
```
```   394
```
```   395 lemma dvd_trans_ring:
```
```   396   fixes a b c :: "'a::ring"
```
```   397   assumes a_dvd_b: "a dvd b"
```
```   398   and b_dvd_c: "b dvd c"
```
```   399   shows "a dvd c"
```
```   400 proof -
```
```   401   from a_dvd_b obtain l where "b = a * l" using dvd_def by blast
```
```   402   moreover from b_dvd_c obtain j where "c = b * j" using dvd_def by blast
```
```   403   ultimately have "c = a * (l * j)" by simp
```
```   404   then have "\<exists>k. c = a * k" ..
```
```   405   then show ?thesis using dvd_def by blast
```
```   406 qed
```
```   407
```
```   408
```
```   409 lemma unit_mult:
```
```   410   "!!a::'a::ring. [| a dvd 1; b dvd 1 |] ==> a * b dvd 1"
```
```   411   apply (unfold dvd_def)
```
```   412   apply clarify
```
```   413   apply (rule_tac x = "k * ka" in exI)
```
```   414   apply simp
```
```   415   done
```
```   416
```
```   417 lemma unit_power: "!!a::'a::ring. a dvd 1 ==> a^n dvd 1"
```
```   418   apply (induct_tac n)
```
```   419    apply simp
```
```   420   apply (simp add: unit_mult)
```
```   421   done
```
```   422
```
```   423 lemma dvd_add_right [simp]:
```
```   424   "!! a::'a::ring. [| a dvd b; a dvd c |] ==> a dvd b + c"
```
```   425   apply (unfold dvd_def)
```
```   426   apply clarify
```
```   427   apply (rule_tac x = "k + ka" in exI)
```
```   428   apply (simp add: r_distr)
```
```   429   done
```
```   430
```
```   431 lemma dvd_uminus_right [simp]:
```
```   432   "!! a::'a::ring. a dvd b ==> a dvd -b"
```
```   433   apply (unfold dvd_def)
```
```   434   apply clarify
```
```   435   apply (rule_tac x = "-k" in exI)
```
```   436   apply (simp add: r_minus)
```
```   437   done
```
```   438
```
```   439 lemma dvd_l_mult_right [simp]:
```
```   440   "!! a::'a::ring. a dvd b ==> a dvd c*b"
```
```   441   apply (unfold dvd_def)
```
```   442   apply clarify
```
```   443   apply (rule_tac x = "c * k" in exI)
```
```   444   apply simp
```
```   445   done
```
```   446
```
```   447 lemma dvd_r_mult_right [simp]:
```
```   448   "!! a::'a::ring. a dvd b ==> a dvd b*c"
```
```   449   apply (unfold dvd_def)
```
```   450   apply clarify
```
```   451   apply (rule_tac x = "k * c" in exI)
```
```   452   apply simp
```
```   453   done
```
```   454
```
```   455
```
```   456 (* Inverse of multiplication *)
```
```   457
```
```   458 section "inverse"
```
```   459
```
```   460 lemma inverse_unique: "!! a::'a::ring. [| a * x = 1; a * y = 1 |] ==> x = y"
```
```   461   apply (rule_tac a = "(a*y) * x" and b = "y * (a*x)" in box_equals)
```
```   462     apply (simp (no_asm))
```
```   463   apply auto
```
```   464   done
```
```   465
```
```   466 lemma r_inverse_ring: "!! a::'a::ring. a dvd 1 ==> a * inverse a = 1"
```
```   467   apply (unfold inverse_def dvd_def)
```
```   468   apply (tactic {* asm_full_simp_tac (simpset () delsimprocs [ring_simproc]) 1 *})
```
```   469   apply clarify
```
```   470   apply (rule theI)
```
```   471    apply assumption
```
```   472   apply (rule inverse_unique)
```
```   473    apply assumption
```
```   474   apply assumption
```
```   475   done
```
```   476
```
```   477 lemma l_inverse_ring: "!! a::'a::ring. a dvd 1 ==> inverse a * a = 1"
```
```   478   by (simp add: r_inverse_ring)
```
```   479
```
```   480
```
```   481 (* Fields *)
```
```   482
```
```   483 section "Fields"
```
```   484
```
```   485 lemma field_unit [simp]: "!! a::'a::field. (a dvd 1) = (a ~= 0)"
```
```   486   by (auto dest: field_ax dvd_zero_left simp add: field_one_not_zero)
```
```   487
```
```   488 lemma r_inverse [simp]: "!! a::'a::field. a ~= 0 ==> a * inverse a = 1"
```
```   489   by (simp add: r_inverse_ring)
```
```   490
```
```   491 lemma l_inverse [simp]: "!! a::'a::field. a ~= 0 ==> inverse a * a= 1"
```
```   492   by (simp add: l_inverse_ring)
```
```   493
```
```   494
```
```   495 (* fields are integral domains *)
```
```   496
```
```   497 lemma field_integral: "!! a::'a::field. a * b = 0 ==> a = 0 | b = 0"
```
```   498   apply (tactic "step_tac @{claset} 1")
```
```   499   apply (rule_tac a = " (a*b) * inverse b" in box_equals)
```
```   500     apply (rule_tac [3] refl)
```
```   501    prefer 2
```
```   502    apply (simp (no_asm))
```
```   503    apply auto
```
```   504   done
```
```   505
```
```   506
```
```   507 (* fields are factorial domains *)
```
```   508
```
```   509 lemma field_fact_prime: "!! a::'a::field. irred a ==> prime a"
```
```   510   unfolding prime_def irred_def by (blast intro: field_ax)
```
```   511
```
```   512 end
```