src/HOL/Algebra/IntRing.thy
 author wenzelm Wed Mar 05 21:51:30 2014 +0100 (2014-03-05) changeset 55926 3ef14caf5637 parent 55242 413ec965f95d child 55991 3fa6e6c81788 permissions -rw-r--r--
more symbols;
```     1 (*  Title:      HOL/Algebra/IntRing.thy
```
```     2     Author:     Stephan Hohe, TU Muenchen
```
```     3     Author:     Clemens Ballarin
```
```     4 *)
```
```     5
```
```     6 theory IntRing
```
```     7 imports QuotRing Lattice Int "~~/src/HOL/Number_Theory/Primes"
```
```     8 begin
```
```     9
```
```    10 section {* The Ring of Integers *}
```
```    11
```
```    12 subsection {* Some properties of @{typ int} *}
```
```    13
```
```    14 lemma dvds_eq_abseq:
```
```    15   "(l dvd k \<and> k dvd l) = (abs l = abs (k::int))"
```
```    16 apply rule
```
```    17  apply (simp add: zdvd_antisym_abs)
```
```    18 apply (simp add: dvd_if_abs_eq)
```
```    19 done
```
```    20
```
```    21
```
```    22 subsection {* @{text "\<Z>"}: The Set of Integers as Algebraic Structure *}
```
```    23
```
```    24 abbreviation
```
```    25   int_ring :: "int ring" ("\<Z>") where
```
```    26   "int_ring == \<lparr>carrier = UNIV, mult = op *, one = 1, zero = 0, add = op +\<rparr>"
```
```    27
```
```    28 lemma int_Zcarr [intro!, simp]:
```
```    29   "k \<in> carrier \<Z>"
```
```    30   by simp
```
```    31
```
```    32 lemma int_is_cring:
```
```    33   "cring \<Z>"
```
```    34 apply (rule cringI)
```
```    35   apply (rule abelian_groupI, simp_all)
```
```    36   defer 1
```
```    37   apply (rule comm_monoidI, simp_all)
```
```    38  apply (rule distrib_right)
```
```    39 apply (fast intro: left_minus)
```
```    40 done
```
```    41
```
```    42 (*
```
```    43 lemma int_is_domain:
```
```    44   "domain \<Z>"
```
```    45 apply (intro domain.intro domain_axioms.intro)
```
```    46   apply (rule int_is_cring)
```
```    47  apply (unfold int_ring_def, simp+)
```
```    48 done
```
```    49 *)
```
```    50
```
```    51
```
```    52 subsection {* Interpretations *}
```
```    53
```
```    54 text {* Since definitions of derived operations are global, their
```
```    55   interpretation needs to be done as early as possible --- that is,
```
```    56   with as few assumptions as possible. *}
```
```    57
```
```    58 interpretation int: monoid \<Z>
```
```    59   where "carrier \<Z> = UNIV"
```
```    60     and "mult \<Z> x y = x * y"
```
```    61     and "one \<Z> = 1"
```
```    62     and "pow \<Z> x n = x^n"
```
```    63 proof -
```
```    64   -- "Specification"
```
```    65   show "monoid \<Z>" by default auto
```
```    66   then interpret int: monoid \<Z> .
```
```    67
```
```    68   -- "Carrier"
```
```    69   show "carrier \<Z> = UNIV" by simp
```
```    70
```
```    71   -- "Operations"
```
```    72   { fix x y show "mult \<Z> x y = x * y" by simp }
```
```    73   note mult = this
```
```    74   show one: "one \<Z> = 1" by simp
```
```    75   show "pow \<Z> x n = x^n" by (induct n) simp_all
```
```    76 qed
```
```    77
```
```    78 interpretation int: comm_monoid \<Z>
```
```    79   where "finprod \<Z> f A = (if finite A then setprod f A else undefined)"
```
```    80 proof -
```
```    81   -- "Specification"
```
```    82   show "comm_monoid \<Z>" by default auto
```
```    83   then interpret int: comm_monoid \<Z> .
```
```    84
```
```    85   -- "Operations"
```
```    86   { fix x y have "mult \<Z> x y = x * y" by simp }
```
```    87   note mult = this
```
```    88   have one: "one \<Z> = 1" by simp
```
```    89   show "finprod \<Z> f A = (if finite A then setprod f A else undefined)"
```
```    90   proof (cases "finite A")
```
```    91     case True then show ?thesis proof induct
```
```    92       case empty show ?case by (simp add: one)
```
```    93     next
```
```    94       case insert then show ?case by (simp add: Pi_def mult)
```
```    95     qed
```
```    96   next
```
```    97     case False then show ?thesis by (simp add: finprod_def)
```
```    98   qed
```
```    99 qed
```
```   100
```
```   101 interpretation int: abelian_monoid \<Z>
```
```   102   where int_carrier_eq: "carrier \<Z> = UNIV"
```
```   103     and int_zero_eq: "zero \<Z> = 0"
```
```   104     and int_add_eq: "add \<Z> x y = x + y"
```
```   105     and int_finsum_eq: "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
```
```   106 proof -
```
```   107   -- "Specification"
```
```   108   show "abelian_monoid \<Z>" by default auto
```
```   109   then interpret int: abelian_monoid \<Z> .
```
```   110
```
```   111   -- "Carrier"
```
```   112   show "carrier \<Z> = UNIV" by simp
```
```   113
```
```   114   -- "Operations"
```
```   115   { fix x y show "add \<Z> x y = x + y" by simp }
```
```   116   note add = this
```
```   117   show zero: "zero \<Z> = 0" by simp
```
```   118   show "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
```
```   119   proof (cases "finite A")
```
```   120     case True then show ?thesis proof induct
```
```   121       case empty show ?case by (simp add: zero)
```
```   122     next
```
```   123       case insert then show ?case by (simp add: Pi_def add)
```
```   124     qed
```
```   125   next
```
```   126     case False then show ?thesis by (simp add: finsum_def finprod_def)
```
```   127   qed
```
```   128 qed
```
```   129
```
```   130 interpretation int: abelian_group \<Z>
```
```   131   (* The equations from the interpretation of abelian_monoid need to be repeated.
```
```   132      Since the morphisms through which the abelian structures are interpreted are
```
```   133      not the identity, the equations of these interpretations are not inherited. *)
```
```   134   (* FIXME *)
```
```   135   where "carrier \<Z> = UNIV"
```
```   136     and "zero \<Z> = 0"
```
```   137     and "add \<Z> x y = x + y"
```
```   138     and "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
```
```   139     and int_a_inv_eq: "a_inv \<Z> x = - x"
```
```   140     and int_a_minus_eq: "a_minus \<Z> x y = x - y"
```
```   141 proof -
```
```   142   -- "Specification"
```
```   143   show "abelian_group \<Z>"
```
```   144   proof (rule abelian_groupI)
```
```   145     show "!!x. x \<in> carrier \<Z> ==> EX y : carrier \<Z>. y \<oplus>\<^bsub>\<Z>\<^esub> x = \<zero>\<^bsub>\<Z>\<^esub>"
```
```   146       by simp arith
```
```   147   qed auto
```
```   148   then interpret int: abelian_group \<Z> .
```
```   149   -- "Operations"
```
```   150   { fix x y have "add \<Z> x y = x + y" by simp }
```
```   151   note add = this
```
```   152   have zero: "zero \<Z> = 0" by simp
```
```   153   { fix x
```
```   154     have "add \<Z> (-x) x = zero \<Z>" by (simp add: add zero)
```
```   155     then show "a_inv \<Z> x = - x" by (simp add: int.minus_equality) }
```
```   156   note a_inv = this
```
```   157   show "a_minus \<Z> x y = x - y" by (simp add: int.minus_eq add a_inv)
```
```   158 qed (simp add: int_carrier_eq int_zero_eq int_add_eq int_finsum_eq)+
```
```   159
```
```   160 interpretation int: "domain" \<Z>
```
```   161   where "carrier \<Z> = UNIV"
```
```   162     and "zero \<Z> = 0"
```
```   163     and "add \<Z> x y = x + y"
```
```   164     and "finsum \<Z> f A = (if finite A then setsum f A else undefined)"
```
```   165     and "a_inv \<Z> x = - x"
```
```   166     and "a_minus \<Z> x y = x - y"
```
```   167 proof -
```
```   168   show "domain \<Z>" by unfold_locales (auto simp: distrib_right distrib_left)
```
```   169 qed (simp
```
```   170     add: int_carrier_eq int_zero_eq int_add_eq int_finsum_eq int_a_inv_eq int_a_minus_eq)+
```
```   171
```
```   172
```
```   173 text {* Removal of occurrences of @{term UNIV} in interpretation result
```
```   174   --- experimental. *}
```
```   175
```
```   176 lemma UNIV:
```
```   177   "x \<in> UNIV = True"
```
```   178   "A \<subseteq> UNIV = True"
```
```   179   "(ALL x : UNIV. P x) = (ALL x. P x)"
```
```   180   "(EX x : UNIV. P x) = (EX x. P x)"
```
```   181   "(True --> Q) = Q"
```
```   182   "(True ==> PROP R) == PROP R"
```
```   183   by simp_all
```
```   184
```
```   185 interpretation int (* FIXME [unfolded UNIV] *) :
```
```   186   partial_order "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
```
```   187   where "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> = UNIV"
```
```   188     and "le \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x \<le> y)"
```
```   189     and "lless \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x < y)"
```
```   190 proof -
```
```   191   show "partial_order \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
```
```   192     by default simp_all
```
```   193   show "carrier \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> = UNIV"
```
```   194     by simp
```
```   195   show "le \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x \<le> y)"
```
```   196     by simp
```
```   197   show "lless \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = (x < y)"
```
```   198     by (simp add: lless_def) auto
```
```   199 qed
```
```   200
```
```   201 interpretation int (* FIXME [unfolded UNIV] *) :
```
```   202   lattice "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
```
```   203   where "join \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = max x y"
```
```   204     and "meet \<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr> x y = min x y"
```
```   205 proof -
```
```   206   let ?Z = "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
```
```   207   show "lattice ?Z"
```
```   208     apply unfold_locales
```
```   209     apply (simp add: least_def Upper_def)
```
```   210     apply arith
```
```   211     apply (simp add: greatest_def Lower_def)
```
```   212     apply arith
```
```   213     done
```
```   214   then interpret int: lattice "?Z" .
```
```   215   show "join ?Z x y = max x y"
```
```   216     apply (rule int.joinI)
```
```   217     apply (simp_all add: least_def Upper_def)
```
```   218     apply arith
```
```   219     done
```
```   220   show "meet ?Z x y = min x y"
```
```   221     apply (rule int.meetI)
```
```   222     apply (simp_all add: greatest_def Lower_def)
```
```   223     apply arith
```
```   224     done
```
```   225 qed
```
```   226
```
```   227 interpretation int (* [unfolded UNIV] *) :
```
```   228   total_order "\<lparr>carrier = UNIV::int set, eq = op =, le = op \<le>\<rparr>"
```
```   229   by default clarsimp
```
```   230
```
```   231
```
```   232 subsection {* Generated Ideals of @{text "\<Z>"} *}
```
```   233
```
```   234 lemma int_Idl:
```
```   235   "Idl\<^bsub>\<Z>\<^esub> {a} = {x * a | x. True}"
```
```   236   apply (subst int.cgenideal_eq_genideal[symmetric]) apply simp
```
```   237   apply (simp add: cgenideal_def)
```
```   238   done
```
```   239
```
```   240 lemma multiples_principalideal:
```
```   241   "principalideal {x * a | x. True } \<Z>"
```
```   242 by (metis UNIV_I int.cgenideal_eq_genideal int.cgenideal_is_principalideal int_Idl)
```
```   243
```
```   244 lemma prime_primeideal:
```
```   245   assumes prime: "prime p"
```
```   246   shows "primeideal (Idl\<^bsub>\<Z>\<^esub> {p}) \<Z>"
```
```   247 apply (rule primeidealI)
```
```   248    apply (rule int.genideal_ideal, simp)
```
```   249   apply (rule int_is_cring)
```
```   250  apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
```
```   251  apply clarsimp defer 1
```
```   252  apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
```
```   253  apply (elim exE)
```
```   254 proof -
```
```   255   fix a b x
```
```   256   assume "a * b = x * int p"
```
```   257   hence "p dvd a * b" by simp
```
```   258   hence "p dvd a | p dvd b"
```
```   259     by (metis prime prime_dvd_mult_eq_int)
```
```   260   thus "(\<exists>x. a = x * int p) \<or> (\<exists>x. b = x * int p)"
```
```   261     by (metis dvd_def mult_commute)
```
```   262 next
```
```   263   assume "UNIV = {uu. EX x. uu = x * int p}"
```
```   264   then obtain x where "1 = x * int p" by best
```
```   265   hence "\<bar>int p * x\<bar> = 1" by (simp add: mult_commute)
```
```   266   then show False
```
```   267     by (metis abs_of_nat int_1 of_nat_eq_iff abs_zmult_eq_1 one_not_prime_nat prime)
```
```   268 qed
```
```   269
```
```   270
```
```   271 subsection {* Ideals and Divisibility *}
```
```   272
```
```   273 lemma int_Idl_subset_ideal:
```
```   274   "Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l} = (k \<in> Idl\<^bsub>\<Z>\<^esub> {l})"
```
```   275 by (rule int.Idl_subset_ideal', simp+)
```
```   276
```
```   277 lemma Idl_subset_eq_dvd:
```
```   278   "(Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l}) = (l dvd k)"
```
```   279 apply (subst int_Idl_subset_ideal, subst int_Idl, simp)
```
```   280 apply (rule, clarify)
```
```   281 apply (simp add: dvd_def)
```
```   282 apply (simp add: dvd_def mult_ac)
```
```   283 done
```
```   284
```
```   285 lemma dvds_eq_Idl:
```
```   286   "(l dvd k \<and> k dvd l) = (Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l})"
```
```   287 proof -
```
```   288   have a: "l dvd k = (Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l})" by (rule Idl_subset_eq_dvd[symmetric])
```
```   289   have b: "k dvd l = (Idl\<^bsub>\<Z>\<^esub> {l} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {k})" by (rule Idl_subset_eq_dvd[symmetric])
```
```   290
```
```   291   have "(l dvd k \<and> k dvd l) = ((Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l}) \<and> (Idl\<^bsub>\<Z>\<^esub> {l} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {k}))"
```
```   292   by (subst a, subst b, simp)
```
```   293   also have "((Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l}) \<and> (Idl\<^bsub>\<Z>\<^esub> {l} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {k})) = (Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l})" by (rule, fast+)
```
```   294   finally
```
```   295     show ?thesis .
```
```   296 qed
```
```   297
```
```   298 lemma Idl_eq_abs:
```
```   299   "(Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l}) = (abs l = abs k)"
```
```   300 apply (subst dvds_eq_abseq[symmetric])
```
```   301 apply (rule dvds_eq_Idl[symmetric])
```
```   302 done
```
```   303
```
```   304
```
```   305 subsection {* Ideals and the Modulus *}
```
```   306
```
```   307 definition
```
```   308   ZMod :: "int => int => int set"
```
```   309   where "ZMod k r = (Idl\<^bsub>\<Z>\<^esub> {k}) +>\<^bsub>\<Z>\<^esub> r"
```
```   310
```
```   311 lemmas ZMod_defs =
```
```   312   ZMod_def genideal_def
```
```   313
```
```   314 lemma rcos_zfact:
```
```   315   assumes kIl: "k \<in> ZMod l r"
```
```   316   shows "EX x. k = x * l + r"
```
```   317 proof -
```
```   318   from kIl[unfolded ZMod_def]
```
```   319       have "\<exists>xl\<in>Idl\<^bsub>\<Z>\<^esub> {l}. k = xl + r" by (simp add: a_r_coset_defs)
```
```   320   then obtain xl
```
```   321       where xl: "xl \<in> Idl\<^bsub>\<Z>\<^esub> {l}"
```
```   322       and k: "k = xl + r"
```
```   323       by auto
```
```   324   from xl obtain x
```
```   325       where "xl = x * l"
```
```   326       by (simp add: int_Idl, fast)
```
```   327   from k and this
```
```   328       have "k = x * l + r" by simp
```
```   329   thus "\<exists>x. k = x * l + r" ..
```
```   330 qed
```
```   331
```
```   332 lemma ZMod_imp_zmod:
```
```   333   assumes zmods: "ZMod m a = ZMod m b"
```
```   334   shows "a mod m = b mod m"
```
```   335 proof -
```
```   336   interpret ideal "Idl\<^bsub>\<Z>\<^esub> {m}" \<Z> by (rule int.genideal_ideal, fast)
```
```   337   from zmods
```
```   338       have "b \<in> ZMod m a"
```
```   339       unfolding ZMod_def
```
```   340       by (simp add: a_repr_independenceD)
```
```   341   then
```
```   342       have "EX x. b = x * m + a" by (rule rcos_zfact)
```
```   343   then obtain x
```
```   344       where "b = x * m + a"
```
```   345       by fast
```
```   346
```
```   347   hence "b mod m = (x * m + a) mod m" by simp
```
```   348   also
```
```   349       have "\<dots> = ((x * m) mod m) + (a mod m)" by (simp add: mod_add_eq)
```
```   350   also
```
```   351       have "\<dots> = a mod m" by simp
```
```   352   finally
```
```   353       have "b mod m = a mod m" .
```
```   354   thus "a mod m = b mod m" ..
```
```   355 qed
```
```   356
```
```   357 lemma ZMod_mod:
```
```   358   shows "ZMod m a = ZMod m (a mod m)"
```
```   359 proof -
```
```   360   interpret ideal "Idl\<^bsub>\<Z>\<^esub> {m}" \<Z> by (rule int.genideal_ideal, fast)
```
```   361   show ?thesis
```
```   362       unfolding ZMod_def
```
```   363   apply (rule a_repr_independence'[symmetric])
```
```   364   apply (simp add: int_Idl a_r_coset_defs)
```
```   365   proof -
```
```   366     have "a = m * (a div m) + (a mod m)" by (simp add: zmod_zdiv_equality)
```
```   367     hence "a = (a div m) * m + (a mod m)" by simp
```
```   368     thus "\<exists>h. (\<exists>x. h = x * m) \<and> a = h + a mod m" by fast
```
```   369   qed simp
```
```   370 qed
```
```   371
```
```   372 lemma zmod_imp_ZMod:
```
```   373   assumes modeq: "a mod m = b mod m"
```
```   374   shows "ZMod m a = ZMod m b"
```
```   375 proof -
```
```   376   have "ZMod m a = ZMod m (a mod m)" by (rule ZMod_mod)
```
```   377   also have "\<dots> = ZMod m (b mod m)" by (simp add: modeq[symmetric])
```
```   378   also have "\<dots> = ZMod m b" by (rule ZMod_mod[symmetric])
```
```   379   finally show ?thesis .
```
```   380 qed
```
```   381
```
```   382 corollary ZMod_eq_mod:
```
```   383   shows "(ZMod m a = ZMod m b) = (a mod m = b mod m)"
```
```   384 by (rule, erule ZMod_imp_zmod, erule zmod_imp_ZMod)
```
```   385
```
```   386
```
```   387 subsection {* Factorization *}
```
```   388
```
```   389 definition
```
```   390   ZFact :: "int \<Rightarrow> int set ring"
```
```   391   where "ZFact k = \<Z> Quot (Idl\<^bsub>\<Z>\<^esub> {k})"
```
```   392
```
```   393 lemmas ZFact_defs = ZFact_def FactRing_def
```
```   394
```
```   395 lemma ZFact_is_cring:
```
```   396   shows "cring (ZFact k)"
```
```   397 apply (unfold ZFact_def)
```
```   398 apply (rule ideal.quotient_is_cring)
```
```   399  apply (intro ring.genideal_ideal)
```
```   400   apply (simp add: cring.axioms[OF int_is_cring] ring.intro)
```
```   401  apply simp
```
```   402 apply (rule int_is_cring)
```
```   403 done
```
```   404
```
```   405 lemma ZFact_zero:
```
```   406   "carrier (ZFact 0) = (\<Union>a. {{a}})"
```
```   407 apply (insert int.genideal_zero)
```
```   408 apply (simp add: ZFact_defs A_RCOSETS_defs r_coset_def ring_record_simps)
```
```   409 done
```
```   410
```
```   411 lemma ZFact_one:
```
```   412   "carrier (ZFact 1) = {UNIV}"
```
```   413 apply (simp only: ZFact_defs A_RCOSETS_defs r_coset_def ring_record_simps)
```
```   414 apply (subst int.genideal_one)
```
```   415 apply (rule, rule, clarsimp)
```
```   416  apply (rule, rule, clarsimp)
```
```   417  apply (rule, clarsimp, arith)
```
```   418 apply (rule, clarsimp)
```
```   419 apply (rule exI[of _ "0"], clarsimp)
```
```   420 done
```
```   421
```
```   422 lemma ZFact_prime_is_domain:
```
```   423   assumes pprime: "prime p"
```
```   424   shows "domain (ZFact p)"
```
```   425 apply (unfold ZFact_def)
```
```   426 apply (rule primeideal.quotient_is_domain)
```
```   427 apply (rule prime_primeideal[OF pprime])
```
```   428 done
```
```   429
```
```   430 end
```