src/HOL/Algebra/IntRing.thy
 author hoelzl Tue Nov 05 09:44:57 2013 +0100 (2013-11-05) changeset 54257 5c7a3b6b05a9 parent 49962 a8cc904a6820 child 55157 06897ea77f78 permissions -rw-r--r--
generalize SUP and INF to the syntactic type classes Sup and Inf
```     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/Old_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 == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
```
```    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 "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
```
```   187   where "carrier (| carrier = UNIV::int set, eq = op =, le = op \<le> |) = UNIV"
```
```   188     and "le (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x \<le> y)"
```
```   189     and "lless (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x < y)"
```
```   190 proof -
```
```   191   show "partial_order (| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
```
```   192     by default simp_all
```
```   193   show "carrier (| carrier = UNIV::int set, eq = op =, le = op \<le> |) = UNIV"
```
```   194     by simp
```
```   195   show "le (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x \<le> y)"
```
```   196     by simp
```
```   197   show "lless (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = (x < y)"
```
```   198     by (simp add: lless_def) auto
```
```   199 qed
```
```   200
```
```   201 interpretation int (* FIXME [unfolded UNIV] *) :
```
```   202   lattice "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
```
```   203   where "join (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = max x y"
```
```   204     and "meet (| carrier = UNIV::int set, eq = op =, le = op \<le> |) x y = min x y"
```
```   205 proof -
```
```   206   let ?Z = "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
```
```   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 "(| carrier = UNIV::int set, eq = op =, le = op \<le> |)"
```
```   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 apply (subst int_Idl[symmetric], rule principalidealI)
```
```   243  apply (rule int.genideal_ideal, simp)
```
```   244 apply fast
```
```   245 done
```
```   246
```
```   247
```
```   248 lemma prime_primeideal:
```
```   249   assumes prime: "prime (nat p)"
```
```   250   shows "primeideal (Idl\<^bsub>\<Z>\<^esub> {p}) \<Z>"
```
```   251 apply (rule primeidealI)
```
```   252    apply (rule int.genideal_ideal, simp)
```
```   253   apply (rule int_is_cring)
```
```   254  apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
```
```   255  apply clarsimp defer 1
```
```   256  apply (simp add: int.cgenideal_eq_genideal[symmetric] cgenideal_def)
```
```   257  apply (elim exE)
```
```   258 proof -
```
```   259   fix a b x
```
```   260
```
```   261   from prime
```
```   262       have ppos: "0 <= p" by (simp add: prime_def)
```
```   263   have unnat: "!!x. nat p dvd nat (abs x) ==> p dvd x"
```
```   264   proof -
```
```   265     fix x
```
```   266     assume "nat p dvd nat (abs x)"
```
```   267     hence "int (nat p) dvd x" by (simp add: int_dvd_iff[symmetric])
```
```   268     thus "p dvd x" by (simp add: ppos)
```
```   269   qed
```
```   270
```
```   271
```
```   272   assume "a * b = x * p"
```
```   273   hence "p dvd a * b" by simp
```
```   274   hence "nat p dvd nat (abs (a * b))" using ppos by (simp add: nat_dvd_iff)
```
```   275   hence "nat p dvd (nat (abs a) * nat (abs b))" by (simp add: nat_abs_mult_distrib)
```
```   276   hence "nat p dvd nat (abs a) | nat p dvd nat (abs b)" by (rule prime_dvd_mult[OF prime])
```
```   277   hence "p dvd a | p dvd b" by (fast intro: unnat)
```
```   278   thus "(EX x. a = x * p) | (EX x. b = x * p)"
```
```   279   proof
```
```   280     assume "p dvd a"
```
```   281     hence "EX x. a = p * x" by (simp add: dvd_def)
```
```   282     from this obtain x
```
```   283         where "a = p * x" by fast
```
```   284     hence "a = x * p" by simp
```
```   285     hence "EX x. a = x * p" by simp
```
```   286     thus "(EX x. a = x * p) | (EX x. b = x * p)" ..
```
```   287   next
```
```   288     assume "p dvd b"
```
```   289     hence "EX x. b = p * x" by (simp add: dvd_def)
```
```   290     from this obtain x
```
```   291         where "b = p * x" by fast
```
```   292     hence "b = x * p" by simp
```
```   293     hence "EX x. b = x * p" by simp
```
```   294     thus "(EX x. a = x * p) | (EX x. b = x * p)" ..
```
```   295   qed
```
```   296 next
```
```   297   assume "UNIV = {uu. EX x. uu = x * p}"
```
```   298   from this obtain x
```
```   299       where "1 = x * p" by best
```
```   300   from this [symmetric]
```
```   301       have "p * x = 1" by (subst mult_commute)
```
```   302   hence "\<bar>p * x\<bar> = 1" by simp
```
```   303   hence "\<bar>p\<bar> = 1" by (rule abs_zmult_eq_1)
```
```   304   from this and prime
```
```   305       show "False" by (simp add: prime_def)
```
```   306 qed
```
```   307
```
```   308
```
```   309 subsection {* Ideals and Divisibility *}
```
```   310
```
```   311 lemma int_Idl_subset_ideal:
```
```   312   "Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l} = (k \<in> Idl\<^bsub>\<Z>\<^esub> {l})"
```
```   313 by (rule int.Idl_subset_ideal', simp+)
```
```   314
```
```   315 lemma Idl_subset_eq_dvd:
```
```   316   "(Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l}) = (l dvd k)"
```
```   317 apply (subst int_Idl_subset_ideal, subst int_Idl, simp)
```
```   318 apply (rule, clarify)
```
```   319 apply (simp add: dvd_def)
```
```   320 apply (simp add: dvd_def mult_ac)
```
```   321 done
```
```   322
```
```   323 lemma dvds_eq_Idl:
```
```   324   "(l dvd k \<and> k dvd l) = (Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l})"
```
```   325 proof -
```
```   326   have a: "l dvd k = (Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l})" by (rule Idl_subset_eq_dvd[symmetric])
```
```   327   have b: "k dvd l = (Idl\<^bsub>\<Z>\<^esub> {l} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {k})" by (rule Idl_subset_eq_dvd[symmetric])
```
```   328
```
```   329   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}))"
```
```   330   by (subst a, subst b, simp)
```
```   331   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+)
```
```   332   finally
```
```   333     show ?thesis .
```
```   334 qed
```
```   335
```
```   336 lemma Idl_eq_abs:
```
```   337   "(Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l}) = (abs l = abs k)"
```
```   338 apply (subst dvds_eq_abseq[symmetric])
```
```   339 apply (rule dvds_eq_Idl[symmetric])
```
```   340 done
```
```   341
```
```   342
```
```   343 subsection {* Ideals and the Modulus *}
```
```   344
```
```   345 definition
```
```   346   ZMod :: "int => int => int set"
```
```   347   where "ZMod k r = (Idl\<^bsub>\<Z>\<^esub> {k}) +>\<^bsub>\<Z>\<^esub> r"
```
```   348
```
```   349 lemmas ZMod_defs =
```
```   350   ZMod_def genideal_def
```
```   351
```
```   352 lemma rcos_zfact:
```
```   353   assumes kIl: "k \<in> ZMod l r"
```
```   354   shows "EX x. k = x * l + r"
```
```   355 proof -
```
```   356   from kIl[unfolded ZMod_def]
```
```   357       have "\<exists>xl\<in>Idl\<^bsub>\<Z>\<^esub> {l}. k = xl + r" by (simp add: a_r_coset_defs)
```
```   358   from this obtain xl
```
```   359       where xl: "xl \<in> Idl\<^bsub>\<Z>\<^esub> {l}"
```
```   360       and k: "k = xl + r"
```
```   361       by auto
```
```   362   from xl obtain x
```
```   363       where "xl = x * l"
```
```   364       by (simp add: int_Idl, fast)
```
```   365   from k and this
```
```   366       have "k = x * l + r" by simp
```
```   367   thus "\<exists>x. k = x * l + r" ..
```
```   368 qed
```
```   369
```
```   370 lemma ZMod_imp_zmod:
```
```   371   assumes zmods: "ZMod m a = ZMod m b"
```
```   372   shows "a mod m = b mod m"
```
```   373 proof -
```
```   374   interpret ideal "Idl\<^bsub>\<Z>\<^esub> {m}" \<Z> by (rule int.genideal_ideal, fast)
```
```   375   from zmods
```
```   376       have "b \<in> ZMod m a"
```
```   377       unfolding ZMod_def
```
```   378       by (simp add: a_repr_independenceD)
```
```   379   from this
```
```   380       have "EX x. b = x * m + a" by (rule rcos_zfact)
```
```   381   from this obtain x
```
```   382       where "b = x * m + a"
```
```   383       by fast
```
```   384
```
```   385   hence "b mod m = (x * m + a) mod m" by simp
```
```   386   also
```
```   387       have "\<dots> = ((x * m) mod m) + (a mod m)" by (simp add: mod_add_eq)
```
```   388   also
```
```   389       have "\<dots> = a mod m" by simp
```
```   390   finally
```
```   391       have "b mod m = a mod m" .
```
```   392   thus "a mod m = b mod m" ..
```
```   393 qed
```
```   394
```
```   395 lemma ZMod_mod:
```
```   396   shows "ZMod m a = ZMod m (a mod m)"
```
```   397 proof -
```
```   398   interpret ideal "Idl\<^bsub>\<Z>\<^esub> {m}" \<Z> by (rule int.genideal_ideal, fast)
```
```   399   show ?thesis
```
```   400       unfolding ZMod_def
```
```   401   apply (rule a_repr_independence'[symmetric])
```
```   402   apply (simp add: int_Idl a_r_coset_defs)
```
```   403   proof -
```
```   404     have "a = m * (a div m) + (a mod m)" by (simp add: zmod_zdiv_equality)
```
```   405     hence "a = (a div m) * m + (a mod m)" by simp
```
```   406     thus "\<exists>h. (\<exists>x. h = x * m) \<and> a = h + a mod m" by fast
```
```   407   qed simp
```
```   408 qed
```
```   409
```
```   410 lemma zmod_imp_ZMod:
```
```   411   assumes modeq: "a mod m = b mod m"
```
```   412   shows "ZMod m a = ZMod m b"
```
```   413 proof -
```
```   414   have "ZMod m a = ZMod m (a mod m)" by (rule ZMod_mod)
```
```   415   also have "\<dots> = ZMod m (b mod m)" by (simp add: modeq[symmetric])
```
```   416   also have "\<dots> = ZMod m b" by (rule ZMod_mod[symmetric])
```
```   417   finally show ?thesis .
```
```   418 qed
```
```   419
```
```   420 corollary ZMod_eq_mod:
```
```   421   shows "(ZMod m a = ZMod m b) = (a mod m = b mod m)"
```
```   422 by (rule, erule ZMod_imp_zmod, erule zmod_imp_ZMod)
```
```   423
```
```   424
```
```   425 subsection {* Factorization *}
```
```   426
```
```   427 definition
```
```   428   ZFact :: "int \<Rightarrow> int set ring"
```
```   429   where "ZFact k = \<Z> Quot (Idl\<^bsub>\<Z>\<^esub> {k})"
```
```   430
```
```   431 lemmas ZFact_defs = ZFact_def FactRing_def
```
```   432
```
```   433 lemma ZFact_is_cring:
```
```   434   shows "cring (ZFact k)"
```
```   435 apply (unfold ZFact_def)
```
```   436 apply (rule ideal.quotient_is_cring)
```
```   437  apply (intro ring.genideal_ideal)
```
```   438   apply (simp add: cring.axioms[OF int_is_cring] ring.intro)
```
```   439  apply simp
```
```   440 apply (rule int_is_cring)
```
```   441 done
```
```   442
```
```   443 lemma ZFact_zero:
```
```   444   "carrier (ZFact 0) = (\<Union>a. {{a}})"
```
```   445 apply (insert int.genideal_zero)
```
```   446 apply (simp add: ZFact_defs A_RCOSETS_defs r_coset_def ring_record_simps)
```
```   447 done
```
```   448
```
```   449 lemma ZFact_one:
```
```   450   "carrier (ZFact 1) = {UNIV}"
```
```   451 apply (simp only: ZFact_defs A_RCOSETS_defs r_coset_def ring_record_simps)
```
```   452 apply (subst int.genideal_one)
```
```   453 apply (rule, rule, clarsimp)
```
```   454  apply (rule, rule, clarsimp)
```
```   455  apply (rule, clarsimp, arith)
```
```   456 apply (rule, clarsimp)
```
```   457 apply (rule exI[of _ "0"], clarsimp)
```
```   458 done
```
```   459
```
```   460 lemma ZFact_prime_is_domain:
```
```   461   assumes pprime: "prime (nat p)"
```
```   462   shows "domain (ZFact p)"
```
```   463 apply (unfold ZFact_def)
```
```   464 apply (rule primeideal.quotient_is_domain)
```
```   465 apply (rule prime_primeideal[OF pprime])
```
```   466 done
```
```   467
```
```   468 end
```