src/HOL/Library/Fraction_Field.thy
author wenzelm
Thu Feb 11 23:00:22 2010 +0100 (2010-02-11)
changeset 35115 446c5063e4fd
parent 31998 2c7a24f74db9
child 35372 ca158c7b1144
permissions -rw-r--r--
modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
minor tuning;
     1 (*  Title:      Fraction_Field.thy
     2     Author:     Amine Chaieb, University of Cambridge
     3 *)
     4 
     5 header{* A formalization of the fraction field of any integral domain 
     6          A generalization of Rational.thy from int to any integral domain *}
     7 
     8 theory Fraction_Field
     9 imports Main (* Equiv_Relations Plain *)
    10 begin
    11 
    12 subsection {* General fractions construction *}
    13 
    14 subsubsection {* Construction of the type of fractions *}
    15 
    16 definition fractrel :: "(('a::idom * 'a ) * ('a * 'a)) set" where
    17   "fractrel == {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
    18 
    19 lemma fractrel_iff [simp]:
    20   "(x, y) \<in> fractrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
    21   by (simp add: fractrel_def)
    22 
    23 lemma refl_fractrel: "refl_on {x. snd x \<noteq> 0} fractrel"
    24   by (auto simp add: refl_on_def fractrel_def)
    25 
    26 lemma sym_fractrel: "sym fractrel"
    27   by (simp add: fractrel_def sym_def)
    28 
    29 lemma trans_fractrel: "trans fractrel"
    30 proof (rule transI, unfold split_paired_all)
    31   fix a b a' b' a'' b'' :: 'a
    32   assume A: "((a, b), (a', b')) \<in> fractrel"
    33   assume B: "((a', b'), (a'', b'')) \<in> fractrel"
    34   have "b' * (a * b'') = b'' * (a * b')" by (simp add: mult_ac)
    35   also from A have "a * b' = a' * b" by auto
    36   also have "b'' * (a' * b) = b * (a' * b'')" by (simp add: mult_ac)
    37   also from B have "a' * b'' = a'' * b'" by auto
    38   also have "b * (a'' * b') = b' * (a'' * b)" by (simp add: mult_ac)
    39   finally have "b' * (a * b'') = b' * (a'' * b)" .
    40   moreover from B have "b' \<noteq> 0" by auto
    41   ultimately have "a * b'' = a'' * b" by simp
    42   with A B show "((a, b), (a'', b'')) \<in> fractrel" by auto
    43 qed
    44   
    45 lemma equiv_fractrel: "equiv {x. snd x \<noteq> 0} fractrel"
    46   by (rule equiv.intro [OF refl_fractrel sym_fractrel trans_fractrel])
    47 
    48 lemmas UN_fractrel = UN_equiv_class [OF equiv_fractrel]
    49 lemmas UN_fractrel2 = UN_equiv_class2 [OF equiv_fractrel equiv_fractrel]
    50 
    51 lemma equiv_fractrel_iff [iff]: 
    52   assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"
    53   shows "fractrel `` {x} = fractrel `` {y} \<longleftrightarrow> (x, y) \<in> fractrel"
    54   by (rule eq_equiv_class_iff, rule equiv_fractrel) (auto simp add: assms)
    55 
    56 typedef 'a fract = "{(x::'a\<times>'a). snd x \<noteq> (0::'a::idom)} // fractrel"
    57 proof
    58   have "(0::'a, 1::'a) \<in> {x. snd x \<noteq> 0}" by simp
    59   then show "fractrel `` {(0::'a, 1)} \<in> {x. snd x \<noteq> 0} // fractrel" by (rule quotientI)
    60 qed
    61 
    62 lemma fractrel_in_fract [simp]: "snd x \<noteq> 0 \<Longrightarrow> fractrel `` {x} \<in> fract"
    63   by (simp add: fract_def quotientI)
    64 
    65 declare Abs_fract_inject [simp] Abs_fract_inverse [simp]
    66 
    67 
    68 subsubsection {* Representation and basic operations *}
    69 
    70 definition
    71   Fract :: "'a::idom \<Rightarrow> 'a \<Rightarrow> 'a fract" where
    72   [code del]: "Fract a b = Abs_fract (fractrel `` {if b = 0 then (0, 1) else (a, b)})"
    73 
    74 code_datatype Fract
    75 
    76 lemma Fract_cases [case_names Fract, cases type: fract]:
    77   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C"
    78   shows C
    79   using assms by (cases q) (clarsimp simp add: Fract_def fract_def quotient_def)
    80 
    81 lemma Fract_induct [case_names Fract, induct type: fract]:
    82   assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)"
    83   shows "P q"
    84   using assms by (cases q) simp
    85 
    86 lemma eq_fract:
    87   shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b"
    88   and "\<And>a. Fract a 0 = Fract 0 1"
    89   and "\<And>a c. Fract 0 a = Fract 0 c"
    90   by (simp_all add: Fract_def)
    91 
    92 instantiation fract :: (idom) "{comm_ring_1, power}"
    93 begin
    94 
    95 definition
    96   Zero_fract_def [code, code_unfold]: "0 = Fract 0 1"
    97 
    98 definition
    99   One_fract_def [code, code_unfold]: "1 = Fract 1 1"
   100 
   101 definition
   102   add_fract_def [code del]:
   103   "q + r = Abs_fract (\<Union>x \<in> Rep_fract q. \<Union>y \<in> Rep_fract r.
   104     fractrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
   105 
   106 lemma add_fract [simp]:
   107   assumes "b \<noteq> (0::'a::idom)" and "d \<noteq> 0"
   108   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
   109 proof -
   110   have "(\<lambda>x y. fractrel``{(fst x * snd y + fst y * snd x, snd x * snd y :: 'a)})
   111     respects2 fractrel"
   112   apply (rule equiv_fractrel [THEN congruent2_commuteI]) apply (auto simp add: algebra_simps)
   113   unfolding mult_assoc[symmetric] .
   114   with assms show ?thesis by (simp add: Fract_def add_fract_def UN_fractrel2)
   115 qed
   116 
   117 definition
   118   minus_fract_def [code del]:
   119   "- q = Abs_fract (\<Union>x \<in> Rep_fract q. fractrel `` {(- fst x, snd x)})"
   120 
   121 lemma minus_fract [simp, code]: "- Fract a b = Fract (- a) (b::'a::idom)"
   122 proof -
   123   have "(\<lambda>x. fractrel `` {(- fst x, snd x :: 'a)}) respects fractrel"
   124     by (simp add: congruent_def)
   125   then show ?thesis by (simp add: Fract_def minus_fract_def UN_fractrel)
   126 qed
   127 
   128 lemma minus_fract_cancel [simp]: "Fract (- a) (- b) = Fract a b"
   129   by (cases "b = 0") (simp_all add: eq_fract)
   130 
   131 definition
   132   diff_fract_def [code del]: "q - r = q + - (r::'a fract)"
   133 
   134 lemma diff_fract [simp]:
   135   assumes "b \<noteq> 0" and "d \<noteq> 0"
   136   shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
   137   using assms by (simp add: diff_fract_def diff_minus)
   138 
   139 definition
   140   mult_fract_def [code del]:
   141   "q * r = Abs_fract (\<Union>x \<in> Rep_fract q. \<Union>y \<in> Rep_fract r.
   142     fractrel``{(fst x * fst y, snd x * snd y)})"
   143 
   144 lemma mult_fract [simp]: "Fract (a::'a::idom) b * Fract c d = Fract (a * c) (b * d)"
   145 proof -
   146   have "(\<lambda>x y. fractrel `` {(fst x * fst y, snd x * snd y :: 'a)}) respects2 fractrel"
   147     apply (rule equiv_fractrel [THEN congruent2_commuteI]) apply (auto simp add: algebra_simps)
   148     unfolding mult_assoc[symmetric] .
   149   then show ?thesis by (simp add: Fract_def mult_fract_def UN_fractrel2)
   150 qed
   151 
   152 lemma mult_fract_cancel:
   153   assumes "c \<noteq> 0"
   154   shows "Fract (c * a) (c * b) = Fract a b"
   155 proof -
   156   from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
   157   then show ?thesis by (simp add: mult_fract [symmetric])
   158 qed
   159 
   160 instance proof
   161   fix q r s :: "'a fract" show "(q * r) * s = q * (r * s)" 
   162     by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
   163 next
   164   fix q r :: "'a fract" show "q * r = r * q"
   165     by (cases q, cases r) (simp add: eq_fract algebra_simps)
   166 next
   167   fix q :: "'a fract" show "1 * q = q"
   168     by (cases q) (simp add: One_fract_def eq_fract)
   169 next
   170   fix q r s :: "'a fract" show "(q + r) + s = q + (r + s)"
   171     by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
   172 next
   173   fix q r :: "'a fract" show "q + r = r + q"
   174     by (cases q, cases r) (simp add: eq_fract algebra_simps)
   175 next
   176   fix q :: "'a fract" show "0 + q = q"
   177     by (cases q) (simp add: Zero_fract_def eq_fract)
   178 next
   179   fix q :: "'a fract" show "- q + q = 0"
   180     by (cases q) (simp add: Zero_fract_def eq_fract)
   181 next
   182   fix q r :: "'a fract" show "q - r = q + - r"
   183     by (cases q, cases r) (simp add: eq_fract)
   184 next
   185   fix q r s :: "'a fract" show "(q + r) * s = q * s + r * s"
   186     by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
   187 next
   188   show "(0::'a fract) \<noteq> 1" by (simp add: Zero_fract_def One_fract_def eq_fract)
   189 qed
   190 
   191 end
   192 
   193 lemma of_nat_fract: "of_nat k = Fract (of_nat k) 1"
   194   by (induct k) (simp_all add: Zero_fract_def One_fract_def)
   195 
   196 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
   197   by (rule of_nat_fract [symmetric])
   198 
   199 lemma fract_collapse [code_post]:
   200   "Fract 0 k = 0"
   201   "Fract 1 1 = 1"
   202   "Fract k 0 = 0"
   203   by (cases "k = 0")
   204     (simp_all add: Zero_fract_def One_fract_def eq_fract Fract_def)
   205 
   206 lemma fract_expand [code_unfold]:
   207   "0 = Fract 0 1"
   208   "1 = Fract 1 1"
   209   by (simp_all add: fract_collapse)
   210 
   211 lemma Fract_cases_nonzero [case_names Fract 0]:
   212   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
   213   assumes 0: "q = 0 \<Longrightarrow> C"
   214   shows C
   215 proof (cases "q = 0")
   216   case True then show C using 0 by auto
   217 next
   218   case False
   219   then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
   220   moreover with False have "0 \<noteq> Fract a b" by simp
   221   with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_fract_def eq_fract)
   222   with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
   223 qed
   224   
   225 
   226 
   227 subsubsection {* The field of rational numbers *}
   228 
   229 context idom
   230 begin
   231 subclass ring_no_zero_divisors ..
   232 thm mult_eq_0_iff
   233 end
   234 
   235 instantiation fract :: (idom) "{field, division_by_zero}"
   236 begin
   237 
   238 definition
   239   inverse_fract_def [code del]:
   240   "inverse q = Abs_fract (\<Union>x \<in> Rep_fract q.
   241      fractrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
   242 
   243 
   244 lemma inverse_fract [simp]: "inverse (Fract a b) = Fract (b::'a::idom) a"
   245 proof -
   246   have stupid: "\<And>x. (0::'a) = x \<longleftrightarrow> x = 0" by auto
   247   have "(\<lambda>x. fractrel `` {if fst x = 0 then (0, 1) else (snd x, fst x :: 'a)}) respects fractrel"
   248     by (auto simp add: congruent_def stupid algebra_simps)
   249   then show ?thesis by (simp add: Fract_def inverse_fract_def UN_fractrel)
   250 qed
   251 
   252 definition
   253   divide_fract_def [code del]: "q / r = q * inverse (r:: 'a fract)"
   254 
   255 lemma divide_fract [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
   256   by (simp add: divide_fract_def)
   257 
   258 instance proof
   259   show "inverse 0 = (0:: 'a fract)" by (simp add: fract_expand)
   260     (simp add: fract_collapse)
   261 next
   262   fix q :: "'a fract"
   263   assume "q \<noteq> 0"
   264   then show "inverse q * q = 1" apply (cases q rule: Fract_cases_nonzero)
   265     by (simp_all add: mult_fract  inverse_fract fract_expand eq_fract mult_commute)
   266 next
   267   fix q r :: "'a fract"
   268   show "q / r = q * inverse r" by (simp add: divide_fract_def)
   269 qed
   270 
   271 end
   272 
   273 
   274 end