author huffman Thu May 10 21:02:36 2012 +0200 (2012-05-10) changeset 47907 54e3847f1669 parent 47906 09a896d295bd child 47908 25686e1e0024
simplify instance proofs for rat
 src/HOL/Rat.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Rat.thy	Thu May 10 21:18:41 2012 +0200
1.2 +++ b/src/HOL/Rat.thy	Thu May 10 21:02:36 2012 +0200
1.3 @@ -400,200 +400,121 @@
1.4
1.5  subsubsection {* The ordered field of rational numbers *}
1.6
1.7 -instantiation rat :: linorder
1.8 +lift_definition positive :: "rat \<Rightarrow> bool"
1.9 +  is "\<lambda>x. 0 < fst x * snd x"
1.10 +proof (clarsimp)
1.11 +  fix a b c d :: int
1.12 +  assume "b \<noteq> 0" and "d \<noteq> 0" and "a * d = c * b"
1.13 +  hence "a * d * b * d = c * b * b * d"
1.14 +    by simp
1.15 +  hence "a * b * d\<twosuperior> = c * d * b\<twosuperior>"
1.16 +    unfolding power2_eq_square by (simp add: mult_ac)
1.17 +  hence "0 < a * b * d\<twosuperior> \<longleftrightarrow> 0 < c * d * b\<twosuperior>"
1.18 +    by simp
1.19 +  thus "0 < a * b \<longleftrightarrow> 0 < c * d"
1.20 +    using `b \<noteq> 0` and `d \<noteq> 0`
1.21 +    by (simp add: zero_less_mult_iff)
1.22 +qed
1.23 +
1.24 +lemma positive_zero: "\<not> positive 0"
1.25 +  by transfer simp
1.26 +
1.27 +lemma positive_add:
1.28 +  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
1.29 +apply transfer
1.30 +apply (simp add: zero_less_mult_iff)
1.31 +apply (elim disjE, simp_all add: add_pos_pos add_neg_neg
1.32 +  mult_pos_pos mult_pos_neg mult_neg_pos mult_neg_neg)
1.33 +done
1.34 +
1.35 +lemma positive_mult:
1.36 +  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
1.37 +by transfer (drule (1) mult_pos_pos, simp add: mult_ac)
1.38 +
1.39 +lemma positive_minus:
1.40 +  "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
1.41 +by transfer (force simp: neq_iff zero_less_mult_iff mult_less_0_iff)
1.42 +
1.43 +instantiation rat :: linordered_field_inverse_zero
1.44  begin
1.45
1.46 -lift_definition less_eq_rat :: "rat \<Rightarrow> rat \<Rightarrow> bool"
1.47 -  is "\<lambda>x y. (fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)"
1.48 -proof (clarsimp)
1.49 -  fix a b a' b' c d c' d'::int
1.50 -  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
1.51 -  assume eq1: "a * b' = a' * b"
1.52 -  assume eq2: "c * d' = c' * d"
1.53 +definition
1.54 +  "x < y \<longleftrightarrow> positive (y - x)"
1.55 +
1.56 +definition
1.57 +  "x \<le> (y::rat) \<longleftrightarrow> x < y \<or> x = y"
1.58 +
1.59 +definition
1.60 +  "abs (a::rat) = (if a < 0 then - a else a)"
1.61 +
1.62 +definition
1.63 +  "sgn (a::rat) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
1.64
1.65 -  let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
1.66 -  {
1.67 -    fix a b c d x :: int assume x: "x \<noteq> 0"
1.68 -    have "?le a b c d = ?le (a * x) (b * x) c d"
1.69 -    proof -
1.70 -      from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)
1.71 -      hence "?le a b c d =
1.72 -        ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
1.73 -        by (simp add: mult_le_cancel_right)
1.74 -      also have "... = ?le (a * x) (b * x) c d"
1.75 -        by (simp add: mult_ac)
1.76 -      finally show ?thesis .
1.77 -    qed
1.78 -  } note le_factor = this
1.79 -
1.80 -  let ?D = "b * d" and ?D' = "b' * d'"
1.81 -  from neq have D: "?D \<noteq> 0" by simp
1.82 -  from neq have "?D' \<noteq> 0" by simp
1.83 -  hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
1.84 -    by (rule le_factor)
1.85 -  also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
1.86 -    by (simp add: mult_ac)
1.87 -  also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
1.88 -    by (simp only: eq1 eq2)
1.89 -  also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
1.90 -    by (simp add: mult_ac)
1.91 -  also from D have "... = ?le a' b' c' d'"
1.92 -    by (rule le_factor [symmetric])
1.93 -  finally show "?le a b c d = ?le a' b' c' d'" .
1.94 +instance proof
1.95 +  fix a b c :: rat
1.96 +  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
1.97 +    by (rule abs_rat_def)
1.98 +  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
1.99 +    unfolding less_eq_rat_def less_rat_def
1.100 +    by (auto, drule (1) positive_add, simp_all add: positive_zero)
1.101 +  show "a \<le> a"
1.102 +    unfolding less_eq_rat_def by simp
1.103 +  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
1.104 +    unfolding less_eq_rat_def less_rat_def
1.105 +    by (auto, drule (1) positive_add, simp add: algebra_simps)
1.106 +  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
1.107 +    unfolding less_eq_rat_def less_rat_def
1.108 +    by (auto, drule (1) positive_add, simp add: positive_zero)
1.109 +  show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
1.110 +    unfolding less_eq_rat_def less_rat_def by (auto simp: diff_minus)
1.111 +  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
1.112 +    by (rule sgn_rat_def)
1.113 +  show "a \<le> b \<or> b \<le> a"
1.114 +    unfolding less_eq_rat_def less_rat_def
1.115 +    by (auto dest!: positive_minus)
1.116 +  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
1.117 +    unfolding less_rat_def
1.118 +    by (drule (1) positive_mult, simp add: algebra_simps)
1.119  qed
1.120
1.121 -lemma le_rat [simp]:
1.122 -  assumes "b \<noteq> 0" and "d \<noteq> 0"
1.123 -  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
1.124 -  using assms by transfer simp
1.125 +end
1.126 +
1.127 +instantiation rat :: distrib_lattice
1.128 +begin
1.129 +
1.130 +definition
1.131 +  "(inf :: rat \<Rightarrow> rat \<Rightarrow> rat) = min"
1.132
1.133  definition
1.134 -  less_rat_def: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
1.135 +  "(sup :: rat \<Rightarrow> rat \<Rightarrow> rat) = max"
1.136 +
1.137 +instance proof
1.138 +qed (auto simp add: inf_rat_def sup_rat_def min_max.sup_inf_distrib1)
1.139 +
1.140 +end
1.141 +
1.142 +lemma positive_rat: "positive (Fract a b) \<longleftrightarrow> 0 < a * b"
1.143 +  by transfer simp
1.144
1.145  lemma less_rat [simp]:
1.146    assumes "b \<noteq> 0" and "d \<noteq> 0"
1.147    shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
1.148 -  using assms by (simp add: less_rat_def eq_rat order_less_le)
1.149 +  using assms unfolding less_rat_def
1.150 +  by (simp add: positive_rat algebra_simps)
1.151
1.152 -instance proof
1.153 -  fix q r s :: rat
1.154 -  {
1.155 -    assume "q \<le> r" and "r \<le> s"
1.156 -    then show "q \<le> s"
1.157 -    proof (induct q, induct r, induct s)
1.158 -      fix a b c d e f :: int
1.159 -      assume neq: "b > 0"  "d > 0"  "f > 0"
1.160 -      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
1.161 -      show "Fract a b \<le> Fract e f"
1.162 -      proof -
1.163 -        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
1.164 -          by (auto simp add: zero_less_mult_iff linorder_neq_iff)
1.165 -        have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
1.166 -        proof -
1.167 -          from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
1.168 -            by simp
1.169 -          with ff show ?thesis by (simp add: mult_le_cancel_right)
1.170 -        qed
1.171 -        also have "... = (c * f) * (d * f) * (b * b)" by algebra
1.172 -        also have "... \<le> (e * d) * (d * f) * (b * b)"
1.173 -        proof -
1.174 -          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
1.175 -            by simp
1.176 -          with bb show ?thesis by (simp add: mult_le_cancel_right)
1.177 -        qed
1.178 -        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
1.179 -          by (simp only: mult_ac)
1.180 -        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
1.181 -          by (simp add: mult_le_cancel_right)
1.182 -        with neq show ?thesis by simp
1.183 -      qed
1.184 -    qed
1.185 -  next
1.186 -    assume "q \<le> r" and "r \<le> q"
1.187 -    then show "q = r"
1.188 -    proof (induct q, induct r)
1.189 -      fix a b c d :: int
1.190 -      assume neq: "b > 0"  "d > 0"
1.191 -      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
1.192 -      show "Fract a b = Fract c d"
1.193 -      proof -
1.194 -        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
1.195 -          by simp
1.196 -        also have "... \<le> (a * d) * (b * d)"
1.197 -        proof -
1.198 -          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
1.199 -            by simp
1.200 -          thus ?thesis by (simp only: mult_ac)
1.201 -        qed
1.202 -        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
1.203 -        moreover from neq have "b * d \<noteq> 0" by simp
1.204 -        ultimately have "a * d = c * b" by simp
1.205 -        with neq show ?thesis by (simp add: eq_rat)
1.206 -      qed
1.207 -    qed
1.208 -  next
1.209 -    show "q \<le> q"
1.210 -      by (induct q) simp
1.211 -    show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
1.212 -      by (induct q, induct r) (auto simp add: le_less mult_commute)
1.213 -    show "q \<le> r \<or> r \<le> q"
1.214 -      by (induct q, induct r)
1.215 -         (simp add: mult_commute, rule linorder_linear)
1.216 -  }
1.217 -qed
1.218 -
1.219 -end
1.220 -
1.221 -instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"
1.222 -begin
1.223 -
1.224 -definition
1.225 -  abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"
1.226 +lemma le_rat [simp]:
1.227 +  assumes "b \<noteq> 0" and "d \<noteq> 0"
1.228 +  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
1.229 +  using assms unfolding le_less by (simp add: eq_rat)
1.230
1.231  lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
1.232    by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
1.233
1.234 -definition
1.235 -  sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
1.236 -
1.237  lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"
1.238    unfolding Fract_of_int_eq
1.239    by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)
1.240      (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)
1.241
1.242 -definition
1.243 -  "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"
1.244 -
1.245 -definition
1.246 -  "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"
1.247 -
1.248 -instance by intro_classes
1.249 -  (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)
1.250 -
1.251 -end
1.252 -
1.253 -instance rat :: linordered_field_inverse_zero
1.254 -proof
1.255 -  fix q r s :: rat
1.256 -  show "q \<le> r ==> s + q \<le> s + r"
1.257 -  proof (induct q, induct r, induct s)
1.258 -    fix a b c d e f :: int
1.259 -    assume neq: "b > 0"  "d > 0"  "f > 0"
1.260 -    assume le: "Fract a b \<le> Fract c d"
1.261 -    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
1.262 -    proof -
1.263 -      let ?F = "f * f" from neq have F: "0 < ?F"
1.264 -        by (auto simp add: zero_less_mult_iff)
1.265 -      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
1.266 -        by simp
1.267 -      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
1.268 -        by (simp add: mult_le_cancel_right)
1.269 -      with neq show ?thesis by (simp add: mult_ac int_distrib)
1.270 -    qed
1.271 -  qed
1.272 -  show "q < r ==> 0 < s ==> s * q < s * r"
1.273 -  proof (induct q, induct r, induct s)
1.274 -    fix a b c d e f :: int
1.275 -    assume neq: "b > 0"  "d > 0"  "f > 0"
1.276 -    assume le: "Fract a b < Fract c d"
1.277 -    assume gt: "0 < Fract e f"
1.278 -    show "Fract e f * Fract a b < Fract e f * Fract c d"
1.279 -    proof -
1.280 -      let ?E = "e * f" and ?F = "f * f"
1.281 -      from neq gt have "0 < ?E"
1.282 -        by (auto simp add: Zero_rat_def order_less_le eq_rat)
1.283 -      moreover from neq have "0 < ?F"
1.284 -        by (auto simp add: zero_less_mult_iff)
1.285 -      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
1.286 -        by simp
1.287 -      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
1.288 -        by (simp add: mult_less_cancel_right)
1.289 -      with neq show ?thesis
1.290 -        by (simp add: mult_ac)
1.291 -    qed
1.292 -  qed
1.293 -qed auto
1.294 -
1.295  lemma Rat_induct_pos [case_names Fract, induct type: rat]:
1.296    assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"
1.297    shows "P q"
1.298 @@ -1194,13 +1115,13 @@
1.299  by simp
1.300
1.301
1.302 -hide_const (open) normalize
1.303 +hide_const (open) normalize positive
1.304
1.305  lemmas [transfer_rule del] =
1.306    rat.All_transfer rat.Ex_transfer rat.rel_eq_transfer forall_rat_transfer
1.307    Fract.transfer zero_rat.transfer one_rat.transfer plus_rat.transfer
1.308    uminus_rat.transfer times_rat.transfer inverse_rat.transfer
1.309 -  less_eq_rat.transfer of_rat.transfer
1.310 +  positive.transfer of_rat.transfer
1.311
1.312  text {* De-register @{text "rat"} as a quotient type: *}
1.313
```