author hoelzl Mon Nov 14 18:36:31 2011 +0100 (2011-11-14) changeset 45495 c55a07526dbe parent 45494 e828ccc5c110 child 45496 5c0444d2abfe child 45500 e2e27909bb66
cleaned up float theories; removed duplicate definitions and theorems
 src/HOL/Library/Float.thy file | annotate | diff | revisions src/HOL/Matrix/ComputeFloat.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Library/Float.thy	Mon Nov 14 12:28:34 2011 +0100
1.2 +++ b/src/HOL/Library/Float.thy	Mon Nov 14 18:36:31 2011 +0100
1.3 @@ -72,78 +72,19 @@
1.4  lemma pow2_1[simp]: "pow2 1 = 2" by simp
1.5  lemma pow2_neg: "pow2 x = inverse (pow2 (-x))" by simp
1.6
1.7 -declare pow2_def[simp del]
1.8 +lemma pow2_powr: "pow2 a = 2 powr a"
1.9 +  by (simp add: powr_realpow[symmetric] powr_minus)
1.10
1.11 -lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
1.12 -proof -
1.13 -  have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
1.14 -  have g: "! a b. a - -1 = a + (1::int)" by arith
1.15 -  have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
1.16 -    apply (auto, induct_tac n)
1.17 -    apply (simp_all add: pow2_def)
1.18 -    apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
1.19 -    by (auto simp add: h)
1.20 -  show ?thesis
1.21 -  proof (induct a)
1.22 -    case (nonneg n)
1.23 -    from pos show ?case by (simp add: algebra_simps)
1.24 -  next
1.25 -    case (neg n)
1.26 -    show ?case
1.27 -      apply (auto)
1.28 -      apply (subst pow2_neg[of "- int n"])
1.29 -      apply (subst pow2_neg[of "-1 - int n"])
1.30 -      apply (auto simp add: g pos)
1.31 -      done
1.32 -  qed
1.33 -qed
1.34 +declare pow2_def[simp del]
1.35
1.36  lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
1.37 -proof (induct b)
1.38 -  case (nonneg n)
1.39 -  show ?case
1.40 -  proof (induct n)
1.41 -    case 0
1.42 -    show ?case by simp
1.43 -  next
1.44 -    case (Suc m)
1.46 -  qed
1.47 -next
1.48 -  case (neg n)
1.49 -  show ?case
1.50 -  proof (induct n)
1.51 -    case 0
1.52 -    show ?case
1.53 -      apply (auto)
1.54 -      apply (subst pow2_neg[of "a + -1"])
1.55 -      apply (subst pow2_neg[of "-1"])
1.56 -      apply (simp)
1.57 -      apply (insert pow2_add1[of "-a"])
1.58 -      apply (simp add: algebra_simps)
1.59 -      apply (subst pow2_neg[of "-a"])
1.60 -      apply (simp)
1.61 -      done
1.62 -  next
1.63 -    case (Suc m)
1.64 -    have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
1.65 -    have b: "int m - -2 = 1 + (int m + 1)" by arith
1.66 -    show ?case
1.67 -      apply (auto)
1.68 -      apply (subst pow2_neg[of "a + (-2 - int m)"])
1.69 -      apply (subst pow2_neg[of "-2 - int m"])
1.70 -      apply (auto simp add: algebra_simps)
1.71 -      apply (subst a)
1.72 -      apply (subst b)
1.73 -      apply (simp only: pow2_add1)
1.74 -      apply (subst pow2_neg[of "int m - a + 1"])
1.75 -      apply (subst pow2_neg[of "int m + 1"])
1.76 -      apply auto
1.77 -      apply (insert Suc)
1.78 -      apply (auto simp add: algebra_simps)
1.79 -      done
1.80 -  qed
1.81 -qed
1.83 +
1.84 +lemma pow2_diff: "pow2 (a - b) = pow2 a / pow2 b"
1.85 +  by (simp add: pow2_powr powr_divide2)
1.86 +
1.87 +lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
1.89
1.90  lemma float_components[simp]: "Float (mantissa f) (scale f) = f" by (cases f) auto
1.91
1.92 @@ -185,23 +126,7 @@
1.93
1.94  lemma zero_less_pow2[simp]:
1.95    "0 < pow2 x"
1.96 -proof -
1.97 -  {
1.98 -    fix y
1.99 -    have "0 <= y \<Longrightarrow> 0 < pow2 y"
1.100 -      apply (induct y)
1.101 -      apply (induct_tac n)
1.103 -      done
1.104 -  }
1.105 -  note helper=this
1.106 -  show ?thesis
1.107 -    apply (case_tac "0 <= x")
1.108 -    apply (simp add: helper)
1.109 -    apply (subst pow2_neg)
1.110 -    apply (simp add: helper)
1.111 -    done
1.112 -qed
1.113 +  by (simp add: pow2_powr)
1.114
1.115  lemma normfloat_imp_odd_or_zero: "normfloat f = Float a b \<Longrightarrow> odd a \<or> (a = 0 \<and> b = 0)"
1.116  proof (induct f rule: normfloat.induct)
1.117 @@ -245,46 +170,19 @@
1.118    and floateq: "real (Float a b) = real (Float a' b')"
1.119    shows "b \<le> b'"
1.120  proof -
1.121 +  from odd have "a' \<noteq> 0" by auto
1.122 +  with floateq have a': "real a' = real a * pow2 (b - b')"
1.123 +    by (simp add: pow2_diff field_simps)
1.124 +
1.125    {
1.126      assume bcmp: "b > b'"
1.127 -    from floateq have eq: "real a * pow2 b = real a' * pow2 b'" by simp
1.128 -    {
1.129 -      fix x y z :: real
1.130 -      assume "y \<noteq> 0"
1.131 -      then have "(x * inverse y = z) = (x = z * y)"
1.132 -        by auto
1.133 -    }
1.134 -    note inverse = this
1.135 -    have eq': "real a * (pow2 (b - b')) = real a'"
1.136 -      apply (subst diff_int_def)
1.138 -      apply (subst pow2_neg[where x = "-b'"])
1.139 -      apply simp
1.140 -      apply (subst mult_assoc[symmetric])
1.141 -      apply (subst inverse)
1.142 -      apply (simp_all add: eq)
1.143 -      done
1.144 -    have "\<exists> z > 0. pow2 (b-b') = 2^z"
1.145 -      apply (rule exI[where x="nat (b - b')"])
1.146 -      apply (auto)
1.147 -      apply (insert bcmp)
1.148 -      apply simp
1.149 -      apply (subst pow2_int[symmetric])
1.150 -      apply auto
1.151 -      done
1.152 -    then obtain z where z: "z > 0 \<and> pow2 (b-b') = 2^z" by auto
1.153 -    with eq' have "real a * 2^z = real a'"
1.154 -      by auto
1.155 -    then have "real a * real ((2::int)^z) = real a'"
1.156 -      by auto
1.157 -    then have "real (a * 2^z) = real a'"
1.158 -      apply (subst real_of_int_mult)
1.159 -      apply simp
1.160 -      done
1.161 -    then have a'_rep: "a * 2^z = a'" by arith
1.162 -    then have "a' = a*2^z" by simp
1.163 -    with z have "even a'" by simp
1.164 -    with odd have False by auto
1.165 +    then have "\<exists>c::nat. b - b' = int c + 1"
1.166 +      by arith
1.167 +    then guess c ..
1.168 +    with a' have "real a' = real (a * 2^c * 2)"
1.170 +    with odd have False
1.171 +      unfolding real_of_int_inject by simp
1.172    }
1.173    then show ?thesis by arith
1.174  qed
```
```     2.1 --- a/src/HOL/Matrix/ComputeFloat.thy	Mon Nov 14 12:28:34 2011 +0100
2.2 +++ b/src/HOL/Matrix/ComputeFloat.thy	Mon Nov 14 18:36:31 2011 +0100
2.3 @@ -9,94 +9,6 @@
2.4  uses "~~/src/Tools/float.ML" ("~~/src/HOL/Tools/float_arith.ML")
2.5  begin
2.6
2.7 -definition pow2 :: "int \<Rightarrow> real"
2.8 -  where "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"
2.9 -
2.10 -definition float :: "int * int \<Rightarrow> real"
2.11 -  where "float x = real (fst x) * pow2 (snd x)"
2.12 -
2.13 -lemma pow2_0[simp]: "pow2 0 = 1"
2.15 -
2.16 -lemma pow2_1[simp]: "pow2 1 = 2"
2.18 -
2.19 -lemma pow2_neg: "pow2 x = inverse (pow2 (-x))"
2.21 -
2.22 -lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
2.23 -proof -
2.24 -  have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
2.25 -  have g: "! a b. a - -1 = a + (1::int)" by arith
2.26 -  have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
2.27 -    apply (auto, induct_tac n)
2.28 -    apply (simp_all add: pow2_def)
2.29 -    apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
2.30 -    by (auto simp add: h)
2.31 -  show ?thesis
2.32 -  proof (induct a)
2.33 -    case (nonneg n)
2.34 -    from pos show ?case by (simp add: algebra_simps)
2.35 -  next
2.36 -    case (neg n)
2.37 -    show ?case
2.38 -      apply (auto)
2.39 -      apply (subst pow2_neg[of "- int n"])
2.40 -      apply (subst pow2_neg[of "-1 - int n"])
2.41 -      apply (auto simp add: g pos)
2.42 -      done
2.43 -  qed
2.44 -qed
2.45 -
2.46 -lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
2.47 -proof (induct b)
2.48 -  case (nonneg n)
2.49 -  show ?case
2.50 -  proof (induct n)
2.51 -    case 0
2.52 -    show ?case by simp
2.53 -  next
2.54 -    case (Suc m)
2.55 -    show ?case by (auto simp add: algebra_simps pow2_add1 nonneg Suc)
2.56 -  qed
2.57 -next
2.58 -  case (neg n)
2.59 -  show ?case
2.60 -  proof (induct n)
2.61 -    case 0
2.62 -    show ?case
2.63 -      apply (auto)
2.64 -      apply (subst pow2_neg[of "a + -1"])
2.65 -      apply (subst pow2_neg[of "-1"])
2.66 -      apply (simp)
2.67 -      apply (insert pow2_add1[of "-a"])
2.68 -      apply (simp add: algebra_simps)
2.69 -      apply (subst pow2_neg[of "-a"])
2.70 -      apply (simp)
2.71 -      done
2.72 -    case (Suc m)
2.73 -    have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
2.74 -    have b: "int m - -2 = 1 + (int m + 1)" by arith
2.75 -    show ?case
2.76 -      apply (auto)
2.77 -      apply (subst pow2_neg[of "a + (-2 - int m)"])
2.78 -      apply (subst pow2_neg[of "-2 - int m"])
2.79 -      apply (auto simp add: algebra_simps)
2.80 -      apply (subst a)
2.81 -      apply (subst b)
2.82 -      apply (simp only: pow2_add1)
2.83 -      apply (subst pow2_neg[of "int m - a + 1"])
2.84 -      apply (subst pow2_neg[of "int m + 1"])
2.85 -      apply auto
2.86 -      apply (insert Suc)
2.87 -      apply (auto simp add: algebra_simps)
2.88 -      done
2.89 -  qed
2.90 -qed
2.91 -
2.92 -lemma "float (a, e) + float (b, e) = float (a + b, e)"
2.93 -by (simp add: float_def algebra_simps)
2.94 -
2.95  definition int_of_real :: "real \<Rightarrow> int"
2.96    where "int_of_real x = (SOME y. real y = x)"
2.97
2.98 @@ -104,16 +16,7 @@
2.99    where "real_is_int x = (EX (u::int). x = real u)"
2.100
2.101  lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
2.102 -by (auto simp add: real_is_int_def int_of_real_def)
2.103 -
2.104 -lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \<Longrightarrow> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)"
2.106 -
2.107 -lemma pow2_int: "pow2 (int c) = 2^c"
2.109 -
2.110 -lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)"
2.112 +  by (auto simp add: real_is_int_def int_of_real_def)
2.113
2.114  lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
2.115  by (auto simp add: real_is_int_def int_of_real_def)
2.116 @@ -146,18 +49,9 @@
2.117  lemma int_of_real_mult:
2.118    assumes "real_is_int a" "real_is_int b"
2.119    shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
2.120 -proof -
2.121 -  from assms have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)
2.122 -  from assms have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)
2.123 -  from a obtain a'::int where a':"a = real a'" by auto
2.124 -  from b obtain b'::int where b':"b = real b'" by auto
2.125 -  have r: "real a' * real b' = real (a' * b')" by auto
2.126 -  show ?thesis
2.127 -    apply (simp add: a' b')
2.128 -    apply (subst r)
2.129 -    apply (simp only: int_of_real_real)
2.130 -    done
2.131 -qed
2.132 +  using assms
2.133 +  by (auto simp add: real_is_int_def real_of_int_mult[symmetric]
2.134 +           simp del: real_of_int_mult)
2.135
2.136  lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"
2.137  apply (subst real_is_int_def2)
2.138 @@ -182,47 +76,7 @@
2.139  qed
2.140
2.141  lemma real_is_int_number_of[simp]: "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
2.142 -proof -
2.143 -  have neg1: "real_is_int (-1::real)"
2.144 -  proof -
2.145 -    have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
2.146 -    also have "\<dots> = True" by (simp only: real_is_int_real)
2.147 -    ultimately show ?thesis by auto
2.148 -  qed
2.149 -
2.150 -  {
2.151 -    fix x :: int
2.152 -    have "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
2.153 -      unfolding number_of_eq
2.154 -      apply (induct x)
2.155 -      apply (induct_tac n)
2.156 -      apply (simp)
2.157 -      apply (simp)
2.158 -      apply (induct_tac n)
2.159 -      apply (simp add: neg1)
2.160 -    proof -
2.161 -      fix n :: nat
2.162 -      assume rn: "(real_is_int (of_int (- (int (Suc n)))))"
2.163 -      have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp
2.164 -      show "real_is_int (of_int (- (int (Suc (Suc n)))))"
2.165 -        apply (simp only: s of_int_add)
2.167 -        apply (simp add: neg1)
2.168 -        apply (simp only: rn)
2.169 -        done
2.170 -    qed
2.171 -  }
2.172 -  note Abs_Bin = this
2.173 -  {
2.174 -    fix x :: int
2.175 -    have "? u. x = u"
2.176 -      apply (rule exI[where x = "x"])
2.177 -      apply (simp)
2.178 -      done
2.179 -  }
2.180 -  then obtain u::int where "x = u" by auto
2.181 -  with Abs_Bin show ?thesis by auto
2.182 -qed
2.183 +  by (auto simp: real_is_int_def intro!: exI[of _ "number_of x"])
2.184
2.185  lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"
2.187 @@ -234,30 +88,9 @@
2.188  qed
2.189
2.190  lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b"
2.191 -proof -
2.192 -  have "real_is_int (number_of b)" by simp
2.193 -  then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep)
2.194 -  then obtain u::int where u:"number_of b = real u" by auto
2.195 -  have "number_of b = real ((number_of b)::int)"
2.196 -    by (simp add: number_of_eq real_of_int_def)
2.197 -  have ub: "number_of b = real ((number_of b)::int)"
2.198 -    by (simp add: number_of_eq real_of_int_def)
2.199 -  from uu u ub have unb: "u = number_of b"
2.200 -    by blast
2.201 -  have "int_of_real (number_of b) = u" by (simp add: u)
2.202 -  with unb show ?thesis by simp
2.203 -qed
2.204 -
2.205 -lemma float_transfer_even: "even a \<Longrightarrow> float (a, b) = float (a div 2, b+1)"
2.206 -  apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified])
2.207 -  apply (simp_all add: pow2_def even_def real_is_int_def algebra_simps)
2.208 -  apply (auto)
2.209 -proof -
2.210 -  fix q::int
2.211 -  have a:"b - (-1\<Colon>int) = (1\<Colon>int) + b" by arith
2.212 -  show "(float (q, (b - (-1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))"
2.213 -    by (simp add: a)
2.214 -qed
2.215 +  unfolding int_of_real_def
2.216 +  by (intro some_equality)
2.217 +     (auto simp add: real_of_int_inject[symmetric] simp del: real_of_int_inject)
2.218
2.219  lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
2.220  by (rule zdiv_int)
2.221 @@ -268,163 +101,6 @@
2.222  lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
2.223  by arith
2.224
2.225 -function norm_float :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
2.226 -  "norm_float a b = (if a \<noteq> 0 \<and> even a then norm_float (a div 2) (b + 1)
2.227 -    else if a = 0 then (0, 0) else (a, b))"
2.228 -by auto
2.229 -
2.230 -termination by (relation "measure (nat o abs o fst)")
2.231 -  (auto intro: abs_div_2_less)
2.232 -
2.233 -lemma norm_float: "float x = float (split norm_float x)"
2.234 -proof -
2.235 -  {
2.236 -    fix a b :: int
2.237 -    have norm_float_pair: "float (a, b) = float (norm_float a b)"
2.238 -    proof (induct a b rule: norm_float.induct)
2.239 -      case (1 u v)
2.240 -      show ?case
2.241 -      proof cases
2.242 -        assume u: "u \<noteq> 0 \<and> even u"
2.243 -        with 1 have ind: "float (u div 2, v + 1) = float (norm_float (u div 2) (v + 1))" by auto
2.244 -        with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even)
2.245 -        then show ?thesis
2.246 -          apply (subst norm_float.simps)
2.247 -          apply (simp add: ind)
2.248 -          done
2.249 -      next
2.250 -        assume nu: "~(u \<noteq> 0 \<and> even u)"
2.251 -        show ?thesis
2.252 -          by (simp add: nu float_def)
2.253 -      qed
2.254 -    qed
2.255 -  }
2.256 -  note helper = this
2.257 -  have "? a b. x = (a,b)" by auto
2.258 -  then obtain a b where "x = (a, b)" by blast
2.259 -  then show ?thesis by (simp add: helper)
2.260 -qed
2.261 -
2.262 -lemma float_add_l0: "float (0, e) + x = x"
2.263 -  by (simp add: float_def)
2.264 -
2.265 -lemma float_add_r0: "x + float (0, e) = x"
2.266 -  by (simp add: float_def)
2.267 -
2.269 -  "float (a1, e1) + float (a2, e2) =
2.270 -  (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1)
2.271 -  else float (a1*2^(nat (e1-e2))+a2, e2))"
2.272 -  apply (simp add: float_def algebra_simps)
2.274 -  done
2.275 -
2.277 -  "(x + float (y1, e1)) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x"
2.278 -  by simp
2.279 -
2.281 -  "(float (y1, e1) + x) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x"
2.282 -  by simp
2.283 -
2.285 -  "float (y1, e1) + (x + float (y2, e2)) = (float (y1, e1) + float (y2, e2)) + x"
2.286 -  by simp
2.287 -
2.289 -  "float (y1, e1) + (float (y2, e2) + x) = (float (y1, e1) + float (y2, e2)) + x"
2.290 -  by simp
2.291 -
2.292 -lemma float_mult_l0: "float (0, e) * x = float (0, 0)"
2.293 -  by (simp add: float_def)
2.294 -
2.295 -lemma float_mult_r0: "x * float (0, e) = float (0, 0)"
2.296 -  by (simp add: float_def)
2.297 -
2.298 -definition lbound :: "real \<Rightarrow> real"
2.299 -  where "lbound x = min 0 x"
2.300 -
2.301 -definition ubound :: "real \<Rightarrow> real"
2.302 -  where "ubound x = max 0 x"
2.303 -
2.304 -lemma lbound: "lbound x \<le> x"
2.305 -  by (simp add: lbound_def)
2.306 -
2.307 -lemma ubound: "x \<le> ubound x"
2.308 -  by (simp add: ubound_def)
2.309 -
2.310 -lemma float_mult:
2.311 -  "float (a1, e1) * float (a2, e2) =
2.312 -  (float (a1 * a2, e1 + e2))"
2.314 -
2.315 -lemma float_minus:
2.316 -  "- (float (a,b)) = float (-a, b)"
2.317 -  by (simp add: float_def)
2.318 -
2.319 -lemma zero_less_pow2:
2.320 -  "0 < pow2 x"
2.321 -proof -
2.322 -  {
2.323 -    fix y
2.324 -    have "0 <= y \<Longrightarrow> 0 < pow2 y"
2.326 -  }
2.327 -  note helper=this
2.328 -  show ?thesis
2.329 -    apply (case_tac "0 <= x")
2.330 -    apply (simp add: helper)
2.331 -    apply (subst pow2_neg)
2.332 -    apply (simp add: helper)
2.333 -    done
2.334 -qed
2.335 -
2.336 -lemma zero_le_float:
2.337 -  "(0 <= float (a,b)) = (0 <= a)"
2.338 -  apply (auto simp add: float_def)
2.339 -  apply (auto simp add: zero_le_mult_iff zero_less_pow2)
2.340 -  apply (insert zero_less_pow2[of b])
2.341 -  apply (simp_all)
2.342 -  done
2.343 -
2.344 -lemma float_le_zero:
2.345 -  "(float (a,b) <= 0) = (a <= 0)"
2.346 -  apply (auto simp add: float_def)
2.347 -  apply (auto simp add: mult_le_0_iff)
2.348 -  apply (insert zero_less_pow2[of b])
2.349 -  apply auto
2.350 -  done
2.351 -
2.352 -lemma float_abs:
2.353 -  "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
2.354 -  apply (auto simp add: abs_if)
2.355 -  apply (simp_all add: zero_le_float[symmetric, of a b] float_minus)
2.356 -  done
2.357 -
2.358 -lemma float_zero:
2.359 -  "float (0, b) = 0"
2.360 -  by (simp add: float_def)
2.361 -
2.362 -lemma float_pprt:
2.363 -  "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"
2.364 -  by (auto simp add: zero_le_float float_le_zero float_zero)
2.365 -
2.366 -lemma pprt_lbound: "pprt (lbound x) = float (0, 0)"
2.367 -  apply (simp add: float_def)
2.368 -  apply (rule pprt_eq_0)
2.369 -  apply (simp add: lbound_def)
2.370 -  done
2.371 -
2.372 -lemma nprt_ubound: "nprt (ubound x) = float (0, 0)"
2.373 -  apply (simp add: float_def)
2.374 -  apply (rule nprt_eq_0)
2.375 -  apply (simp add: ubound_def)
2.376 -  done
2.377 -
2.378 -lemma float_nprt:
2.379 -  "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"
2.380 -  by (auto simp add: zero_le_float float_le_zero float_zero)
2.381 -
2.382  lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
2.383    by auto
2.384
2.385 @@ -549,6 +225,79 @@
2.386    zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
2.387    zpower_Pls zpower_Min
2.388
2.389 +definition float :: "(int \<times> int) \<Rightarrow> real" where
2.390 +  "float = (\<lambda>(a, b). real a * 2 powr real b)"
2.391 +
2.392 +lemma float_add_l0: "float (0, e) + x = x"
2.393 +  by (simp add: float_def)
2.394 +
2.395 +lemma float_add_r0: "x + float (0, e) = x"
2.396 +  by (simp add: float_def)
2.397 +
2.399 +  "float (a1, e1) + float (a2, e2) =
2.400 +  (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1) else float (a1*2^(nat (e1-e2))+a2, e2))"
2.401 +  by (simp add: float_def algebra_simps powr_realpow[symmetric] powr_divide2[symmetric])
2.402 +
2.403 +lemma float_mult_l0: "float (0, e) * x = float (0, 0)"
2.404 +  by (simp add: float_def)
2.405 +
2.406 +lemma float_mult_r0: "x * float (0, e) = float (0, 0)"
2.407 +  by (simp add: float_def)
2.408 +
2.409 +lemma float_mult:
2.410 +  "float (a1, e1) * float (a2, e2) = (float (a1 * a2, e1 + e2))"
2.412 +
2.413 +lemma float_minus:
2.414 +  "- (float (a,b)) = float (-a, b)"
2.415 +  by (simp add: float_def)
2.416 +
2.417 +lemma zero_le_float:
2.418 +  "(0 <= float (a,b)) = (0 <= a)"
2.419 +  using powr_gt_zero[of 2 "real b", arith]
2.420 +  by (simp add: float_def zero_le_mult_iff)
2.421 +
2.422 +lemma float_le_zero:
2.423 +  "(float (a,b) <= 0) = (a <= 0)"
2.424 +  using powr_gt_zero[of 2 "real b", arith]
2.425 +  by (simp add: float_def mult_le_0_iff)
2.426 +
2.427 +lemma float_abs:
2.428 +  "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
2.429 +  using powr_gt_zero[of 2 "real b", arith]
2.430 +  by (simp add: float_def abs_if mult_less_0_iff)
2.431 +
2.432 +lemma float_zero:
2.433 +  "float (0, b) = 0"
2.434 +  by (simp add: float_def)
2.435 +
2.436 +lemma float_pprt:
2.437 +  "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"
2.438 +  by (auto simp add: zero_le_float float_le_zero float_zero)
2.439 +
2.440 +lemma float_nprt:
2.441 +  "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"
2.442 +  by (auto simp add: zero_le_float float_le_zero float_zero)
2.443 +
2.444 +definition lbound :: "real \<Rightarrow> real"
2.445 +  where "lbound x = min 0 x"
2.446 +
2.447 +definition ubound :: "real \<Rightarrow> real"
2.448 +  where "ubound x = max 0 x"
2.449 +
2.450 +lemma lbound: "lbound x \<le> x"
2.451 +  by (simp add: lbound_def)
2.452 +
2.453 +lemma ubound: "x \<le> ubound x"
2.454 +  by (simp add: ubound_def)
2.455 +
2.456 +lemma pprt_lbound: "pprt (lbound x) = float (0, 0)"
2.457 +  by (auto simp: float_def lbound_def)
2.458 +
2.459 +lemma nprt_ubound: "nprt (ubound x) = float (0, 0)"
2.460 +  by (auto simp: float_def ubound_def)
2.461 +