src/HOL/Real/Float.thy
changeset 19765 dfe940911617
parent 16890 c4e5afaba440
child 20217 25b068a99d2b
     1.1 --- a/src/HOL/Real/Float.thy	Fri Jun 02 20:12:59 2006 +0200
     1.2 +++ b/src/HOL/Real/Float.thy	Fri Jun 02 23:22:29 2006 +0200
     1.3 @@ -5,11 +5,11 @@
     1.4  
     1.5  theory Float imports Real begin
     1.6  
     1.7 -constdefs  
     1.8 +definition
     1.9    pow2 :: "int \<Rightarrow> real"
    1.10 -  "pow2 a == if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a))))" 
    1.11 +  "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"
    1.12    float :: "int * int \<Rightarrow> real"
    1.13 -  "float x == (real (fst x)) * (pow2 (snd x))"
    1.14 +  "float x = real (fst x) * pow2 (snd x)"
    1.15  
    1.16  lemma pow2_0[simp]: "pow2 0 = 1"
    1.17  by (simp add: pow2_def)
    1.18 @@ -20,7 +20,7 @@
    1.19  lemma pow2_neg: "pow2 x = inverse (pow2 (-x))"
    1.20  by (simp add: pow2_def)
    1.21  
    1.22 -lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)" 
    1.23 +lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
    1.24  proof -
    1.25    have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
    1.26    have g: "! a b. a - -1 = a + (1::int)" by arith
    1.27 @@ -30,7 +30,7 @@
    1.28      apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
    1.29      apply (auto simp add: h)
    1.30      apply arith
    1.31 -    done  
    1.32 +    done
    1.33    show ?thesis
    1.34    proof (induct a)
    1.35      case (1 n)
    1.36 @@ -43,12 +43,12 @@
    1.37        apply (subst pow2_neg[of "-1 - int n"])
    1.38        apply (auto simp add: g pos)
    1.39        done
    1.40 -  qed  
    1.41 +  qed
    1.42  qed
    1.43 -  
    1.44 +
    1.45  lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
    1.46  proof (induct b)
    1.47 -  case (1 n) 
    1.48 +  case (1 n)
    1.49    show ?case
    1.50    proof (induct n)
    1.51      case 0
    1.52 @@ -59,10 +59,10 @@
    1.53    qed
    1.54  next
    1.55    case (2 n)
    1.56 -  show ?case 
    1.57 +  show ?case
    1.58    proof (induct n)
    1.59      case 0
    1.60 -    show ?case 
    1.61 +    show ?case
    1.62        apply (auto)
    1.63        apply (subst pow2_neg[of "a + -1"])
    1.64        apply (subst pow2_neg[of "-1"])
    1.65 @@ -73,7 +73,7 @@
    1.66        apply (simp)
    1.67        done
    1.68      case (Suc m)
    1.69 -    have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith	
    1.70 +    have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
    1.71      have b: "int m - -2 = 1 + (int m + 1)" by arith
    1.72      show ?case
    1.73        apply (auto)
    1.74 @@ -92,14 +92,14 @@
    1.75    qed
    1.76  qed
    1.77  
    1.78 -lemma "float (a, e) + float (b, e) = float (a + b, e)"  
    1.79 +lemma "float (a, e) + float (b, e) = float (a + b, e)"
    1.80  by (simp add: float_def ring_eq_simps)
    1.81  
    1.82 -constdefs 
    1.83 +definition
    1.84    int_of_real :: "real \<Rightarrow> int"
    1.85 -  "int_of_real x == SOME y. real y = x"  
    1.86 +  "int_of_real x = (SOME y. real y = x)"
    1.87    real_is_int :: "real \<Rightarrow> bool"
    1.88 -  "real_is_int x == ? (u::int). x = real u" 
    1.89 +  "real_is_int x = (EX (u::int). x = real u)"
    1.90  
    1.91  lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
    1.92  by (auto simp add: real_is_int_def int_of_real_def)
    1.93 @@ -110,7 +110,7 @@
    1.94  lemma pow2_int: "pow2 (int c) = (2::real)^c"
    1.95  by (simp add: pow2_def)
    1.96  
    1.97 -lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)" 
    1.98 +lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)"
    1.99  by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric])
   1.100  
   1.101  lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
   1.102 @@ -141,7 +141,7 @@
   1.103  lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"
   1.104  by (auto simp add: real_is_int_def)
   1.105  
   1.106 -lemma int_of_real_mult: 
   1.107 +lemma int_of_real_mult:
   1.108    assumes "real_is_int a" "real_is_int b"
   1.109    shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
   1.110  proof -
   1.111 @@ -187,8 +187,8 @@
   1.112      also have "\<dots> = True" by (simp only: real_is_int_real)
   1.113      ultimately show ?thesis by auto
   1.114    qed
   1.115 -  
   1.116 -  { 
   1.117 +
   1.118 +  {
   1.119      fix x::int
   1.120      have "!! y. real_is_int ((number_of::bin\<Rightarrow>real) (Abs_Bin x))"
   1.121        apply (simp add: number_of_eq)
   1.122 @@ -205,11 +205,11 @@
   1.123        assume rn: "(real_is_int (of_int (- (int (Suc n)))))"
   1.124        have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp
   1.125        show "real_is_int (of_int (- (int (Suc (Suc n)))))"
   1.126 -	apply (simp only: s of_int_add)
   1.127 -	apply (rule real_is_int_add)
   1.128 -	apply (simp add: neg1)
   1.129 -	apply (simp only: rn)
   1.130 -	done
   1.131 +        apply (simp only: s of_int_add)
   1.132 +        apply (rule real_is_int_add)
   1.133 +        apply (simp add: neg1)
   1.134 +        apply (simp only: rn)
   1.135 +        done
   1.136      qed
   1.137    }
   1.138    note Abs_Bin = this
   1.139 @@ -228,7 +228,7 @@
   1.140  by (simp add: int_of_real_def)
   1.141  
   1.142  lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"
   1.143 -proof - 
   1.144 +proof -
   1.145    have 1: "(1::real) = real (1::int)" by auto
   1.146    show ?thesis by (simp only: 1 int_of_real_real)
   1.147  qed
   1.148 @@ -238,9 +238,9 @@
   1.149    have "real_is_int (number_of b)" by simp
   1.150    then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep)
   1.151    then obtain u::int where u:"number_of b = real u" by auto
   1.152 -  have "number_of b = real ((number_of b)::int)" 
   1.153 +  have "number_of b = real ((number_of b)::int)"
   1.154      by (simp add: number_of_eq real_of_int_def)
   1.155 -  have ub: "number_of b = real ((number_of b)::int)" 
   1.156 +  have ub: "number_of b = real ((number_of b)::int)"
   1.157      by (simp add: number_of_eq real_of_int_def)
   1.158    from uu u ub have unb: "u = number_of b"
   1.159      by blast
   1.160 @@ -255,10 +255,10 @@
   1.161  proof -
   1.162    fix q::int
   1.163    have a:"b - (-1\<Colon>int) = (1\<Colon>int) + b" by arith
   1.164 -  show "(float (q, (b - (-1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))" 
   1.165 +  show "(float (q, (b - (-1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))"
   1.166      by (simp add: a)
   1.167  qed
   1.168 -    
   1.169 +
   1.170  consts
   1.171    norm_float :: "int*int \<Rightarrow> int*int"
   1.172  
   1.173 @@ -285,7 +285,7 @@
   1.174  apply (auto)
   1.175  done
   1.176  
   1.177 -ML {* simp_depth_limit := 2 *} 
   1.178 +ML {* simp_depth_limit := 2 *}
   1.179  recdef norm_float "measure (% (a,b). nat (abs a))"
   1.180    "norm_float (a,b) = (if (a \<noteq> 0) & (even a) then norm_float (a div 2, b+1) else (if a=0 then (0,0) else (a,b)))"
   1.181  (hints simp: terminating_norm_float)
   1.182 @@ -294,23 +294,23 @@
   1.183  lemma norm_float: "float x = float (norm_float x)"
   1.184  proof -
   1.185    {
   1.186 -    fix a b :: int 
   1.187 -    have norm_float_pair: "float (a,b) = float (norm_float (a,b))" 
   1.188 +    fix a b :: int
   1.189 +    have norm_float_pair: "float (a,b) = float (norm_float (a,b))"
   1.190      proof (induct a b rule: norm_float.induct)
   1.191        case (1 u v)
   1.192 -      show ?case 
   1.193 +      show ?case
   1.194        proof cases
   1.195 -	assume u: "u \<noteq> 0 \<and> even u"
   1.196 -	with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2, v + 1))" by auto
   1.197 -	with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even) 
   1.198 -	then show ?thesis
   1.199 -	  apply (subst norm_float.simps)
   1.200 -	  apply (simp add: ind)
   1.201 -	  done
   1.202 +        assume u: "u \<noteq> 0 \<and> even u"
   1.203 +        with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2, v + 1))" by auto
   1.204 +        with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even)
   1.205 +        then show ?thesis
   1.206 +          apply (subst norm_float.simps)
   1.207 +          apply (simp add: ind)
   1.208 +          done
   1.209        next
   1.210 -	assume "~(u \<noteq> 0 \<and> even u)"
   1.211 -	then show ?thesis
   1.212 -	  by (simp add: prems float_def)
   1.213 +        assume "~(u \<noteq> 0 \<and> even u)"
   1.214 +        then show ?thesis
   1.215 +          by (simp add: prems float_def)
   1.216        qed
   1.217      qed
   1.218    }
   1.219 @@ -323,16 +323,16 @@
   1.220  lemma pow2_int: "pow2 (int n) = 2^n"
   1.221    by (simp add: pow2_def)
   1.222  
   1.223 -lemma float_add: 
   1.224 -  "float (a1, e1) + float (a2, e2) = 
   1.225 -  (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1) 
   1.226 +lemma float_add:
   1.227 +  "float (a1, e1) + float (a2, e2) =
   1.228 +  (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1)
   1.229    else float (a1*2^(nat (e1-e2))+a2, e2))"
   1.230    apply (simp add: float_def ring_eq_simps)
   1.231    apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric])
   1.232    done
   1.233  
   1.234  lemma float_mult:
   1.235 -  "float (a1, e1) * float (a2, e2) = 
   1.236 +  "float (a1, e1) * float (a2, e2) =
   1.237    (float (a1 * a2, e1 + e2))"
   1.238    by (simp add: float_def pow2_add)
   1.239  
   1.240 @@ -345,7 +345,7 @@
   1.241  proof -
   1.242    {
   1.243      fix y
   1.244 -    have "0 <= y \<Longrightarrow> 0 < pow2 y"    
   1.245 +    have "0 <= y \<Longrightarrow> 0 < pow2 y"
   1.246        by (induct y, induct_tac n, simp_all add: pow2_add)
   1.247    }
   1.248    note helper=this
   1.249 @@ -360,7 +360,7 @@
   1.250  lemma zero_le_float:
   1.251    "(0 <= float (a,b)) = (0 <= a)"
   1.252    apply (auto simp add: float_def)
   1.253 -  apply (auto simp add: zero_le_mult_iff zero_less_pow2) 
   1.254 +  apply (auto simp add: zero_le_mult_iff zero_less_pow2)
   1.255    apply (insert zero_less_pow2[of b])
   1.256    apply (simp_all)
   1.257    done
   1.258 @@ -393,7 +393,7 @@
   1.259  
   1.260  lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
   1.261    by auto
   1.262 -  
   1.263 +
   1.264  lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
   1.265    by simp
   1.266  
   1.267 @@ -451,19 +451,19 @@
   1.268  lemma not_true_eq_false: "(~ True) = False" by simp
   1.269  
   1.270  
   1.271 -lemmas binarith = 
   1.272 +lemmas binarith =
   1.273    Pls_0_eq Min_1_eq
   1.274 -  bin_pred_Pls bin_pred_Min bin_pred_1 bin_pred_0     
   1.275 +  bin_pred_Pls bin_pred_Min bin_pred_1 bin_pred_0
   1.276    bin_succ_Pls bin_succ_Min bin_succ_1 bin_succ_0
   1.277    bin_add_Pls bin_add_Min bin_add_BIT_0 bin_add_BIT_10
   1.278 -  bin_add_BIT_11 bin_minus_Pls bin_minus_Min bin_minus_1 
   1.279 -  bin_minus_0 bin_mult_Pls bin_mult_Min bin_mult_1 bin_mult_0 
   1.280 +  bin_add_BIT_11 bin_minus_Pls bin_minus_Min bin_minus_1
   1.281 +  bin_minus_0 bin_mult_Pls bin_mult_Min bin_mult_1 bin_mult_0
   1.282    bin_add_Pls_right bin_add_Min_right
   1.283  
   1.284  lemma int_eq_number_of_eq: "(((number_of v)::int)=(number_of w)) = iszero ((number_of (bin_add v (bin_minus w)))::int)"
   1.285    by simp
   1.286  
   1.287 -lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)" 
   1.288 +lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
   1.289    by (simp only: iszero_number_of_Pls)
   1.290  
   1.291  lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
   1.292 @@ -472,13 +472,13 @@
   1.293  lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
   1.294    by simp
   1.295  
   1.296 -lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)" 
   1.297 +lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)"
   1.298    by simp
   1.299  
   1.300  lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (bin_add x (bin_minus y)))::int)"
   1.301    by simp
   1.302  
   1.303 -lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))" 
   1.304 +lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
   1.305    by simp
   1.306  
   1.307  lemma int_neg_number_of_Min: "neg (-1::int)"
   1.308 @@ -490,9 +490,9 @@
   1.309  lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (bin_add y (bin_minus x)))::int))"
   1.310    by simp
   1.311  
   1.312 -lemmas intarithrel = 
   1.313 -  int_eq_number_of_eq 
   1.314 -  lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_0 
   1.315 +lemmas intarithrel =
   1.316 +  int_eq_number_of_eq
   1.317 +  lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_0
   1.318    lift_bool[OF int_iszero_number_of_1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min]
   1.319    int_neg_number_of_BIT int_le_number_of_eq
   1.320  
   1.321 @@ -512,8 +512,8 @@
   1.322  
   1.323  lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
   1.324  
   1.325 -lemmas powerarith = nat_number_of zpower_number_of_even 
   1.326 -  zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]   
   1.327 +lemmas powerarith = nat_number_of zpower_number_of_even
   1.328 +  zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
   1.329    zpower_Pls zpower_Min
   1.330  
   1.331  lemmas floatarith[simplified norm_0_1] = float_add float_mult float_minus float_abs zero_le_float float_pprt float_nprt
   1.332 @@ -522,4 +522,3 @@
   1.333  lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false
   1.334  
   1.335  end
   1.336 -