(*  Title: HOL/Real/Float.thy

     ID:    $Id$

     Author: Steven Obua

 *)

 header {* Floating Point Representation of the Reals *}

 theory Float

 imports Real Parity

 uses "~~/src/Tools/float.ML" ("float_arith.ML")

 begin

 definition

   pow2 :: "int \<Rightarrow> real" where

   "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"

 definition

   float :: "int * int \<Rightarrow> real" where

   "float x = real (fst x) * pow2 (snd x)"

 lemma pow2_0[simp]: "pow2 0 = 1"

 by (simp add: pow2_def)

 lemma pow2_1[simp]: "pow2 1 = 2"

 by (simp add: pow2_def)

 lemma pow2_neg: "pow2 x = inverse (pow2 (-x))"

 by (simp add: pow2_def)

 lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"

 proof -

   have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith

   have g: "! a b. a - -1 = a + (1::int)" by arith

   have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"

     apply (auto, induct_tac n)

     apply (simp_all add: pow2_def)

     apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])

     by (auto simp add: h)

   show ?thesis

   proof (induct a)

     case (1 n)

     from pos show ?case by (simp add: ring_simps)

   next

     case (2 n)

     show ?case

       apply (auto)

       apply (subst pow2_neg[of "- int n"])

       apply (subst pow2_neg[of "-1 - int n"])

       apply (auto simp add: g pos)

       done

   qed

 qed

 lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"

 proof (induct b)

   case (1 n)

   show ?case

   proof (induct n)

     case 0

     show ?case by simp

   next

     case (Suc m)

     show ?case by (auto simp add: ring_simps pow2_add1 prems)

   qed

 next

   case (2 n)

   show ?case

   proof (induct n)

     case 0

     show ?case

       apply (auto)

       apply (subst pow2_neg[of "a + -1"])

       apply (subst pow2_neg[of "-1"])

       apply (simp)

       apply (insert pow2_add1[of "-a"])

       apply (simp add: ring_simps)

       apply (subst pow2_neg[of "-a"])

       apply (simp)

       done

     case (Suc m)

     have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith

     have b: "int m - -2 = 1 + (int m + 1)" by arith

     show ?case

       apply (auto)

       apply (subst pow2_neg[of "a + (-2 - int m)"])

       apply (subst pow2_neg[of "-2 - int m"])

       apply (auto simp add: ring_simps)

       apply (subst a)

       apply (subst b)

       apply (simp only: pow2_add1)

       apply (subst pow2_neg[of "int m - a + 1"])

       apply (subst pow2_neg[of "int m + 1"])

       apply auto

       apply (insert prems)

       apply (auto simp add: ring_simps)

       done

   qed

 qed

 lemma "float (a, e) + float (b, e) = float (a + b, e)"

 by (simp add: float_def ring_simps)

 definition

   int_of_real :: "real \<Rightarrow> int" where

   "int_of_real x = (SOME y. real y = x)"

 definition

   real_is_int :: "real \<Rightarrow> bool" where

   "real_is_int x = (EX (u::int). x = real u)"

 lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"

 by (auto simp add: real_is_int_def int_of_real_def)

 lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \<Longrightarrow> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)"

 by (simp add: float_def real_is_int_def2 pow2_add[symmetric])

 lemma pow2_int: "pow2 (int c) = 2^c"

 by (simp add: pow2_def)

 lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)"

 by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric])

 lemma real_is_int_real[simp]: "real_is_int (real (x::int))"

 by (auto simp add: real_is_int_def int_of_real_def)

 lemma int_of_real_real[simp]: "int_of_real (real x) = x"

 by (simp add: int_of_real_def)

 lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x"

 by (auto simp add: int_of_real_def real_is_int_def)

 lemma real_is_int_add_int_of_real: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a+b)) = (int_of_real a) + (int_of_real b)"

 by (auto simp add: int_of_real_def real_is_int_def)

 lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)"

 apply (subst real_is_int_def2)

 apply (simp add: real_is_int_add_int_of_real real_int_of_real)

 done

 lemma int_of_real_sub: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a-b)) = (int_of_real a) - (int_of_real b)"

 by (auto simp add: int_of_real_def real_is_int_def)

 lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a-b)"

 apply (subst real_is_int_def2)

 apply (simp add: int_of_real_sub real_int_of_real)

 done

 lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"

 by (auto simp add: real_is_int_def)

 lemma int_of_real_mult:

   assumes "real_is_int a" "real_is_int b"

   shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"

 proof -

   from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)

   from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)

   from a obtain a'::int where a':"a = real a'" by auto

   from b obtain b'::int where b':"b = real b'" by auto

   have r: "real a' * real b' = real (a' * b')" by auto

   show ?thesis

     apply (simp add: a' b')

     apply (subst r)

     apply (simp only: int_of_real_real)

     done

 qed

 lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"

 apply (subst real_is_int_def2)

 apply (simp add: int_of_real_mult)

 done

 lemma real_is_int_0[simp]: "real_is_int (0::real)"

 by (simp add: real_is_int_def int_of_real_def)

 lemma real_is_int_1[simp]: "real_is_int (1::real)"

 proof -

   have "real_is_int (1::real) = real_is_int(real (1::int))" by auto

   also have "\<dots> = True" by (simp only: real_is_int_real)

   ultimately show ?thesis by auto

 qed

 lemma real_is_int_n1: "real_is_int (-1::real)"

 proof -

   have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto

   also have "\<dots> = True" by (simp only: real_is_int_real)

   ultimately show ?thesis by auto

 qed

 lemma real_is_int_number_of[simp]: "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"

 proof -

   have neg1: "real_is_int (-1::real)"

   proof -

     have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto

     also have "\<dots> = True" by (simp only: real_is_int_real)

     ultimately show ?thesis by auto

   qed

   {

     fix x :: int

     have "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"

       unfolding number_of_eq

       apply (induct x)

       apply (induct_tac n)

       apply (simp)

       apply (simp)

       apply (induct_tac n)

       apply (simp add: neg1)

     proof -

       fix n :: nat

       assume rn: "(real_is_int (of_int (- (int (Suc n)))))"

       have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp

       show "real_is_int (of_int (- (int (Suc (Suc n)))))"

         apply (simp only: s of_int_add)

         apply (rule real_is_int_add)

         apply (simp add: neg1)

         apply (simp only: rn)

         done

     qed

   }

   note Abs_Bin = this

   {

     fix x :: int

     have "? u. x = u"

       apply (rule exI[where x = "x"])

       apply (simp)

       done

   }

   then obtain u::int where "x = u" by auto

   with Abs_Bin show ?thesis by auto

 qed

 lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"

 by (simp add: int_of_real_def)

 lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"

 proof -

   have 1: "(1::real) = real (1::int)" by auto

   show ?thesis by (simp only: 1 int_of_real_real)

 qed

 lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b"

 proof -

   have "real_is_int (number_of b)" by simp

   then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep)

   then obtain u::int where u:"number_of b = real u" by auto

   have "number_of b = real ((number_of b)::int)"

     by (simp add: number_of_eq real_of_int_def)

   have ub: "number_of b = real ((number_of b)::int)"

     by (simp add: number_of_eq real_of_int_def)

   from uu u ub have unb: "u = number_of b"

     by blast

   have "int_of_real (number_of b) = u" by (simp add: u)

   with unb show ?thesis by simp

 qed

 lemma float_transfer_even: "even a \<Longrightarrow> float (a, b) = float (a div 2, b+1)"

   apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified])

   apply (simp_all add: pow2_def even_def real_is_int_def ring_simps)

   apply (auto)

 proof -

   fix q::int

   have a:"b - (-1\<Colon>int) = (1\<Colon>int) + b" by arith

   show "(float (q, (b - (-1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))"

     by (simp add: a)

 qed

 lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"

 by (rule zdiv_int)

 lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)"

 by (rule zmod_int)

 lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"

 by arith

 function norm_float :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

   "norm_float a b = (if a \<noteq> 0 \<and> even a then norm_float (a div 2) (b + 1)

     else if a = 0 then (0, 0) else (a, b))"

 by auto

 termination by (relation "measure (nat o abs o fst)")

   (auto intro: abs_div_2_less)

 lemma norm_float: "float x = float (split norm_float x)"

 proof -

   {

     fix a b :: int

     have norm_float_pair: "float (a, b) = float (norm_float a b)"

     proof (induct a b rule: norm_float.induct)

       case (1 u v)

       show ?case

       proof cases

         assume u: "u \<noteq> 0 \<and> even u"

         with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2) (v + 1))" by auto

         with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even)

         then show ?thesis

           apply (subst norm_float.simps)

           apply (simp add: ind)

           done

       next

         assume "~(u \<noteq> 0 \<and> even u)"

         then show ?thesis

           by (simp add: prems float_def)

       qed

     qed

   }

   note helper = this

   have "? a b. x = (a,b)" by auto

   then obtain a b where "x = (a, b)" by blast

   then show ?thesis by (simp add: helper)

 qed

 lemma float_add_l0: "float (0, e) + x = x"

   by (simp add: float_def)

 lemma float_add_r0: "x + float (0, e) = x"

   by (simp add: float_def)

 lemma float_add:

   "float (a1, e1) + float (a2, e2) =

   (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1)

   else float (a1*2^(nat (e1-e2))+a2, e2))"

   apply (simp add: float_def ring_simps)

   apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric])

   done

 lemma float_add_assoc1:

   "(x + float (y1, e1)) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x"

   by simp

 lemma float_add_assoc2:

   "(float (y1, e1) + x) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x"

   by simp

 lemma float_add_assoc3:

   "float (y1, e1) + (x + float (y2, e2)) = (float (y1, e1) + float (y2, e2)) + x"

   by simp

 lemma float_add_assoc4:

   "float (y1, e1) + (float (y2, e2) + x) = (float (y1, e1) + float (y2, e2)) + x"

   by simp

 lemma float_mult_l0: "float (0, e) * x = float (0, 0)"

   by (simp add: float_def)

 lemma float_mult_r0: "x * float (0, e) = float (0, 0)"

   by (simp add: float_def)

 definition 

   lbound :: "real \<Rightarrow> real"

 where

   "lbound x = min 0 x"

 definition

   ubound :: "real \<Rightarrow> real"

 where

   "ubound x = max 0 x"

 lemma lbound: "lbound x \<le> x"   

   by (simp add: lbound_def)

 lemma ubound: "x \<le> ubound x"

   by (simp add: ubound_def)

 lemma float_mult:

   "float (a1, e1) * float (a2, e2) =

   (float (a1 * a2, e1 + e2))"

   by (simp add: float_def pow2_add)

 lemma float_minus:

   "- (float (a,b)) = float (-a, b)"

   by (simp add: float_def)

 lemma zero_less_pow2:

   "0 < pow2 x"

 proof -

   {

     fix y

     have "0 <= y \<Longrightarrow> 0 < pow2 y"

       by (induct y, induct_tac n, simp_all add: pow2_add)

   }

   note helper=this

   show ?thesis

     apply (case_tac "0 <= x")

     apply (simp add: helper)

     apply (subst pow2_neg)

     apply (simp add: helper)

     done

 qed

 lemma zero_le_float:

   "(0 <= float (a,b)) = (0 <= a)"

   apply (auto simp add: float_def)

   apply (auto simp add: zero_le_mult_iff zero_less_pow2)

   apply (insert zero_less_pow2[of b])

   apply (simp_all)

   done

 lemma float_le_zero:

   "(float (a,b) <= 0) = (a <= 0)"

   apply (auto simp add: float_def)

   apply (auto simp add: mult_le_0_iff)

   apply (insert zero_less_pow2[of b])

   apply auto

   done

 lemma float_abs:

   "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"

   apply (auto simp add: abs_if)

   apply (simp_all add: zero_le_float[symmetric, of a b] float_minus)

   done

 lemma float_zero:

   "float (0, b) = 0"

   by (simp add: float_def)

 lemma float_pprt:

   "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"

   by (auto simp add: zero_le_float float_le_zero float_zero)

 lemma pprt_lbound: "pprt (lbound x) = float (0, 0)"

   apply (simp add: float_def)

   apply (rule pprt_eq_0)

   apply (simp add: lbound_def)

   done

 lemma nprt_ubound: "nprt (ubound x) = float (0, 0)"

   apply (simp add: float_def)

   apply (rule nprt_eq_0)

   apply (simp add: ubound_def)

   done

 lemma float_nprt:

   "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"

   by (auto simp add: zero_le_float float_le_zero float_zero)

 lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"

   by auto

 lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"

   by simp

 lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"

   by simp

 lemma mult_left_one: "1 * a = (a::'a::semiring_1)"

   by simp

 lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"

   by simp

 lemma int_pow_0: "(a::int)^(Numeral0) = 1"

   by simp

 lemma int_pow_1: "(a::int)^(Numeral1) = a"

   by simp

 lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"

   by simp

 lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"

   by simp

 lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"

   by simp

 lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"

   by simp

 lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"

   by simp

 lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"

 proof -

   have 1:"((-1)::nat) = 0"

     by simp

   show ?thesis by (simp add: 1)

 qed

 lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)"

   by simp

 lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')"

   by simp

 lemma lift_bool: "x \<Longrightarrow> x=True"

   by simp

 lemma nlift_bool: "~x \<Longrightarrow> x=False"

   by simp

 lemma not_false_eq_true: "(~ False) = True" by simp

 lemma not_true_eq_false: "(~ True) = False" by simp

 lemmas binarith =

   normalize_bin_simps

   pred_bin_simps succ_bin_simps

   add_bin_simps minus_bin_simps mult_bin_simps

 lemma int_eq_number_of_eq:

   "(((number_of v)::int)=(number_of w)) = iszero ((number_of (v + uminus w))::int)"

   by simp

 lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"

   by (simp only: iszero_number_of_Pls)

 lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"

   by simp

 lemma int_iszero_number_of_Bit0: "iszero ((number_of (Int.Bit0 w))::int) = iszero ((number_of w)::int)"

   by simp

 lemma int_iszero_number_of_Bit1: "\<not> iszero ((number_of (Int.Bit1 w))::int)"

   by simp

 lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)"

   by simp

 lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"

   by simp

 lemma int_neg_number_of_Min: "neg (-1::int)"

   by simp

 lemma int_neg_number_of_Bit0: "neg ((number_of (Int.Bit0 w))::int) = neg ((number_of w)::int)"

   by simp

 lemma int_neg_number_of_Bit1: "neg ((number_of (Int.Bit1 w))::int) = neg ((number_of w)::int)"

   by simp

 lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))"

   by simp

 lemmas intarithrel =

   int_eq_number_of_eq

   lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_Bit0

   lift_bool[OF int_iszero_number_of_Bit1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min]

   int_neg_number_of_Bit0 int_neg_number_of_Bit1 int_le_number_of_eq

 lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)"

   by simp

 lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (v + (uminus w))"

   by simp

 lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (v * w)"

   by simp

 lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)"

   by simp

 lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym

 lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of

 lemmas powerarith = nat_number_of zpower_number_of_even

   zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]

   zpower_Pls zpower_Min

 lemmas floatarith[simplified norm_0_1] = float_add float_add_l0 float_add_r0 float_mult float_mult_l0 float_mult_r0 

           float_minus float_abs zero_le_float float_pprt float_nprt pprt_lbound nprt_ubound

 (* for use with the compute oracle *)

 lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false

 use "float_arith.ML";

 end