obua@16782: (*  Title: HOL/Real/Float.thy
obua@16782:     ID:    $Id$
obua@16782:     Author: Steven Obua
obua@16782: *)
obua@16782: 
huffman@20717: header {* Floating Point Representation of the Reals *}
huffman@20717: 
haftmann@20485: theory Float
wenzelm@21256: imports Real Parity
wenzelm@20771: uses ("float.ML")
haftmann@20485: begin
obua@16782: 
wenzelm@19765: definition
obua@16782:   pow2 :: "int \<Rightarrow> real"
wenzelm@19765:   "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"
obua@16782:   float :: "int * int \<Rightarrow> real"
wenzelm@19765:   "float x = real (fst x) * pow2 (snd x)"
obua@16782: 
obua@16782: lemma pow2_0[simp]: "pow2 0 = 1"
obua@16782: by (simp add: pow2_def)
obua@16782: 
obua@16782: lemma pow2_1[simp]: "pow2 1 = 2"
obua@16782: by (simp add: pow2_def)
obua@16782: 
obua@16782: lemma pow2_neg: "pow2 x = inverse (pow2 (-x))"
obua@16782: by (simp add: pow2_def)
obua@16782: 
wenzelm@19765: lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
obua@16782: proof -
obua@16782:   have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
obua@16782:   have g: "! a b. a - -1 = a + (1::int)" by arith
obua@16782:   have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
obua@16782:     apply (auto, induct_tac n)
obua@16782:     apply (simp_all add: pow2_def)
obua@16782:     apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
webertj@20217:     by (auto simp add: h)
obua@16782:   show ?thesis
obua@16782:   proof (induct a)
obua@16782:     case (1 n)
obua@16782:     from pos show ?case by (simp add: ring_eq_simps)
obua@16782:   next
obua@16782:     case (2 n)
obua@16782:     show ?case
obua@16782:       apply (auto)
obua@16782:       apply (subst pow2_neg[of "- int n"])
obua@16782:       apply (subst pow2_neg[of "-1 - int n"])
obua@16782:       apply (auto simp add: g pos)
obua@16782:       done
wenzelm@19765:   qed
obua@16782: qed
wenzelm@19765: 
obua@16782: lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
obua@16782: proof (induct b)
wenzelm@19765:   case (1 n)
obua@16782:   show ?case
obua@16782:   proof (induct n)
obua@16782:     case 0
obua@16782:     show ?case by simp
obua@16782:   next
obua@16782:     case (Suc m)
obua@16782:     show ?case by (auto simp add: ring_eq_simps pow2_add1 prems)
obua@16782:   qed
obua@16782: next
obua@16782:   case (2 n)
wenzelm@19765:   show ?case
obua@16782:   proof (induct n)
obua@16782:     case 0
wenzelm@19765:     show ?case
obua@16782:       apply (auto)
obua@16782:       apply (subst pow2_neg[of "a + -1"])
obua@16782:       apply (subst pow2_neg[of "-1"])
obua@16782:       apply (simp)
obua@16782:       apply (insert pow2_add1[of "-a"])
obua@16782:       apply (simp add: ring_eq_simps)
obua@16782:       apply (subst pow2_neg[of "-a"])
obua@16782:       apply (simp)
obua@16782:       done
obua@16782:     case (Suc m)
wenzelm@19765:     have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
obua@16782:     have b: "int m - -2 = 1 + (int m + 1)" by arith
obua@16782:     show ?case
obua@16782:       apply (auto)
obua@16782:       apply (subst pow2_neg[of "a + (-2 - int m)"])
obua@16782:       apply (subst pow2_neg[of "-2 - int m"])
obua@16782:       apply (auto simp add: ring_eq_simps)
obua@16782:       apply (subst a)
obua@16782:       apply (subst b)
obua@16782:       apply (simp only: pow2_add1)
obua@16782:       apply (subst pow2_neg[of "int m - a + 1"])
obua@16782:       apply (subst pow2_neg[of "int m + 1"])
obua@16782:       apply auto
obua@16782:       apply (insert prems)
obua@16782:       apply (auto simp add: ring_eq_simps)
obua@16782:       done
obua@16782:   qed
obua@16782: qed
obua@16782: 
wenzelm@19765: lemma "float (a, e) + float (b, e) = float (a + b, e)"
obua@16782: by (simp add: float_def ring_eq_simps)
obua@16782: 
wenzelm@19765: definition
obua@16782:   int_of_real :: "real \<Rightarrow> int"
wenzelm@19765:   "int_of_real x = (SOME y. real y = x)"
obua@16782:   real_is_int :: "real \<Rightarrow> bool"
wenzelm@19765:   "real_is_int x = (EX (u::int). x = real u)"
obua@16782: 
obua@16782: lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
obua@16782: by (auto simp add: real_is_int_def int_of_real_def)
obua@16782: 
obua@16782: lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \<Longrightarrow> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)"
obua@16782: by (simp add: float_def real_is_int_def2 pow2_add[symmetric])
obua@16782: 
obua@16782: lemma pow2_int: "pow2 (int c) = (2::real)^c"
obua@16782: by (simp add: pow2_def)
obua@16782: 
wenzelm@19765: lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)"
obua@16782: by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric])
obua@16782: 
obua@16782: lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
obua@16782: by (auto simp add: real_is_int_def int_of_real_def)
obua@16782: 
obua@16782: lemma int_of_real_real[simp]: "int_of_real (real x) = x"
obua@16782: by (simp add: int_of_real_def)
obua@16782: 
obua@16782: lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x"
obua@16782: by (auto simp add: int_of_real_def real_is_int_def)
obua@16782: 
obua@16782: 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)"
obua@16782: by (auto simp add: int_of_real_def real_is_int_def)
obua@16782: 
obua@16782: lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)"
obua@16782: apply (subst real_is_int_def2)
obua@16782: apply (simp add: real_is_int_add_int_of_real real_int_of_real)
obua@16782: done
obua@16782: 
obua@16782: 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)"
obua@16782: by (auto simp add: int_of_real_def real_is_int_def)
obua@16782: 
obua@16782: lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a-b)"
obua@16782: apply (subst real_is_int_def2)
obua@16782: apply (simp add: int_of_real_sub real_int_of_real)
obua@16782: done
obua@16782: 
obua@16782: lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"
obua@16782: by (auto simp add: real_is_int_def)
obua@16782: 
wenzelm@19765: lemma int_of_real_mult:
obua@16782:   assumes "real_is_int a" "real_is_int b"
obua@16782:   shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
obua@16782: proof -
obua@16782:   from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)
obua@16782:   from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)
obua@16782:   from a obtain a'::int where a':"a = real a'" by auto
obua@16782:   from b obtain b'::int where b':"b = real b'" by auto
obua@16782:   have r: "real a' * real b' = real (a' * b')" by auto
obua@16782:   show ?thesis
obua@16782:     apply (simp add: a' b')
obua@16782:     apply (subst r)
obua@16782:     apply (simp only: int_of_real_real)
obua@16782:     done
obua@16782: qed
obua@16782: 
obua@16782: lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"
obua@16782: apply (subst real_is_int_def2)
obua@16782: apply (simp add: int_of_real_mult)
obua@16782: done
obua@16782: 
obua@16782: lemma real_is_int_0[simp]: "real_is_int (0::real)"
obua@16782: by (simp add: real_is_int_def int_of_real_def)
obua@16782: 
obua@16782: lemma real_is_int_1[simp]: "real_is_int (1::real)"
obua@16782: proof -
obua@16782:   have "real_is_int (1::real) = real_is_int(real (1::int))" by auto
obua@16782:   also have "\<dots> = True" by (simp only: real_is_int_real)
obua@16782:   ultimately show ?thesis by auto
obua@16782: qed
obua@16782: 
obua@16782: lemma real_is_int_n1: "real_is_int (-1::real)"
obua@16782: proof -
obua@16782:   have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
obua@16782:   also have "\<dots> = True" by (simp only: real_is_int_real)
obua@16782:   ultimately show ?thesis by auto
obua@16782: qed
obua@16782: 
haftmann@20485: lemma real_is_int_number_of[simp]: "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
obua@16782: proof -
obua@16782:   have neg1: "real_is_int (-1::real)"
obua@16782:   proof -
obua@16782:     have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
obua@16782:     also have "\<dots> = True" by (simp only: real_is_int_real)
obua@16782:     ultimately show ?thesis by auto
obua@16782:   qed
wenzelm@19765: 
wenzelm@19765:   {
haftmann@20485:     fix x :: int
haftmann@20485:     have "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
haftmann@20485:       unfolding number_of_eq
obua@16782:       apply (induct x)
obua@16782:       apply (induct_tac n)
obua@16782:       apply (simp)
obua@16782:       apply (simp)
obua@16782:       apply (induct_tac n)
obua@16782:       apply (simp add: neg1)
obua@16782:     proof -
obua@16782:       fix n :: nat
obua@16782:       assume rn: "(real_is_int (of_int (- (int (Suc n)))))"
obua@16782:       have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp
obua@16782:       show "real_is_int (of_int (- (int (Suc (Suc n)))))"
wenzelm@19765:         apply (simp only: s of_int_add)
wenzelm@19765:         apply (rule real_is_int_add)
wenzelm@19765:         apply (simp add: neg1)
wenzelm@19765:         apply (simp only: rn)
wenzelm@19765:         done
obua@16782:     qed
obua@16782:   }
obua@16782:   note Abs_Bin = this
obua@16782:   {
haftmann@20485:     fix x :: int
haftmann@20485:     have "? u. x = u"
haftmann@20485:       apply (rule exI[where x = "x"])
haftmann@20485:       apply (simp)
obua@16782:       done
obua@16782:   }
haftmann@20485:   then obtain u::int where "x = u" by auto
obua@16782:   with Abs_Bin show ?thesis by auto
obua@16782: qed
obua@16782: 
obua@16782: lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"
obua@16782: by (simp add: int_of_real_def)
obua@16782: 
obua@16782: lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"
wenzelm@19765: proof -
obua@16782:   have 1: "(1::real) = real (1::int)" by auto
obua@16782:   show ?thesis by (simp only: 1 int_of_real_real)
obua@16782: qed
obua@16782: 
obua@16782: lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b"
obua@16782: proof -
obua@16782:   have "real_is_int (number_of b)" by simp
obua@16782:   then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep)
obua@16782:   then obtain u::int where u:"number_of b = real u" by auto
wenzelm@19765:   have "number_of b = real ((number_of b)::int)"
obua@16782:     by (simp add: number_of_eq real_of_int_def)
wenzelm@19765:   have ub: "number_of b = real ((number_of b)::int)"
obua@16782:     by (simp add: number_of_eq real_of_int_def)
obua@16782:   from uu u ub have unb: "u = number_of b"
obua@16782:     by blast
obua@16782:   have "int_of_real (number_of b) = u" by (simp add: u)
obua@16782:   with unb show ?thesis by simp
obua@16782: qed
obua@16782: 
obua@16782: lemma float_transfer_even: "even a \<Longrightarrow> float (a, b) = float (a div 2, b+1)"
obua@16782:   apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified])
obua@16782:   apply (simp_all add: pow2_def even_def real_is_int_def ring_eq_simps)
obua@16782:   apply (auto)
obua@16782: proof -
obua@16782:   fix q::int
obua@16782:   have a:"b - (-1\<Colon>int) = (1\<Colon>int) + b" by arith
wenzelm@19765:   show "(float (q, (b - (-1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))"
obua@16782:     by (simp add: a)
obua@16782: qed
wenzelm@19765: 
obua@16782: consts
obua@16782:   norm_float :: "int*int \<Rightarrow> int*int"
obua@16782: 
obua@16782: lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
obua@16782: apply (subst split_div, auto)
obua@16782: apply (subst split_zdiv, auto)
obua@16782: apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
obua@16782: apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
obua@16782: done
obua@16782: 
obua@16782: lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)"
obua@16782: apply (subst split_mod, auto)
obua@16782: apply (subst split_zmod, auto)
obua@16782: apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in IntDiv.unique_remainder)
obua@16782: apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
obua@16782: done
obua@16782: 
obua@16782: lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
obua@16782: by arith
obua@16782: 
obua@16782: lemma terminating_norm_float: "\<forall>a. (a::int) \<noteq> 0 \<and> even a \<longrightarrow> a \<noteq> 0 \<and> \<bar>a div 2\<bar> < \<bar>a\<bar>"
obua@16782: apply (auto)
obua@16782: apply (rule abs_div_2_less)
obua@16782: apply (auto)
obua@16782: done
obua@16782: 
wenzelm@19765: ML {* simp_depth_limit := 2 *}
obua@16782: recdef norm_float "measure (% (a,b). nat (abs a))"
obua@16782:   "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)))"
obua@16782: (hints simp: terminating_norm_float)
obua@16782: ML {* simp_depth_limit := 1000 *}
obua@16782: 
obua@16782: lemma norm_float: "float x = float (norm_float x)"
obua@16782: proof -
obua@16782:   {
wenzelm@19765:     fix a b :: int
wenzelm@19765:     have norm_float_pair: "float (a,b) = float (norm_float (a,b))"
obua@16782:     proof (induct a b rule: norm_float.induct)
obua@16782:       case (1 u v)
wenzelm@19765:       show ?case
obua@16782:       proof cases
wenzelm@19765:         assume u: "u \<noteq> 0 \<and> even u"
wenzelm@19765:         with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2, v + 1))" by auto
wenzelm@19765:         with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even)
wenzelm@19765:         then show ?thesis
wenzelm@19765:           apply (subst norm_float.simps)
wenzelm@19765:           apply (simp add: ind)
wenzelm@19765:           done
obua@16782:       next
wenzelm@19765:         assume "~(u \<noteq> 0 \<and> even u)"
wenzelm@19765:         then show ?thesis
wenzelm@19765:           by (simp add: prems float_def)
obua@16782:       qed
obua@16782:     qed
obua@16782:   }
obua@16782:   note helper = this
obua@16782:   have "? a b. x = (a,b)" by auto
obua@16782:   then obtain a b where "x = (a, b)" by blast
obua@16782:   then show ?thesis by (simp only: helper)
obua@16782: qed
obua@16782: 
obua@16782: lemma pow2_int: "pow2 (int n) = 2^n"
obua@16782:   by (simp add: pow2_def)
obua@16782: 
wenzelm@19765: lemma float_add:
wenzelm@19765:   "float (a1, e1) + float (a2, e2) =
wenzelm@19765:   (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1)
obua@16782:   else float (a1*2^(nat (e1-e2))+a2, e2))"
obua@16782:   apply (simp add: float_def ring_eq_simps)
obua@16782:   apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric])
obua@16782:   done
obua@16782: 
obua@16782: lemma float_mult:
wenzelm@19765:   "float (a1, e1) * float (a2, e2) =
obua@16782:   (float (a1 * a2, e1 + e2))"
obua@16782:   by (simp add: float_def pow2_add)
obua@16782: 
obua@16782: lemma float_minus:
obua@16782:   "- (float (a,b)) = float (-a, b)"
obua@16782:   by (simp add: float_def)
obua@16782: 
obua@16782: lemma zero_less_pow2:
obua@16782:   "0 < pow2 x"
obua@16782: proof -
obua@16782:   {
obua@16782:     fix y
wenzelm@19765:     have "0 <= y \<Longrightarrow> 0 < pow2 y"
obua@16782:       by (induct y, induct_tac n, simp_all add: pow2_add)
obua@16782:   }
obua@16782:   note helper=this
obua@16782:   show ?thesis
obua@16782:     apply (case_tac "0 <= x")
obua@16782:     apply (simp add: helper)
obua@16782:     apply (subst pow2_neg)
obua@16782:     apply (simp add: helper)
obua@16782:     done
obua@16782: qed
obua@16782: 
obua@16782: lemma zero_le_float:
obua@16782:   "(0 <= float (a,b)) = (0 <= a)"
obua@16782:   apply (auto simp add: float_def)
wenzelm@19765:   apply (auto simp add: zero_le_mult_iff zero_less_pow2)
obua@16782:   apply (insert zero_less_pow2[of b])
obua@16782:   apply (simp_all)
obua@16782:   done
obua@16782: 
obua@16782: lemma float_le_zero:
obua@16782:   "(float (a,b) <= 0) = (a <= 0)"
obua@16782:   apply (auto simp add: float_def)
obua@16782:   apply (auto simp add: mult_le_0_iff)
obua@16782:   apply (insert zero_less_pow2[of b])
obua@16782:   apply auto
obua@16782:   done
obua@16782: 
obua@16782: lemma float_abs:
obua@16782:   "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
obua@16782:   apply (auto simp add: abs_if)
obua@16782:   apply (simp_all add: zero_le_float[symmetric, of a b] float_minus)
obua@16782:   done
obua@16782: 
obua@16782: lemma float_zero:
obua@16782:   "float (0, b) = 0"
obua@16782:   by (simp add: float_def)
obua@16782: 
obua@16782: lemma float_pprt:
obua@16782:   "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"
obua@16782:   by (auto simp add: zero_le_float float_le_zero float_zero)
obua@16782: 
obua@16782: lemma float_nprt:
obua@16782:   "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"
obua@16782:   by (auto simp add: zero_le_float float_le_zero float_zero)
obua@16782: 
obua@16782: lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
obua@16782:   by auto
wenzelm@19765: 
obua@16782: lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma int_pow_0: "(a::int)^(Numeral0) = 1"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma int_pow_1: "(a::int)^(Numeral1) = a"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
obua@16782: proof -
obua@16782:   have 1:"((-1)::nat) = 0"
obua@16782:     by simp
obua@16782:   show ?thesis by (simp add: 1)
obua@16782: qed
obua@16782: 
obua@16782: lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma lift_bool: "x \<Longrightarrow> x=True"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma nlift_bool: "~x \<Longrightarrow> x=False"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma not_false_eq_true: "(~ False) = True" by simp
obua@16782: 
obua@16782: lemma not_true_eq_false: "(~ True) = False" by simp
obua@16782: 
wenzelm@19765: lemmas binarith =
obua@16782:   Pls_0_eq Min_1_eq
haftmann@20485:   pred_Pls pred_Min pred_1 pred_0
haftmann@20485:   succ_Pls succ_Min succ_1 succ_0
haftmann@20485:   add_Pls add_Min add_BIT_0 add_BIT_10
haftmann@20485:   add_BIT_11 minus_Pls minus_Min minus_1
haftmann@20485:   minus_0 mult_Pls mult_Min mult_num1 mult_num0
haftmann@20485:   add_Pls_right add_Min_right
obua@16782: 
haftmann@20485: lemma int_eq_number_of_eq:
haftmann@20485:   "(((number_of v)::int)=(number_of w)) = iszero ((number_of (v + uminus w))::int)"
obua@16782:   by simp
obua@16782: 
wenzelm@19765: lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
obua@16782:   by (simp only: iszero_number_of_Pls)
obua@16782: 
obua@16782: lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
obua@16782:   by simp
obua@16782: 
wenzelm@19765: lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)"
obua@16782:   by simp
obua@16782: 
haftmann@20485: lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)"
obua@16782:   by simp
obua@16782: 
wenzelm@19765: lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma int_neg_number_of_Min: "neg (-1::int)"
obua@16782:   by simp
obua@16782: 
obua@16782: lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
obua@16782:   by simp
obua@16782: 
haftmann@20485: lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))"
obua@16782:   by simp
obua@16782: 
wenzelm@19765: lemmas intarithrel =
wenzelm@19765:   int_eq_number_of_eq
wenzelm@19765:   lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_0
obua@16782:   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]
obua@16782:   int_neg_number_of_BIT int_le_number_of_eq
obua@16782: 
haftmann@20485: lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)"
obua@16782:   by simp
obua@16782: 
haftmann@20485: lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (v + (uminus w))"
obua@16782:   by simp
obua@16782: 
haftmann@20485: lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (v * w)"
obua@16782:   by simp
obua@16782: 
haftmann@20485: lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)"
obua@16782:   by simp
obua@16782: 
obua@16782: lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym
obua@16782: 
obua@16782: lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
obua@16782: 
wenzelm@19765: lemmas powerarith = nat_number_of zpower_number_of_even
wenzelm@19765:   zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
obua@16782:   zpower_Pls zpower_Min
obua@16782: 
obua@16782: lemmas floatarith[simplified norm_0_1] = float_add float_mult float_minus float_abs zero_le_float float_pprt float_nprt
obua@16782: 
obua@16782: (* for use with the compute oracle *)
obua@16782: lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false
obua@16782: 
wenzelm@20771: use "float.ML";
wenzelm@20771: 
obua@16782: end