src/HOL/Library/Float.thy
author hoelzl
Wed Apr 18 14:29:21 2012 +0200 (2012-04-18)
changeset 47599 400b158f1589
parent 47230 6584098d5378
child 47600 e12289b5796b
permissions -rw-r--r--
replace the float datatype by a type with unique representation
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header {* Floating-Point Numbers *}
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theory Float
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imports Complex_Main "~~/src/HOL/Library/Lattice_Algebras"
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begin
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typedef float = "{m * 2 powr e | (m :: int) (e :: int). True }"
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  morphisms real_of_float float_of
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  by auto
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declare [[coercion "real::float\<Rightarrow>real"]]
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lemmas float_of_inject[simp]
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lemmas float_of_cases2 = float_of_cases[case_product float_of_cases]
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lemmas float_of_cases3 = float_of_cases2[case_product float_of_cases]
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defs (overloaded)
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  real_of_float_def[code_unfold]: "real == real_of_float"
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lemma real_of_float_eq[simp]:
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  fixes f1 f2 :: float shows "real f1 = real f2 \<longleftrightarrow> f1 = f2"
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  unfolding real_of_float_def real_of_float_inject ..
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lemma float_of_real[simp]: "float_of (real x) = x"
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  unfolding real_of_float_def by (rule real_of_float_inverse)
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lemma real_float[simp]: "x \<in> float \<Longrightarrow> real (float_of x) = x"
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  unfolding real_of_float_def by (rule float_of_inverse)
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subsection {* Real operations preserving the representation as floating point number *}
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lemma floatI: fixes m e :: int shows "m * 2 powr e = x \<Longrightarrow> x \<in> float"
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  by (auto simp: float_def)
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lemma zero_float[simp]: "0 \<in> float" by (auto simp: float_def)
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lemma one_float[simp]: "1 \<in> float" by (intro floatI[of 1 0]) simp
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lemma numeral_float[simp]: "numeral i \<in> float" by (intro floatI[of "numeral i" 0]) simp  
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lemma neg_numeral_float[simp]: "neg_numeral i \<in> float" by (intro floatI[of "neg_numeral i" 0]) simp
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lemma real_of_int_float[simp]: "real (x :: int) \<in> float" by (intro floatI[of x 0]) simp
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lemma real_of_nat_float[simp]: "real (x ::nat) \<in> float" by (intro floatI[of x 0]) simp
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lemma two_powr_int_float[simp]: "2 powr (real (i::int)) \<in> float" by (intro floatI[of 1 i]) simp
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lemma two_powr_nat_float[simp]: "2 powr (real (i::nat)) \<in> float" by (intro floatI[of 1 i]) simp
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lemma two_powr_minus_int_float[simp]: "2 powr - (real (i::int)) \<in> float" by (intro floatI[of 1 "-i"]) simp
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lemma two_powr_minus_nat_float[simp]: "2 powr - (real (i::nat)) \<in> float" by (intro floatI[of 1 "-i"]) simp
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lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float" by (intro floatI[of 1 "numeral i"]) simp
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lemma two_powr_neg_numeral_float[simp]: "2 powr neg_numeral i \<in> float" by (intro floatI[of 1 "neg_numeral i"]) simp
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lemma two_pow_float[simp]: "2 ^ n \<in> float" by (intro floatI[of 1 "n"]) (simp add: powr_realpow)
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lemma real_of_float_float[simp]: "real (f::float) \<in> float" by (cases f) simp
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lemma plus_float[simp]: "r \<in> float \<Longrightarrow> p \<in> float \<Longrightarrow> r + p \<in> float"
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  unfolding float_def
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proof (safe, simp)
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  fix e1 m1 e2 m2 :: int
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  { fix e1 m1 e2 m2 :: int assume "e1 \<le> e2"
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    then have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"
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      by (simp add: powr_realpow[symmetric] powr_divide2[symmetric] field_simps)
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    then have "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
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      by blast }
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  note * = this
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  show "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
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  proof (cases e1 e2 rule: linorder_le_cases)
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    assume "e2 \<le> e1" from *[OF this, of m2 m1] show ?thesis by (simp add: ac_simps)
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  qed (rule *)
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qed
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lemma uminus_float[simp]: "x \<in> float \<Longrightarrow> -x \<in> float"
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  apply (auto simp: float_def)
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  apply (rule_tac x="-x" in exI)
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  apply (rule_tac x="xa" in exI)
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  apply (simp add: field_simps)
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  done
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lemma times_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x * y \<in> float"
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  apply (auto simp: float_def)
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  apply (rule_tac x="x * xa" in exI)
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  apply (rule_tac x="xb + xc" in exI)
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  apply (simp add: powr_add)
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  done
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lemma minus_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x - y \<in> float"
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  unfolding ab_diff_minus by (intro uminus_float plus_float)
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lemma abs_float[simp]: "x \<in> float \<Longrightarrow> abs x \<in> float"
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  by (cases x rule: linorder_cases[of 0]) auto
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lemma sgn_of_float[simp]: "x \<in> float \<Longrightarrow> sgn x \<in> float"
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  by (cases x rule: linorder_cases[of 0]) (auto intro!: uminus_float)
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lemma div_power_2_float[simp]: "x \<in> float \<Longrightarrow> x / 2^d \<in> float"
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  apply (auto simp add: float_def)
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  apply (rule_tac x="x" in exI)
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  apply (rule_tac x="xa - d" in exI)
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  apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
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  done
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lemma div_power_2_int_float[simp]: "x \<in> float \<Longrightarrow> x / (2::int)^d \<in> float"
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  apply (auto simp add: float_def)
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  apply (rule_tac x="x" in exI)
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  apply (rule_tac x="xa - d" in exI)
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  apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
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  done
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lemma div_numeral_Bit0_float[simp]:
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  assumes x: "x / numeral n \<in> float" shows "x / (numeral (Num.Bit0 n)) \<in> float"
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proof -
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  have "(x / numeral n) / 2^1 \<in> float"
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    by (intro x div_power_2_float)
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  also have "(x / numeral n) / 2^1 = x / (numeral (Num.Bit0 n))"
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    by (induct n) auto
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  finally show ?thesis .
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qed
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lemma div_neg_numeral_Bit0_float[simp]:
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  assumes x: "x / numeral n \<in> float" shows "x / (neg_numeral (Num.Bit0 n)) \<in> float"
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proof -
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  have "- (x / numeral (Num.Bit0 n)) \<in> float" using x by simp
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  also have "- (x / numeral (Num.Bit0 n)) = x / neg_numeral (Num.Bit0 n)"
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    unfolding neg_numeral_def by (simp del: minus_numeral)
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  finally show ?thesis .
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qed
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subsection {* Arithmetic operations on floating point numbers *}
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instantiation float :: ring_1
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begin
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definition [simp]: "(0::float) = float_of 0"
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definition [simp]: "(1::float) = float_of 1"
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definition "(x + y::float) = float_of (real x + real y)"
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lemma float_plus_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> float_of x + float_of y = float_of (x + y)"
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  by (simp add: plus_float_def)
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definition "(-x::float) = float_of (- real x)"
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lemma uminus_of_float[simp]: "x \<in> float \<Longrightarrow> - float_of x = float_of (- x)"
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  by (simp add: uminus_float_def)
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definition "(x - y::float) = float_of (real x - real y)"
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lemma float_minus_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> float_of x - float_of y = float_of (x - y)"
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  by (simp add: minus_float_def)
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definition "(x * y::float) = float_of (real x * real y)"
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lemma float_times_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> float_of x * float_of y = float_of (x * y)"
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  by (simp add: times_float_def)
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instance
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proof
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  fix a b c :: float
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  show "0 + a = a"
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    by (cases a rule: float_of_cases) simp
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  show "1 * a = a"
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    by (cases a rule: float_of_cases) simp
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  show "a * 1 = a"
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    by (cases a rule: float_of_cases) simp
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  show "-a + a = 0"
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    by (cases a rule: float_of_cases) simp
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  show "a + b = b + a"
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    by (cases a b rule: float_of_cases2) (simp add: ac_simps)
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  show "a - b = a + -b"
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    by (cases a b rule: float_of_cases2) (simp add: field_simps)
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  show "a + b + c = a + (b + c)"
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    by (cases a b c rule: float_of_cases3) (simp add: ac_simps)
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  show "a * b * c = a * (b * c)"
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    by (cases a b c rule: float_of_cases3) (simp add: ac_simps)
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  show "(a + b) * c = a * c + b * c"
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    by (cases a b c rule: float_of_cases3) (simp add: field_simps)
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  show "a * (b + c) = a * b + a * c"
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    by (cases a b c rule: float_of_cases3) (simp add: field_simps)
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  show "0 \<noteq> (1::float)" by simp
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qed
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end
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lemma real_of_float_uminus[simp]:
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  fixes f g::float shows "real (- g) = - real g"
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  by (simp add: uminus_float_def)
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lemma real_of_float_plus[simp]:
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  fixes f g::float shows "real (f + g) = real f + real g"
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  by (simp add: plus_float_def)
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lemma real_of_float_minus[simp]:
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  fixes f g::float shows "real (f - g) = real f - real g"
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  by (simp add: minus_float_def)
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lemma real_of_float_times[simp]:
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  fixes f g::float shows "real (f * g) = real f * real g"
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  by (simp add: times_float_def)
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lemma real_of_float_zero[simp]: "real (0::float) = 0" by simp
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lemma real_of_float_one[simp]: "real (1::float) = 1" by simp
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lemma real_of_float_power[simp]: fixes f::float shows "real (f^n) = real f^n"
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  by (induct n) simp_all
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instantiation float :: linorder
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begin
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definition "x \<le> (y::float) \<longleftrightarrow> real x \<le> real y"
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lemma float_le_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> float_of x \<le> float_of y \<longleftrightarrow> x \<le> y"
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  by (simp add: less_eq_float_def)
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definition "x < (y::float) \<longleftrightarrow> real x < real y"
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lemma float_less_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> float_of x < float_of y \<longleftrightarrow> x < y"
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  by (simp add: less_float_def)
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instance
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proof
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  fix a b c :: float
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  show "a \<le> a"
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    by (cases a rule: float_of_cases) simp
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  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
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    by (cases a b rule: float_of_cases2) auto
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  show "a \<le> b \<or> b \<le> a"
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    by (cases a b rule: float_of_cases2) auto
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  { assume "a \<le> b" "b \<le> a" then show "a = b"
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    by (cases a b rule: float_of_cases2) auto }
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  { assume "a \<le> b" "b \<le> c" then show "a \<le> c"
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    by (cases a b c rule: float_of_cases3) auto }
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qed
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end
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lemma real_of_float_min: fixes a b::float shows "real (min a b) = min (real a) (real b)"
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  by (simp add: min_def less_eq_float_def)
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lemma real_of_float_max: fixes a b::float shows "real (max a b) = max (real a) (real b)"
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  by (simp add: max_def less_eq_float_def)
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instantiation float :: linordered_ring
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begin
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definition "(abs x :: float) = float_of (abs (real x))"
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lemma float_abs[simp]: "x \<in> float \<Longrightarrow> abs (float_of x) = float_of (abs x)"
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  by (simp add: abs_float_def)
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instance
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proof
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  fix a b c :: float
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  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
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    by (cases a rule: float_of_cases) simp
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  assume "a \<le> b"
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  then show "c + a \<le> c + b"
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    by (cases a b c rule: float_of_cases3) simp
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  assume "0 \<le> c"
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  with `a \<le> b` show "c * a \<le> c * b"
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    by (cases a b c rule: float_of_cases3) (auto intro: mult_left_mono)
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  from `0 \<le> c` `a \<le> b` show "a * c \<le> b * c"
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    by (cases a b c rule: float_of_cases3) (auto intro: mult_right_mono)
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qed
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end
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lemma real_of_abs_float[simp]: fixes f::float shows "real (abs f) = abs (real f)"
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  unfolding abs_float_def by simp
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instance float :: dense_linorder
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proof
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  fix a b :: float
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  show "\<exists>c. a < c"
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    apply (intro exI[of _ "a + 1"])
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    apply (cases a rule: float_of_cases)
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    apply simp
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    done
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  show "\<exists>c. c < a"
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    apply (intro exI[of _ "a - 1"])
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    apply (cases a rule: float_of_cases)
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    apply simp
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    done
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  assume "a < b"
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  then show "\<exists>c. a < c \<and> c < b"
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    apply (intro exI[of _ "(b + a) * float_of (1/2)"])
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    apply (cases a b rule: float_of_cases2)
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    apply simp
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    done
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qed
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instantiation float :: linordered_idom
wenzelm@46573
   284
begin
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   285
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   286
definition "sgn x = float_of (sgn (real x))"
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   287
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   288
lemma sgn_float[simp]: "x \<in> float \<Longrightarrow> sgn (float_of x) = float_of (sgn x)"
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   289
  by (simp add: sgn_float_def)
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   290
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   291
instance
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   292
proof
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   293
  fix a b c :: float
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   294
  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
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   295
    by (cases a rule: float_of_cases) simp
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   296
  show "a * b = b * a"
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   297
    by (cases a b rule: float_of_cases2) (simp add: field_simps)
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   298
  show "1 * a = a" "(a + b) * c = a * c + b * c"
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   299
    by (simp_all add: field_simps del: one_float_def)
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   300
  assume "a < b" "0 < c" then show "c * a < c * b"
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   301
    by (cases a b c rule: float_of_cases3) (simp add: field_simps)
hoelzl@47599
   302
qed
hoelzl@29804
   303
end
hoelzl@29804
   304
hoelzl@47599
   305
definition Float :: "int \<Rightarrow> int \<Rightarrow> float" where
hoelzl@47599
   306
  [simp, code del]: "Float m e = float_of (m * 2 powr e)"
hoelzl@29804
   307
hoelzl@47599
   308
lemma real_of_float_Float[code]: "real_of_float (Float m e) =
hoelzl@47599
   309
  (if e \<ge> 0 then m * 2 ^ nat e else m * inverse (2 ^ nat (- e)))"
hoelzl@47599
   310
by (auto simp add: powr_realpow[symmetric] powr_minus real_of_float_def[symmetric])
hoelzl@29804
   311
hoelzl@47599
   312
code_datatype Float
obua@16782
   313
hoelzl@47599
   314
lemma real_Float: "real (Float m e) = m * 2 powr e" by simp
obua@16782
   315
hoelzl@47599
   316
definition normfloat :: "float \<Rightarrow> float" where
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   317
  [simp]: "normfloat x = x"
hoelzl@29804
   318
hoelzl@47599
   319
lemma compute_normfloat[code]: "normfloat (Float m e) =
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   320
  (if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
hoelzl@47599
   321
                           else if m = 0 then 0 else Float m e)"
hoelzl@47599
   322
  by (simp del: real_of_int_add split: prod.split)
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   323
     (auto simp add: powr_add zmod_eq_0_iff)
hoelzl@29804
   324
hoelzl@47599
   325
lemma compute_zero[code_unfold, code]: "0 = Float 0 0"
hoelzl@47599
   326
  by simp
hoelzl@47599
   327
hoelzl@47599
   328
lemma compute_one[code_unfold, code]: "1 = Float 1 0"
hoelzl@47599
   329
  by simp
hoelzl@29804
   330
huffman@47108
   331
instance float :: numeral ..
huffman@47108
   332
hoelzl@47599
   333
lemma float_of_numeral[simp]: "numeral k = float_of (numeral k)"
hoelzl@47599
   334
  by (induct k)
hoelzl@47599
   335
     (simp_all only: numeral.simps one_float_def float_plus_float numeral_float one_float plus_float)
hoelzl@47599
   336
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   337
lemma float_of_neg_numeral[simp]: "neg_numeral k = float_of (neg_numeral k)"
hoelzl@47599
   338
  by (simp add: minus_numeral[symmetric] del: minus_numeral)
huffman@47108
   339
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   340
lemma
hoelzl@47599
   341
  shows float_numeral[simp]: "real (numeral x :: float) = numeral x"
hoelzl@47599
   342
    and float_neg_numeral[simp]: "real (neg_numeral x :: float) = neg_numeral x"
hoelzl@47599
   343
  by simp_all
huffman@47108
   344
hoelzl@47599
   345
subsection {* Represent floats as unique mantissa and exponent *}
huffman@47108
   346
huffman@47108
   347
hoelzl@47599
   348
lemma int_induct_abs[case_names less]:
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   349
  fixes j :: int
hoelzl@47599
   350
  assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n"
hoelzl@47599
   351
  shows "P j"
hoelzl@47599
   352
proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct)
hoelzl@47599
   353
  case less show ?case by (rule H[OF less]) simp
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   354
qed
hoelzl@47599
   355
hoelzl@47599
   356
lemma int_cancel_factors:
hoelzl@47599
   357
  fixes n :: int assumes "1 < r" shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
hoelzl@47599
   358
proof (induct n rule: int_induct_abs)
hoelzl@47599
   359
  case (less n)
hoelzl@47599
   360
  { fix m assume n: "n \<noteq> 0" "n = m * r"
hoelzl@47599
   361
    then have "\<bar>m \<bar> < \<bar>n\<bar>"
hoelzl@47599
   362
      by (metis abs_dvd_iff abs_ge_self assms comm_semiring_1_class.normalizing_semiring_rules(7)
hoelzl@47599
   363
                dvd_imp_le_int dvd_refl dvd_triv_right linorder_neq_iff linorder_not_le
hoelzl@47599
   364
                mult_eq_0_iff zdvd_mult_cancel1)
hoelzl@47599
   365
    from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto }
hoelzl@47599
   366
  then show ?case
hoelzl@47599
   367
    by (metis comm_semiring_1_class.normalizing_semiring_rules(12,7) dvdE power_0)
hoelzl@47599
   368
qed
hoelzl@47599
   369
hoelzl@47599
   370
lemma mult_powr_eq_mult_powr_iff_asym:
hoelzl@47599
   371
  fixes m1 m2 e1 e2 :: int
hoelzl@47599
   372
  assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2"
hoelzl@47599
   373
  shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
hoelzl@47599
   374
proof
hoelzl@47599
   375
  have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto
hoelzl@47599
   376
  assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"
hoelzl@47599
   377
  with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2 - e1)"
hoelzl@47599
   378
    by (simp add: powr_divide2[symmetric] field_simps)
hoelzl@47599
   379
  also have "\<dots> = m2 * 2^nat (e2 - e1)"
hoelzl@47599
   380
    by (simp add: powr_realpow)
hoelzl@47599
   381
  finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
hoelzl@47599
   382
    unfolding real_of_int_inject .
hoelzl@47599
   383
  with m1 have "m1 = m2"
hoelzl@47599
   384
    by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
hoelzl@47599
   385
  then show "m1 = m2 \<and> e1 = e2"
hoelzl@47599
   386
    using eq `m1 \<noteq> 0` by (simp add: powr_inj)
hoelzl@47599
   387
qed simp
hoelzl@47599
   388
hoelzl@47599
   389
lemma mult_powr_eq_mult_powr_iff:
hoelzl@47599
   390
  fixes m1 m2 e1 e2 :: int
hoelzl@47599
   391
  shows "\<not> 2 dvd m1 \<Longrightarrow> \<not> 2 dvd m2 \<Longrightarrow> m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
hoelzl@47599
   392
  using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]
hoelzl@47599
   393
  using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]
hoelzl@47599
   394
  by (cases e1 e2 rule: linorder_le_cases) auto
hoelzl@47599
   395
hoelzl@47599
   396
lemma floatE_normed:
hoelzl@47599
   397
  assumes x: "x \<in> float"
hoelzl@47599
   398
  obtains (zero) "x = 0"
hoelzl@47599
   399
   | (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
hoelzl@47599
   400
proof atomize_elim
hoelzl@47599
   401
  { assume "x \<noteq> 0"
hoelzl@47599
   402
    from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
hoelzl@47599
   403
    with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
hoelzl@47599
   404
      by auto
hoelzl@47599
   405
    with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"
hoelzl@47599
   406
      by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
hoelzl@47599
   407
         (simp add: powr_add powr_realpow) }
hoelzl@47599
   408
  then show "x = 0 \<or> (\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m \<and> x \<noteq> 0)"
hoelzl@47599
   409
    by blast
hoelzl@47599
   410
qed
hoelzl@47599
   411
hoelzl@47599
   412
lemma float_normed_cases:
hoelzl@47599
   413
  fixes f :: float
hoelzl@47599
   414
  obtains (zero) "f = 0"
hoelzl@47599
   415
   | (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
hoelzl@47599
   416
proof (atomize_elim, induct f)
hoelzl@47599
   417
  case (float_of y) then show ?case
hoelzl@47599
   418
    by (cases rule: floatE_normed) auto
hoelzl@47599
   419
qed
hoelzl@47599
   420
hoelzl@47599
   421
definition mantissa :: "float \<Rightarrow> int" where
hoelzl@47599
   422
  "mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   423
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   424
hoelzl@47599
   425
definition exponent :: "float \<Rightarrow> int" where
hoelzl@47599
   426
  "exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   427
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   428
hoelzl@47599
   429
lemma 
hoelzl@47599
   430
  shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
hoelzl@47599
   431
    and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
hoelzl@47599
   432
proof -
hoelzl@47599
   433
  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto
hoelzl@47599
   434
  then show ?E ?M
hoelzl@47599
   435
    by (auto simp add: mantissa_def exponent_def)
hoelzl@29804
   436
qed
hoelzl@29804
   437
hoelzl@47599
   438
lemma
hoelzl@47599
   439
  shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
hoelzl@47599
   440
    and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
hoelzl@47599
   441
proof cases
hoelzl@47599
   442
  assume [simp]: "f \<noteq> (float_of 0)"
hoelzl@47599
   443
  have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
hoelzl@47599
   444
  proof (cases f rule: float_normed_cases)
hoelzl@47599
   445
    case (powr m e)
hoelzl@47599
   446
    then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   447
     \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"
hoelzl@47599
   448
      by auto
hoelzl@47599
   449
    then show ?thesis
hoelzl@47599
   450
      unfolding exponent_def mantissa_def
hoelzl@47599
   451
      by (rule someI2_ex) simp
hoelzl@47599
   452
  qed simp
hoelzl@47599
   453
  then show ?E ?D by auto
hoelzl@47599
   454
qed simp
hoelzl@47599
   455
hoelzl@47599
   456
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"
hoelzl@47599
   457
  using mantissa_not_dvd[of f] by auto
hoelzl@47599
   458
hoelzl@47599
   459
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
hoelzl@47599
   460
  unfolding real_of_float_eq[symmetric] mantissa_exponent[of f] by simp
hoelzl@47599
   461
hoelzl@47599
   462
lemma Float_cases[case_names Float, cases type: float]:
hoelzl@47599
   463
  fixes f :: float
hoelzl@47599
   464
  obtains (Float) m e :: int where "f = Float m e"
hoelzl@47599
   465
  using Float_mantissa_exponent[symmetric]
hoelzl@47599
   466
  by (atomize_elim) auto
hoelzl@29804
   467
hoelzl@47599
   468
lemma 
hoelzl@47599
   469
  fixes m e :: int
hoelzl@47599
   470
  defines "f \<equiv> float_of (m * 2 powr e)"
hoelzl@47599
   471
  assumes dvd: "\<not> 2 dvd m"
hoelzl@47599
   472
  shows mantissa_float: "mantissa f = m" (is "?M")
hoelzl@47599
   473
    and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
hoelzl@47599
   474
proof cases
hoelzl@47599
   475
  assume "m = 0" with dvd show "mantissa f = m" by auto
hoelzl@47599
   476
next
hoelzl@47599
   477
  assume "m \<noteq> 0"
hoelzl@47599
   478
  then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)
hoelzl@47599
   479
  from mantissa_exponent[of f]
hoelzl@47599
   480
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   481
    by (auto simp add: f_def)
hoelzl@47599
   482
  then show "?M" "?E"
hoelzl@47599
   483
    using mantissa_not_dvd[OF f_not_0] dvd
hoelzl@47599
   484
    by (auto simp: mult_powr_eq_mult_powr_iff)
hoelzl@47599
   485
qed
hoelzl@47599
   486
hoelzl@47599
   487
lemma denormalize_shift:
hoelzl@47599
   488
  assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0"
hoelzl@47599
   489
  obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
hoelzl@47599
   490
proof
hoelzl@47599
   491
  from mantissa_exponent[of f] f_def
hoelzl@47599
   492
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   493
    by simp
hoelzl@47599
   494
  then have eq: "m = mantissa f * 2 powr (exponent f - e)"
hoelzl@47599
   495
    by (simp add: powr_divide2[symmetric] field_simps)
hoelzl@47599
   496
  moreover
hoelzl@47599
   497
  have "e \<le> exponent f"
hoelzl@47599
   498
  proof (rule ccontr)
hoelzl@47599
   499
    assume "\<not> e \<le> exponent f"
hoelzl@47599
   500
    then have pos: "exponent f < e" by simp
hoelzl@47599
   501
    then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"
hoelzl@47599
   502
      by simp
hoelzl@47599
   503
    also have "\<dots> = 1 / 2^nat (e - exponent f)"
hoelzl@47599
   504
      using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47599
   505
    finally have "m * 2^nat (e - exponent f) = real (mantissa f)"
hoelzl@47599
   506
      using eq by simp
hoelzl@47599
   507
    then have "mantissa f = m * 2^nat (e - exponent f)"
hoelzl@47599
   508
      unfolding real_of_int_inject by simp
hoelzl@47599
   509
    with `exponent f < e` have "2 dvd mantissa f"
hoelzl@47599
   510
      apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
hoelzl@47599
   511
      apply (cases "nat (e - exponent f)")
hoelzl@47599
   512
      apply auto
hoelzl@47599
   513
      done
hoelzl@47599
   514
    then show False using mantissa_not_dvd[OF not_0] by simp
hoelzl@47599
   515
  qed
hoelzl@47599
   516
  ultimately have "real m = mantissa f * 2^nat (exponent f - e)"
hoelzl@47599
   517
    by (simp add: powr_realpow[symmetric])
hoelzl@47599
   518
  with `e \<le> exponent f`
hoelzl@47599
   519
  show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
hoelzl@47599
   520
    unfolding real_of_int_inject by auto
hoelzl@29804
   521
qed
hoelzl@29804
   522
hoelzl@47599
   523
subsection {* Compute arithmetic operations *}
hoelzl@47599
   524
hoelzl@47599
   525
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
hoelzl@47599
   526
  by simp
hoelzl@47599
   527
hoelzl@47599
   528
lemma compute_float_neg_numeral[code_abbrev]: "Float (neg_numeral k) 0 = neg_numeral k"
hoelzl@47599
   529
  by simp
hoelzl@47599
   530
hoelzl@47599
   531
lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
hoelzl@47599
   532
  by simp
hoelzl@47599
   533
hoelzl@47599
   534
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
hoelzl@47599
   535
  by (simp add: field_simps powr_add)
hoelzl@47599
   536
hoelzl@47599
   537
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
hoelzl@47599
   538
  (if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
hoelzl@47599
   539
              else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
hoelzl@47599
   540
  by (simp add: field_simps)
hoelzl@47599
   541
     (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47599
   542
hoelzl@47599
   543
lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)" by simp
hoelzl@47599
   544
hoelzl@47599
   545
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
hoelzl@47599
   546
  by (simp add: sgn_times)
hoelzl@47599
   547
hoelzl@47599
   548
definition is_float_pos :: "float \<Rightarrow> bool" where
hoelzl@47599
   549
  "is_float_pos f \<longleftrightarrow> 0 < f"
hoelzl@47599
   550
hoelzl@47599
   551
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
hoelzl@47599
   552
  by (auto simp add: is_float_pos_def zero_less_mult_iff) (simp add: not_le[symmetric])
hoelzl@47599
   553
hoelzl@47599
   554
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
hoelzl@47599
   555
  by (simp add: is_float_pos_def field_simps del: zero_float_def)
hoelzl@47599
   556
hoelzl@47599
   557
definition is_float_nonneg :: "float \<Rightarrow> bool" where
hoelzl@47599
   558
  "is_float_nonneg f \<longleftrightarrow> 0 \<le> f"
hoelzl@47599
   559
hoelzl@47599
   560
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
hoelzl@47599
   561
  by (auto simp add: is_float_nonneg_def zero_le_mult_iff) (simp add: not_less[symmetric])
hoelzl@47599
   562
hoelzl@47599
   563
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
hoelzl@47599
   564
  by (simp add: is_float_nonneg_def field_simps del: zero_float_def)
hoelzl@47599
   565
hoelzl@47599
   566
definition is_float_zero :: "float \<Rightarrow> bool" where
hoelzl@47599
   567
  "is_float_zero f \<longleftrightarrow> 0 = f"
hoelzl@47599
   568
hoelzl@47599
   569
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
hoelzl@47599
   570
  by (auto simp add: is_float_zero_def)
hoelzl@29804
   571
hoelzl@47599
   572
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e" by (simp add: abs_mult)
hoelzl@47599
   573
hoelzl@47599
   574
instantiation float :: equal
hoelzl@47599
   575
begin
hoelzl@47599
   576
hoelzl@47599
   577
definition "equal_float (f1 :: float) f2 \<longleftrightarrow> is_float_zero (f1 - f2)"
hoelzl@47599
   578
hoelzl@47599
   579
instance proof qed (auto simp: equal_float_def is_float_zero_def simp del: zero_float_def)
hoelzl@47599
   580
end
hoelzl@47599
   581
hoelzl@47599
   582
subsection {* Rounding Real numbers *}
hoelzl@47599
   583
hoelzl@47599
   584
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   585
  "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   586
hoelzl@47599
   587
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   588
  "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   589
hoelzl@47599
   590
lemma round_down_float[simp]: "round_down prec x \<in> float"
hoelzl@47599
   591
  unfolding round_down_def
hoelzl@47599
   592
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   593
hoelzl@47599
   594
lemma round_up_float[simp]: "round_up prec x \<in> float"
hoelzl@47599
   595
  unfolding round_up_def
hoelzl@47599
   596
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   597
hoelzl@47599
   598
lemma round_up: "x \<le> round_up prec x"
hoelzl@47599
   599
  by (simp add: powr_minus_divide le_divide_eq round_up_def)
hoelzl@47599
   600
hoelzl@47599
   601
lemma round_down: "round_down prec x \<le> x"
hoelzl@47599
   602
  by (simp add: powr_minus_divide divide_le_eq round_down_def)
hoelzl@47599
   603
hoelzl@47599
   604
lemma round_up_0[simp]: "round_up p 0 = 0"
hoelzl@47599
   605
  unfolding round_up_def by simp
hoelzl@47599
   606
hoelzl@47599
   607
lemma round_down_0[simp]: "round_down p 0 = 0"
hoelzl@47599
   608
  unfolding round_down_def by simp
hoelzl@47599
   609
hoelzl@47599
   610
lemma round_up_diff_round_down:
hoelzl@47599
   611
  "round_up prec x - round_down prec x \<le> 2 powr -prec"
hoelzl@47599
   612
proof -
hoelzl@47599
   613
  have "round_up prec x - round_down prec x =
hoelzl@47599
   614
    (ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec"
hoelzl@47599
   615
    by (simp add: round_up_def round_down_def field_simps)
hoelzl@47599
   616
  also have "\<dots> \<le> 1 * 2 powr -prec"
hoelzl@47599
   617
    by (rule mult_mono)
hoelzl@47599
   618
       (auto simp del: real_of_int_diff
hoelzl@47599
   619
             simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1)
hoelzl@47599
   620
  finally show ?thesis by simp
hoelzl@29804
   621
qed
hoelzl@29804
   622
hoelzl@47599
   623
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"
hoelzl@47599
   624
  unfolding round_down_def
hoelzl@47599
   625
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   626
    (simp add: powr_add[symmetric])
hoelzl@29804
   627
hoelzl@47599
   628
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"
hoelzl@47599
   629
  unfolding round_up_def
hoelzl@47599
   630
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   631
    (simp add: powr_add[symmetric])
hoelzl@47599
   632
hoelzl@47599
   633
subsection {* Rounding Floats *}
hoelzl@29804
   634
hoelzl@47599
   635
definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" where
hoelzl@47599
   636
  "float_up prec x = float_of (round_up prec (real x))"
hoelzl@47599
   637
hoelzl@47599
   638
lemma float_up_float: 
hoelzl@47599
   639
  "x \<in> float \<Longrightarrow> float_up prec (float_of x) = float_of (round_up prec x)"
hoelzl@47599
   640
  unfolding float_up_def by simp
hoelzl@29804
   641
hoelzl@47599
   642
lemma float_up_correct:
hoelzl@47599
   643
  shows "real (float_up e f) - real f \<in> {0..2 powr -e}"
hoelzl@47599
   644
unfolding atLeastAtMost_iff
hoelzl@47599
   645
proof
hoelzl@47599
   646
  have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp
hoelzl@47599
   647
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
hoelzl@47599
   648
  finally show "real (float_up e f) - real f \<le> 2 powr real (- e)"
hoelzl@47599
   649
    by (simp add: float_up_def)
hoelzl@47599
   650
qed (simp add: algebra_simps float_up_def round_up)
hoelzl@29804
   651
hoelzl@47599
   652
definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" where
hoelzl@47599
   653
  "float_down prec x = float_of (round_down prec (real x))"
hoelzl@47599
   654
hoelzl@47599
   655
lemma float_down_float: 
hoelzl@47599
   656
  "x \<in> float \<Longrightarrow> float_down prec (float_of x) = float_of (round_down prec x)"
hoelzl@47599
   657
  unfolding float_down_def by simp
obua@16782
   658
hoelzl@47599
   659
lemma float_down_correct:
hoelzl@47599
   660
  shows "real f - real (float_down e f) \<in> {0..2 powr -e}"
hoelzl@47599
   661
unfolding atLeastAtMost_iff
hoelzl@47599
   662
proof
hoelzl@47599
   663
  have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp
hoelzl@47599
   664
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
hoelzl@47599
   665
  finally show "real f - real (float_down e f) \<le> 2 powr real (- e)"
hoelzl@47599
   666
    by (simp add: float_down_def)
hoelzl@47599
   667
qed (simp add: algebra_simps float_down_def round_down)
hoelzl@47599
   668
hoelzl@47599
   669
lemma round_down_Float_id:
hoelzl@47599
   670
  assumes "p + e \<ge> 0"
hoelzl@47599
   671
  shows "round_down p (Float m e) = Float m e"
hoelzl@47599
   672
proof -
hoelzl@47599
   673
  from assms have r: "real e + real p = real (nat (e + p))" by simp
hoelzl@47599
   674
  have r: "\<lfloor>real (Float m e) * 2 powr real p\<rfloor> = real (Float m e) * 2 powr real p"
hoelzl@47599
   675
    by (auto intro: exI[where x="m*2^nat (e+p)"]
hoelzl@47599
   676
      simp add: ac_simps powr_add[symmetric] r powr_realpow)
hoelzl@47599
   677
  show ?thesis using assms
hoelzl@47599
   678
    unfolding round_down_def floor_divide_eq_div r
hoelzl@47599
   679
    by (simp add: ac_simps powr_add[symmetric])
hoelzl@47599
   680
qed
obua@24301
   681
hoelzl@47599
   682
lemma compute_float_down[code]:
hoelzl@47599
   683
  "float_down p (Float m e) =
hoelzl@47599
   684
    (if p + e < 0 then Float (m div 2^nat (-(p + e))) (-p) else Float m e)"
hoelzl@47599
   685
proof cases
hoelzl@47599
   686
  assume "p + e < 0"
hoelzl@47599
   687
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
hoelzl@47599
   688
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
hoelzl@47599
   689
  also have "... = 1 / 2 powr p / 2 powr e"
hoelzl@47599
   690
  unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
hoelzl@47599
   691
  finally show ?thesis
hoelzl@47599
   692
    unfolding float_down_def round_down_def floor_divide_eq_div[symmetric]
hoelzl@47599
   693
    using `p + e < 0` by (simp add: ac_simps)
hoelzl@47599
   694
next
hoelzl@47599
   695
  assume "\<not> p + e < 0" with round_down_Float_id show ?thesis by (simp add: float_down_def)
hoelzl@47599
   696
qed
obua@24301
   697
hoelzl@47599
   698
lemma ceil_divide_floor_conv:
hoelzl@47599
   699
assumes "b \<noteq> 0"
hoelzl@47599
   700
shows "\<lceil>real a / real b\<rceil> = (if b dvd a then a div b else \<lfloor>real a / real b\<rfloor> + 1)"
hoelzl@47599
   701
proof cases
hoelzl@47599
   702
  assume "\<not> b dvd a"
hoelzl@47599
   703
  hence "a mod b \<noteq> 0" by auto
hoelzl@47599
   704
  hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto
hoelzl@47599
   705
  have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
hoelzl@47599
   706
  apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
hoelzl@47599
   707
  proof -
hoelzl@47599
   708
    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
hoelzl@47599
   709
    moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
hoelzl@47599
   710
    apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto
hoelzl@47599
   711
    ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
hoelzl@47599
   712
  qed
hoelzl@47599
   713
  thus ?thesis using `\<not> b dvd a` by simp
hoelzl@47599
   714
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
hoelzl@47599
   715
  floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)
wenzelm@19765
   716
hoelzl@47599
   717
lemma round_up_Float_id:
hoelzl@47599
   718
  assumes "p + e \<ge> 0"
hoelzl@47599
   719
  shows "round_up p (Float m e) = Float m e"
hoelzl@29804
   720
proof -
hoelzl@47599
   721
  from assms have r1: "real e + real p = real (nat (e + p))" by simp
hoelzl@47599
   722
  have r: "\<lceil>real (Float m e) * 2 powr real p\<rceil> = real (Float m e) * 2 powr real p"
hoelzl@47599
   723
    by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow
hoelzl@47599
   724
      intro: exI[where x="m*2^nat (e+p)"])
hoelzl@47599
   725
  show ?thesis using assms
hoelzl@47599
   726
    unfolding float_up_def round_up_def floor_divide_eq_div Let_def r
hoelzl@47599
   727
    by (simp add: ac_simps powr_add[symmetric])
hoelzl@47599
   728
qed
hoelzl@47599
   729
hoelzl@47599
   730
lemma compute_float_up[code]:
hoelzl@47599
   731
  "float_up p (Float m e) =
hoelzl@47599
   732
    (let P = 2^nat (-(p + e)); r = m mod P in
hoelzl@47599
   733
      if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (-p) else Float m e)"
hoelzl@47599
   734
proof cases
hoelzl@47599
   735
  assume "p + e < 0"
hoelzl@47599
   736
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
hoelzl@47599
   737
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
hoelzl@47599
   738
  also have "... = 1 / 2 powr p / 2 powr e"
hoelzl@47599
   739
  unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
hoelzl@47599
   740
  finally have twopow_rewrite:
hoelzl@47599
   741
    "real ((2::int) ^ nat (- (p + e))) = 1 / 2 powr real p / 2 powr real e" .
hoelzl@47599
   742
  with `p + e < 0` have powr_rewrite:
hoelzl@47599
   743
    "2 powr real e * 2 powr real p = 1 / real ((2::int) ^ nat (- (p + e)))"
hoelzl@47599
   744
    unfolding powr_divide2 by simp
hoelzl@47599
   745
  show ?thesis
hoelzl@47599
   746
  proof cases
hoelzl@47599
   747
    assume "2^nat (-(p + e)) dvd m"
hoelzl@47599
   748
    with `p + e < 0` twopow_rewrite show ?thesis
hoelzl@47599
   749
      by (auto simp: ac_simps float_up_def round_up_def floor_divide_eq_div dvd_eq_mod_eq_0)
hoelzl@47599
   750
  next
hoelzl@47599
   751
    assume ndvd: "\<not> 2 ^ nat (- (p + e)) dvd m"
hoelzl@47599
   752
    have one_div: "real m * (1 / real ((2::int) ^ nat (- (p + e)))) =
hoelzl@47599
   753
      real m / real ((2::int) ^ nat (- (p + e)))"
hoelzl@47599
   754
      by (simp add: field_simps)
hoelzl@47599
   755
    have "real \<lceil>real m * (2 powr real e * 2 powr real p)\<rceil> =
hoelzl@47599
   756
      real \<lfloor>real m * (2 powr real e * 2 powr real p)\<rfloor> + 1"
hoelzl@47599
   757
      using ndvd unfolding powr_rewrite one_div
hoelzl@47599
   758
      by (subst ceil_divide_floor_conv) (auto simp: field_simps)
hoelzl@47599
   759
    thus ?thesis using `p + e < 0` twopow_rewrite
hoelzl@47599
   760
      by (auto simp: ac_simps Let_def float_up_def round_up_def floor_divide_eq_div[symmetric])
hoelzl@29804
   761
  qed
hoelzl@47599
   762
next
hoelzl@47599
   763
  assume "\<not> p + e < 0" with round_up_Float_id show ?thesis by (simp add: float_up_def)
hoelzl@29804
   764
qed
hoelzl@29804
   765
hoelzl@47599
   766
lemmas real_of_ints =
hoelzl@47599
   767
  real_of_int_zero
hoelzl@47599
   768
  real_of_one
hoelzl@47599
   769
  real_of_int_add
hoelzl@47599
   770
  real_of_int_minus
hoelzl@47599
   771
  real_of_int_diff
hoelzl@47599
   772
  real_of_int_mult
hoelzl@47599
   773
  real_of_int_power
hoelzl@47599
   774
  real_numeral
hoelzl@47599
   775
lemmas real_of_nats =
hoelzl@47599
   776
  real_of_nat_zero
hoelzl@47599
   777
  real_of_nat_one
hoelzl@47599
   778
  real_of_nat_1
hoelzl@47599
   779
  real_of_nat_add
hoelzl@47599
   780
  real_of_nat_mult
hoelzl@47599
   781
  real_of_nat_power
hoelzl@47599
   782
hoelzl@47599
   783
lemmas int_of_reals = real_of_ints[symmetric]
hoelzl@47599
   784
lemmas nat_of_reals = real_of_nats[symmetric]
hoelzl@47599
   785
hoelzl@47599
   786
lemma two_real_int: "(2::real) = real (2::int)" by simp
hoelzl@47599
   787
lemma two_real_nat: "(2::real) = real (2::nat)" by simp
hoelzl@47599
   788
hoelzl@47599
   789
lemma mult_cong: "a = c ==> b = d ==> a*b = c*d" by simp
hoelzl@47599
   790
hoelzl@47599
   791
subsection {* Compute bitlen of integers *}
hoelzl@47599
   792
hoelzl@47599
   793
definition bitlen::"int => int"
hoelzl@47599
   794
where "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
hoelzl@47599
   795
hoelzl@47599
   796
lemma bitlen_nonneg: "0 \<le> bitlen x"
hoelzl@29804
   797
proof -
hoelzl@47599
   798
  {
hoelzl@47599
   799
    assume "0 > x"
hoelzl@47599
   800
    have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
hoelzl@47599
   801
    also have "... < log 2 (-x)" using `0 > x` by auto
hoelzl@47599
   802
    finally have "-1 < log 2 (-x)" .
hoelzl@47599
   803
  } thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
hoelzl@47599
   804
qed
hoelzl@47599
   805
hoelzl@47599
   806
lemma bitlen_bounds:
hoelzl@47599
   807
  assumes "x > 0"
hoelzl@47599
   808
  shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
hoelzl@47599
   809
proof
hoelzl@47599
   810
  have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
hoelzl@47599
   811
    using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0`
hoelzl@47599
   812
    using real_nat_eq_real[of "floor (log 2 (real x))"]
hoelzl@47599
   813
    by simp
hoelzl@47599
   814
  also have "... \<le> 2 powr log 2 (real x)"
hoelzl@47599
   815
    by simp
hoelzl@47599
   816
  also have "... = real x"
hoelzl@47599
   817
    using `0 < x` by simp
hoelzl@47599
   818
  finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
hoelzl@47599
   819
  thus "2 ^ nat (bitlen x - 1) \<le> x" using `x > 0`
hoelzl@47599
   820
    by (simp add: bitlen_def)
hoelzl@47599
   821
next
hoelzl@47599
   822
  have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp
hoelzl@47599
   823
  also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
hoelzl@47599
   824
    apply (simp add: powr_realpow[symmetric])
hoelzl@47599
   825
    using `x > 0` by simp
hoelzl@47599
   826
  finally show "x < 2 ^ nat (bitlen x)" using `x > 0`
hoelzl@47599
   827
    by (simp add: bitlen_def ac_simps int_of_reals del: real_of_ints)
hoelzl@47599
   828
qed
hoelzl@47599
   829
hoelzl@47599
   830
lemma bitlen_pow2[simp]:
hoelzl@47599
   831
  assumes "b > 0"
hoelzl@47599
   832
  shows "bitlen (b * 2 ^ c) = bitlen b + c"
hoelzl@47599
   833
proof -
hoelzl@47599
   834
  from assms have "b * 2 ^ c > 0" by (auto intro: mult_pos_pos)
hoelzl@47599
   835
  thus ?thesis
hoelzl@47599
   836
    using floor_add[of "log 2 b" c] assms
hoelzl@47599
   837
    by (auto simp add: log_mult log_nat_power bitlen_def)
hoelzl@29804
   838
qed
hoelzl@29804
   839
hoelzl@47599
   840
lemma bitlen_Float:
hoelzl@47599
   841
fixes m e
hoelzl@47599
   842
defines "f \<equiv> Float m e"
hoelzl@47599
   843
shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
hoelzl@47599
   844
proof cases
hoelzl@47599
   845
  assume "m \<noteq> 0" hence "f \<noteq> float_of 0" by (simp add: f_def) hence "mantissa f \<noteq> 0"
hoelzl@47599
   846
    by (simp add: mantissa_noteq_0)
hoelzl@47599
   847
  moreover
hoelzl@47599
   848
  from f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`] guess i .
hoelzl@47599
   849
  ultimately show ?thesis by (simp add: abs_mult)
hoelzl@47599
   850
qed (simp add: f_def bitlen_def)
hoelzl@29804
   851
hoelzl@47599
   852
lemma compute_bitlen[code]:
hoelzl@47599
   853
  shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
hoelzl@47599
   854
proof -
hoelzl@47599
   855
  { assume "2 \<le> x"
hoelzl@47599
   856
    then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
hoelzl@47599
   857
      by (simp add: log_mult zmod_zdiv_equality')
hoelzl@47599
   858
    also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   859
    proof cases
hoelzl@47599
   860
      assume "x mod 2 = 0" then show ?thesis by simp
hoelzl@47599
   861
    next
hoelzl@47599
   862
      def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   863
      then have "0 \<le> n"
hoelzl@47599
   864
        using `2 \<le> x` by simp
hoelzl@47599
   865
      assume "x mod 2 \<noteq> 0"
hoelzl@47599
   866
      with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
hoelzl@47599
   867
      with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto
hoelzl@47599
   868
      moreover
hoelzl@47599
   869
      { have "real (2^nat n :: int) = 2 powr (nat n)"
hoelzl@47599
   870
          by (simp add: powr_realpow)
hoelzl@47599
   871
        also have "\<dots> \<le> 2 powr (log 2 x)"
hoelzl@47599
   872
          using `2 \<le> x` by (simp add: n_def del: powr_log_cancel)
hoelzl@47599
   873
        finally have "2^nat n \<le> x" using `2 \<le> x` by simp }
hoelzl@47599
   874
      ultimately have "2^nat n \<le> x - 1" by simp
hoelzl@47599
   875
      then have "2^nat n \<le> real (x - 1)"
hoelzl@47599
   876
        unfolding real_of_int_le_iff[symmetric] by simp
hoelzl@47599
   877
      { have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
hoelzl@47599
   878
          using `0 \<le> n` by (simp add: log_nat_power)
hoelzl@47599
   879
        also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
hoelzl@47599
   880
          using `2^nat n \<le> real (x - 1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono)
hoelzl@47599
   881
        finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
hoelzl@47599
   882
      moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
hoelzl@47599
   883
        using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono)
hoelzl@47599
   884
      ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"
hoelzl@47599
   885
        unfolding n_def `x mod 2 = 1` by auto
hoelzl@47599
   886
    qed
hoelzl@47599
   887
    finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }
hoelzl@47599
   888
  moreover
hoelzl@47599
   889
  { assume "x < 2" "0 < x"
hoelzl@47599
   890
    then have "x = 1" by simp
hoelzl@47599
   891
    then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }
hoelzl@47599
   892
  ultimately show ?thesis
hoelzl@47599
   893
    unfolding bitlen_def
hoelzl@47599
   894
    by (auto simp: pos_imp_zdiv_pos_iff not_le)
hoelzl@47599
   895
qed
hoelzl@29804
   896
hoelzl@47599
   897
lemma float_gt1_scale: assumes "1 \<le> Float m e"
hoelzl@47599
   898
  shows "0 \<le> e + (bitlen m - 1)"
hoelzl@47599
   899
proof -
hoelzl@47599
   900
  have "0 < Float m e" using assms by auto
hoelzl@47599
   901
  hence "0 < m" using powr_gt_zero[of 2 e]
hoelzl@47599
   902
    by (auto simp: less_float_def less_eq_float_def zero_less_mult_iff)
hoelzl@47599
   903
  hence "m \<noteq> 0" by auto
hoelzl@47599
   904
  show ?thesis
hoelzl@47599
   905
  proof (cases "0 \<le> e")
hoelzl@47599
   906
    case True thus ?thesis using `0 < m`  by (simp add: bitlen_def)
hoelzl@29804
   907
  next
hoelzl@47599
   908
    have "(1::int) < 2" by simp
hoelzl@47599
   909
    case False let ?S = "2^(nat (-e))"
hoelzl@47599
   910
    have "inverse (2 ^ nat (- e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (-e)"]
hoelzl@47599
   911
      by (auto simp: powr_minus field_simps inverse_eq_divide)
hoelzl@47599
   912
    hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
hoelzl@47599
   913
      by (auto simp: powr_minus)
hoelzl@47599
   914
    hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
hoelzl@47599
   915
    hence "?S \<le> real m" unfolding mult_assoc by auto
hoelzl@47599
   916
    hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@47599
   917
    from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
hoelzl@47599
   918
    have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
hoelzl@47599
   919
    hence "-e < bitlen m" using False by auto
hoelzl@47599
   920
    thus ?thesis by auto
hoelzl@29804
   921
  qed
hoelzl@47599
   922
qed
hoelzl@29804
   923
hoelzl@29804
   924
lemma bitlen_div: assumes "0 < m" shows "1 \<le> real m / 2^nat (bitlen m - 1)" and "real m / 2^nat (bitlen m - 1) < 2"
hoelzl@29804
   925
proof -
hoelzl@29804
   926
  let ?B = "2^nat(bitlen m - 1)"
hoelzl@29804
   927
hoelzl@29804
   928
  have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
hoelzl@29804
   929
  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   930
  thus "1 \<le> real m / ?B" by auto
hoelzl@29804
   931
hoelzl@29804
   932
  have "m \<noteq> 0" using assms by auto
hoelzl@47599
   933
  have "0 \<le> bitlen m - 1" using `0 < m` by (auto simp: bitlen_def)
obua@16782
   934
hoelzl@29804
   935
  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
hoelzl@47599
   936
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)
hoelzl@29804
   937
  also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
hoelzl@29804
   938
  finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
hoelzl@29804
   939
  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
hoelzl@29804
   940
  thus "real m / ?B < 2" by auto
hoelzl@29804
   941
qed
hoelzl@29804
   942
hoelzl@47599
   943
subsection {* Approximation of positive rationals *}
hoelzl@47599
   944
hoelzl@47599
   945
lemma zdiv_zmult_twopow_eq: fixes a b::int shows "a div b div (2 ^ n) = a div (b * 2 ^ n)"
hoelzl@47599
   946
by (simp add: zdiv_zmult2_eq)
hoelzl@29804
   947
hoelzl@47599
   948
lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
hoelzl@47599
   949
  by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
hoelzl@29804
   950
hoelzl@47599
   951
lemma real_div_nat_eq_floor_of_divide:
hoelzl@47599
   952
  fixes a b::nat
hoelzl@47599
   953
  shows "a div b = real (floor (a/b))"
hoelzl@47599
   954
by (metis floor_divide_eq_div real_of_int_of_nat_eq zdiv_int)
hoelzl@29804
   955
hoelzl@47599
   956
definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"
hoelzl@29804
   957
hoelzl@47599
   958
definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" where
hoelzl@47599
   959
  "lapprox_posrat prec x y = float_of (round_down (rat_precision prec x y) (x / y))"
obua@16782
   960
hoelzl@47599
   961
lemma compute_lapprox_posrat[code]:
hoelzl@47599
   962
  fixes prec x y 
hoelzl@47599
   963
  shows "lapprox_posrat prec x y = 
hoelzl@47599
   964
   (let 
hoelzl@47599
   965
       l = rat_precision prec x y;
hoelzl@47599
   966
       d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
hoelzl@47599
   967
    in normfloat (Float d (- l)))"
hoelzl@47599
   968
    unfolding lapprox_posrat_def div_mult_twopow_eq
hoelzl@47599
   969
    by (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide
hoelzl@47599
   970
                  field_simps Let_def
hoelzl@47599
   971
             del: two_powr_minus_int_float)
hoelzl@29804
   972
hoelzl@47599
   973
definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" where
hoelzl@47599
   974
  "rapprox_posrat prec x y = float_of (round_up (rat_precision prec x y) (x / y))"
hoelzl@29804
   975
hoelzl@47599
   976
(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *)
hoelzl@47599
   977
lemma compute_rapprox_posrat[code]:
hoelzl@47599
   978
  fixes prec x y
hoelzl@47599
   979
  defines "l \<equiv> rat_precision prec x y"
hoelzl@47599
   980
  shows "rapprox_posrat prec x y = (let
hoelzl@47599
   981
     l = l ;
hoelzl@47599
   982
     X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l)) ;
hoelzl@47599
   983
     d = fst X div snd X ;
hoelzl@47599
   984
     m = fst X mod snd X
hoelzl@47599
   985
   in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
hoelzl@47599
   986
proof (cases "y = 0")
hoelzl@47599
   987
  assume "y = 0" thus ?thesis by (simp add: rapprox_posrat_def Let_def)
hoelzl@47599
   988
next
hoelzl@47599
   989
  assume "y \<noteq> 0"
hoelzl@29804
   990
  show ?thesis
hoelzl@47599
   991
  proof (cases "0 \<le> l")
hoelzl@47599
   992
    assume "0 \<le> l"
hoelzl@47599
   993
    def x' == "x * 2 ^ nat l"
hoelzl@47599
   994
    have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
hoelzl@47599
   995
    moreover have "real x * 2 powr real l = real x'"
hoelzl@47599
   996
      by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def)
hoelzl@47599
   997
    ultimately show ?thesis
hoelzl@47599
   998
      unfolding rapprox_posrat_def round_up_def l_def[symmetric]
hoelzl@47599
   999
      using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0`
hoelzl@47599
  1000
      by (simp add: Let_def floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 int_of_reals
hoelzl@47599
  1001
               del: real_of_ints)
hoelzl@47599
  1002
   next
hoelzl@47599
  1003
    assume "\<not> 0 \<le> l"
hoelzl@47599
  1004
    def y' == "y * 2 ^ nat (- l)"
hoelzl@47599
  1005
    from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def)
hoelzl@47599
  1006
    have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
hoelzl@47599
  1007
    moreover have "real x * real (2::int) powr real l / real y = x / real y'"
hoelzl@47599
  1008
      using `\<not> 0 \<le> l`
hoelzl@47599
  1009
      by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps inverse_eq_divide)
hoelzl@47599
  1010
    ultimately show ?thesis
hoelzl@47599
  1011
      using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0`
hoelzl@47599
  1012
      by (simp add: rapprox_posrat_def l_def round_up_def ceil_divide_floor_conv
hoelzl@47599
  1013
                    floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 int_of_reals
hoelzl@47599
  1014
               del: real_of_ints)
hoelzl@29804
  1015
  qed
hoelzl@29804
  1016
qed
hoelzl@29804
  1017
hoelzl@47599
  1018
hoelzl@47599
  1019
lemma rat_precision_pos:
hoelzl@47599
  1020
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@47599
  1021
  shows "rat_precision n (int x) (int y) > 0"
hoelzl@29804
  1022
proof -
hoelzl@47599
  1023
  { assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }
hoelzl@47599
  1024
  hence "bitlen (int x) < bitlen (int y)" using assms
hoelzl@47599
  1025
    by (simp add: bitlen_def del: floor_add_one)
hoelzl@47599
  1026
      (auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one)
hoelzl@47599
  1027
  thus ?thesis
hoelzl@47599
  1028
    using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
hoelzl@29804
  1029
qed
obua@16782
  1030
hoelzl@47599
  1031
lemma power_aux: assumes "x > 0" shows "(2::int) ^ nat (x - 1) \<le> 2 ^ nat x - 1"
hoelzl@47599
  1032
proof -
hoelzl@47599
  1033
  def y \<equiv> "nat (x - 1)" moreover
hoelzl@47599
  1034
  have "(2::int) ^ y \<le> (2 ^ (y + 1)) - 1" by simp
hoelzl@47599
  1035
  ultimately show ?thesis using assms by simp
hoelzl@29804
  1036
qed
hoelzl@29804
  1037
hoelzl@29804
  1038
lemma rapprox_posrat_less1: assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@31098
  1039
  shows "real (rapprox_posrat n x y) < 1"
hoelzl@47599
  1040
proof -
hoelzl@47599
  1041
  have powr1: "2 powr real (rat_precision n (int x) (int y)) = 
hoelzl@47599
  1042
    2 ^ nat (rat_precision n (int x) (int y))" using rat_precision_pos[of x y n] assms
hoelzl@47599
  1043
    by (simp add: powr_realpow[symmetric])
hoelzl@47599
  1044
  have "x * 2 powr real (rat_precision n (int x) (int y)) / y = (x / y) *
hoelzl@47599
  1045
     2 powr real (rat_precision n (int x) (int y))" by simp
hoelzl@47599
  1046
  also have "... < (1 / 2) * 2 powr real (rat_precision n (int x) (int y))"
hoelzl@47599
  1047
    apply (rule mult_strict_right_mono) by (insert assms) auto
hoelzl@47599
  1048
  also have "\<dots> = 2 powr real (rat_precision n (int x) (int y) - 1)"
hoelzl@47599
  1049
    by (simp add: powr_add diff_def powr_neg_numeral)
hoelzl@47599
  1050
  also have "\<dots> = 2 ^ nat (rat_precision n (int x) (int y) - 1)"
hoelzl@47599
  1051
    using rat_precision_pos[of x y n] assms by (simp add: powr_realpow[symmetric])
hoelzl@47599
  1052
  also have "\<dots> \<le> 2 ^ nat (rat_precision n (int x) (int y)) - 1"
hoelzl@47599
  1053
    unfolding int_of_reals real_of_int_le_iff
hoelzl@47599
  1054
    using rat_precision_pos[OF assms] by (rule power_aux)
hoelzl@47599
  1055
  finally show ?thesis unfolding rapprox_posrat_def
hoelzl@47599
  1056
    apply (simp add: round_up_def)
hoelzl@47599
  1057
    apply (simp add: round_up_def field_simps powr_minus inverse_eq_divide)
hoelzl@47599
  1058
    unfolding powr1
hoelzl@47599
  1059
    unfolding int_of_reals real_of_int_less_iff
hoelzl@47599
  1060
    unfolding ceiling_less_eq using rat_precision_pos[of x y n] assms apply simp done
hoelzl@29804
  1061
qed
hoelzl@29804
  1062
hoelzl@47599
  1063
definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" where
hoelzl@47599
  1064
  "lapprox_rat prec x y = float_of (round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y))"
obua@16782
  1065
hoelzl@29804
  1066
lemma compute_lapprox_rat[code]:
hoelzl@47599
  1067
  "lapprox_rat prec x y =
hoelzl@47599
  1068
    (if y = 0 then 0
hoelzl@47599
  1069
    else if 0 \<le> x then
hoelzl@47599
  1070
      (if 0 < y then lapprox_posrat prec (nat x) (nat y)
hoelzl@47599
  1071
      else - (rapprox_posrat prec (nat x) (nat (-y)))) 
hoelzl@47599
  1072
      else (if 0 < y
hoelzl@47599
  1073
        then - (rapprox_posrat prec (nat (-x)) (nat y))
hoelzl@47599
  1074
        else lapprox_posrat prec (nat (-x)) (nat (-y))))"
hoelzl@47599
  1075
  apply (cases "y = 0")
hoelzl@47599
  1076
  apply (simp add: lapprox_posrat_def rapprox_posrat_def round_down_def lapprox_rat_def)
hoelzl@47599
  1077
  apply (auto simp: lapprox_rat_def lapprox_posrat_def rapprox_posrat_def round_up_def round_down_def
hoelzl@47599
  1078
        ceiling_def real_of_float_uminus[symmetric] ac_simps int_of_reals simp del: real_of_ints)
hoelzl@47599
  1079
  apply (auto simp: ac_simps)
hoelzl@47599
  1080
  done
hoelzl@47599
  1081
hoelzl@47599
  1082
definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" where
hoelzl@47599
  1083
  "rapprox_rat prec x y = float_of (round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y))"
hoelzl@47599
  1084
hoelzl@47599
  1085
lemma compute_rapprox_rat[code]:
hoelzl@47599
  1086
  "rapprox_rat prec x y =
hoelzl@47599
  1087
    (if y = 0 then 0
hoelzl@47599
  1088
    else if 0 \<le> x then
hoelzl@47599
  1089
      (if 0 < y then rapprox_posrat prec (nat x) (nat y)
hoelzl@47599
  1090
      else - (lapprox_posrat prec (nat x) (nat (-y)))) 
hoelzl@47599
  1091
      else (if 0 < y
hoelzl@47599
  1092
        then - (lapprox_posrat prec (nat (-x)) (nat y))
hoelzl@47599
  1093
        else rapprox_posrat prec (nat (-x)) (nat (-y))))"
hoelzl@47599
  1094
  apply (cases "y = 0", simp add: lapprox_posrat_def rapprox_posrat_def round_up_def rapprox_rat_def)
hoelzl@47599
  1095
  apply (auto simp: rapprox_rat_def lapprox_posrat_def rapprox_posrat_def round_up_def round_down_def
hoelzl@47599
  1096
        ceiling_def real_of_float_uminus[symmetric] ac_simps int_of_reals simp del: real_of_ints)
hoelzl@47599
  1097
  apply (auto simp: ac_simps)
hoelzl@47599
  1098
  done
hoelzl@47599
  1099
hoelzl@47599
  1100
subsection {* Division *}
hoelzl@47599
  1101
hoelzl@47599
  1102
definition div_precision
hoelzl@47599
  1103
where "div_precision prec x y =
hoelzl@47599
  1104
  rat_precision prec \<bar>mantissa x\<bar> \<bar>mantissa y\<bar> - exponent x + exponent y"
hoelzl@47599
  1105
hoelzl@47599
  1106
definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
hoelzl@47599
  1107
where "float_divl prec a b =
hoelzl@47599
  1108
  float_of (round_down (div_precision prec a b) (a / b))"
hoelzl@47599
  1109
hoelzl@47599
  1110
lemma compute_float_divl[code]:
hoelzl@47599
  1111
  fixes m1 s1 m2 s2
hoelzl@47599
  1112
  defines "f1 \<equiv> Float m1 s1" and "f2 \<equiv> Float m2 s2"
hoelzl@47599
  1113
  shows "float_divl prec f1 f2 = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
hoelzl@47599
  1114
proof cases
hoelzl@47599
  1115
  assume "f1 \<noteq> 0 \<and> f2 \<noteq> 0"
hoelzl@47599
  1116
  then have "f1 \<noteq> float_of 0" "f2 \<noteq> float_of 0" by auto
hoelzl@47599
  1117
  with mantissa_not_dvd[of f1] mantissa_not_dvd[of f2]
hoelzl@47599
  1118
  have "mantissa f1 \<noteq> 0" "mantissa f2 \<noteq> 0"
hoelzl@47599
  1119
    by (auto simp add: dvd_def)  
hoelzl@47599
  1120
  then have pos: "0 < \<bar>mantissa f1\<bar>" "0 < \<bar>mantissa f2\<bar>"
hoelzl@47599
  1121
    by simp_all
hoelzl@47599
  1122
  moreover from f1_def[THEN denormalize_shift, OF `f1 \<noteq> float_of 0`] guess i . note i = this
hoelzl@47599
  1123
  moreover from f2_def[THEN denormalize_shift, OF `f2 \<noteq> float_of 0`] guess j . note j = this
hoelzl@47599
  1124
  moreover have "(real (mantissa f1) * 2 ^ i / (real (mantissa f2) * 2 ^ j))
hoelzl@47599
  1125
    = (real (mantissa f1) / real (mantissa f2)) * 2 powr (int i - int j)"
hoelzl@47599
  1126
    by (simp add: powr_divide2[symmetric] powr_realpow)
hoelzl@47599
  1127
  moreover have "real f1 / real f2 = real (mantissa f1) / real (mantissa f2) * 2 powr real (exponent f1 - exponent f2)"
hoelzl@47599
  1128
    unfolding mantissa_exponent by (simp add: powr_divide2[symmetric])
hoelzl@47599
  1129
  moreover have "rat_precision prec (\<bar>mantissa f1\<bar> * 2 ^ i) (\<bar>mantissa f2\<bar> * 2 ^ j) =
hoelzl@47599
  1130
    rat_precision prec \<bar>mantissa f1\<bar> \<bar>mantissa f2\<bar> + j - i"
hoelzl@47599
  1131
    using pos by (simp add: rat_precision_def)
hoelzl@47599
  1132
  ultimately show ?thesis
hoelzl@47599
  1133
    unfolding float_divl_def lapprox_rat_def div_precision_def
hoelzl@47599
  1134
    by (simp add: abs_mult round_down_shift powr_divide2[symmetric]
hoelzl@47599
  1135
                del: int_nat_eq real_of_int_diff times_divide_eq_left )
hoelzl@47599
  1136
       (simp add: field_simps powr_divide2[symmetric] powr_add)
hoelzl@47599
  1137
next
hoelzl@47599
  1138
  assume "\<not> (f1 \<noteq> 0 \<and> f2 \<noteq> 0)" then show ?thesis
hoelzl@47599
  1139
    by (auto simp add: float_divl_def f1_def f2_def lapprox_rat_def)
hoelzl@47599
  1140
qed  
hoelzl@47599
  1141
hoelzl@47599
  1142
definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
hoelzl@47599
  1143
where "float_divr prec a b =
hoelzl@47599
  1144
  float_of (round_up (div_precision prec a b) (a / b))"
hoelzl@47599
  1145
hoelzl@47599
  1146
lemma compute_float_divr[code]:
hoelzl@47599
  1147
  fixes m1 s1 m2 s2
hoelzl@47599
  1148
  defines "f1 \<equiv> Float m1 s1" and "f2 \<equiv> Float m2 s2"
hoelzl@47599
  1149
  shows "float_divr prec f1 f2 = rapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
hoelzl@47599
  1150
proof cases
hoelzl@47599
  1151
  assume "f1 \<noteq> 0 \<and> f2 \<noteq> 0"
hoelzl@47599
  1152
  then have "f1 \<noteq> float_of 0" "f2 \<noteq> float_of 0" by auto
hoelzl@47599
  1153
  with mantissa_not_dvd[of f1] mantissa_not_dvd[of f2]
hoelzl@47599
  1154
  have "mantissa f1 \<noteq> 0" "mantissa f2 \<noteq> 0"
hoelzl@47599
  1155
    by (auto simp add: dvd_def)  
hoelzl@47599
  1156
  then have pos: "0 < \<bar>mantissa f1\<bar>" "0 < \<bar>mantissa f2\<bar>"
hoelzl@47599
  1157
    by simp_all
hoelzl@47599
  1158
  moreover from f1_def[THEN denormalize_shift, OF `f1 \<noteq> float_of 0`] guess i . note i = this
hoelzl@47599
  1159
  moreover from f2_def[THEN denormalize_shift, OF `f2 \<noteq> float_of 0`] guess j . note j = this
hoelzl@47599
  1160
  moreover have "(real (mantissa f1) * 2 ^ i / (real (mantissa f2) * 2 ^ j))
hoelzl@47599
  1161
    = (real (mantissa f1) / real (mantissa f2)) * 2 powr (int i - int j)"
hoelzl@47599
  1162
    by (simp add: powr_divide2[symmetric] powr_realpow)
hoelzl@47599
  1163
  moreover have "real f1 / real f2 = real (mantissa f1) / real (mantissa f2) * 2 powr real (exponent f1 - exponent f2)"
hoelzl@47599
  1164
    unfolding mantissa_exponent by (simp add: powr_divide2[symmetric])
hoelzl@47599
  1165
  moreover have "rat_precision prec (\<bar>mantissa f1\<bar> * 2 ^ i) (\<bar>mantissa f2\<bar> * 2 ^ j) =
hoelzl@47599
  1166
    rat_precision prec \<bar>mantissa f1\<bar> \<bar>mantissa f2\<bar> + j - i"
hoelzl@47599
  1167
    using pos by (simp add: rat_precision_def)
hoelzl@47599
  1168
  ultimately show ?thesis
hoelzl@47599
  1169
    unfolding float_divr_def rapprox_rat_def div_precision_def
hoelzl@47599
  1170
    by (simp add: abs_mult round_up_shift powr_divide2[symmetric]
hoelzl@47599
  1171
                del: int_nat_eq real_of_int_diff times_divide_eq_left)
hoelzl@47599
  1172
       (simp add: field_simps powr_divide2[symmetric] powr_add)
hoelzl@47599
  1173
next
hoelzl@47599
  1174
  assume "\<not> (f1 \<noteq> 0 \<and> f2 \<noteq> 0)" then show ?thesis
hoelzl@47599
  1175
    by (auto simp add: float_divr_def f1_def f2_def rapprox_rat_def)
hoelzl@29804
  1176
qed
obua@16782
  1177
hoelzl@47599
  1178
subsection {* Lemmas needed by Approximate *}
hoelzl@47599
  1179
hoelzl@47599
  1180
declare one_float_def[simp del] zero_float_def[simp del]
hoelzl@47599
  1181
hoelzl@47599
  1182
lemma Float_num[simp]: shows
hoelzl@47599
  1183
   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
hoelzl@47599
  1184
   "real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and
hoelzl@47599
  1185
   "real (Float -1 0) = -1" and "real (Float (number_of n) 0) = number_of n"
hoelzl@47599
  1186
using two_powr_int_float[of 2] two_powr_int_float[of "-1"] two_powr_int_float[of "-2"] two_powr_int_float[of "-3"]
hoelzl@47599
  1187
using powr_realpow[of 2 2] powr_realpow[of 2 3]
hoelzl@47599
  1188
using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3]
hoelzl@47599
  1189
by auto
hoelzl@47599
  1190
hoelzl@47599
  1191
lemma real_of_Float_int[simp]: "real (Float n 0) = real n" by simp
hoelzl@47599
  1192
hoelzl@47599
  1193
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
hoelzl@47599
  1194
hoelzl@47599
  1195
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
hoelzl@47599
  1196
by arith
hoelzl@29804
  1197
hoelzl@47599
  1198
lemma lapprox_rat:
hoelzl@47599
  1199
  shows "real (lapprox_rat prec x y) \<le> real x / real y"
hoelzl@47599
  1200
  using round_down by (simp add: lapprox_rat_def)
obua@16782
  1201
hoelzl@47599
  1202
lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"
hoelzl@47599
  1203
proof -
hoelzl@47599
  1204
  from zmod_zdiv_equality'[of a b]
hoelzl@47599
  1205
  have "a = b * (a div b) + a mod b" by simp
hoelzl@47599
  1206
  also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)
hoelzl@47599
  1207
  using assms by simp
hoelzl@47599
  1208
  finally show ?thesis by simp
hoelzl@47599
  1209
qed
hoelzl@47599
  1210
hoelzl@47599
  1211
lemma lapprox_rat_nonneg:
hoelzl@47599
  1212
  fixes n x y
hoelzl@47599
  1213
  defines "p == int n - ((bitlen \<bar>x\<bar>) - (bitlen \<bar>y\<bar>))"
hoelzl@47599
  1214
  assumes "0 \<le> x" "0 < y"
hoelzl@47599
  1215
  shows "0 \<le> real (lapprox_rat n x y)"
hoelzl@47599
  1216
using assms unfolding lapprox_rat_def p_def[symmetric] round_down_def real_of_int_minus[symmetric]
hoelzl@47599
  1217
   powr_int[of 2, simplified]
hoelzl@47599
  1218
  by (auto simp add: inverse_eq_divide intro!: mult_nonneg_nonneg divide_nonneg_pos mult_pos_pos)
obua@16782
  1219
hoelzl@31098
  1220
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
hoelzl@47599
  1221
  using round_up by (simp add: rapprox_rat_def)
hoelzl@47599
  1222
hoelzl@47599
  1223
lemma rapprox_rat_le1:
hoelzl@47599
  1224
  fixes n x y
hoelzl@47599
  1225
  assumes xy: "0 \<le> x" "0 < y" "x \<le> y"
hoelzl@47599
  1226
  shows "real (rapprox_rat n x y) \<le> 1"
hoelzl@47599
  1227
proof -
hoelzl@47599
  1228
  have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"
hoelzl@47599
  1229
    using xy unfolding bitlen_def by (auto intro!: floor_mono)
hoelzl@47599
  1230
  then have "0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>" by (simp add: rat_precision_def)
hoelzl@47599
  1231
  have "real \<lceil>real x / real y * 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>
hoelzl@47599
  1232
      \<le> real \<lceil>2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>"
hoelzl@47599
  1233
    using xy by (auto intro!: ceiling_mono simp: field_simps)
hoelzl@47599
  1234
  also have "\<dots> = 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"
hoelzl@47599
  1235
    using `0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>`
hoelzl@47599
  1236
    by (auto intro!: exI[of _ "2^nat (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"] simp: powr_int)
hoelzl@47599
  1237
  finally show ?thesis
hoelzl@47599
  1238
    by (simp add: rapprox_rat_def round_up_def)
hoelzl@47599
  1239
       (simp add: powr_minus inverse_eq_divide)
hoelzl@29804
  1240
qed
obua@16782
  1241
hoelzl@47599
  1242
lemma rapprox_rat_nonneg_neg: 
hoelzl@47599
  1243
  "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
hoelzl@47599
  1244
  unfolding rapprox_rat_def round_up_def
hoelzl@47599
  1245
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
obua@16782
  1246
hoelzl@47599
  1247
lemma rapprox_rat_neg:
hoelzl@47599
  1248
  "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
hoelzl@47599
  1249
  unfolding rapprox_rat_def round_up_def
hoelzl@47599
  1250
  by (auto simp: field_simps mult_le_0_iff)
hoelzl@29804
  1251
hoelzl@47599
  1252
lemma rapprox_rat_nonpos_pos:
hoelzl@47599
  1253
  "x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
hoelzl@47599
  1254
  unfolding rapprox_rat_def round_up_def
hoelzl@47599
  1255
  by (auto simp: field_simps mult_le_0_iff)
obua@16782
  1256
hoelzl@31098
  1257
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
hoelzl@47599
  1258
  using round_down by (simp add: float_divl_def)
hoelzl@47599
  1259
hoelzl@47599
  1260
lemma float_divl_lower_bound:
hoelzl@47599
  1261
  fixes x y prec
hoelzl@47599
  1262
  defines "p == rat_precision prec \<bar>mantissa x\<bar> \<bar>mantissa y\<bar> - exponent x + exponent y"
hoelzl@47599
  1263
  assumes xy: "0 \<le> x" "0 < y" shows "0 \<le> real (float_divl prec x y)"
hoelzl@47599
  1264
  using xy unfolding float_divl_def p_def[symmetric] round_down_def
hoelzl@47599
  1265
  by (simp add: zero_le_mult_iff zero_le_divide_iff less_eq_float_def less_float_def)
hoelzl@47599
  1266
hoelzl@47599
  1267
lemma exponent_1: "exponent 1 = 0"
hoelzl@47599
  1268
  using exponent_float[of 1 0] by (simp add: one_float_def)
hoelzl@47599
  1269
hoelzl@47599
  1270
lemma mantissa_1: "mantissa 1 = 1"
hoelzl@47599
  1271
  using mantissa_float[of 1 0] by (simp add: one_float_def)
obua@16782
  1272
hoelzl@47599
  1273
lemma bitlen_1: "bitlen 1 = 1"
hoelzl@47599
  1274
  by (simp add: bitlen_def)
hoelzl@47599
  1275
hoelzl@47599
  1276
lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
hoelzl@47599
  1277
proof
hoelzl@47599
  1278
  assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp
hoelzl@47599
  1279
  show "x = 0" by (simp add: zero_float_def z)
hoelzl@47599
  1280
qed (simp add: zero_float_def)
obua@16782
  1281
hoelzl@47599
  1282
lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
hoelzl@47599
  1283
proof (cases "x = 0", simp)
hoelzl@47599
  1284
  assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto
hoelzl@47599
  1285
  have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
hoelzl@47599
  1286
  also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
hoelzl@47599
  1287
  also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
hoelzl@47599
  1288
    using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0`
hoelzl@47599
  1289
    by (simp add: powr_int) (simp only: two_real_int int_of_reals real_of_int_abs[symmetric]
hoelzl@47599
  1290
      real_of_int_le_iff less_imp_le)
hoelzl@47599
  1291
  finally show ?thesis by (simp add: powr_add)
hoelzl@29804
  1292
qed
hoelzl@29804
  1293
wenzelm@41528
  1294
lemma float_divl_pos_less1_bound:
hoelzl@47599
  1295
  assumes "0 < real x" and "real x < 1" and "prec \<ge> 1"
hoelzl@47599
  1296
  shows "1 \<le> real (float_divl prec 1 x)"
hoelzl@47599
  1297
proof cases
hoelzl@47599
  1298
  assume nonneg: "div_precision prec 1 x \<ge> 0"
hoelzl@47599
  1299
  hence "2 powr real (div_precision prec 1 x) =
hoelzl@47599
  1300
    floor (real ((2::int) ^ nat (div_precision prec 1 x))) * floor (1::real)"
hoelzl@47599
  1301
    by (simp add: powr_int del: real_of_int_power) simp
hoelzl@47599
  1302
  also have "floor (1::real) \<le> floor (1 / x)" using assms by simp
hoelzl@47599
  1303
  also have "floor (real ((2::int) ^ nat (div_precision prec 1 x))) * floor (1 / x) \<le>
hoelzl@47599
  1304
    floor (real ((2::int) ^ nat (div_precision prec 1 x)) * (1 / x))"
hoelzl@47599
  1305
    by (rule le_mult_floor) (auto simp: assms less_imp_le)
hoelzl@47599
  1306
  finally have "2 powr real (div_precision prec 1 x) <=
hoelzl@47599
  1307
    floor (2 powr nat (div_precision prec 1 x) / x)" by (simp add: powr_realpow)
hoelzl@47599
  1308
  thus ?thesis
hoelzl@47599
  1309
    using assms nonneg
hoelzl@47599
  1310
    unfolding float_divl_def round_down_def
hoelzl@47599
  1311
    by simp (simp add: powr_minus inverse_eq_divide)
hoelzl@47599
  1312
next
hoelzl@47599
  1313
  assume neg: "\<not> 0 \<le> div_precision prec 1 x"
hoelzl@47599
  1314
  have "1 \<le> 1 * 2 powr (prec - 1)" using assms by (simp add: powr_realpow)
hoelzl@47599
  1315
  also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x) / x * 2 powr (prec - 1)"
hoelzl@47599
  1316
    apply (rule mult_mono) using assms float_upper_bound
hoelzl@47599
  1317
    by (auto intro!: divide_nonneg_pos)
hoelzl@47599
  1318
  also have "2 powr (bitlen \<bar>mantissa x\<bar> + exponent x) / x * 2 powr (prec - 1) =
hoelzl@47599
  1319
    2 powr real (div_precision prec 1 x) / real x"
hoelzl@47599
  1320
    using assms
hoelzl@47599
  1321
    apply (simp add: div_precision_def rat_precision_def diff_diff_eq2
hoelzl@47599
  1322
    mantissa_1 exponent_1 bitlen_1 powr_add powr_minus real_of_nat_diff)
hoelzl@47599
  1323
    apply (simp only: diff_def powr_add)
hoelzl@47599
  1324
    apply simp
hoelzl@47599
  1325
    done
hoelzl@47599
  1326
  finally have "1 \<le> \<lfloor>2 powr real (div_precision prec 1 x) / real x\<rfloor>"
hoelzl@47599
  1327
    using floor_mono[of "1::real"] by simp thm mult_mono
hoelzl@47599
  1328
  hence "1 \<le> real \<lfloor>2 powr real (div_precision prec 1 x) / real x\<rfloor>"
hoelzl@47599
  1329
    by (metis floor_real_of_int one_le_floor)
hoelzl@47599
  1330
  hence "1 * 1 \<le>
hoelzl@47599
  1331
    real \<lfloor>2 powr real (div_precision prec 1 x) / real x\<rfloor> * 2 powr - real (div_precision prec 1 x)"
hoelzl@47599
  1332
  apply (rule mult_mono)
hoelzl@47599
  1333
    using assms neg
hoelzl@47599
  1334
    by (auto intro: divide_nonneg_pos mult_nonneg_nonneg simp: real_of_int_minus[symmetric] powr_int simp del: real_of_int_minus) find_theorems "real (- _)"
hoelzl@47599
  1335
  thus ?thesis
hoelzl@47599
  1336
    using assms neg
hoelzl@47599
  1337
    unfolding float_divl_def round_down_def
hoelzl@47599
  1338
    by simp
hoelzl@29804
  1339
qed
obua@16782
  1340
hoelzl@31098
  1341
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
hoelzl@47599
  1342
  using round_up by (simp add: float_divr_def)
obua@16782
  1343
hoelzl@29804
  1344
lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x"
hoelzl@29804
  1345
proof -
hoelzl@31098
  1346
  have "1 \<le> 1 / real x" using `0 < x` and `x < 1` unfolding less_float_def by auto
hoelzl@31098
  1347
  also have "\<dots> \<le> real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
hoelzl@47599
  1348
  finally show ?thesis unfolding less_eq_float_def by auto
hoelzl@29804
  1349
qed
hoelzl@29804
  1350
hoelzl@47599
  1351
lemma float_divr_nonpos_pos_upper_bound:
hoelzl@47599
  1352
  assumes "real x \<le> 0" and "0 < real y"
hoelzl@47599
  1353
  shows "real (float_divr prec x y) \<le> 0"
hoelzl@47599
  1354
using assms
hoelzl@47599
  1355
unfolding float_divr_def round_up_def
hoelzl@47599
  1356
by (auto simp: field_simps mult_le_0_iff divide_le_0_iff)
obua@16782
  1357
hoelzl@47599
  1358
lemma float_divr_nonneg_neg_upper_bound:
hoelzl@47599
  1359
  assumes "0 \<le> real x" and "real y < 0"
hoelzl@47599
  1360
  shows "real (float_divr prec x y) \<le> 0"
hoelzl@47599
  1361
using assms
hoelzl@47599
  1362
unfolding float_divr_def round_up_def
hoelzl@47599
  1363
by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff divide_le_0_iff)
hoelzl@47599
  1364
hoelzl@47599
  1365
definition "round_prec p f = int p - (bitlen \<bar>mantissa f\<bar> + exponent f)"
hoelzl@47599
  1366
hoelzl@47599
  1367
definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@47599
  1368
"float_round_down prec f = float_of (round_down (round_prec prec f) f)"
hoelzl@47599
  1369
hoelzl@47599
  1370
definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@47599
  1371
"float_round_up prec f = float_of (round_up (round_prec prec f) f)"
obua@16782
  1372
hoelzl@47599
  1373
lemma compute_float_round_down[code]:
hoelzl@47599
  1374
fixes prec m e
hoelzl@47599
  1375
defines "d == bitlen (abs m) - int prec"
hoelzl@47599
  1376
defines "P == 2^nat d"
hoelzl@47599
  1377
defines "f == Float m e"
hoelzl@47599
  1378
shows "float_round_down prec f = (let d = d in
hoelzl@47599
  1379
    if 0 < d then let P = P ; n = m div P in Float n (e + d)
hoelzl@47599
  1380
             else f)"
hoelzl@47599
  1381
  unfolding float_round_down_def float_down_def[symmetric]
hoelzl@47599
  1382
    compute_float_down f_def Let_def P_def round_prec_def d_def bitlen_Float
hoelzl@47599
  1383
  by (simp add: field_simps)
hoelzl@47599
  1384
  
hoelzl@47599
  1385
lemma compute_float_round_up[code]:
hoelzl@47599
  1386
fixes prec m e
hoelzl@47599
  1387
defines "d == bitlen (abs m) - int prec"
hoelzl@47599
  1388
defines "P == 2^nat d"
hoelzl@47599
  1389
defines "f == Float m e"
hoelzl@47599
  1390
shows "float_round_up prec f = (let d = d in
hoelzl@47599
  1391
  if 0 < d then let P = P ; n = m div P ; r = m mod P in Float (n + (if r = 0 then 0 else 1)) (e + d)
hoelzl@47599
  1392
           else f)"
hoelzl@47599
  1393
  unfolding float_round_up_def float_up_def[symmetric]
hoelzl@47599
  1394
    compute_float_up f_def Let_def P_def round_prec_def d_def bitlen_Float
hoelzl@47599
  1395
  by (simp add: field_simps)
hoelzl@47599
  1396
hoelzl@47599
  1397
lemma float_round_up: "real x \<le> real (float_round_up prec x)"
hoelzl@47599
  1398
  using round_up
hoelzl@47599
  1399
  by (simp add: float_round_up_def)
hoelzl@47599
  1400
hoelzl@47599
  1401
lemma float_round_down: "real (float_round_down prec x) \<le> real x"
hoelzl@47599
  1402
  using round_down
hoelzl@47599
  1403
  by (simp add: float_round_down_def)
hoelzl@47599
  1404
hoelzl@47599
  1405
instantiation float :: lattice_ab_group_add
hoelzl@47599
  1406
begin
hoelzl@47599
  1407
hoelzl@47599
  1408
definition inf_float::"float\<Rightarrow>float\<Rightarrow>float"
hoelzl@47599
  1409
where "inf_float a b = min a b"
hoelzl@47599
  1410
hoelzl@47599
  1411
definition sup_float::"float\<Rightarrow>float\<Rightarrow>float"
hoelzl@47599
  1412
where "sup_float a b = max a b"
hoelzl@47599
  1413
hoelzl@47599
  1414
instance
hoelzl@47599
  1415
proof
hoelzl@47599
  1416
  fix x y :: float show "inf x y \<le> x" unfolding inf_float_def by simp
hoelzl@47599
  1417
  show "inf x y \<le> y" unfolding inf_float_def by simp
hoelzl@47599
  1418
  show "x \<le> sup x y" unfolding sup_float_def by simp
hoelzl@47599
  1419
  show "y \<le> sup x y" unfolding sup_float_def by simp
hoelzl@47599
  1420
  fix z::float
hoelzl@47599
  1421
  assume "x \<le> y" "x \<le> z" thus "x \<le> inf y z" unfolding inf_float_def by simp
hoelzl@47599
  1422
  next fix x y z :: float
hoelzl@47599
  1423
  assume "y \<le> x" "z \<le> x" thus "sup y z \<le> x" unfolding sup_float_def by simp
hoelzl@29804
  1424
qed
obua@16782
  1425
hoelzl@29804
  1426
end
obua@16782
  1427
hoelzl@47599
  1428
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
hoelzl@47599
  1429
 apply (auto simp: zero_float_def mult_le_0_iff)
hoelzl@47599
  1430
 using powr_gt_zero[of 2 b] by simp
hoelzl@47599
  1431
hoelzl@47599
  1432
(* TODO: how to use as code equation? -> pprt_float?! *)
hoelzl@47599
  1433
lemma compute_pprt[code]: "pprt (Float a e) = (if a <= 0 then 0 else (Float a e))"
hoelzl@47599
  1434
unfolding pprt_def sup_float_def max_def Float_le_zero_iff ..
hoelzl@29804
  1435
hoelzl@47599
  1436
(* TODO: how to use as code equation? *)
hoelzl@47599
  1437
lemma compute_nprt[code]: "nprt (Float a e) = (if a <= 0 then (Float a e) else 0)"
hoelzl@47599
  1438
unfolding nprt_def inf_float_def min_def Float_le_zero_iff ..
hoelzl@47599
  1439
hoelzl@47599
  1440
lemma of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"
hoelzl@47599
  1441
  unfolding pprt_def sup_float_def max_def sup_real_def by (auto simp: less_eq_float_def)
hoelzl@47599
  1442
hoelzl@47599
  1443
lemma of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)"
hoelzl@47599
  1444
  unfolding nprt_def inf_float_def min_def inf_real_def by (auto simp: less_eq_float_def)
hoelzl@47599
  1445
hoelzl@47599
  1446
definition int_floor_fl :: "float \<Rightarrow> int" where
hoelzl@47599
  1447
  "int_floor_fl f = floor f"
obua@16782
  1448
hoelzl@47599
  1449
lemma compute_int_floor_fl[code]:
hoelzl@47599
  1450
  shows "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e
hoelzl@47599
  1451
                                  else m div (2 ^ (nat (-e))))"
hoelzl@47599
  1452
  by (simp add: int_floor_fl_def powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hoelzl@47599
  1453
hoelzl@47599
  1454
definition floor_fl :: "float \<Rightarrow> float" where
hoelzl@47599
  1455
  "floor_fl f = float_of (floor f)"
hoelzl@47599
  1456
hoelzl@47599
  1457
lemma compute_floor_fl[code]:
hoelzl@47599
  1458
  shows "floor_fl (Float m e) = (if 0 \<le> e then Float m e
hoelzl@47599
  1459
                                  else Float (m div (2 ^ (nat (-e)))) 0)"
hoelzl@47599
  1460
  by (simp add: floor_fl_def powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
obua@16782
  1461
hoelzl@47599
  1462
lemma floor_fl: "real (floor_fl x) \<le> real x" by (simp add: floor_fl_def)
hoelzl@47599
  1463
lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by (simp add: int_floor_fl_def)
hoelzl@29804
  1464
hoelzl@47599
  1465
lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
hoelzl@47599
  1466
proof cases
hoelzl@47599
  1467
  assume nzero: "floor_fl x \<noteq> float_of 0"
hoelzl@47599
  1468
  have "floor_fl x \<equiv> Float \<lfloor>real x\<rfloor> 0" by (simp add: floor_fl_def)
hoelzl@47599
  1469
  from denormalize_shift[OF this nzero] guess i . note i = this
hoelzl@47599
  1470
  thus ?thesis by simp
hoelzl@47599
  1471
qed (simp add: floor_fl_def)
obua@16782
  1472
hoelzl@47599
  1473
(* TODO: not used in approximation
hoelzl@47599
  1474
definition ceiling_fl :: "float_of \<Rightarrow> float" where
hoelzl@47599
  1475
  "ceiling_fl f = float_of (ceiling f)"
hoelzl@47599
  1476
hoelzl@47599
  1477
lemma compute_ceiling_fl:
haftmann@30960
  1478
  "ceiling_fl (Float m e) = (if 0 \<le> e then Float m e
hoelzl@29804
  1479
                                    else Float (m div (2 ^ (nat (-e))) + 1) 0)"
obua@16782
  1480
hoelzl@31098
  1481
lemma ceiling_fl: "real x \<le> real (ceiling_fl x)"
hoelzl@29804
  1482
hoelzl@47599
  1483
definition lb_mod :: "nat \<Rightarrow> float_of \<Rightarrow> float_of \<Rightarrow> float_of \<Rightarrow> float" where
hoelzl@47599
  1484
"lb_mod prec x ub lb = x - ceiling_fl (float_divr prec x lb) * ub"
hoelzl@29804
  1485
hoelzl@47599
  1486
definition ub_mod :: "nat \<Rightarrow> float_of \<Rightarrow> float_of \<Rightarrow> float_of \<Rightarrow> float" where
hoelzl@47599
  1487
"ub_mod prec x ub lb = x - floor_fl (float_divl prec x ub) * lb"
obua@16782
  1488
hoelzl@31098
  1489
lemma lb_mod: fixes k :: int assumes "0 \<le> real x" and "real k * y \<le> real x" (is "?k * y \<le> ?x")
hoelzl@31098
  1490
  assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")
hoelzl@31098
  1491
  shows "real (lb_mod prec x ub lb) \<le> ?x - ?k * y"
obua@16782
  1492
hoelzl@47599
  1493
lemma ub_mod: fixes k :: int and x :: float_of assumes "0 \<le> real x" and "real x \<le> real k * y" (is "?x \<le> ?k * y")
hoelzl@31098
  1494
  assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")
hoelzl@31098
  1495
  shows "?x - ?k * y \<le> real (ub_mod prec x ub lb)"
obua@16782
  1496
hoelzl@47599
  1497
*)
wenzelm@20771
  1498
obua@16782
  1499
end
hoelzl@47599
  1500