author  hoelzl 
Wed, 18 Apr 2012 14:29:21 +0200  
changeset 47599  400b158f1589 
parent 47230  6584098d5378 
child 47600  e12289b5796b 
permissions  rwrr 
29988  1 
header {* FloatingPoint 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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137 

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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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140 

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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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145 

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definition "(x * y::float) = float_of (real x * real y)" 
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147 

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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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177 

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lemma real_of_float_uminus[simp]: 
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179 
fixes f g::float shows "real ( g) =  real g" 
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180 
by (simp add: uminus_float_def) 
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181 

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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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184 
by (simp add: plus_float_def) 
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185 

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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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189 

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lemma real_of_float_times[simp]: 
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191 
fixes f g::float shows "real (f * g) = real f * real g" 
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192 
by (simp add: times_float_def) 
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193 

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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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196 

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lemma real_of_float_power[simp]: fixes f::float shows "real (f^n) = real f^n" 
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198 
by (induct n) simp_all 
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199 

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instantiation float :: linorder 
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begin 
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202 

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definition "x \<le> (y::float) \<longleftrightarrow> real x \<le> real y" 
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204 

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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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207 

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definition "x < (y::float) \<longleftrightarrow> real x < real y" 
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209 

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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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211 
by (simp add: less_float_def) 
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213 
instance 
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214 
proof 
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215 
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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226 
qed 
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227 
end 
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228 

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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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230 
by (simp add: min_def less_eq_float_def) 
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231 

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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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233 
by (simp add: max_def less_eq_float_def) 
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234 

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235 
instantiation float :: linordered_ring 
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236 
begin 
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237 

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definition "(abs x :: float) = float_of (abs (real x))" 
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239 

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lemma float_abs[simp]: "x \<in> float \<Longrightarrow> abs (float_of x) = float_of (abs x)" 
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241 
by (simp add: abs_float_def) 
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242 

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243 
instance 
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244 
proof 
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245 
fix a b c :: float 
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246 
show "\<bar>a\<bar> = (if a < 0 then  a else a)" 
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247 
by (cases a rule: float_of_cases) simp 
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248 
assume "a \<le> b" 
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249 
then show "c + a \<le> c + b" 
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250 
by (cases a b c rule: float_of_cases3) simp 
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assume "0 \<le> c" 
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252 
with `a \<le> b` show "c * a \<le> c * b" 
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253 
by (cases a b c rule: float_of_cases3) (auto intro: mult_left_mono) 
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254 
from `0 \<le> c` `a \<le> b` show "a * c \<le> b * c" 
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255 
by (cases a b c rule: float_of_cases3) (auto intro: mult_right_mono) 
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256 
qed 
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257 
end 
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258 

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lemma real_of_abs_float[simp]: fixes f::float shows "real (abs f) = abs (real f)" 
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260 
unfolding abs_float_def by simp 
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261 

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262 
instance float :: dense_linorder 
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263 
proof 
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264 
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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282 

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instantiation float :: linordered_idom 
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begin 
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285 

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definition "sgn x = float_of (sgn (real x))" 
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287 

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lemma sgn_float[simp]: "x \<in> float \<Longrightarrow> sgn (float_of x) = float_of (sgn x)" 
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by (simp add: sgn_float_def) 
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instance 
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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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by (cases a b rule: float_of_cases2) (simp add: field_simps) 
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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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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) 
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qed 
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end 
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definition Float :: "int \<Rightarrow> int \<Rightarrow> float" where 
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[simp, code del]: "Float m e = float_of (m * 2 powr e)" 
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307 

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lemma real_of_float_Float[code]: "real_of_float (Float m e) = 
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(if e \<ge> 0 then m * 2 ^ nat e else m * inverse (2 ^ nat ( e)))" 
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by (auto simp add: powr_realpow[symmetric] powr_minus real_of_float_def[symmetric]) 
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311 

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code_datatype Float 
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lemma real_Float: "real (Float m e) = m * 2 powr e" by simp 
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315 

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definition normfloat :: "float \<Rightarrow> float" where 
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[simp]: "normfloat x = x" 
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lemma compute_normfloat[code]: "normfloat (Float m e) = 
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(if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1)) 
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else if m = 0 then 0 else Float m e)" 
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by (simp del: real_of_int_add split: prod.split) 
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323 
(auto simp add: powr_add zmod_eq_0_iff) 
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324 

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lemma compute_zero[code_unfold, code]: "0 = Float 0 0" 
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326 
by simp 
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327 

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lemma compute_one[code_unfold, code]: "1 = Float 1 0" 
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329 
by simp 
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330 

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instance float :: numeral .. 
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332 

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lemma float_of_numeral[simp]: "numeral k = float_of (numeral k)" 
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334 
by (induct k) 
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335 
(simp_all only: numeral.simps one_float_def float_plus_float numeral_float one_float plus_float) 
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336 

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lemma float_of_neg_numeral[simp]: "neg_numeral k = float_of (neg_numeral k)" 
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338 
by (simp add: minus_numeral[symmetric] del: minus_numeral) 
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339 

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340 
lemma 
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341 
shows float_numeral[simp]: "real (numeral x :: float) = numeral x" 
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and float_neg_numeral[simp]: "real (neg_numeral x :: float) = neg_numeral x" 
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343 
by simp_all 
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344 

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subsection {* Represent floats as unique mantissa and exponent *} 
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346 

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347 

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348 
lemma int_induct_abs[case_names less]: 
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349 
fixes j :: int 
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assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n" 
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351 
shows "P j" 
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352 
proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct) 
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353 
case less show ?case by (rule H[OF less]) simp 
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354 
qed 
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355 

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356 
lemma int_cancel_factors: 
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357 
fixes n :: int assumes "1 < r" shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)" 
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358 
proof (induct n rule: int_induct_abs) 
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359 
case (less n) 
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360 
{ fix m assume n: "n \<noteq> 0" "n = m * r" 
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361 
then have "\<bar>m \<bar> < \<bar>n\<bar>" 
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362 
by (metis abs_dvd_iff abs_ge_self assms comm_semiring_1_class.normalizing_semiring_rules(7) 
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363 
dvd_imp_le_int dvd_refl dvd_triv_right linorder_neq_iff linorder_not_le 
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364 
mult_eq_0_iff zdvd_mult_cancel1) 
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365 
from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto } 
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366 
then show ?case 
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367 
by (metis comm_semiring_1_class.normalizing_semiring_rules(12,7) dvdE power_0) 
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368 
qed 
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369 

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370 
lemma mult_powr_eq_mult_powr_iff_asym: 
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371 
fixes m1 m2 e1 e2 :: int 
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372 
assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2" 
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373 
shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2" 
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374 
proof 
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375 
have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto 
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376 
assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2" 
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377 
with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2  e1)" 
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378 
by (simp add: powr_divide2[symmetric] field_simps) 
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379 
also have "\<dots> = m2 * 2^nat (e2  e1)" 
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380 
by (simp add: powr_realpow) 
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381 
finally have m1_eq: "m1 = m2 * 2^nat (e2  e1)" 
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382 
unfolding real_of_int_inject . 
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383 
with m1 have "m1 = m2" 
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384 
by (cases "nat (e2  e1)") (auto simp add: dvd_def) 
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385 
then show "m1 = m2 \<and> e1 = e2" 
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386 
using eq `m1 \<noteq> 0` by (simp add: powr_inj) 
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387 
qed simp 
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388 

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389 
lemma mult_powr_eq_mult_powr_iff: 
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390 
fixes m1 m2 e1 e2 :: int 
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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" 
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392 
using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2] 
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393 
using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1] 
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394 
by (cases e1 e2 rule: linorder_le_cases) auto 
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395 

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396 
lemma floatE_normed: 
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397 
assumes x: "x \<in> float" 
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398 
obtains (zero) "x = 0" 
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399 
 (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0" 
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400 
proof atomize_elim 
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401 
{ assume "x \<noteq> 0" 
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402 
from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def) 
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hoelzl
parents:
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diff
changeset

403 
with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

404 
by auto 
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hoelzl
parents:
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diff
changeset

405 
with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

406 
by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"]) 
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replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

407 
(simp add: powr_add powr_realpow) } 
400b158f1589
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parents:
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changeset

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)" 
400b158f1589
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hoelzl
parents:
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diff
changeset

409 
by blast 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

410 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

411 

400b158f1589
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hoelzl
parents:
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diff
changeset

412 
lemma float_normed_cases: 
400b158f1589
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parents:
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diff
changeset

413 
fixes f :: float 
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hoelzl
parents:
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diff
changeset

414 
obtains (zero) "f = 0" 
400b158f1589
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hoelzl
parents:
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diff
changeset

415 
 (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0" 
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replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

416 
proof (atomize_elim, induct f) 
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changeset

417 
case (float_of y) then show ?case 
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changeset

418 
by (cases rule: floatE_normed) auto 
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parents:
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diff
changeset

419 
qed 
400b158f1589
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hoelzl
parents:
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diff
changeset

420 

400b158f1589
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parents:
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diff
changeset

421 
definition mantissa :: "float \<Rightarrow> int" where 
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changeset

422 
"mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) 
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parents:
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423 
\<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))" 
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parents:
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changeset

424 

400b158f1589
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parents:
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changeset

425 
definition exponent :: "float \<Rightarrow> int" where 
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426 
"exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) 
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parents:
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diff
changeset

427 
\<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))" 
400b158f1589
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hoelzl
parents:
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diff
changeset

428 

400b158f1589
replace the float datatype by a type with unique representation
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parents:
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diff
changeset

429 
lemma 
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changeset

430 
shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E) 
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parents:
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diff
changeset

431 
and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M) 
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parents:
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changeset

432 
proof  
400b158f1589
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parents:
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diff
changeset

433 
have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto 
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parents:
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changeset

434 
then show ?E ?M 
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parents:
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diff
changeset

435 
by (auto simp add: mantissa_def exponent_def) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
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diff
changeset

436 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

437 

47599
400b158f1589
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parents:
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diff
changeset

438 
lemma 
400b158f1589
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parents:
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diff
changeset

439 
shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E) 
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hoelzl
parents:
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diff
changeset

440 
and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

441 
proof cases 
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parents:
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changeset

442 
assume [simp]: "f \<noteq> (float_of 0)" 
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parents:
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diff
changeset

443 
have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f" 
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replace the float datatype by a type with unique representation
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parents:
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diff
changeset

444 
proof (cases f rule: float_normed_cases) 
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parents:
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changeset

445 
case (powr m e) 
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parents:
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changeset

446 
then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) 
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hoelzl
parents:
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diff
changeset

447 
\<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)" 
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hoelzl
parents:
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diff
changeset

448 
by auto 
400b158f1589
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hoelzl
parents:
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diff
changeset

449 
then show ?thesis 
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parents:
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diff
changeset

450 
unfolding exponent_def mantissa_def 
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parents:
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diff
changeset

451 
by (rule someI2_ex) simp 
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parents:
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diff
changeset

452 
qed simp 
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hoelzl
parents:
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diff
changeset

453 
then show ?E ?D by auto 
400b158f1589
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hoelzl
parents:
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diff
changeset

454 
qed simp 
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hoelzl
parents:
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diff
changeset

455 

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parents:
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changeset

456 
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0" 
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hoelzl
parents:
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diff
changeset

457 
using mantissa_not_dvd[of f] by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

458 

400b158f1589
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parents:
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changeset

459 
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f" 
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hoelzl
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diff
changeset

460 
unfolding real_of_float_eq[symmetric] mantissa_exponent[of f] by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

461 

400b158f1589
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parents:
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diff
changeset

462 
lemma Float_cases[case_names Float, cases type: float]: 
400b158f1589
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changeset

463 
fixes f :: float 
400b158f1589
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parents:
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changeset

464 
obtains (Float) m e :: int where "f = Float m e" 
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parents:
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diff
changeset

465 
using Float_mantissa_exponent[symmetric] 
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hoelzl
parents:
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diff
changeset

466 
by (atomize_elim) auto 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

467 

47599
400b158f1589
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hoelzl
parents:
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diff
changeset

468 
lemma 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

469 
fixes m e :: int 
400b158f1589
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hoelzl
parents:
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changeset

470 
defines "f \<equiv> float_of (m * 2 powr e)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

471 
assumes dvd: "\<not> 2 dvd m" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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changeset

472 
shows mantissa_float: "mantissa f = m" (is "?M") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

473 
and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E") 
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hoelzl
parents:
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diff
changeset

474 
proof cases 
400b158f1589
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hoelzl
parents:
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diff
changeset

475 
assume "m = 0" with dvd show "mantissa f = m" by auto 
400b158f1589
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hoelzl
parents:
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diff
changeset

476 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

477 
assume "m \<noteq> 0" 
400b158f1589
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hoelzl
parents:
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diff
changeset

478 
then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def) 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

479 
from mantissa_exponent[of f] 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

480 
have "m * 2 powr e = mantissa f * 2 powr exponent f" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

481 
by (auto simp add: f_def) 
400b158f1589
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hoelzl
parents:
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diff
changeset

482 
then show "?M" "?E" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

483 
using mantissa_not_dvd[OF f_not_0] dvd 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

484 
by (auto simp: mult_powr_eq_mult_powr_iff) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

485 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

486 

400b158f1589
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hoelzl
parents:
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diff
changeset

487 
lemma denormalize_shift: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

488 
assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0" 
400b158f1589
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hoelzl
parents:
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diff
changeset

489 
obtains i where "m = mantissa f * 2 ^ i" "e = exponent f  i" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

490 
proof 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

491 
from mantissa_exponent[of f] f_def 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

492 
have "m * 2 powr e = mantissa f * 2 powr exponent f" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

493 
by simp 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

494 
then have eq: "m = mantissa f * 2 powr (exponent f  e)" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

495 
by (simp add: powr_divide2[symmetric] field_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

496 
moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

497 
have "e \<le> exponent f" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

498 
proof (rule ccontr) 
400b158f1589
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hoelzl
parents:
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diff
changeset

499 
assume "\<not> e \<le> exponent f" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

500 
then have pos: "exponent f < e" by simp 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

501 
then have "2 powr (exponent f  e) = 2 powr  real (e  exponent f)" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

502 
by simp 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

503 
also have "\<dots> = 1 / 2^nat (e  exponent f)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

504 
using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

505 
finally have "m * 2^nat (e  exponent f) = real (mantissa f)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

506 
using eq by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

507 
then have "mantissa f = m * 2^nat (e  exponent f)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

508 
unfolding real_of_int_inject by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

509 
with `exponent f < e` have "2 dvd mantissa f" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

510 
apply (intro dvdI[where k="m * 2^(nat (eexponent f)) div 2"]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

511 
apply (cases "nat (e  exponent f)") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

512 
apply auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

513 
done 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

514 
then show False using mantissa_not_dvd[OF not_0] by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

515 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

516 
ultimately have "real m = mantissa f * 2^nat (exponent f  e)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

517 
by (simp add: powr_realpow[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

518 
with `e \<le> exponent f` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

519 
show "m = mantissa f * 2 ^ nat (exponent f  e)" "e = exponent f  nat (exponent f  e)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

520 
unfolding real_of_int_inject by auto 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

521 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

522 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

523 
subsection {* Compute arithmetic operations *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

524 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

525 
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

526 
by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

527 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

528 
lemma compute_float_neg_numeral[code_abbrev]: "Float (neg_numeral k) 0 = neg_numeral k" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

529 
by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

530 

400b158f1589
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hoelzl
parents:
47230
diff
changeset

531 
lemma compute_float_uminus[code]: " Float m1 e1 = Float ( m1) e1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

532 
by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

533 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

534 
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

535 
by (simp add: field_simps powr_add) 
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536 

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537 
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 = 
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538 
(if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2  e1)) e1 
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539 
else Float (m2 + m1 * 2^nat (e1  e2)) e2)" 
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540 
by (simp add: field_simps) 
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541 
(simp add: powr_realpow[symmetric] powr_divide2[symmetric]) 
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changeset

542 

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diff
changeset

543 
lemma compute_float_minus[code]: fixes f g::float shows "f  g = f + (g)" by simp 
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544 

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545 
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then 1 else 0)" 
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546 
by (simp add: sgn_times) 
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changeset

547 

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548 
definition is_float_pos :: "float \<Rightarrow> bool" where 
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549 
"is_float_pos f \<longleftrightarrow> 0 < f" 
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changeset

550 

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551 
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m" 
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552 
by (auto simp add: is_float_pos_def zero_less_mult_iff) (simp add: not_le[symmetric]) 
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553 

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554 
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b  a)" 
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555 
by (simp add: is_float_pos_def field_simps del: zero_float_def) 
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changeset

556 

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557 
definition is_float_nonneg :: "float \<Rightarrow> bool" where 
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558 
"is_float_nonneg f \<longleftrightarrow> 0 \<le> f" 
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559 

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560 
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m" 
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561 
by (auto simp add: is_float_nonneg_def zero_le_mult_iff) (simp add: not_less[symmetric]) 
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562 

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563 
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b  a)" 
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564 
by (simp add: is_float_nonneg_def field_simps del: zero_float_def) 
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changeset

565 

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566 
definition is_float_zero :: "float \<Rightarrow> bool" where 
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567 
"is_float_zero f \<longleftrightarrow> 0 = f" 
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changeset

568 

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569 
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m" 
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570 
by (auto simp add: is_float_zero_def) 
29804
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Added new Float theory and moved old Library/Float.thy to ComputeFloat
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571 

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572 
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e" by (simp add: abs_mult) 
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573 

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574 
instantiation float :: equal 
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575 
begin 
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576 

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577 
definition "equal_float (f1 :: float) f2 \<longleftrightarrow> is_float_zero (f1  f2)" 
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578 

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579 
instance proof qed (auto simp: equal_float_def is_float_zero_def simp del: zero_float_def) 
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580 
end 
400b158f1589
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581 

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582 
subsection {* Rounding Real numbers *} 
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583 

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584 
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where 
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585 
"round_down prec x = floor (x * 2 powr prec) * 2 powr prec" 
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changeset

586 

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587 
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where 
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588 
"round_up prec x = ceiling (x * 2 powr prec) * 2 powr prec" 
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changeset

589 

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590 
lemma round_down_float[simp]: "round_down prec x \<in> float" 
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591 
unfolding round_down_def 
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592 
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus) 
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changeset

593 

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594 
lemma round_up_float[simp]: "round_up prec x \<in> float" 
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changeset

595 
unfolding round_up_def 
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changeset

596 
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus) 
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changeset

597 

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598 
lemma round_up: "x \<le> round_up prec x" 
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599 
by (simp add: powr_minus_divide le_divide_eq round_up_def) 
400b158f1589
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parents:
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diff
changeset

600 

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changeset

601 
lemma round_down: "round_down prec x \<le> x" 
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changeset

602 
by (simp add: powr_minus_divide divide_le_eq round_down_def) 
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diff
changeset

603 

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changeset

604 
lemma round_up_0[simp]: "round_up p 0 = 0" 
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changeset

605 
unfolding round_up_def by simp 
400b158f1589
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hoelzl
parents:
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diff
changeset

606 

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changeset

607 
lemma round_down_0[simp]: "round_down p 0 = 0" 
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changeset

608 
unfolding round_down_def by simp 
400b158f1589
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hoelzl
parents:
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diff
changeset

609 

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610 
lemma round_up_diff_round_down: 
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611 
"round_up prec x  round_down prec x \<le> 2 powr prec" 
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hoelzl
parents:
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diff
changeset

612 
proof  
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parents:
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changeset

613 
have "round_up prec x  round_down prec x = 
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changeset

614 
(ceiling (x * 2 powr prec)  floor (x * 2 powr prec)) * 2 powr prec" 
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changeset

615 
by (simp add: round_up_def round_down_def field_simps) 
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hoelzl
parents:
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diff
changeset

616 
also have "\<dots> \<le> 1 * 2 powr prec" 
400b158f1589
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hoelzl
parents:
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diff
changeset

617 
by (rule mult_mono) 
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changeset

618 
(auto simp del: real_of_int_diff 
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changeset

619 
simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1) 
400b158f1589
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hoelzl
parents:
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diff
changeset

620 
finally show ?thesis by simp 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

621 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
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diff
changeset

622 

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623 
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x" 
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changeset

624 
unfolding round_down_def 
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hoelzl
parents:
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diff
changeset

625 
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric]) 
400b158f1589
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hoelzl
parents:
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diff
changeset

626 
(simp add: powr_add[symmetric]) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

627 

47599
400b158f1589
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hoelzl
parents:
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diff
changeset

628 
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x" 
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hoelzl
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diff
changeset

629 
unfolding round_up_def 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

630 
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

631 
(simp add: powr_add[symmetric]) 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

632 

400b158f1589
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diff
changeset

633 
subsection {* Rounding Floats *} 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
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diff
changeset

634 

47599
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changeset

635 
definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" where 
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changeset

636 
"float_up prec x = float_of (round_up prec (real x))" 
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hoelzl
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diff
changeset

637 

400b158f1589
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hoelzl
parents:
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diff
changeset

638 
lemma float_up_float: 
400b158f1589
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changeset

639 
"x \<in> float \<Longrightarrow> float_up prec (float_of x) = float_of (round_up prec x)" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

640 
unfolding float_up_def by simp 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

641 

47599
400b158f1589
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parents:
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diff
changeset

642 
lemma float_up_correct: 
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diff
changeset

643 
shows "real (float_up e f)  real f \<in> {0..2 powr e}" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

644 
unfolding atLeastAtMost_iff 
400b158f1589
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hoelzl
parents:
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diff
changeset

645 
proof 
400b158f1589
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hoelzl
parents:
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changeset

646 
have "round_up e f  f \<le> round_up e f  round_down e f" using round_down by simp 
400b158f1589
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hoelzl
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diff
changeset

647 
also have "\<dots> \<le> 2 powr e" using round_up_diff_round_down by simp 
400b158f1589
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hoelzl
parents:
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diff
changeset

648 
finally show "real (float_up e f)  real f \<le> 2 powr real ( e)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

649 
by (simp add: float_up_def) 
400b158f1589
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hoelzl
parents:
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diff
changeset

650 
qed (simp add: algebra_simps float_up_def round_up) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

651 

47599
400b158f1589
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changeset

652 
definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" where 
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changeset

653 
"float_down prec x = float_of (round_down prec (real x))" 
400b158f1589
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hoelzl
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47230
diff
changeset

654 

400b158f1589
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changeset

655 
lemma float_down_float: 
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changeset

656 
"x \<in> float \<Longrightarrow> float_down prec (float_of x) = float_of (round_down prec x)" 
400b158f1589
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hoelzl
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diff
changeset

657 
unfolding float_down_def by simp 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

658 

47599
400b158f1589
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hoelzl
parents:
47230
diff
changeset

659 
lemma float_down_correct: 
400b158f1589
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parents:
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changeset

660 
shows "real f  real (float_down e f) \<in> {0..2 powr e}" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

661 
unfolding atLeastAtMost_iff 
400b158f1589
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hoelzl
parents:
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diff
changeset

662 
proof 
400b158f1589
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hoelzl
parents:
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changeset

663 
have "f  round_down e f \<le> round_up e f  round_down e f" using round_up by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
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diff
changeset

664 
also have "\<dots> \<le> 2 powr e" using round_up_diff_round_down by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

665 
finally show "real f  real (float_down e f) \<le> 2 powr real ( e)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

666 
by (simp add: float_down_def) 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

667 
qed (simp add: algebra_simps float_down_def round_down) 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

668 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

669 
lemma round_down_Float_id: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

670 
assumes "p + e \<ge> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

671 
shows "round_down p (Float m e) = Float m e" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

672 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

673 
from assms have r: "real e + real p = real (nat (e + p))" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

674 
have r: "\<lfloor>real (Float m e) * 2 powr real p\<rfloor> = real (Float m e) * 2 powr real p" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

675 
by (auto intro: exI[where x="m*2^nat (e+p)"] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

676 
simp add: ac_simps powr_add[symmetric] r powr_realpow) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

677 
show ?thesis using assms 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

678 
unfolding round_down_def floor_divide_eq_div r 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

679 
by (simp add: ac_simps powr_add[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

680 
qed 
24301  681 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

682 
lemma compute_float_down[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

683 
"float_down p (Float m e) = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

684 
(if p + e < 0 then Float (m div 2^nat ((p + e))) (p) else Float m e)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

685 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

686 
assume "p + e < 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

687 
hence "real ((2::int) ^ nat ((p + e))) = 2 powr ((p + e))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

688 
using powr_realpow[of 2 "nat ((p + e))"] by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

689 
also have "... = 1 / 2 powr p / 2 powr e" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

690 
unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

691 
finally show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

692 
unfolding float_down_def round_down_def floor_divide_eq_div[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

693 
using `p + e < 0` by (simp add: ac_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

694 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

695 
assume "\<not> p + e < 0" with round_down_Float_id show ?thesis by (simp add: float_down_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

696 
qed 
24301  697 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

698 
lemma ceil_divide_floor_conv: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

699 
assumes "b \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

700 
shows "\<lceil>real a / real b\<rceil> = (if b dvd a then a div b else \<lfloor>real a / real b\<rfloor> + 1)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

701 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

702 
assume "\<not> b dvd a" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

703 
hence "a mod b \<noteq> 0" by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

704 
hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

705 
have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

706 
apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

707 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

708 
have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

709 
moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

710 
apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

711 
ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

712 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

713 
thus ?thesis using `\<not> b dvd a` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

714 
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

715 
floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus) 
19765  716 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

717 
lemma round_up_Float_id: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

718 
assumes "p + e \<ge> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

719 
shows "round_up p (Float m e) = Float m e" 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

720 
proof  
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

721 
from assms have r1: "real e + real p = real (nat (e + p))" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

722 
have r: "\<lceil>real (Float m e) * 2 powr real p\<rceil> = real (Float m e) * 2 powr real p" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

723 
by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

724 
intro: exI[where x="m*2^nat (e+p)"]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

725 
show ?thesis using assms 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

726 
unfolding float_up_def round_up_def floor_divide_eq_div Let_def r 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

727 
by (simp add: ac_simps powr_add[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

728 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

729 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

730 
lemma compute_float_up[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

731 
"float_up p (Float m e) = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

732 
(let P = 2^nat ((p + e)); r = m mod P in 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

733 
if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (p) else Float m e)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

734 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

735 
assume "p + e < 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

736 
hence "real ((2::int) ^ nat ((p + e))) = 2 powr ((p + e))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

737 
using powr_realpow[of 2 "nat ((p + e))"] by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

738 
also have "... = 1 / 2 powr p / 2 powr e" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

739 
unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

740 
finally have twopow_rewrite: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

741 
"real ((2::int) ^ nat ( (p + e))) = 1 / 2 powr real p / 2 powr real e" . 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

742 
with `p + e < 0` have powr_rewrite: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

743 
"2 powr real e * 2 powr real p = 1 / real ((2::int) ^ nat ( (p + e)))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

744 
unfolding powr_divide2 by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

745 
show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

746 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

747 
assume "2^nat ((p + e)) dvd m" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

748 
with `p + e < 0` twopow_rewrite show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

749 
by (auto simp: ac_simps float_up_def round_up_def floor_divide_eq_div dvd_eq_mod_eq_0) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

750 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

751 
assume ndvd: "\<not> 2 ^ nat ( (p + e)) dvd m" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

752 
have one_div: "real m * (1 / real ((2::int) ^ nat ( (p + e)))) = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

753 
real m / real ((2::int) ^ nat ( (p + e)))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

754 
by (simp add: field_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

755 
have "real \<lceil>real m * (2 powr real e * 2 powr real p)\<rceil> = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

756 
real \<lfloor>real m * (2 powr real e * 2 powr real p)\<rfloor> + 1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

757 
using ndvd unfolding powr_rewrite one_div 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

758 
by (subst ceil_divide_floor_conv) (auto simp: field_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

759 
thus ?thesis using `p + e < 0` twopow_rewrite 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

760 
by (auto simp: ac_simps Let_def float_up_def round_up_def floor_divide_eq_div[symmetric]) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

761 
qed 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

762 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

763 
assume "\<not> p + e < 0" with round_up_Float_id show ?thesis by (simp add: float_up_def) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

764 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

765 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

766 
lemmas real_of_ints = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

767 
real_of_int_zero 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

768 
real_of_one 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

769 
real_of_int_add 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

770 
real_of_int_minus 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

771 
real_of_int_diff 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

772 
real_of_int_mult 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

773 
real_of_int_power 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

774 
real_numeral 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

775 
lemmas real_of_nats = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

776 
real_of_nat_zero 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

777 
real_of_nat_one 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

778 
real_of_nat_1 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

779 
real_of_nat_add 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

780 
real_of_nat_mult 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

781 
real_of_nat_power 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

782 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

783 
lemmas int_of_reals = real_of_ints[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

784 
lemmas nat_of_reals = real_of_nats[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

785 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

786 
lemma two_real_int: "(2::real) = real (2::int)" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

787 
lemma two_real_nat: "(2::real) = real (2::nat)" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

788 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

789 
lemma mult_cong: "a = c ==> b = d ==> a*b = c*d" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

790 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

791 
subsection {* Compute bitlen of integers *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

792 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

793 
definition bitlen::"int => int" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

794 
where "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

795 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

796 
lemma bitlen_nonneg: "0 \<le> bitlen x" 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

797 
proof  
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

798 
{ 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

799 
assume "0 > x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

800 
have "1 = log 2 (inverse 2)" by (subst log_inverse) simp_all 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

801 
also have "... < log 2 (x)" using `0 > x` by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

802 
finally have "1 < log 2 (x)" . 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

803 
} thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

804 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

805 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

806 
lemma bitlen_bounds: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

807 
assumes "x > 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

808 
shows "2 ^ nat (bitlen x  1) \<le> x \<and> x < 2 ^ nat (bitlen x)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

809 
proof 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

810 
have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

811 
using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

812 
using real_nat_eq_real[of "floor (log 2 (real x))"] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

813 
by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

814 
also have "... \<le> 2 powr log 2 (real x)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

815 
by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

816 
also have "... = real x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

817 
using `0 < x` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

818 
finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

819 
thus "2 ^ nat (bitlen x  1) \<le> x" using `x > 0` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

820 
by (simp add: bitlen_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

821 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

822 
have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

823 
also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

824 
apply (simp add: powr_realpow[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

825 
using `x > 0` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

826 
finally show "x < 2 ^ nat (bitlen x)" using `x > 0` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

827 
by (simp add: bitlen_def ac_simps int_of_reals del: real_of_ints) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

828 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

829 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

830 
lemma bitlen_pow2[simp]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

831 
assumes "b > 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

832 
shows "bitlen (b * 2 ^ c) = bitlen b + c" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

833 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

834 
from assms have "b * 2 ^ c > 0" by (auto intro: mult_pos_pos) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

835 
thus ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

836 
using floor_add[of "log 2 b" c] assms 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

837 
by (auto simp add: log_mult log_nat_power bitlen_def) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

838 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

839 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

840 
lemma bitlen_Float: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

841 
fixes m e 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

842 
defines "f \<equiv> Float m e" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

843 
shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

844 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

845 
assume "m \<noteq> 0" hence "f \<noteq> float_of 0" by (simp add: f_def) hence "mantissa f \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

846 
by (simp add: mantissa_noteq_0) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

847 
moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

848 
from f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`] guess i . 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

849 
ultimately show ?thesis by (simp add: abs_mult) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

850 
qed (simp add: f_def bitlen_def) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

851 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

852 
lemma compute_bitlen[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

853 
shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

854 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

855 
{ assume "2 \<le> x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

856 
then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x  x mod 2)\<rfloor>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

857 
by (simp add: log_mult zmod_zdiv_equality') 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

858 
also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

859 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

860 
assume "x mod 2 = 0" then show ?thesis by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

861 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

862 
def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

863 
then have "0 \<le> n" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

864 
using `2 \<le> x` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

865 
assume "x mod 2 \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

866 
with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

867 
with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

868 
moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

869 
{ have "real (2^nat n :: int) = 2 powr (nat n)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

870 
by (simp add: powr_realpow) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

871 
also have "\<dots> \<le> 2 powr (log 2 x)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

872 
using `2 \<le> x` by (simp add: n_def del: powr_log_cancel) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

873 
finally have "2^nat n \<le> x" using `2 \<le> x` by simp } 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

874 
ultimately have "2^nat n \<le> x  1" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

875 
then have "2^nat n \<le> real (x  1)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

876 
unfolding real_of_int_le_iff[symmetric] by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

877 
{ have "n = \<lfloor>log 2 (2^nat n)\<rfloor>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

878 
using `0 \<le> n` by (simp add: log_nat_power) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

879 
also have "\<dots> \<le> \<lfloor>log 2 (x  1)\<rfloor>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

880 
using `2^nat n \<le> real (x  1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

881 
finally have "n \<le> \<lfloor>log 2 (x  1)\<rfloor>" . } 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

882 
moreover have "\<lfloor>log 2 (x  1)\<rfloor> \<le> n" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

883 
using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

884 
ultimately show "\<lfloor>log 2 (x  x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

885 
unfolding n_def `x mod 2 = 1` by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

886 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

887 
finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . } 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

888 
moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

889 
{ assume "x < 2" "0 < x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

890 
then have "x = 1" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

891 
then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp } 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

892 
ultimately show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

893 
unfolding bitlen_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

894 
by (auto simp: pos_imp_zdiv_pos_iff not_le) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

895 
qed 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

896 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

897 
lemma float_gt1_scale: assumes "1 \<le> Float m e" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

898 
shows "0 \<le> e + (bitlen m  1)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

899 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

900 
have "0 < Float m e" using assms by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

901 
hence "0 < m" using powr_gt_zero[of 2 e] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

902 
by (auto simp: less_float_def less_eq_float_def zero_less_mult_iff) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

903 
hence "m \<noteq> 0" by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

904 
show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

905 
proof (cases "0 \<le> e") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

906 
case True thus ?thesis using `0 < m` by (simp add: bitlen_def) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

907 
next 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

908 
have "(1::int) < 2" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

909 
case False let ?S = "2^(nat (e))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

910 
have "inverse (2 ^ nat ( e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (e)"] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

911 
by (auto simp: powr_minus field_simps inverse_eq_divide) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

912 
hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (e)"] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

913 
by (auto simp: powr_minus) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

914 
hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

915 
hence "?S \<le> real m" unfolding mult_assoc by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

916 
hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

917 
from this bitlen_bounds[OF `0 < m`, THEN conjunct2] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

918 
have "nat (e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

919 
hence "e < bitlen m" using False by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

920 
thus ?thesis by auto 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

921 
qed 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

922 
qed 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

923 

e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

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" 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

925 
proof  
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

926 
let ?B = "2^nat(bitlen m  1)" 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

927 

e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

928 
have "?B \<le> m" using bitlen_bounds[OF `0 <m`] .. 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

929 
hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

930 
thus "1 \<le> real m / ?B" by auto 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

931 

e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

932 
have "m \<noteq> 0" using assms by auto 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

933 
have "0 \<le> bitlen m  1" using `0 < m` by (auto simp: bitlen_def) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

934 

29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

935 
have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] .. 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

936 
also have "\<dots> = 2^nat(bitlen m  1 + 1)" using `0 < m` by (auto simp: bitlen_def) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

937 
also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m  1` zero_le_one] by auto 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

938 
finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

939 
hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto) 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

940 
thus "real m / ?B < 2" by auto 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

941 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

942 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

943 
subsection {* Approximation of positive rationals *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

944 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

945 
lemma zdiv_zmult_twopow_eq: fixes a b::int shows "a div b div (2 ^ n) = a div (b * 2 ^ n)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

946 
by (simp add: zdiv_zmult2_eq) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

947 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

948 
lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

949 
by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

950 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

951 
lemma real_div_nat_eq_floor_of_divide: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

952 
fixes a b::nat 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

953 
shows "a div b = real (floor (a/b))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

954 
by (metis floor_divide_eq_div real_of_int_of_nat_eq zdiv_int) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

955 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

956 
definition "rat_precision prec x y = int prec  (bitlen x  bitlen y)" 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

957 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

958 
definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" where 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

959 
"lapprox_posrat prec x y = float_of (round_down (rat_precision prec x y) (x / y))" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

960 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

961 
lemma compute_lapprox_posrat[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

962 
fixes prec x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

963 
shows "lapprox_posrat prec x y = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

964 
(let 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

965 
l = rat_precision prec x y; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

966 
d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat ( l) div y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

967 
in normfloat (Float d ( l)))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

968 
unfolding lapprox_posrat_def div_mult_twopow_eq 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

969 
by (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

970 
field_simps Let_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

971 
del: two_powr_minus_int_float) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

972 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

973 
definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" where 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

974 
"rapprox_posrat prec x y = float_of (round_up (rat_precision prec x y) (x / y))" 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

975 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

976 
(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

977 
lemma compute_rapprox_posrat[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

978 
fixes prec x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

979 
defines "l \<equiv> rat_precision prec x y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

980 
shows "rapprox_posrat prec x y = (let 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

981 
l = l ; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

982 
X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(l)) ; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

983 
d = fst X div snd X ; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

984 
m = fst X mod snd X 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

985 
in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) ( l)))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

986 
proof (cases "y = 0") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

987 
assume "y = 0" thus ?thesis by (simp add: rapprox_posrat_def Let_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

988 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

989 
assume "y \<noteq> 0" 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

990 
show ?thesis 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

991 
proof (cases "0 \<le> l") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

992 
assume "0 \<le> l" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

993 
def x' == "x * 2 ^ nat l" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

994 
have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

995 
moreover have "real x * 2 powr real l = real x'" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

996 
by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

997 
ultimately show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

998 
unfolding rapprox_posrat_def round_up_def l_def[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

999 
using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1000 
by (simp add: Let_def floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 int_of_reals 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1001 
del: real_of_ints) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1002 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1003 
assume "\<not> 0 \<le> l" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1004 
def y' == "y * 2 ^ nat ( l)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1005 
from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1006 
have "int y * 2 ^ nat ( l) = y'" by (simp add: y'_def int_mult int_power) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1007 
moreover have "real x * real (2::int) powr real l / real y = x / real y'" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1008 
using `\<not> 0 \<le> l` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1009 
by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps inverse_eq_divide) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1010 
ultimately show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1011 
using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1012 
by (simp add: rapprox_posrat_def l_def round_up_def ceil_divide_floor_conv 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1013 
floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 int_of_reals 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1014 
del: real_of_ints) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1015 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1016 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1017 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1018 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1019 
lemma rat_precision_pos: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1020 
assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1021 
shows "rat_precision n (int x) (int y) > 0" 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1022 
proof  
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1023 
{ assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) } 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1024 
hence "bitlen (int x) < bitlen (int y)" using assms 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1025 
by (simp add: bitlen_def del: floor_add_one) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1026 
(auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1027 
thus ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1028 
using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1029 
qed 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1030 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1031 
lemma power_aux: assumes "x > 0" shows "(2::int) ^ nat (x  1) \<le> 2 ^ nat x  1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1032 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1033 
def y \<equiv> "nat (x  1)" moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1034 
have "(2::int) ^ y \<le> (2 ^ (y + 1))  1" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1035 
ultimately show ?thesis using assms by simp 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1036 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1037 

e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1038 
lemma rapprox_posrat_less1: assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n" 
31098
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
31021
diff
changeset

1039 
shows "real (rapprox_posrat n x y) < 1" 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1040 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1041 
have powr1: "2 powr real (rat_precision n (int x) (int y)) = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1042 
2 ^ nat (rat_precision n (int x) (int y))" using rat_precision_pos[of x y n] assms 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1043 
by (simp add: powr_realpow[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1044 
have "x * 2 powr real (rat_precision n (int x) (int y)) / y = (x / y) * 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1045 
2 powr real (rat_precision n (int x) (int y))" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1046 
also have "... < (1 / 2) * 2 powr real (rat_precision n (int x) (int y))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1047 
apply (rule mult_strict_right_mono) by (insert assms) auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1048 
also have "\<dots> = 2 powr real (rat_precision n (int x) (int y)  1)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1049 
by (simp add: powr_add diff_def powr_neg_numeral) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1050 
also have "\<dots> = 2 ^ nat (rat_precision n (int x) (int y)  1)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1051 
using rat_precision_pos[of x y n] assms by (simp add: powr_realpow[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1052 
also have "\<dots> \<le> 2 ^ nat (rat_precision n (int x) (int y))  1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1053 
unfolding int_of_reals real_of_int_le_iff 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1054 
using rat_precision_pos[OF assms] by (rule power_aux) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1055 
finally show ?thesis unfolding rapprox_posrat_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1056 
apply (simp add: round_up_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1057 
apply (simp add: round_up_def field_simps powr_minus inverse_eq_divide) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1058 
unfolding powr1 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1059 
unfolding int_of_reals real_of_int_less_iff 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1060 
unfolding ceiling_less_eq using rat_precision_pos[of x y n] assms apply simp done 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1061 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1062 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1063 
definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" where 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1064 
"lapprox_rat prec x y = float_of (round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y))" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1065 

29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1066 
lemma compute_lapprox_rat[code]: 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1067 
"lapprox_rat prec x y = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1068 
(if y = 0 then 0 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1069 
else if 0 \<le> x then 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1070 
(if 0 < y then lapprox_posrat prec (nat x) (nat y) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1071 
else  (rapprox_posrat prec (nat x) (nat (y)))) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1072 
else (if 0 < y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1073 
then  (rapprox_posrat prec (nat (x)) (nat y)) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1074 
else lapprox_posrat prec (nat (x)) (nat (y))))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1075 
apply (cases "y = 0") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1076 
apply (simp add: lapprox_posrat_def rapprox_posrat_def round_down_def lapprox_rat_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1077 
apply (auto simp: lapprox_rat_def lapprox_posrat_def rapprox_posrat_def round_up_def round_down_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1078 
ceiling_def real_of_float_uminus[symmetric] ac_simps int_of_reals simp del: real_of_ints) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1079 
apply (auto simp: ac_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1080 
done 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1081 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1082 
definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" where 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1083 
"rapprox_rat prec x y = float_of (round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1084 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1085 
lemma compute_rapprox_rat[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1086 
"rapprox_rat prec x y = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1087 
(if y = 0 then 0 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1088 
else if 0 \<le> x then 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1089 
(if 0 < y then rapprox_posrat prec (nat x) (nat y) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1090 
else  (lapprox_posrat prec (nat x) (nat (y)))) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1091 
else (if 0 < y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1092 
then  (lapprox_posrat prec (nat (x)) (nat y)) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1093 
else rapprox_posrat prec (nat (x)) (nat (y))))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1094 
apply (cases "y = 0", simp add: lapprox_posrat_def rapprox_posrat_def round_up_def rapprox_rat_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1095 
apply (auto simp: rapprox_rat_def lapprox_posrat_def rapprox_posrat_def round_up_def round_down_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1096 
ceiling_def real_of_float_uminus[symmetric] ac_simps int_of_reals simp del: real_of_ints) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1097 
apply (auto simp: ac_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1098 
done 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1099 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1100 
subsection {* Division *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1101 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1102 
definition div_precision 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1103 
where "div_precision prec x y = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1104 
rat_precision prec \<bar>mantissa x\<bar> \<bar>mantissa y\<bar>  exponent x + exponent y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1105 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1106 
definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1107 
where "float_divl prec a b = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1108 
float_of (round_down (div_precision prec a b) (a / b))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1109 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1110 
lemma compute_float_divl[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1111 
fixes m1 s1 m2 s2 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1112 
defines "f1 \<equiv> Float m1 s1" and "f2 \<equiv> Float m2 s2" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1113 
shows "float_divl prec f1 f2 = lapprox_rat prec m1 m2 * Float 1 (s1  s2)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1114 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1115 
assume "f1 \<noteq> 0 \<and> f2 \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1116 
then have "f1 \<noteq> float_of 0" "f2 \<noteq> float_of 0" by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1117 
with mantissa_not_dvd[of f1] mantissa_not_dvd[of f2] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1118 
have "mantissa f1 \<noteq> 0" "mantissa f2 \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1119 
by (auto simp add: dvd_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1120 
then have pos: "0 < \<bar>mantissa f1\<bar>" "0 < \<bar>mantissa f2\<bar>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1121 
by simp_all 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1122 
moreover from f1_def[THEN denormalize_shift, OF `f1 \<noteq> float_of 0`] guess i . note i = this 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1123 
moreover from f2_def[THEN denormalize_shift, OF `f2 \<noteq> float_of 0`] guess j . note j = this 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1124 
moreover have "(real (mantissa f1) * 2 ^ i / (real (mantissa f2) * 2 ^ j)) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1125 
= (real (mantissa f1) / real (mantissa f2)) * 2 powr (int i  int j)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1126 
by (simp add: powr_divide2[symmetric] powr_realpow) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1127 
moreover have "real f1 / real f2 = real (mantissa f1) / real (mantissa f2) * 2 powr real (exponent f1  exponent f2)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1128 
unfolding mantissa_exponent by (simp add: powr_divide2[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1129 
moreover have "rat_precision prec (\<bar>mantissa f1\<bar> * 2 ^ i) (\<bar>mantissa f2\<bar> * 2 ^ j) = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1130 
rat_precision prec \<bar>mantissa f1\<bar> \<bar>mantissa f2\<bar> + j  i" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1131 
using pos by (simp add: rat_precision_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1132 
ultimately show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1133 
unfolding float_divl_def lapprox_rat_def div_precision_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1134 
by (simp add: abs_mult round_down_shift powr_divide2[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1135 
del: int_nat_eq real_of_int_diff times_divide_eq_left ) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1136 
(simp add: field_simps powr_divide2[symmetric] powr_add) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1137 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1138 
assume "\<not> (f1 \<noteq> 0 \<and> f2 \<noteq> 0)" then show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1139 
by (auto simp add: float_divl_def f1_def f2_def lapprox_rat_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1140 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1141 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1142 
definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1143 
where "float_divr prec a b = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1144 
float_of (round_up (div_precision prec a b) (a / b))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1145 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1146 
lemma compute_float_divr[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1147 
fixes m1 s1 m2 s2 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1148 
defines "f1 \<equiv> Float m1 s1" and "f2 \<equiv> Float m2 s2" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1149 
shows "float_divr prec f1 f2 = rapprox_rat prec m1 m2 * Float 1 (s1  s2)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1150 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1151 
assume "f1 \<noteq> 0 \<and> f2 \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1152 
then have "f1 \<noteq> float_of 0" "f2 \<noteq> float_of 0" by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1153 
with mantissa_not_dvd[of f1] mantissa_not_dvd[of f2] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1154 
have "mantissa f1 \<noteq> 0" "mantissa f2 \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1155 
by (auto simp add: dvd_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1156 
then have pos: "0 < \<bar>mantissa f1\<bar>" "0 < \<bar>mantissa f2\<bar>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1157 
by simp_all 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1158 
moreover from f1_def[THEN denormalize_shift, OF `f1 \<noteq> float_of 0`] guess i . note i = this 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1159 
moreover from f2_def[THEN denormalize_shift, OF `f2 \<noteq> float_of 0`] guess j . note j = this 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1160 
moreover have "(real (mantissa f1) * 2 ^ i / (real (mantissa f2) * 2 ^ j)) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1161 
= (real (mantissa f1) / real (mantissa f2)) * 2 powr (int i  int j)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1162 
by (simp add: powr_divide2[symmetric] powr_realpow) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1163 
moreover have "real f1 / real f2 = real (mantissa f1) / real (mantissa f2) * 2 powr real (exponent f1  exponent f2)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1164 
unfolding mantissa_exponent by (simp add: powr_divide2[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1165 
moreover have "rat_precision prec (\<bar>mantissa f1\<bar> * 2 ^ i) (\<bar>mantissa f2\<bar> * 2 ^ j) = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1166 
rat_precision prec \<bar>mantissa f1\<bar> \<bar>mantissa f2\<bar> + j  i" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1167 
using pos by (simp add: rat_precision_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1168 
ultimately show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1169 
unfolding float_divr_def rapprox_rat_def div_precision_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1170 
by (simp add: abs_mult round_up_shift powr_divide2[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1171 
del: int_nat_eq real_of_int_diff times_divide_eq_left) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1172 
(simp add: field_simps powr_divide2[symmetric] powr_add) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1173 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1174 
assume "\<not> (f1 \<noteq> 0 \<and> f2 \<noteq> 0)" then show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1175 
by (auto simp add: float_divr_def f1_def f2_def rapprox_rat_def) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1176 
qed 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1177 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1178 
subsection {* Lemmas needed by Approximate *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1179 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1180 
declare one_float_def[simp del] zero_float_def[simp del] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1181 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1182 
lemma Float_num[simp]: shows 
400b158f1589
replace the float datatype by a type with unique representation
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parents:
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diff
changeset

1183 
"real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1184 
"real (Float 1 1) = 1/2" and "real (Float 1 2) = 1/4" and "real (Float 1 3) = 1/8" and 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1185 
"real (Float 1 0) = 1" and "real (Float (number_of n) 0) = number_of n" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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"] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1187 
using powr_realpow[of 2 2] powr_realpow[of 2 3] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1188 
using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1189 
by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1190 

400b158f1589
replace the float datatype by a type with unique representation
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parents:
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diff
changeset

1191 
lemma real_of_Float_int[simp]: "real (Float n 0) = real n" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1192 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1193 
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1194 

400b158f1589
replace the float datatype by a type with unique representation
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parents:
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diff
changeset

1195 
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> abs((a::int) div 2) < abs a" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1196 
by arith 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1197 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1198 
lemma lapprox_rat: 
400b158f1589
replace the float datatype by a type with unique representation
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parents:
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diff
changeset

1199 
shows "real (lapprox_rat prec x y) \<le> real x / real y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1200 
using round_down by (simp add: lapprox_rat_def) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1201 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1202 
lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1203 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1204 
from zmod_zdiv_equality'[of a b] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1205 
have "a = b * (a div b) + a mod b" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1206 
also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1207 
using assms by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1208 
finally show ?thesis by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1209 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1210 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1211 
lemma lapprox_rat_nonneg: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1212 
fixes n x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1213 
defines "p == int n  ((bitlen \<bar>x\<bar>)  (bitlen \<bar>y\<bar>))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1214 
assumes "0 \<le> x" "0 < y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1215 
shows "0 \<le> real (lapprox_rat n x y)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1216 
using assms unfolding lapprox_rat_def p_def[symmetric] round_down_def real_of_int_minus[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1217 
powr_int[of 2, simplified] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1218 
by (auto simp add: inverse_eq_divide intro!: mult_nonneg_nonneg divide_nonneg_pos mult_pos_pos) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1219 

31098
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
31021
diff
changeset

1220 
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)" 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1221 
using round_up by (simp add: rapprox_rat_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1222 

400b158f1589
replace the float datatype by a type with unique representation
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parents:
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diff
changeset

1223 
lemma rapprox_rat_le1: 
400b158f1589
replace the float datatype by a type with unique representation
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parents:
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diff
changeset

1224 
fixes n x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1225 
assumes xy: "0 \<le> x" "0 < y" "x \<le> y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1226 
shows "real (rapprox_rat n x y) \<le> 1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1227 
proof  
400b158f1589
replace the float datatype by a type with unique representation
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parents:
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diff
changeset

1228 
have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1229 
using xy unfolding bitlen_def by (auto intro!: floor_mono) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1230 
then have "0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>" by (simp add: rat_precision_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1231 
have "real \<lceil>real x / real y * 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil> 
400b158f1589
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parents:
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diff
changeset

1232 
\<le> real \<lceil>2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>" 
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replace the float datatype by a type with unique representation
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parents:
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diff
changeset

1233 
using xy by (auto intro!: ceiling_mono simp: field_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1234 
also have "\<dots> = 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1235 
using `0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1236 
by (auto intro!: exI[of _ "2^nat (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"] simp: powr_int) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1237 
finally show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1238 
by (simp add: rapprox_rat_def round_up_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1239 
(simp add: powr_minus inverse_eq_divide) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1240 
qed 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1241 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1242 
lemma rapprox_rat_nonneg_neg: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1243 
"0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1244 
unfolding rapprox_rat_def round_up_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1245 
by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1246 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1247 
lemma rapprox_rat_neg: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1248 
"x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1249 
unfolding rapprox_rat_def round_up_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1250 
by (auto simp: field_simps mult_le_0_iff) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1251 

47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1252 
lemma rapprox_rat_nonpos_pos: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1253 
"x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1254 
unfolding rapprox_rat_def round_up_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1255 
by (auto simp: field_simps mult_le_0_iff) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1256 

31098
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
31021
diff
changeset

1257 
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y" 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1258 
using round_down by (simp add: float_divl_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1259 

400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1260 
lemma float_divl_lower_bound: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1261 
fixes x y prec 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1262 
defines "p == rat_precision prec \<bar>mantissa x\<bar> \<bar>mantissa y\<bar>  exponent x + exponent y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl< 