author  hoelzl 
Thu, 19 Apr 2012 11:55:30 +0200  
changeset 47601  050718fe6eee 
parent 47600  e12289b5796b 
child 47608  572d7e51de4d 
permissions  rwrr 
29988  1 
header {* FloatingPoint Numbers *} 
2 

20485  3 
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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defs (overloaded) 
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real_of_float_def[code_unfold]: "real \<equiv> real_of_float" 
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lemma type_definition_float': "type_definition real float_of float" 
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using type_definition_float unfolding real_of_float_def . 
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setup_lifting (no_code) type_definition_float' 
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lemmas float_of_inject[simp] 
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declare [[coercion "real :: float \<Rightarrow> real"]] 
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lemma real_of_float_eq: 

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fixes f1 f2 :: float shows "f1 = f2 \<longleftrightarrow> real f1 = real 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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lift_definition Float :: "int \<Rightarrow> int \<Rightarrow> float" is "\<lambda>(m::int) (e::int). m * 2 powr e" by simp 
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declare Float.rep_eq[simp] 
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code_datatype Float 
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subsection {* Arithmetic operations on floating point numbers *} 
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instantiation float :: "{ring_1, linorder, linordered_ring, linordered_idom, numeral, equal}" 
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begin 
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lift_definition zero_float :: float is 0 by simp 
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declare zero_float.rep_eq[simp] 
47600  137 
lift_definition one_float :: float is 1 by simp 
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declare one_float.rep_eq[simp] 
47600  139 
lift_definition plus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op +" by simp 
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declare plus_float.rep_eq[simp] 
47600  141 
lift_definition times_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op *" by simp 
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declare times_float.rep_eq[simp] 
47600  143 
lift_definition minus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op " by simp 
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declare minus_float.rep_eq[simp] 
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lift_definition uminus_float :: "float \<Rightarrow> float" is "uminus" by simp 
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declare uminus_float.rep_eq[simp] 
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47600  148 
lift_definition abs_float :: "float \<Rightarrow> float" is abs by simp 
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declare abs_float.rep_eq[simp] 
47600  150 
lift_definition sgn_float :: "float \<Rightarrow> float" is sgn by simp 
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declare sgn_float.rep_eq[simp] 
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47600  153 
lift_definition equal_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op = :: real \<Rightarrow> real \<Rightarrow> bool" .. 
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47600  155 
lift_definition less_eq_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op \<le>" .. 
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declare less_eq_float.rep_eq[simp] 
47600  157 
lift_definition less_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op <" .. 
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declare less_float.rep_eq[simp] 
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instance 
47600  161 
proof qed (transfer, fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+ 
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end 
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163 

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lemma real_of_float_power[simp]: fixes f::float shows "real (f^n) = real f^n" 
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by (induct n) simp_all 
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47600  167 
lemma fixes x y::float 
168 
shows real_of_float_min: "real (min x y) = min (real x) (real y)" 

169 
and real_of_float_max: "real (max x y) = max (real x) (real y)" 

170 
by (simp_all add: min_def max_def) 

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instance float :: dense_linorder 
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proof 
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fix a b :: float 
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show "\<exists>c. a < c" 
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apply (intro exI[of _ "a + 1"]) 
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apply transfer 
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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 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 _ "(a + b) * Float 1 1"]) 
188 
apply transfer 

189 
apply (simp add: powr_neg_numeral) 

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done 
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qed 
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47600  193 
instantiation float :: lattice_ab_group_add 
46573  194 
begin 
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47600  196 
definition inf_float::"float\<Rightarrow>float\<Rightarrow>float" 
197 
where "inf_float a b = min a b" 

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47600  199 
definition sup_float::"float\<Rightarrow>float\<Rightarrow>float" 
200 
where "sup_float a b = max a b" 

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instance 
47600  203 
by default 
204 
(transfer, simp_all add: inf_float_def sup_float_def real_of_float_min real_of_float_max)+ 

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end 
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47600  207 
lemma float_numeral[simp]: "real (numeral x :: float) = numeral x" 
208 
apply (induct x) 

209 
apply simp 

210 
apply (simp_all only: numeral_Bit0 numeral_Bit1 real_of_float_eq real_float 

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plus_float.rep_eq one_float.rep_eq plus_float numeral_float one_float) 
47600  212 
done 
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47600  214 
lemma transfer_numeral [transfer_rule]: 
215 
"fun_rel (op =) cr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)" 

216 
unfolding fun_rel_def cr_float_def by (simp add: real_of_float_def[symmetric]) 

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47600  218 
lemma float_neg_numeral[simp]: "real (neg_numeral x :: float) = neg_numeral x" 
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by (simp add: minus_numeral[symmetric] del: minus_numeral) 
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47600  221 
lemma transfer_neg_numeral [transfer_rule]: 
222 
"fun_rel (op =) cr_float (neg_numeral :: _ \<Rightarrow> real) (neg_numeral :: _ \<Rightarrow> float)" 

223 
unfolding fun_rel_def cr_float_def by (simp add: real_of_float_def[symmetric]) 

224 

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lemma 
47600  226 
shows float_of_numeral[simp]: "numeral k = float_of (numeral k)" 
227 
and float_of_neg_numeral[simp]: "neg_numeral k = float_of (neg_numeral k)" 

228 
unfolding real_of_float_eq by simp_all 

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subsection {* Represent floats as unique mantissa and exponent *} 
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231 

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lemma int_induct_abs[case_names less]: 
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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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235 
shows "P j" 
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proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct) 
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case less show ?case by (rule H[OF less]) simp 
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238 
qed 
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239 

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lemma int_cancel_factors: 
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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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proof (induct n rule: int_induct_abs) 
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case (less n) 
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{ fix m assume n: "n \<noteq> 0" "n = m * r" 
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then have "\<bar>m \<bar> < \<bar>n\<bar>" 
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by (metis abs_dvd_iff abs_ge_self assms comm_semiring_1_class.normalizing_semiring_rules(7) 
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dvd_imp_le_int dvd_refl dvd_triv_right linorder_neq_iff linorder_not_le 
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mult_eq_0_iff zdvd_mult_cancel1) 
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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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250 
then show ?case 
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by (metis comm_semiring_1_class.normalizing_semiring_rules(12,7) dvdE power_0) 
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252 
qed 
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253 

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lemma mult_powr_eq_mult_powr_iff_asym: 
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fixes m1 m2 e1 e2 :: int 
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assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2" 
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shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2" 
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258 
proof 
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have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto 
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assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2" 
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with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2  e1)" 
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by (simp add: powr_divide2[symmetric] field_simps) 
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also have "\<dots> = m2 * 2^nat (e2  e1)" 
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264 
by (simp add: powr_realpow) 
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finally have m1_eq: "m1 = m2 * 2^nat (e2  e1)" 
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unfolding real_of_int_inject . 
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with m1 have "m1 = m2" 
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by (cases "nat (e2  e1)") (auto simp add: dvd_def) 
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then show "m1 = m2 \<and> e1 = e2" 
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using eq `m1 \<noteq> 0` by (simp add: powr_inj) 
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qed simp 
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272 

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lemma mult_powr_eq_mult_powr_iff: 
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fixes m1 m2 e1 e2 :: int 
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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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using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2] 
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using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1] 
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by (cases e1 e2 rule: linorder_le_cases) auto 
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279 

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lemma floatE_normed: 
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281 
assumes x: "x \<in> float" 
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obtains (zero) "x = 0" 
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 (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0" 
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284 
proof atomize_elim 
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{ assume "x \<noteq> 0" 
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from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def) 
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with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k" 
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288 
by auto 
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with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m" 
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by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"]) 
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(simp add: powr_add powr_realpow) } 
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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)" 
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293 
by blast 
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294 
qed 
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295 

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296 
lemma float_normed_cases: 
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297 
fixes f :: float 
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298 
obtains (zero) "f = 0" 
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299 
 (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0" 
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300 
proof (atomize_elim, induct f) 
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301 
case (float_of y) then show ?case 
47600  302 
by (cases rule: floatE_normed) (auto simp: zero_float_def) 
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303 
qed 
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304 

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305 
definition mantissa :: "float \<Rightarrow> int" where 
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306 
"mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) 
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307 
\<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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308 

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309 
definition exponent :: "float \<Rightarrow> int" where 
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310 
"exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) 
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311 
\<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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312 

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313 
lemma 
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314 
shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E) 
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315 
and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M) 
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316 
proof  
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317 
have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto 
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318 
then show ?E ?M 
47600  319 
by (auto simp add: mantissa_def exponent_def zero_float_def) 
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320 
qed 
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321 

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322 
lemma 
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323 
shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E) 
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324 
and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D") 
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325 
proof cases 
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326 
assume [simp]: "f \<noteq> (float_of 0)" 
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327 
have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f" 
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328 
proof (cases f rule: float_normed_cases) 
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329 
case (powr m e) 
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330 
then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) 
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331 
\<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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332 
by auto 
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333 
then show ?thesis 
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334 
unfolding exponent_def mantissa_def 
47600  335 
by (rule someI2_ex) (simp add: zero_float_def) 
336 
qed (simp add: zero_float_def) 

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337 
then show ?E ?D by auto 
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338 
qed simp 
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339 

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340 
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0" 
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341 
using mantissa_not_dvd[of f] by auto 
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342 

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343 
lemma 
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344 
fixes m e :: int 
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345 
defines "f \<equiv> float_of (m * 2 powr e)" 
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346 
assumes dvd: "\<not> 2 dvd m" 
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347 
shows mantissa_float: "mantissa f = m" (is "?M") 
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348 
and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E") 
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349 
proof cases 
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350 
assume "m = 0" with dvd show "mantissa f = m" by auto 
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351 
next 
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352 
assume "m \<noteq> 0" 
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353 
then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def) 
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354 
from mantissa_exponent[of f] 
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355 
have "m * 2 powr e = mantissa f * 2 powr exponent f" 
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356 
by (auto simp add: f_def) 
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357 
then show "?M" "?E" 
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358 
using mantissa_not_dvd[OF f_not_0] dvd 
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359 
by (auto simp: mult_powr_eq_mult_powr_iff) 
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360 
qed 
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361 

47600  362 
subsection {* Compute arithmetic operations *} 
363 

364 
lemma real_of_float_Float[code]: "real_of_float (Float m e) = 

365 
(if e \<ge> 0 then m * 2 ^ nat e else m * inverse (2 ^ nat ( e)))" 

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366 
by (auto simp add: powr_realpow[symmetric] powr_minus real_of_float_def[symmetric]) 
47600  367 

368 
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f" 

369 
unfolding real_of_float_eq mantissa_exponent[of f] by simp 

370 

371 
lemma Float_cases[case_names Float, cases type: float]: 

372 
fixes f :: float 

373 
obtains (Float) m e :: int where "f = Float m e" 

374 
using Float_mantissa_exponent[symmetric] 

375 
by (atomize_elim) auto 

376 

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377 
lemma denormalize_shift: 
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378 
assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0" 
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379 
obtains i where "m = mantissa f * 2 ^ i" "e = exponent f  i" 
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380 
proof 
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381 
from mantissa_exponent[of f] f_def 
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382 
have "m * 2 powr e = mantissa f * 2 powr exponent f" 
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383 
by simp 
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384 
then have eq: "m = mantissa f * 2 powr (exponent f  e)" 
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385 
by (simp add: powr_divide2[symmetric] field_simps) 
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386 
moreover 
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387 
have "e \<le> exponent f" 
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388 
proof (rule ccontr) 
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389 
assume "\<not> e \<le> exponent f" 
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390 
then have pos: "exponent f < e" by simp 
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391 
then have "2 powr (exponent f  e) = 2 powr  real (e  exponent f)" 
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392 
by simp 
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393 
also have "\<dots> = 1 / 2^nat (e  exponent f)" 
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394 
using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric]) 
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395 
finally have "m * 2^nat (e  exponent f) = real (mantissa f)" 
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396 
using eq by simp 
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397 
then have "mantissa f = m * 2^nat (e  exponent f)" 
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398 
unfolding real_of_int_inject by simp 
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399 
with `exponent f < e` have "2 dvd mantissa f" 
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400 
apply (intro dvdI[where k="m * 2^(nat (eexponent f)) div 2"]) 
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401 
apply (cases "nat (e  exponent f)") 
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402 
apply auto 
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403 
done 
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404 
then show False using mantissa_not_dvd[OF not_0] by simp 
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405 
qed 
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406 
ultimately have "real m = mantissa f * 2^nat (exponent f  e)" 
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407 
by (simp add: powr_realpow[symmetric]) 
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408 
with `e \<le> exponent f` 
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409 
show "m = mantissa f * 2 ^ nat (exponent f  e)" "e = exponent f  nat (exponent f  e)" 
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410 
unfolding real_of_int_inject by auto 
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411 
qed 
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412 

47600  413 
lemma compute_zero[code_unfold, code]: "0 = Float 0 0" 
414 
by transfer simp 

415 

416 
lemma compute_one[code_unfold, code]: "1 = Float 1 0" 

417 
by transfer simp 

418 

419 
definition normfloat :: "float \<Rightarrow> float" where 

420 
[simp]: "normfloat x = x" 

421 

422 
lemma compute_normfloat[code]: "normfloat (Float m e) = 

423 
(if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1)) 

424 
else if m = 0 then 0 else Float m e)" 

425 
unfolding normfloat_def 

426 
by transfer (auto simp add: powr_add zmod_eq_0_iff) 

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427 

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428 
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k" 
47600  429 
by transfer simp 
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430 

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431 
lemma compute_float_neg_numeral[code_abbrev]: "Float (neg_numeral k) 0 = neg_numeral k" 
47600  432 
by transfer simp 
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433 

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434 
lemma compute_float_uminus[code]: " Float m1 e1 = Float ( m1) e1" 
47600  435 
by transfer simp 
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436 

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437 
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)" 
47600  438 
by transfer (simp add: field_simps powr_add) 
47599
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439 

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440 
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 = 
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441 
(if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2  e1)) e1 
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442 
else Float (m2 + m1 * 2^nat (e1  e2)) e2)" 
47600  443 
by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric]) 
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444 

47600  445 
lemma compute_float_minus[code]: fixes f g::float shows "f  g = f + (g)" 
446 
by simp 

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447 

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448 
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then 1 else 0)" 
47600  449 
by transfer (simp add: sgn_times) 
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450 

47600  451 
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" .. 
47599
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452 

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453 
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m" 
47600  454 
by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0]) 
47599
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455 

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456 
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b  a)" 
47600  457 
by transfer (simp add: field_simps) 
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458 

47600  459 
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" .. 
47599
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changeset

460 

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461 
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m" 
47600  462 
by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0]) 
47599
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463 

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464 
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b  a)" 
47600  465 
by transfer (simp add: field_simps) 
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466 

47600  467 
lift_definition is_float_zero :: "float \<Rightarrow> bool" is "op = 0 :: real \<Rightarrow> bool" by simp 
47599
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changeset

468 

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469 
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m" 
47600  470 
by transfer (auto simp add: is_float_zero_def) 
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471 

47600  472 
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e" 
473 
by transfer (simp add: abs_mult) 

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474 

47600  475 
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f  g)" 
476 
by transfer simp 

47599
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changeset

477 

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changeset

478 
subsection {* Rounding Real numbers *} 
400b158f1589
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479 

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480 
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where 
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481 
"round_down prec x = floor (x * 2 powr prec) * 2 powr prec" 
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changeset

482 

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483 
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where 
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484 
"round_up prec x = ceiling (x * 2 powr prec) * 2 powr prec" 
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changeset

485 

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486 
lemma round_down_float[simp]: "round_down prec x \<in> float" 
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changeset

487 
unfolding round_down_def 
400b158f1589
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changeset

488 
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus) 
400b158f1589
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parents:
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changeset

489 

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490 
lemma round_up_float[simp]: "round_up prec x \<in> float" 
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491 
unfolding round_up_def 
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492 
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus) 
400b158f1589
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changeset

493 

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494 
lemma round_up: "x \<le> round_up prec x" 
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changeset

495 
by (simp add: powr_minus_divide le_divide_eq round_up_def) 
400b158f1589
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hoelzl
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changeset

496 

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changeset

497 
lemma round_down: "round_down prec x \<le> x" 
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changeset

498 
by (simp add: powr_minus_divide divide_le_eq round_down_def) 
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changeset

499 

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500 
lemma round_up_0[simp]: "round_up p 0 = 0" 
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501 
unfolding round_up_def by simp 
400b158f1589
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changeset

502 

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503 
lemma round_down_0[simp]: "round_down p 0 = 0" 
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changeset

504 
unfolding round_down_def by simp 
400b158f1589
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changeset

505 

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506 
lemma round_up_diff_round_down: 
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507 
"round_up prec x  round_down prec x \<le> 2 powr prec" 
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changeset

508 
proof  
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changeset

509 
have "round_up prec x  round_down prec x = 
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changeset

510 
(ceiling (x * 2 powr prec)  floor (x * 2 powr prec)) * 2 powr prec" 
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changeset

511 
by (simp add: round_up_def round_down_def field_simps) 
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changeset

512 
also have "\<dots> \<le> 1 * 2 powr prec" 
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changeset

513 
by (rule mult_mono) 
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changeset

514 
(auto simp del: real_of_int_diff 
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515 
simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1) 
400b158f1589
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516 
finally show ?thesis by simp 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
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517 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
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changeset

518 

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519 
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x" 
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520 
unfolding round_down_def 
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521 
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric]) 
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522 
(simp add: powr_add[symmetric]) 
29804
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Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
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changeset

523 

47599
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524 
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x" 
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changeset

525 
unfolding round_up_def 
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changeset

526 
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric]) 
400b158f1589
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hoelzl
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changeset

527 
(simp add: powr_add[symmetric]) 
400b158f1589
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hoelzl
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changeset

528 

400b158f1589
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changeset

529 
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

530 

47600  531 
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

532 
declare float_up.rep_eq[simp] 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
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diff
changeset

533 

47599
400b158f1589
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changeset

534 
lemma float_up_correct: 
400b158f1589
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changeset

535 
shows "real (float_up e f)  real f \<in> {0..2 powr e}" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

536 
unfolding atLeastAtMost_iff 
400b158f1589
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hoelzl
parents:
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diff
changeset

537 
proof 
400b158f1589
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hoelzl
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changeset

538 
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

539 
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

540 
finally show "real (float_up e f)  real f \<le> 2 powr real ( e)" 
47600  541 
by simp 
542 
qed (simp add: algebra_simps round_up) 

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

543 

47600  544 
lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

545 
declare float_down.rep_eq[simp] 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

546 

47599
400b158f1589
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hoelzl
parents:
47230
diff
changeset

547 
lemma float_down_correct: 
400b158f1589
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parents:
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diff
changeset

548 
shows "real f  real (float_down e f) \<in> {0..2 powr e}" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

549 
unfolding atLeastAtMost_iff 
400b158f1589
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hoelzl
parents:
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diff
changeset

550 
proof 
400b158f1589
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hoelzl
parents:
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diff
changeset

551 
have "f  round_down e f \<le> round_up e f  round_down e f" using round_up by simp 
400b158f1589
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hoelzl
parents:
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diff
changeset

552 
also have "\<dots> \<le> 2 powr e" using round_up_diff_round_down by simp 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

553 
finally show "real f  real (float_down e f) \<le> 2 powr real ( e)" 
47600  554 
by simp 
555 
qed (simp add: algebra_simps round_down) 

24301  556 

47599
400b158f1589
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hoelzl
parents:
47230
diff
changeset

557 
lemma compute_float_down[code]: 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

558 
"float_down p (Float m e) = 
400b158f1589
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hoelzl
parents:
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diff
changeset

559 
(if p + e < 0 then Float (m div 2^nat ((p + e))) (p) else Float m e)" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

560 
proof cases 
400b158f1589
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hoelzl
parents:
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diff
changeset

561 
assume "p + e < 0" 
400b158f1589
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hoelzl
parents:
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diff
changeset

562 
hence "real ((2::int) ^ nat ((p + e))) = 2 powr ((p + e))" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

563 
using powr_realpow[of 2 "nat ((p + e))"] by simp 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

564 
also have "... = 1 / 2 powr p / 2 powr e" 
47600  565 
unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add) 
47599
400b158f1589
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hoelzl
parents:
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diff
changeset

566 
finally show ?thesis 
47600  567 
using `p + e < 0` 
568 
by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric]) 

47599
400b158f1589
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hoelzl
parents:
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diff
changeset

569 
next 
47600  570 
assume "\<not> p + e < 0" 
571 
then have r: "real e + real p = real (nat (e + p))" by simp 

572 
have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p" 

573 
by (auto intro: exI[where x="m*2^nat (e+p)"] 

574 
simp add: ac_simps powr_add[symmetric] r powr_realpow) 

575 
with `\<not> p + e < 0` show ?thesis 

576 
by transfer 

577 
(auto simp add: round_down_def field_simps powr_add powr_minus inverse_eq_divide) 

47599
400b158f1589
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hoelzl
parents:
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diff
changeset

578 
qed 
24301  579 

47599
400b158f1589
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hoelzl
parents:
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diff
changeset

580 
lemma ceil_divide_floor_conv: 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

581 
assumes "b \<noteq> 0" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

582 
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

583 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

584 
assume "\<not> b dvd a" 
400b158f1589
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hoelzl
parents:
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diff
changeset

585 
hence "a mod b \<noteq> 0" by auto 
400b158f1589
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hoelzl
parents:
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diff
changeset

586 
hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto 
400b158f1589
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hoelzl
parents:
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diff
changeset

587 
have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

588 
apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric]) 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

589 
proof  
400b158f1589
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hoelzl
parents:
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diff
changeset

590 
have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp 
400b158f1589
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hoelzl
parents:
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diff
changeset

591 
moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

592 
apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto 
400b158f1589
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hoelzl
parents:
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diff
changeset

593 
ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

594 
qed 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

595 
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

596 
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric] 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

597 
floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus) 
19765  598 

47599
400b158f1589
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hoelzl
parents:
47230
diff
changeset

599 
lemma compute_float_up[code]: 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

600 
"float_up p (Float m e) = 
400b158f1589
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hoelzl
parents:
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diff
changeset

601 
(let P = 2^nat ((p + e)); r = m mod P in 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

602 
if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (p) else Float m e)" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

603 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

604 
assume "p + e < 0" 
400b158f1589
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hoelzl
parents:
47230
diff
changeset

605 
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

606 
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

607 
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

608 
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

609 
finally have twopow_rewrite: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

610 
"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

611 
with `p + e < 0` have powr_rewrite: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

612 
"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

613 
unfolding powr_divide2 by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

614 
show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

615 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

616 
assume "2^nat ((p + e)) dvd m" 
47600  617 
with `p + e < 0` twopow_rewrite show ?thesis unfolding Let_def 
618 
by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div dvd_eq_mod_eq_0) 

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

619 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

620 
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

621 
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

622 
real m / real ((2::int) ^ nat ( (p + e)))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

623 
by (simp add: field_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

624 
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

625 
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

626 
using ndvd unfolding powr_rewrite one_div 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

627 
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

628 
thus ?thesis using `p + e < 0` twopow_rewrite 
47600  629 
unfolding Let_def 
630 
by transfer (auto simp: ac_simps 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

631 
qed 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

632 
next 
47600  633 
assume "\<not> p + e < 0" 
634 
then have r1: "real e + real p = real (nat (e + p))" by simp 

635 
have r: "\<lceil>(m * 2 powr e) * 2 powr real p\<rceil> = (m * 2 powr e) * 2 powr real p" 

636 
by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow 

637 
intro: exI[where x="m*2^nat (e+p)"]) 

638 
then show ?thesis using `\<not> p + e < 0` 

639 
unfolding Let_def 

640 
by transfer 

641 
(simp add: round_up_def floor_divide_eq_div field_simps powr_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

642 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

643 

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

644 
lemmas real_of_ints = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

645 
real_of_int_zero 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

646 
real_of_one 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

647 
real_of_int_add 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

648 
real_of_int_minus 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

649 
real_of_int_diff 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

650 
real_of_int_mult 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

651 
real_of_int_power 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

652 
real_numeral 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

653 
lemmas real_of_nats = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

654 
real_of_nat_zero 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

655 
real_of_nat_one 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

656 
real_of_nat_1 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

657 
real_of_nat_add 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

658 
real_of_nat_mult 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

659 
real_of_nat_power 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

660 

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

661 
lemmas int_of_reals = real_of_ints[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

662 
lemmas nat_of_reals = real_of_nats[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

663 

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

664 
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

665 
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

666 

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

667 
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

668 

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

669 
subsection {* Compute bitlen of integers *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

670 

47600  671 
definition bitlen :: "int \<Rightarrow> int" where 
672 
"bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)" 

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

673 

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

674 
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

675 
proof  
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

676 
{ 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

677 
assume "0 > x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

678 
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

679 
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

680 
finally have "1 < log 2 (x)" . 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

681 
} 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

682 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

683 

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

684 
lemma bitlen_bounds: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

685 
assumes "x > 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

686 
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

687 
proof 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

688 
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

689 
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

690 
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

691 
by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

692 
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

693 
by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

694 
also have "... = real x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

695 
using `0 < x` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

696 
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

697 
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

698 
by (simp add: bitlen_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

699 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

700 
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

701 
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

702 
apply (simp add: powr_realpow[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

703 
using `x > 0` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

704 
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

705 
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

706 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

707 

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

708 
lemma bitlen_pow2[simp]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

709 
assumes "b > 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

710 
shows "bitlen (b * 2 ^ c) = bitlen b + c" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

711 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

712 
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

713 
thus ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

714 
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

715 
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

716 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

717 

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

718 
lemma bitlen_Float: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

719 
fixes m e 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

720 
defines "f \<equiv> Float m e" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

721 
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

722 
proof cases 
47600  723 
assume "m \<noteq> 0" 
724 
hence "f \<noteq> float_of 0" 

725 
unfolding real_of_float_eq by (simp add: f_def) 

726 
hence "mantissa f \<noteq> 0" 

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

727 
by (simp add: mantissa_noteq_0) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

729 
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

730 
ultimately show ?thesis by (simp add: abs_mult) 
47600  731 
qed (simp add: f_def bitlen_def Float_def) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

732 

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

733 
lemma compute_bitlen[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

734 
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

735 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

736 
{ assume "2 \<le> x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

737 
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

738 
by (simp add: log_mult zmod_zdiv_equality') 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

739 
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

740 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

741 
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

742 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

743 
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

744 
then have "0 \<le> n" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

745 
using `2 \<le> x` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

746 
assume "x mod 2 \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

747 
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

748 
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

749 
moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

750 
{ 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

751 
by (simp add: powr_realpow) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

752 
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

753 
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

754 
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

755 
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

756 
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

757 
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

758 
{ 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

759 
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

760 
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

761 
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

762 
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

763 
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

764 
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

765 
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

766 
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

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

768 
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

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

770 
{ assume "x < 2" "0 < x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

771 
then have "x = 1" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

772 
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

773 
ultimately show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

775 
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

776 
qed 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

777 

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

778 
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

779 
shows "0 \<le> e + (bitlen m  1)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

781 
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

782 
hence "0 < m" using powr_gt_zero[of 2 e] 
47600  783 
by (auto simp: zero_less_mult_iff) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

784 
hence "m \<noteq> 0" by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

785 
show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

786 
proof (cases "0 \<le> e") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

787 
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

788 
next 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

789 
have "(1::int) < 2" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

790 
case False let ?S = "2^(nat (e))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

791 
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

792 
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

793 
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

794 
by (auto simp: powr_minus) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

795 
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

796 
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

797 
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

798 
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

799 
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

800 
hence "e < bitlen m" using False by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

801 
thus ?thesis by auto 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

802 
qed 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

803 
qed 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

804 

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

805 
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

806 
proof  
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

807 
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

808 

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

809 
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

810 
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

811 
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

812 

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

813 
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

814 
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

815 

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

816 
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

817 
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

818 
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

819 
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

820 
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

821 
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

822 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

823 

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

824 
subsection {* Approximation of positive rationals *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

825 

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

826 
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

827 
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

828 

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

829 
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

830 
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

831 

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

832 
lemma real_div_nat_eq_floor_of_divide: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

834 
shows "a div b = real (floor (a/b))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

835 
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

836 

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

837 
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

838 

47600  839 
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" 
840 
is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp 

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

841 

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

842 
lemma compute_lapprox_posrat[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

843 
fixes prec x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

844 
shows "lapprox_posrat prec x y = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

845 
(let 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

846 
l = rat_precision prec x y; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

847 
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

848 
in normfloat (Float d ( l)))" 
47600  849 
unfolding div_mult_twopow_eq Let_def normfloat_def 
850 
by transfer 

851 
(simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps 

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

852 
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

853 

47600  854 
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" 
855 
is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp 

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

856 

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

857 
(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

858 
lemma compute_rapprox_posrat[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

859 
fixes prec x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

860 
defines "l \<equiv> rat_precision prec x y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

861 
shows "rapprox_posrat prec x y = (let 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

862 
l = l ; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

863 
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

864 
d = fst X div snd X ; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

865 
m = fst X mod snd X 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

866 
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

867 
proof (cases "y = 0") 
47600  868 
assume "y = 0" thus ?thesis unfolding Let_def normfloat_def by transfer simp 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

869 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

870 
assume "y \<noteq> 0" 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

871 
show ?thesis 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

872 
proof (cases "0 \<le> l") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

873 
assume "0 \<le> l" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

874 
def x' == "x * 2 ^ nat l" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

875 
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

876 
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

877 
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

878 
ultimately show ?thesis 
47600  879 
unfolding Let_def normfloat_def 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

880 
using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0` 
47600  881 
l_def[symmetric, THEN meta_eq_to_obj_eq] 
882 
by transfer 

883 
(simp add: floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 round_up_def) 

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

884 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

885 
assume "\<not> 0 \<le> l" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

886 
def y' == "y * 2 ^ nat ( l)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

887 
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

888 
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

889 
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

890 
using `\<not> 0 \<le> l` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

891 
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

892 
ultimately show ?thesis 
47600  893 
unfolding Let_def normfloat_def 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

894 
using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0` 
47600  895 
l_def[symmetric, THEN meta_eq_to_obj_eq] 
896 
by transfer 

897 
(simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0) 

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

898 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

899 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

900 

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

901 
lemma rat_precision_pos: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

902 
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

903 
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

904 
proof  
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

905 
{ 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

906 
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

907 
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

908 
(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

909 
thus ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

910 
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

911 
qed 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

912 

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

913 
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

914 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

915 
def y \<equiv> "nat (x  1)" moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

916 
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

917 
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

918 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

919 

47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

920 
lemma rapprox_posrat_less1: 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

921 
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

922 
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

923 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

924 
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

925 
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

926 
by (simp add: powr_realpow[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

927 
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

928 
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

929 
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

930 
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

931 
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

932 
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

933 
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

934 
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

935 
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

936 
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

937 
using rat_precision_pos[OF assms] by (rule power_aux) 
47600  938 
finally show ?thesis 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

939 
apply (transfer fixing: n x y) 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

940 
apply (simp add: round_up_def field_simps powr_minus inverse_eq_divide powr1) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

941 
unfolding int_of_reals real_of_int_less_iff 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

942 
apply (simp add: ceiling_less_eq) 
47600  943 
done 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

944 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

945 

47600  946 
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is 
947 
"\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp 

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

948 

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

949 
lemma compute_lapprox_rat[code]: 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

950 
"lapprox_rat prec x y = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

951 
(if y = 0 then 0 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

952 
else if 0 \<le> x then 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

953 
(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

954 
else  (rapprox_posrat prec (nat x) (nat (y)))) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

955 
else (if 0 < y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

956 
then  (rapprox_posrat prec (nat (x)) (nat y)) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

957 
else lapprox_posrat prec (nat (x)) (nat (y))))" 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

958 
by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

959 

47600  960 
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is 
961 
"\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp 

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

962 

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

963 
lemma compute_rapprox_rat[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

964 
"rapprox_rat prec x y = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

965 
(if y = 0 then 0 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

966 
else if 0 \<le> x then 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

967 
(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

968 
else  (lapprox_posrat prec (nat x) (nat (y)))) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

969 
else (if 0 < y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

970 
then  (lapprox_posrat prec (nat (x)) (nat y)) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

971 
else rapprox_posrat prec (nat (x)) (nat (y))))" 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

972 
by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

973 

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

974 
subsection {* Division *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

975 

47600  976 
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is 
977 
"\<lambda>(prec::nat) a b. round_down (prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor>  \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)" by simp 

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

978 

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

979 
lemma compute_float_divl[code]: 
47600  980 
"float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1  s2)" 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

981 
proof cases 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

982 
let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

983 
let ?m = "real m1 / real m2" and ?s = "2 powr real (s1  s2)" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

984 
assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

985 
then have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor>  \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) = rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2  s1)" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

986 
by (simp add: abs_mult log_mult rat_precision_def bitlen_def) 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

987 
have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

988 
by (simp add: field_simps powr_divide2[symmetric]) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

989 

47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

990 
show ?thesis 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

991 
using not_0 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

992 
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift, simp add: field_simps) 
47600  993 
qed (transfer, auto) 
994 

995 
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is 

996 
"\<lambda>(prec::nat) a b. round_up (prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor>  \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)" by simp 

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

997 

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

998 
lemma compute_float_divr[code]: 
47600  999 
"float_divr prec (Float m1 s1) (Float m2 s2) = rapprox_rat prec m1 m2 * Float 1 (s1  s2)" 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1000 
proof cases 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1001 
let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1002 
let ?m = "real m1 / real m2" and ?s = "2 powr real (s1  s2)" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1003 
assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1004 
then have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor>  \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) = rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2  s1)" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1005 
by (simp add: abs_mult log_mult rat_precision_def bitlen_def) 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1006 
have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s" 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1007 
by (simp add: field_simps powr_divide2[symmetric]) 
47600  1008 

47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1009 
show ?thesis 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1010 
using not_0 
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1011 
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift, simp add: field_simps) 
47600  1012 
qed (transfer, auto) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1013 

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

1014 
subsection {* Lemmas needed by Approximate *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1015 

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

1016 
lemma Float_num[simp]: shows 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1017 
"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

1018 
"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

1019 
"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

1020 
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

1021 
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

1022 
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

1023 
by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1024 

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

1025 
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:
47230
diff
changeset

1026 

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

1027 
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1028 

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

1029 
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:
47230
diff
changeset

1030 
by arith 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1031 

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

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

1033 
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

1034 
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

1035 

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

1036 
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

1037 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1038 
from zmod_zdiv_equality'[of a b] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1039 
have "a = b * (a div b) + a mod b" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1040 
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

1041 
using assms by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1042 
finally show ?thesis by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1043 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1044 

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

1045 
lemma lapprox_rat_nonneg: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1046 
fixes n x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1047 
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:
47230
diff
changeset

1048 
assumes "0 \<le> x" "0 < y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1049 
shows "0 \<le> real (lapprox_rat n x y)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1050 
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

1051 
powr_int[of 2, simplified] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1052 
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

1053 

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

1054 
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:
47230
diff
changeset

1055 
using round_up by (simp add: rapprox_rat_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1056 

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

1057 
lemma rapprox_rat_le1: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1058 
fixes n x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1059 
assumes xy: "0 \<le> x" "0 < y" "x \<le> y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1060 
shows "real (rapprox_rat n x y) \<le> 1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1061 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1062 
have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1063 
using xy unfolding bitlen_def by (auto intro!: floor_mono) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1064 
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:
47230
diff
changeset

1065 
have "real \<lceil>real x / real y * 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil> 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1066 
\<le> real \<lceil>2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1067 
using xy by (auto intro!: ceiling_mono simp: field_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1068 
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

1069 
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

1070 
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:
47230
diff
changeset

1071 
finally show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1072 
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

1073 
(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

1074 
qed 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1075 

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

1076 
lemma rapprox_rat_nonneg_neg: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1077 
"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

1078 
unfolding rapprox_rat_def round_up_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1079 
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

1080 

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

1081 
lemma rapprox_rat_neg: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1082 
"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

1083 
unfolding rapprox_rat_def round_up_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1084 
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

1085 

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

1086 
lemma rapprox_rat_nonpos_pos: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1087 
"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

1088 
unfolding rapprox_rat_def round_up_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1089 
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

1090 

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

1091 
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y" 
47600  1092 
by transfer (simp add: round_down) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1093 

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

1094 
lemma float_divl_lower_bound: 
47600  1095 
"0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 0 \<le> real (float_divl prec x y)" 
1096 
by transfer (simp add: round_down_def zero_le_mult_iff zero_le_divide_iff) 

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

1097 

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

1098 
lemma exponent_1: "exponent 1 = 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1099 
using exponent_float[of 1 0] by (simp add: one_float_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1100 

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

1101 
lemma mantissa_1: "mantissa 1 = 1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1102 
using mantissa_float[of 1 0] by (simp add: one_float_def) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1103 

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

1104 
lemma bitlen_1: "bitlen 1 = 1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1105 
by (simp add: bitlen_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1106 

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

1107 
lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1108 
proof 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1109 
assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1110 
show "x = 0" by (simp add: zero_float_def z) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1111 
qed (simp add: zero_float_def) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1112 

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

1113 
lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1114 
proof (cases "x = 0", simp) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1115 
assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1116 
have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1117 
also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1118 
also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1119 
using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1120 
by (simp add: powr_int) (simp only: two_real_int int_of_reals real_of_int_abs[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

1122 
finally show ?thesis by (simp add: powr_add) 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1123 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1124 

41528  1125 
lemma float_divl_pos_less1_bound: 
47600  1126 
"0 < real x \<Longrightarrow> real x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)" 
1127 
proof transfer 

1128 
fix prec :: nat and x :: real assume x: "0 < x" "x < 1" "x \<in> float" and prec: "1 \<le> prec" 

1129 
def p \<equiv> "int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>" 

1130 
show "1 \<le> round_down (int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>  \<lfloor>log 2 \<bar>1\<bar>\<rfloor>) (1 / x) " 

1131 
proof cases 

1132 
assume nonneg: "0 \<le> p" 

1133 
hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)" 

1134 
by (simp add: powr_int del: real_of_int_power) simp 

1135 
also have "floor (1::real) \<le> floor (1 / x)" using x prec by simp 

1136 
also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) \<le> 

1137 
floor (real ((2::int) ^ nat p) * (1 / x))" 

1138 
by (rule le_mult_floor) (auto simp: x prec less_imp_le) 

1139 
finally have "2 powr real p \<le> floor (2 powr nat p / x)" by (simp add: powr_realpow) 

1140 
thus ?thesis unfolding p_def[symmetric] 

1141 
using x prec nonneg by (simp add: powr_minus inverse_eq_divide round_down_def) 

1142 
next 

1143 
assume neg: "\<not> 0 \<le> p" 

1144 

1145 
have "x = 2 powr (log 2 x)" 

1146 
using x by simp 

1147 
also have "2 powr (log 2 x) \<le> 2 powr p" 

1148 
proof (rule powr_mono) 

1149 
have "log 2 x \<le> \<lceil>log 2 x\<rceil>" 

1150 
by simp 

1151 
also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + 1" 

1152 
using ceiling_diff_floor_le_1[of "log 2 x"] by simp 

1153 
also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + prec" 

1154 
using prec by simp 

1155 
finally show "log 2 x \<le> real p" 

1156 
using x by (simp add: p_def) 

1157 
qed simp 

1158 
finally have x_le: "x \<le> 2 powr p" . 

1159 

1160 
from neg have "2 powr real p \<le> 2 powr 0" 

1161 
by (intro powr_mono) auto 

1162 
also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp 

1163 
also have "\<dots> \<le> \<lfloor>2 powr real p / x\<rfloor>" unfolding real_of_int_le_iff 

1164 
using x x_le by (intro floor_mono) (simp add: pos_le_divide_eq mult_pos_pos) 

1165 
finally show ?thesis 

1166 
using prec x unfolding p_def[symmetric] 

1167 
by (simp add: round_down_def powr_minus_divide pos_le_divide_eq mult_pos_pos) 

1168 
qed 

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

1169 
qed 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1170 

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

1171 
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)" 
47600  1172 
using round_up by transfer simp 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1173 

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

1174 
lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x" 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1175 
proof  
47600  1176 
have "1 \<le> 1 / real x" using `0 < x` and `x < 1` by auto 
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

1177 
also have "\<dots> \<le> real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto 
47600  1178 
finally show ?thesis by auto 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1179 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1180 

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

1181 
lemma float_divr_nonpos_pos_upper_bound: 
47600  1182 
"real x \<le> 0 \<Longrightarrow> 0 < real y \<Longrightarrow> real (float_divr prec x y) \<le> 0" 
1183 
by transfer (auto simp: field_simps mult_le_0_iff divide_le_0_iff round_up_def) 

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

1184 

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

1185 
lemma float_divr_nonneg_neg_upper_bound: 
47600  1186 
"0 \<le> real x \<Longrightarrow> real y < 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0" 
1187 
by transfer (auto simp: field_simps mult_le_0_iff zero_le_mult_iff divide_le_0_iff round_up_def) 

1188 

1189 
lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is 

1190 
"\<lambda>(prec::nat) x. round_up (prec  \<lfloor>log 2 \<bar>x\<bar>\<rfloor>  1) x" by simp 

1191 

1192 
lemma float_round_up: "real x \<le> real (float_round_up prec x)" 

1193 
using round_up by transfer simp 

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

1194 

47600  1195 
lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is 
1196 
"\<lambda>(prec::nat) x. round_down (prec  \<lfloor>log 2 \<bar>x\<bar>\<rfloor>  1) x" by simp 

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

1197 

47600  1198 
lemma float_round_down: "real (float_round_down prec x) \<le> real x" 
1199 
using round_down by transfer simp 

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

1200 

47600  1201 
lemma floor_add2[simp]: "\<lfloor> real i + x \<rfloor> = i + \<lfloor> x \<rfloor>" 
1202 
using floor_add[of x i] by (simp del: floor_add add: ac_simps) 

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

1203 

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

1204 
lemma compute_float_round_down[code]: 
47600  1205 
"float_round_down prec (Float m e) = (let d = bitlen (abs m)  int prec in 
1206 
if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d) 

1207 
else Float m e)" 

47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1208 
using compute_float_down[of "prec  bitlen \<bar>m\<bar>  e" m e, symmetric] 
47600  1209 
unfolding Let_def 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1210 
by transfer (simp add: field_simps abs_mult log_mult bitlen_def cong del: if_weak_cong) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1211 

47600  1212 
lemma compute_float_round_up[code]: 
1213 
"float_round_up prec (Float m e) = (let d = (bitlen (abs m)  int prec) in 

1214 
if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P 

1215 
in Float (n + (if r = 0 then 0 else 1)) (e + d) 

1216 
else Float m e)" 

1217 
using compute_float_up[of "prec  bitlen \<bar>m\<bar>  e" m e, symmetric] 

1218 
unfolding Let_def 

47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1219 
by transfer (simp add: field_simps abs_mult log_mult bitlen_def cong del: if_weak_cong) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1220 

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

1221 
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0" 
400b158f1589
replace the float datatype by a type with unique representation
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parents:
47230
diff
changeset

1222 
apply (auto simp: zero_float_def mult_le_0_iff) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1223 
using powr_gt_zero[of 2 b] by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1224 

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

1225 
(* TODO: how to use as code equation? > pprt_float?! *) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1226 
lemma compute_pprt[code]: "pprt (Float a e) = (if a <= 0 then 0 else (Float a e))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1227 
unfolding pprt_def sup_float_def max_def Float_le_zero_iff .. 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1228 

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

1229 
(* TODO: how to use as code equation? *) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1230 
lemma compute_nprt[code]: "nprt (Float a e) = (if a <= 0 then (Float a e) else 0)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset

1231 
unfolding nprt_def inf_float_def min_def Float_le_zero_iff .. 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1232 

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

1233 
lemma of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)" 
47600  1234 
unfolding pprt_def sup_float_def max_def sup_real_def by auto 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1235 

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

1236 
lemma of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)" 
47600  1237 
unfolding nprt_def inf_float_def min_def inf_real_def by auto 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1238 

47600  1239 
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor by simp 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1240 

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

1241 
lemma compute_int_floor_fl[code]: 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1242 
"int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (e))))" 
47600  1243 
by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1244 

47600  1245 
lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1246 

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

1247 
lemma compute_floor_fl[code]: 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1248 
"floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (e)))) 0)" 
47600  1249 
by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1250 

47600  1251 
lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp 
1252 

1253 
lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp 

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