author  immler 
Mon, 16 Dec 2013 17:08:22 +0100  
changeset 54782  cd8f55c358c5 
parent 54489  03ff4d1e6784 
child 54783  25860d89a044 
permissions  rwrr 
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(* Title: HOL/Library/Float.thy 
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Author: Johannes Hölzl, Fabian Immler 

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Copyright 2012 TU München 

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*) 

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header {* FloatingPoint Numbers *} 
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theory Float 
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imports Complex_Main Lattice_Algebras 
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begin 
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definition "float = {m * 2 powr e  (m :: int) (e :: int). True}" 
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typedef float = float 
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morphisms real_of_float float_of 
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unfolding float_def 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]: " numeral i \<in> float" by (intro floatI[of " 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  numeral i \<in> float" by (intro floatI[of 1 " 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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using plus_float [of x " y"] by simp 
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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 / ( 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 /  numeral (Num.Bit0 n)" 
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by simp 
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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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lemma compute_real_of_float[code]: 
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"real_of_float (Float m e) = (if e \<ge> 0 then m * 2 ^ nat e else m / 2 ^ (nat (e)))" 

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by (simp add: real_of_float_def[symmetric] powr_int) 

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code_datatype Float 
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subsection {* Arithmetic operations on floating point numbers *} 
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47600  143 
instantiation float :: "{ring_1, linorder, linordered_ring, linordered_idom, numeral, equal}" 
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begin 
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47600  146 
lift_definition zero_float :: float is 0 by simp 
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declare zero_float.rep_eq[simp] 
47600  148 
lift_definition one_float :: float is 1 by simp 
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declare one_float.rep_eq[simp] 
47600  150 
lift_definition plus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op +" by simp 
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declare plus_float.rep_eq[simp] 
47600  152 
lift_definition times_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op *" by simp 
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declare times_float.rep_eq[simp] 
47600  154 
lift_definition minus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op " by simp 
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declare minus_float.rep_eq[simp] 
47600  156 
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  159 
lift_definition abs_float :: "float \<Rightarrow> float" is abs by simp 
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declare abs_float.rep_eq[simp] 
47600  161 
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  164 
lift_definition equal_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op = :: real \<Rightarrow> real \<Rightarrow> bool" .. 
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47600  166 
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  168 
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  172 
proof qed (transfer, fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+ 
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end 
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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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53381  178 
lemma fixes x y::float 
47600  179 
shows real_of_float_min: "real (min x y) = min (real x) (real y)" 
180 
and real_of_float_max: "real (max x y) = max (real x) (real y)" 

181 
by (simp_all add: min_def max_def) 

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instance float :: unbounded_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"]) 
47600  188 
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"]) 
47600  193 
apply transfer 
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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" 
47600  198 
apply (intro exI[of _ "(a + b) * Float 1 1"]) 
199 
apply transfer 

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apply (simp add: powr_minus) 
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done 
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qed 
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47600  204 
instantiation float :: lattice_ab_group_add 
46573  205 
begin 
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47600  207 
definition inf_float::"float\<Rightarrow>float\<Rightarrow>float" 
208 
where "inf_float a b = min a b" 

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47600  210 
definition sup_float::"float\<Rightarrow>float\<Rightarrow>float" 
211 
where "sup_float a b = max a b" 

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instance 
47600  214 
by default 
215 
(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  218 
lemma float_numeral[simp]: "real (numeral x :: float) = numeral x" 
219 
apply (induct x) 

220 
apply simp 

221 
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  223 
done 
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53381  225 
lemma transfer_numeral [transfer_rule]: 
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"fun_rel (op =) pcr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)" 
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unfolding fun_rel_def float.pcr_cr_eq cr_float_def by simp 
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lemma float_neg_numeral[simp]: "real ( numeral x :: float) =  numeral x" 
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by simp 
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53381  232 
lemma transfer_neg_numeral [transfer_rule]: 
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"fun_rel (op =) pcr_float ( numeral :: _ \<Rightarrow> real) ( numeral :: _ \<Rightarrow> float)" 
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unfolding fun_rel_def float.pcr_cr_eq cr_float_def by simp 
47600  235 

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lemma 
47600  237 
shows float_of_numeral[simp]: "numeral k = float_of (numeral k)" 
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and float_of_neg_numeral[simp]: " numeral k = float_of ( numeral k)" 
47600  239 
unfolding real_of_float_eq by simp_all 
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240 

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subsection {* Represent floats as unique mantissa and exponent *} 
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242 

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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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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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qed 
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250 

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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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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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qed 
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264 

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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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269 
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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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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283 

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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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290 

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lemma floatE_normed: 
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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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proof atomize_elim 
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{ assume "x \<noteq> 0" 
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297 
from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def) 
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298 
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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299 
by auto 
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300 
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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301 
by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"]) 
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302 
(simp add: powr_add powr_realpow) } 
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303 
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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304 
by blast 
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305 
qed 
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306 

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307 
lemma float_normed_cases: 
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308 
fixes f :: float 
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309 
obtains (zero) "f = 0" 
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310 
 (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0" 
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311 
proof (atomize_elim, induct f) 
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312 
case (float_of y) then show ?case 
47600  313 
by (cases rule: floatE_normed) (auto simp: zero_float_def) 
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314 
qed 
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315 

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316 
definition mantissa :: "float \<Rightarrow> int" where 
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317 
"mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) 
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318 
\<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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319 

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320 
definition exponent :: "float \<Rightarrow> int" where 
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321 
"exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) 
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322 
\<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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323 

53381  324 
lemma 
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325 
shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E) 
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326 
and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M) 
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327 
proof  
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328 
have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto 
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329 
then show ?E ?M 
47600  330 
by (auto simp add: mantissa_def exponent_def zero_float_def) 
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331 
qed 
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332 

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333 
lemma 
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334 
shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E) 
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335 
and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D") 
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336 
proof cases 
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337 
assume [simp]: "f \<noteq> (float_of 0)" 
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338 
have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f" 
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339 
proof (cases f rule: float_normed_cases) 
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340 
case (powr m e) 
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341 
then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) 
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342 
\<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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343 
by auto 
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344 
then show ?thesis 
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345 
unfolding exponent_def mantissa_def 
47600  346 
by (rule someI2_ex) (simp add: zero_float_def) 
347 
qed (simp add: zero_float_def) 

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348 
then show ?E ?D by auto 
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349 
qed simp 
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350 

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351 
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0" 
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352 
using mantissa_not_dvd[of f] by auto 
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353 

53381  354 
lemma 
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355 
fixes m e :: int 
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356 
defines "f \<equiv> float_of (m * 2 powr e)" 
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357 
assumes dvd: "\<not> 2 dvd m" 
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358 
shows mantissa_float: "mantissa f = m" (is "?M") 
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359 
and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E") 
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360 
proof cases 
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361 
assume "m = 0" with dvd show "mantissa f = m" by auto 
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362 
next 
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363 
assume "m \<noteq> 0" 
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364 
then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def) 
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365 
from mantissa_exponent[of f] 
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366 
have "m * 2 powr e = mantissa f * 2 powr exponent f" 
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367 
by (auto simp add: f_def) 
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368 
then show "?M" "?E" 
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369 
using mantissa_not_dvd[OF f_not_0] dvd 
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370 
by (auto simp: mult_powr_eq_mult_powr_iff) 
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371 
qed 
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372 

47600  373 
subsection {* Compute arithmetic operations *} 
374 

375 
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f" 

376 
unfolding real_of_float_eq mantissa_exponent[of f] by simp 

377 

378 
lemma Float_cases[case_names Float, cases type: float]: 

379 
fixes f :: float 

380 
obtains (Float) m e :: int where "f = Float m e" 

381 
using Float_mantissa_exponent[symmetric] 

382 
by (atomize_elim) auto 

383 

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384 
lemma denormalize_shift: 
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385 
assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0" 
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386 
obtains i where "m = mantissa f * 2 ^ i" "e = exponent f  i" 
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387 
proof 
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388 
from mantissa_exponent[of f] f_def 
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389 
have "m * 2 powr e = mantissa f * 2 powr exponent f" 
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390 
by simp 
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391 
then have eq: "m = mantissa f * 2 powr (exponent f  e)" 
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392 
by (simp add: powr_divide2[symmetric] field_simps) 
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393 
moreover 
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394 
have "e \<le> exponent f" 
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395 
proof (rule ccontr) 
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396 
assume "\<not> e \<le> exponent f" 
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397 
then have pos: "exponent f < e" by simp 
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398 
then have "2 powr (exponent f  e) = 2 powr  real (e  exponent f)" 
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399 
by simp 
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400 
also have "\<dots> = 1 / 2^nat (e  exponent f)" 
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401 
using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric]) 
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402 
finally have "m * 2^nat (e  exponent f) = real (mantissa f)" 
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403 
using eq by simp 
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404 
then have "mantissa f = m * 2^nat (e  exponent f)" 
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405 
unfolding real_of_int_inject by simp 
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406 
with `exponent f < e` have "2 dvd mantissa f" 
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407 
apply (intro dvdI[where k="m * 2^(nat (eexponent f)) div 2"]) 
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408 
apply (cases "nat (e  exponent f)") 
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409 
apply auto 
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410 
done 
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411 
then show False using mantissa_not_dvd[OF not_0] by simp 
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412 
qed 
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413 
ultimately have "real m = mantissa f * 2^nat (exponent f  e)" 
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414 
by (simp add: powr_realpow[symmetric]) 
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415 
with `e \<le> exponent f` 
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416 
show "m = mantissa f * 2 ^ nat (exponent f  e)" "e = exponent f  nat (exponent f  e)" 
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417 
unfolding real_of_int_inject by auto 
29804
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418 
qed 
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419 

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420 
lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0" 
47600  421 
by transfer simp 
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422 
hide_fact (open) compute_float_zero 
47600  423 

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424 
lemma compute_float_one[code_unfold, code]: "1 = Float 1 0" 
47600  425 
by transfer simp 
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426 
hide_fact (open) compute_float_one 
47600  427 

428 
definition normfloat :: "float \<Rightarrow> float" where 

429 
[simp]: "normfloat x = x" 

430 

431 
lemma compute_normfloat[code]: "normfloat (Float m e) = 

432 
(if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1)) 

433 
else if m = 0 then 0 else Float m e)" 

434 
unfolding normfloat_def 

435 
by transfer (auto simp add: powr_add zmod_eq_0_iff) 

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436 
hide_fact (open) compute_normfloat 
47599
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437 

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438 
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k" 
47600  439 
by transfer simp 
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440 
hide_fact (open) compute_float_numeral 
47599
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441 

54489
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
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54230
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442 
lemma compute_float_neg_numeral[code_abbrev]: "Float ( numeral k) 0 =  numeral k" 
47600  443 
by transfer simp 
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444 
hide_fact (open) compute_float_neg_numeral 
47599
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445 

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446 
lemma compute_float_uminus[code]: " Float m1 e1 = Float ( m1) e1" 
47600  447 
by transfer simp 
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448 
hide_fact (open) compute_float_uminus 
47599
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449 

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450 
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)" 
47600  451 
by transfer (simp add: field_simps powr_add) 
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452 
hide_fact (open) compute_float_times 
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453 

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454 
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 = 
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455 
(if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2  e1)) e1 
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456 
else Float (m2 + m1 * 2^nat (e1  e2)) e2)" 
47600  457 
by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric]) 
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458 
hide_fact (open) compute_float_plus 
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459 

47600  460 
lemma compute_float_minus[code]: fixes f g::float shows "f  g = f + (g)" 
461 
by simp 

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462 
hide_fact (open) compute_float_minus 
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463 

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464 
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then 1 else 0)" 
47600  465 
by transfer (simp add: sgn_times) 
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466 
hide_fact (open) compute_float_sgn 
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467 

47600  468 
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" .. 
47599
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469 

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470 
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m" 
47600  471 
by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0]) 
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472 
hide_fact (open) compute_is_float_pos 
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473 

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474 
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b  a)" 
47600  475 
by transfer (simp add: field_simps) 
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476 
hide_fact (open) compute_float_less 
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477 

47600  478 
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" .. 
47599
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479 

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480 
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m" 
47600  481 
by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0]) 
47621
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482 
hide_fact (open) compute_is_float_nonneg 
47599
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changeset

483 

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484 
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b  a)" 
47600  485 
by transfer (simp add: field_simps) 
47621
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486 
hide_fact (open) compute_float_le 
47599
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487 

47600  488 
lift_definition is_float_zero :: "float \<Rightarrow> bool" is "op = 0 :: real \<Rightarrow> bool" by simp 
47599
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489 

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490 
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m" 
47600  491 
by transfer (auto simp add: is_float_zero_def) 
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492 
hide_fact (open) compute_is_float_zero 
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493 

47600  494 
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e" 
495 
by transfer (simp add: abs_mult) 

47621
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changeset

496 
hide_fact (open) compute_float_abs 
47599
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497 

47600  498 
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f  g)" 
499 
by transfer simp 

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500 
hide_fact (open) compute_float_eq 
47599
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changeset

501 

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502 
subsection {* Rounding Real numbers *} 
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503 

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504 
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where 
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505 
"round_down prec x = floor (x * 2 powr prec) * 2 powr prec" 
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506 

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507 
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where 
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508 
"round_up prec x = ceiling (x * 2 powr prec) * 2 powr prec" 
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changeset

509 

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510 
lemma round_down_float[simp]: "round_down prec x \<in> float" 
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511 
unfolding round_down_def 
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512 
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus) 
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513 

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514 
lemma round_up_float[simp]: "round_up prec x \<in> float" 
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515 
unfolding round_up_def 
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516 
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus) 
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517 

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518 
lemma round_up: "x \<le> round_up prec x" 
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519 
by (simp add: powr_minus_divide le_divide_eq round_up_def) 
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changeset

520 

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521 
lemma round_down: "round_down prec x \<le> x" 
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522 
by (simp add: powr_minus_divide divide_le_eq round_down_def) 
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523 

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524 
lemma round_up_0[simp]: "round_up p 0 = 0" 
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525 
unfolding round_up_def by simp 
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changeset

526 

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527 
lemma round_down_0[simp]: "round_down p 0 = 0" 
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528 
unfolding round_down_def by simp 
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changeset

529 

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530 
lemma round_up_diff_round_down: 
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531 
"round_up prec x  round_down prec x \<le> 2 powr prec" 
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532 
proof  
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533 
have "round_up prec x  round_down prec x = 
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534 
(ceiling (x * 2 powr prec)  floor (x * 2 powr prec)) * 2 powr prec" 
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535 
by (simp add: round_up_def round_down_def field_simps) 
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changeset

536 
also have "\<dots> \<le> 1 * 2 powr prec" 
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changeset

537 
by (rule mult_mono) 
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538 
(auto simp del: real_of_int_diff 
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539 
simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1) 
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changeset

540 
finally show ?thesis by simp 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
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diff
changeset

541 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
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changeset

542 

47599
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543 
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x" 
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changeset

544 
unfolding round_down_def 
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changeset

545 
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric]) 
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changeset

546 
(simp add: powr_add[symmetric]) 
29804
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hoelzl
parents:
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diff
changeset

547 

47599
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548 
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x" 
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changeset

549 
unfolding round_up_def 
400b158f1589
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changeset

550 
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric]) 
400b158f1589
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changeset

551 
(simp add: powr_add[symmetric]) 
400b158f1589
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changeset

552 

400b158f1589
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changeset

553 
subsection {* Rounding Floats *} 
29804
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changeset

554 

47600  555 
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

556 
declare float_up.rep_eq[simp] 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
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changeset

557 

54782  558 
lemma round_up_correct: 
559 
shows "round_up e f  f \<in> {0..2 powr e}" 

47599
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changeset

560 
unfolding atLeastAtMost_iff 
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changeset

561 
proof 
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changeset

562 
have "round_up e f  f \<le> round_up e f  round_down e f" using round_down by simp 
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changeset

563 
also have "\<dots> \<le> 2 powr e" using round_up_diff_round_down by simp 
54782  564 
finally show "round_up e f  f \<le> 2 powr real ( e)" 
47600  565 
by simp 
566 
qed (simp add: algebra_simps round_up) 

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

567 

54782  568 
lemma float_up_correct: 
569 
shows "real (float_up e f)  real f \<in> {0..2 powr e}" 

570 
by transfer (rule round_up_correct) 

571 

47600  572 
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

573 
declare float_down.rep_eq[simp] 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

574 

54782  575 
lemma round_down_correct: 
576 
shows "f  (round_down e f) \<in> {0..2 powr e}" 

47599
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changeset

577 
unfolding atLeastAtMost_iff 
400b158f1589
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changeset

578 
proof 
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changeset

579 
have "f  round_down e f \<le> round_up e f  round_down e f" using round_up by simp 
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changeset

580 
also have "\<dots> \<le> 2 powr e" using round_up_diff_round_down by simp 
54782  581 
finally show "f  round_down e f \<le> 2 powr real ( e)" 
47600  582 
by simp 
583 
qed (simp add: algebra_simps round_down) 

24301  584 

54782  585 
lemma float_down_correct: 
586 
shows "real f  real (float_down e f) \<in> {0..2 powr e}" 

587 
by transfer (rule round_down_correct) 

588 

47599
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changeset

589 
lemma compute_float_down[code]: 
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changeset

590 
"float_down p (Float m e) = 
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changeset

591 
(if p + e < 0 then Float (m div 2^nat ((p + e))) (p) else Float m e)" 
400b158f1589
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changeset

592 
proof cases 
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changeset

593 
assume "p + e < 0" 
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changeset

594 
hence "real ((2::int) ^ nat ((p + e))) = 2 powr ((p + e))" 
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changeset

595 
using powr_realpow[of 2 "nat ((p + e))"] by simp 
400b158f1589
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hoelzl
parents:
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changeset

596 
also have "... = 1 / 2 powr p / 2 powr e" 
47600  597 
unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add) 
47599
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changeset

598 
finally show ?thesis 
47600  599 
using `p + e < 0` 
600 
by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric]) 

47599
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changeset

601 
next 
47600  602 
assume "\<not> p + e < 0" 
603 
then have r: "real e + real p = real (nat (e + p))" by simp 

604 
have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p" 

605 
by (auto intro: exI[where x="m*2^nat (e+p)"] 

606 
simp add: ac_simps powr_add[symmetric] r powr_realpow) 

607 
with `\<not> p + e < 0` show ?thesis 

608 
by transfer 

609 
(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

610 
qed 
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset

611 
hide_fact (open) compute_float_down 
24301  612 

54782  613 
lemma abs_round_down_le: "\<bar>f  (round_down e f)\<bar> \<le> 2 powr e" 
614 
using round_down_correct[of f e] by simp 

615 

616 
lemma abs_round_up_le: "\<bar>f  (round_up e f)\<bar> \<le> 2 powr e" 

617 
using round_up_correct[of e f] by simp 

618 

619 
lemma round_down_nonneg: "0 \<le> s \<Longrightarrow> 0 \<le> round_down p s" 

620 
by (auto simp: round_down_def intro!: mult_nonneg_nonneg) 

621 

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

622 
lemma ceil_divide_floor_conv: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

623 
assumes "b \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

624 
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

625 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

626 
assume "\<not> b dvd a" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

627 
hence "a mod b \<noteq> 0" by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

628 
hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

629 
have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

630 
apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

632 
have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

633 
moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

634 
apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

635 
ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

636 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

637 
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

638 
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

639 
floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus) 
19765  640 

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

641 
lemma compute_float_up[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

642 
"float_up p (Float m e) = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

643 
(let P = 2^nat ((p + e)); r = m mod P in 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

644 
if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (p) else Float m e)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

646 
assume "p + e < 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

647 
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

648 
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

649 
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

650 
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

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

652 
"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

653 
with `p + e < 0` have powr_rewrite: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

654 
"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

655 
unfolding powr_divide2 by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

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

658 
assume "2^nat ((p + e)) dvd m" 
47615  659 
with `p + e < 0` twopow_rewrite show ?thesis 
47600  660 
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

661 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

662 
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

663 
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

664 
real m / real ((2::int) ^ nat ( (p + e)))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

665 
by (simp add: field_simps) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

666 
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

667 
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

668 
using ndvd unfolding powr_rewrite one_div 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

669 
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

670 
thus ?thesis using `p + e < 0` twopow_rewrite 
47600  671 
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

672 
qed 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

673 
next 
47600  674 
assume "\<not> p + e < 0" 
675 
then have r1: "real e + real p = real (nat (e + p))" by simp 

676 
have r: "\<lceil>(m * 2 powr e) * 2 powr real p\<rceil> = (m * 2 powr e) * 2 powr real p" 

677 
by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow 

678 
intro: exI[where x="m*2^nat (e+p)"]) 

679 
then show ?thesis using `\<not> p + e < 0` 

680 
by transfer 

681 
(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

682 
qed 
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset

683 
hide_fact (open) compute_float_up 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

684 

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

685 
lemmas real_of_ints = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

686 
real_of_int_zero 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

688 
real_of_int_add 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

689 
real_of_int_minus 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

690 
real_of_int_diff 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

692 
real_of_int_power 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

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

695 
real_of_nat_zero 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

696 
real_of_nat_one 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

697 
real_of_nat_1 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

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

700 
real_of_nat_power 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

701 

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

702 
lemmas int_of_reals = real_of_ints[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

703 
lemmas nat_of_reals = real_of_nats[symmetric] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

704 

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

705 
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

706 
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

707 

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

708 
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

709 

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

710 
subsection {* Compute bitlen of integers *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

711 

47600  712 
definition bitlen :: "int \<Rightarrow> int" where 
713 
"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

714 

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

715 
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

716 
proof  
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

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

718 
assume "0 > x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

719 
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

720 
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

721 
finally have "1 < log 2 (x)" . 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

722 
} 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

723 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

724 

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

725 
lemma bitlen_bounds: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

726 
assumes "x > 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

727 
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

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

729 
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

730 
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

731 
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

732 
by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

733 
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

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

735 
also have "... = real x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

736 
using `0 < x` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

737 
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

738 
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

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

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

741 
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

742 
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

743 
apply (simp add: powr_realpow[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

744 
using `x > 0` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

745 
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

746 
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

747 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

748 

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

749 
lemma bitlen_pow2[simp]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

750 
assumes "b > 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

751 
shows "bitlen (b * 2 ^ c) = bitlen b + c" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

752 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

753 
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

754 
thus ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

755 
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

756 
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

757 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

758 

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

759 
lemma bitlen_Float: 
53381  760 
fixes m e 
761 
defines "f \<equiv> Float m e" 

762 
shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)" 

763 
proof (cases "m = 0") 

764 
case True 

765 
then show ?thesis by (simp add: f_def bitlen_def Float_def) 

766 
next 

767 
case False 

47600  768 
hence "f \<noteq> float_of 0" 
769 
unfolding real_of_float_eq by (simp add: f_def) 

770 
hence "mantissa f \<noteq> 0" 

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

771 
by (simp add: mantissa_noteq_0) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

772 
moreover 
53381  773 
obtain i where "m = mantissa f * 2 ^ i" "e = exponent f  int i" 
774 
by (rule f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`]) 

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

775 
ultimately show ?thesis by (simp add: abs_mult) 
53381  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 compute_bitlen[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

779 
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

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

781 
{ assume "2 \<le> x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

782 
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

783 
by (simp add: log_mult zmod_zdiv_equality') 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

784 
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

785 
proof cases 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

786 
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

787 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

788 
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

789 
then have "0 \<le> n" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

790 
using `2 \<le> x` by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

791 
assume "x mod 2 \<noteq> 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

792 
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

793 
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

794 
moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

795 
{ 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

796 
by (simp add: powr_realpow) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

797 
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

798 
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

799 
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

800 
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

801 
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

802 
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

803 
{ 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

804 
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

805 
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

806 
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

807 
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

808 
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

809 
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

810 
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

811 
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

812 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

813 
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

814 
moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

815 
{ assume "x < 2" "0 < x" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

816 
then have "x = 1" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

817 
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

818 
ultimately show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

819 
unfolding bitlen_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

820 
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

821 
qed 
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset

822 
hide_fact (open) compute_bitlen 
29804
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 
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

825 
shows "0 \<le> e + (bitlen m  1)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

826 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

827 
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

828 
hence "0 < m" using powr_gt_zero[of 2 e] 
47600  829 
by (auto simp: zero_less_mult_iff) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

830 
hence "m \<noteq> 0" by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

831 
show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

832 
proof (cases "0 \<le> e") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

833 
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

834 
next 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

835 
have "(1::int) < 2" by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

836 
case False let ?S = "2^(nat (e))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

837 
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

838 
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

839 
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

840 
by (auto simp: powr_minus) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

841 
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

842 
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

843 
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

844 
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

845 
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

846 
hence "e < bitlen m" using False by auto 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

847 
thus ?thesis by auto 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

848 
qed 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

849 
qed 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

850 

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

851 
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

852 
proof  
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

853 
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

854 

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

855 
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

856 
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

857 
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

858 

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

859 
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

860 
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

861 

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

862 
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

863 
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

864 
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

865 
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

866 
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

867 
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

868 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

869 

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

870 
subsection {* Approximation of positive rationals *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

871 

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

872 
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

873 
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

874 

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

875 
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

876 
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

877 

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

878 
lemma real_div_nat_eq_floor_of_divide: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

879 
fixes a b::nat 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

880 
shows "a div b = real (floor (a/b))" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

881 
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

882 

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

883 
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

884 

47600  885 
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" 
886 
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

887 

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

888 
lemma compute_lapprox_posrat[code]: 
53381  889 
fixes prec x y 
890 
shows "lapprox_posrat prec x y = 

891 
(let 

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

892 
l = rat_precision prec x y; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

893 
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

894 
in normfloat (Float d ( l)))" 
47615  895 
unfolding div_mult_twopow_eq normfloat_def 
47600  896 
by transfer 
47615  897 
(simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

898 
del: two_powr_minus_int_float) 
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset

899 
hide_fact (open) compute_lapprox_posrat 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

900 

47600  901 
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" 
902 
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

903 

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

904 
(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

905 
lemma compute_rapprox_posrat[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

906 
fixes prec x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

907 
defines "l \<equiv> rat_precision prec x y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

908 
shows "rapprox_posrat prec x y = (let 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

909 
l = l ; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

910 
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

911 
d = fst X div snd X ; 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

912 
m = fst X mod snd X 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

913 
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

914 
proof (cases "y = 0") 
47615  915 
assume "y = 0" thus ?thesis unfolding normfloat_def by transfer simp 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

916 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

917 
assume "y \<noteq> 0" 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

918 
show ?thesis 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

919 
proof (cases "0 \<le> l") 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

920 
assume "0 \<le> l" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

921 
def x' == "x * 2 ^ nat l" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

922 
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

923 
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

924 
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

925 
ultimately show ?thesis 
47615  926 
unfolding normfloat_def 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

927 
using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0` 
47600  928 
l_def[symmetric, THEN meta_eq_to_obj_eq] 
929 
by transfer 

930 
(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

931 
next 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

932 
assume "\<not> 0 \<le> l" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

933 
def y' == "y * 2 ^ nat ( l)" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

934 
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

935 
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

936 
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

937 
using `\<not> 0 \<le> l` 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

938 
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

939 
ultimately show ?thesis 
47615  940 
unfolding normfloat_def 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

941 
using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0` 
47600  942 
l_def[symmetric, THEN meta_eq_to_obj_eq] 
943 
by transfer 

944 
(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

945 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

946 
qed 
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset

947 
hide_fact (open) compute_rapprox_posrat 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

948 

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

949 
lemma rat_precision_pos: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

950 
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

951 
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

952 
proof  
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

953 
{ 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

954 
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

955 
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

956 
(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

957 
thus ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

958 
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

959 
qed 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

960 

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

961 
lemma 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

962 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

963 
def y \<equiv> "nat (x  1)" moreover 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

964 
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

965 
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

966 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

967 

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

968 
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

969 
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

970 
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

971 
proof  
53381  972 
have powr1: "2 powr real (rat_precision n (int x) (int y)) = 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

973 
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

974 
by (simp add: powr_realpow[symmetric]) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

975 
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

976 
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

977 
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

978 
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

979 
also have "\<dots> = 2 powr real (rat_precision n (int x) (int y)  1)" 
54489
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54230
diff
changeset

980 
using powr_add [of 2 _ " 1", simplified add_uminus_conv_diff] by (simp add: powr_minus) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

981 
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

982 
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

983 
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

984 
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

985 
using rat_precision_pos[OF assms] by (rule power_aux) 
47600  986 
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

987 
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

988 
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

989 
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

990 
apply (simp add: ceiling_less_eq) 
47600  991 
done 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

992 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

993 

47600  994 
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is 
995 
"\<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

996 

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

997 
lemma compute_lapprox_rat[code]: 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

998 
"lapprox_rat prec x y = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

999 
(if y = 0 then 0 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1000 
else if 0 \<le> x then 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1001 
(if 0 < y then lapprox_posrat prec (nat x) (nat y) 
53381  1002 
else  (rapprox_posrat prec (nat x) (nat (y)))) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1003 
else (if 0 < y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1004 
then  (rapprox_posrat prec (nat (x)) (nat y)) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1005 
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

1006 
by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps) 
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset

1007 
hide_fact (open) compute_lapprox_rat 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1008 

47600  1009 
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is 
1010 
"\<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

1011 

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

1012 
lemma compute_rapprox_rat[code]: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1013 
"rapprox_rat prec x y = 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1014 
(if y = 0 then 0 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1015 
else if 0 \<le> x then 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1016 
(if 0 < y then rapprox_posrat prec (nat x) (nat y) 
53381  1017 
else  (lapprox_posrat prec (nat x) (nat (y)))) 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1018 
else (if 0 < y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1019 
then  (lapprox_posrat prec (nat (x)) (nat y)) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1020 
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

1021 
by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps) 
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset

1022 
hide_fact (open) compute_rapprox_rat 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1023 

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

1024 
subsection {* Division *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1025 

54782  1026 
definition "real_divl prec a b = round_down (int prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor>  \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)" 
1027 

1028 
definition "real_divr prec a b = round_up (int prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor>  \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)" 

1029 

1030 
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl 

1031 
by (simp add: real_divl_def) 

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

1032 

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

1033 
lemma compute_float_divl[code]: 
47600  1034 
"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

1035 
proof cases 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1036 
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

1037 
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

1038 
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

1039 
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

1040 
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

1041 
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

1042 
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

1043 

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

1044 
show ?thesis 
53381  1045 
using not_0 
54782  1046 
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def, 
1047 
simp add: field_simps) 

1048 
qed (transfer, auto simp: real_divl_def) 

47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset

1049 
hide_fact (open) compute_float_divl 
47600  1050 

54782  1051 
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr 
1052 
by (simp add: real_divr_def) 

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

1053 

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

1054 
lemma compute_float_divr[code]: 
47600  1055 
"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

1056 
proof cases 
47601
050718fe6eee
use real :: float => real as liftingmorphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset

1057 
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

1058 
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

1059 
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

1060 
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

1061 
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

1062 
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

1063 
by (simp add: field_simps powr_divide2[symmetric]) 
47600  1064 

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

1065 
show ?thesis 
53381  1066 
using not_0 
54782  1067 
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift real_divr_def, 
1068 
simp add: field_simps) 

1069 
qed (transfer, auto simp: real_divr_def) 

47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset

1070 
hide_fact (open) compute_float_divr 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1071 

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

1072 
subsection {* Lemmas needed by Approximate *} 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1073 

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

1074 
lemma Float_num[simp]: shows 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1075 
"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

1076 
"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

1077 
"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

1078 
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

1079 
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

1080 
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

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

1082 

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

1083 
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

1084 

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

1085 
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

1086 

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

1087 
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

1088 
by arith 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1089 

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

1090 
lemma lapprox_rat: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1091 
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

1092 
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

1093 

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

1094 
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

1095 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1096 
from zmod_zdiv_equality'[of a b] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1097 
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

1098 
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

1099 
using assms by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1100 
finally show ?thesis by simp 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1101 
qed 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1102 

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

1103 
lemma lapprox_rat_nonneg: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1104 
fixes n x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1105 
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

1106 
assumes "0 \<le> x" "0 < y" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1107 
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

1108 
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

1109 
powr_int[of 2, simplified] 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1110 
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

1111 

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

1112 
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

1113 
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

1114 

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

1115 
lemma rapprox_rat_le1: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1116 
fixes n x y 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1117 
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

1118 
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

1119 
proof  
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1120 
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

1121 
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

1122 
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

1123 
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

1124 
\<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

1125 
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

1126 
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

1127 
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

1128 
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

1129 
finally show ?thesis 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1130 
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

1131 
(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

1132 
qed 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1133 

53381  1134 
lemma rapprox_rat_nonneg_neg: 
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1135 
"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

1136 
unfolding rapprox_rat_def round_up_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1137 
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

1138 

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

1139 
lemma rapprox_rat_neg: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1140 
"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

1141 
unfolding rapprox_rat_def round_up_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1142 
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

1143 

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

1144 
lemma rapprox_rat_nonpos_pos: 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1145 
"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

1146 
unfolding rapprox_rat_def round_up_def 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1147 
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

1148 

54782  1149 
lemma real_divl: "real_divl prec x y \<le> x / y" 
1150 
by (simp add: real_divl_def round_down) 

1151 

1152 
lemma real_divr: "x / y \<le> real_divr prec x y" 

1153 
using round_up by (simp add: real_divr_def) 

1154 

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

1155 
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y" 
54782  1156 
by transfer (rule real_divl) 
1157 

1158 
lemma real_divl_lower_bound: 

1159 
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_divl prec x y" 

1160 
by (simp add: real_divl_def 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

1161 

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

1162 
lemma float_divl_lower_bound: 
54782  1163 
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real (float_divl prec x y)" 
1164 
by transfer (rule real_divl_lower_bound) 

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

1165 

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

1166 
lemma exponent_1: "exponent 1 = 0" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1167 
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

1168 

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

1169 
lemma mantissa_1: "mantissa 1 = 1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1170 
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

1171 

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

1172 
lemma bitlen_1: "bitlen 1 = 1" 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1173 
by (simp add: bitlen_def) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1174 

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

1175 
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

1176 
proof 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1177 
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

1178 
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

1179 
qed (simp add: zero_float_def) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1180 

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

1181 
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

1182 
proof (cases "x = 0", simp) 
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset

1183 
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

1184 
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

1185 
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

1186 
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

1187 
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

1188 
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

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

1190 
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

1191 
qed 
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1192 

54782  1193 
lemma real_divl_pos_less1_bound: 
1194 
"0 < x \<Longrightarrow> x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real_divl prec 1 x" 

1195 
proof (unfold real_divl_def) 

1196 
fix prec :: nat and x :: real assume x: "0 < x" "x < 1" and prec: "1 \<le> prec" 

53381  1197 
def p \<equiv> "int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>" 
47600  1198 
show "1 \<le> round_down (int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>  \<lfloor>log 2 \<bar>1\<bar>\<rfloor>) (1 / x) " 
1199 
proof cases 

1200 
assume nonneg: "0 \<le> p" 

1201 
hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)" 

1202 
by (simp add: powr_int del: real_of_int_power) simp 

1203 
also have "floor (1::real) \<le> floor (1 / x)" using x prec by simp 

1204 
also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) \<le> 

1205 
floor (real ((2::int) ^ nat p) * (1 / x))" 

1206 
by (rule le_mult_floor) (auto simp: x prec less_imp_le) 

1207 
finally have "2 powr real p \<le> floor (2 powr nat p / x)" by (simp add: powr_realpow) 

1208 
thus ?thesis unfolding p_def[symmetric] 

1209 
using x prec nonneg by (simp add: powr_minus inverse_eq_divide round_down_def) 

1210 
next 

1211 
assume neg: "\<not> 0 \<le> p" 

1212 

1213 
have "x = 2 powr (log 2 x)" 

1214 
using x by simp 

1215 
also have "2 powr (log 2 x) \<le> 2 powr p" 

1216 
proof (rule powr_mono) 

1217 
have "log 2 x \<le> \<lceil>log 2 x\<rceil>" 

1218 
by simp 

1219 
also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + 1" 

1220 
using ceiling_diff_floor_le_1[of "log 2 x"] by simp 

1221 
also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + prec" 

1222 
using prec by simp 

1223 
finally show "log 2 x \<le> real p" 

1224 
using x by (simp add: p_def) 

1225 
qed simp 

1226 
finally have x_le: "x \<le> 2 powr p" . 

1227 

1228 
from neg have "2 powr real p \<le> 2 powr 0" 

1229 
by (intro powr_mono) auto 

1230 
also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp 

1231 
also have "\<dots> \<le> \<lfloor>2 powr real p / x\<rfloor>" unfolding real_of_int_le_iff 

1232 
using x x_le by (intro floor_mono) (simp add: pos_le_divide_eq mult_pos_pos) 

1233 
finally show ?thesis 

1234 
using prec x unfolding p_def[symmetric] 

1235 
by (simp add: round_down_def powr_minus_divide pos_le_divide_eq mult_pos_pos) 

1236 
qed 

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

1237 
qed 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

1238 

54782  1239 
lemma float_divl_pos_less1_bound: 
1240 
"0 < real x \<Longrightarrow> real x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)" 

1241 
by (transfer, rule real_divl_pos_less1_bound) 

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

1242 

54782  1243 
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)" 
1244 
by transfer (rule real_divr) 

1245 

1246 
lemma real_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> real_divr prec 1 x" 

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

1247 
proof  
54782  1248 
have "1 \<le> 1 / x" using `0 < x` and `x < 1` by auto 
1249 
also have "\<dots> \<le> real_divr prec 1 x" using real_divr[where x=1 and y=x] by auto 

47600  1250 
finally show ?thesis by auto 
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset

1251 