author | haftmann |
Fri, 04 Jul 2014 20:18:47 +0200 | |
changeset 57512 | cc97b347b301 |
parent 57492 | 74bf65a1910a |
child 57862 | 8f074e6e22fc |
permissions | -rw-r--r-- |
47615 | 1 |
(* 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 {* Floating-Point 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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|
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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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|
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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 hypsubst_thin |
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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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|
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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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86 |
apply hypsubst_thin |
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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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90 |
done |
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|
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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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|
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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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|
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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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100 |
|
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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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103 |
apply hypsubst_thin |
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104 |
apply (rule_tac x="x" in exI) |
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105 |
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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107 |
done |
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108 |
|
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109 |
lemma div_power_2_int_float[simp]: "x \<in> float \<Longrightarrow> x / (2::int)^d \<in> float" |
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110 |
apply (auto simp add: float_def) |
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|
111 |
apply hypsubst_thin |
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112 |
apply (rule_tac x="x" in exI) |
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113 |
apply (rule_tac x="xa - d" in exI) |
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114 |
apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric]) |
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115 |
done |
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116 |
|
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117 |
lemma div_numeral_Bit0_float[simp]: |
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118 |
assumes x: "x / numeral n \<in> float" shows "x / (numeral (Num.Bit0 n)) \<in> float" |
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119 |
proof - |
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120 |
have "(x / numeral n) / 2^1 \<in> float" |
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121 |
by (intro x div_power_2_float) |
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122 |
also have "(x / numeral n) / 2^1 = x / (numeral (Num.Bit0 n))" |
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123 |
by (induct n) auto |
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124 |
finally show ?thesis . |
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125 |
qed |
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126 |
|
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127 |
lemma div_neg_numeral_Bit0_float[simp]: |
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changeset
|
128 |
assumes x: "x / numeral n \<in> float" shows "x / (- numeral (Num.Bit0 n)) \<in> float" |
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|
129 |
proof - |
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|
130 |
have "- (x / numeral (Num.Bit0 n)) \<in> float" using x by simp |
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|
131 |
also have "- (x / numeral (Num.Bit0 n)) = x / - numeral (Num.Bit0 n)" |
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|
132 |
by simp |
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|
133 |
finally show ?thesis . |
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|
134 |
qed |
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|
135 |
|
47600 | 136 |
lift_definition Float :: "int \<Rightarrow> int \<Rightarrow> float" is "\<lambda>(m::int) (e::int). m * 2 powr e" by simp |
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|
137 |
declare Float.rep_eq[simp] |
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|
138 |
|
47780 | 139 |
lemma compute_real_of_float[code]: |
140 |
"real_of_float (Float m e) = (if e \<ge> 0 then m * 2 ^ nat e else m / 2 ^ (nat (-e)))" |
|
141 |
by (simp add: real_of_float_def[symmetric] powr_int) |
|
142 |
||
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|
143 |
code_datatype Float |
47600 | 144 |
|
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|
145 |
subsection {* Arithmetic operations on floating point numbers *} |
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|
146 |
|
47600 | 147 |
instantiation float :: "{ring_1, linorder, linordered_ring, linordered_idom, numeral, equal}" |
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148 |
begin |
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|
149 |
|
47600 | 150 |
lift_definition zero_float :: float is 0 by simp |
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|
151 |
declare zero_float.rep_eq[simp] |
47600 | 152 |
lift_definition one_float :: float is 1 by simp |
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153 |
declare one_float.rep_eq[simp] |
47600 | 154 |
lift_definition plus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op +" by simp |
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|
155 |
declare plus_float.rep_eq[simp] |
47600 | 156 |
lift_definition times_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op *" by simp |
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|
157 |
declare times_float.rep_eq[simp] |
47600 | 158 |
lift_definition minus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op -" by simp |
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|
159 |
declare minus_float.rep_eq[simp] |
47600 | 160 |
lift_definition uminus_float :: "float \<Rightarrow> float" is "uminus" by simp |
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|
161 |
declare uminus_float.rep_eq[simp] |
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162 |
|
47600 | 163 |
lift_definition abs_float :: "float \<Rightarrow> float" is abs by simp |
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|
164 |
declare abs_float.rep_eq[simp] |
47600 | 165 |
lift_definition sgn_float :: "float \<Rightarrow> float" is sgn by simp |
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|
166 |
declare sgn_float.rep_eq[simp] |
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|
167 |
|
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|
168 |
lift_definition equal_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op = :: real \<Rightarrow> real \<Rightarrow> bool" . |
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|
169 |
|
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|
170 |
lift_definition less_eq_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op \<le>" . |
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|
171 |
declare less_eq_float.rep_eq[simp] |
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|
172 |
lift_definition less_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op <" . |
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|
173 |
declare less_float.rep_eq[simp] |
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|
174 |
|
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|
175 |
instance |
47600 | 176 |
proof qed (transfer, fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+ |
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|
177 |
end |
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|
178 |
|
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|
179 |
lemma real_of_float_power[simp]: fixes f::float shows "real (f^n) = real f^n" |
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180 |
by (induct n) simp_all |
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|
181 |
|
53381 | 182 |
lemma fixes x y::float |
47600 | 183 |
shows real_of_float_min: "real (min x y) = min (real x) (real y)" |
184 |
and real_of_float_max: "real (max x y) = max (real x) (real y)" |
|
185 |
by (simp_all add: min_def max_def) |
|
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|
186 |
|
53215
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|
187 |
instance float :: unbounded_dense_linorder |
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|
188 |
proof |
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|
189 |
fix a b :: float |
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|
190 |
show "\<exists>c. a < c" |
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|
191 |
apply (intro exI[of _ "a + 1"]) |
47600 | 192 |
apply transfer |
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|
193 |
apply simp |
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|
194 |
done |
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|
195 |
show "\<exists>c. c < a" |
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|
196 |
apply (intro exI[of _ "a - 1"]) |
47600 | 197 |
apply transfer |
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|
198 |
apply simp |
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|
199 |
done |
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|
200 |
assume "a < b" |
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|
201 |
then show "\<exists>c. a < c \<and> c < b" |
47600 | 202 |
apply (intro exI[of _ "(a + b) * Float 1 -1"]) |
203 |
apply transfer |
|
54489
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|
204 |
apply (simp add: powr_minus) |
29804
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parents:
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changeset
|
205 |
done |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
206 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
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changeset
|
207 |
|
47600 | 208 |
instantiation float :: lattice_ab_group_add |
46573 | 209 |
begin |
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|
210 |
|
47600 | 211 |
definition inf_float::"float\<Rightarrow>float\<Rightarrow>float" |
212 |
where "inf_float a b = min a b" |
|
29804
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|
213 |
|
47600 | 214 |
definition sup_float::"float\<Rightarrow>float\<Rightarrow>float" |
215 |
where "sup_float a b = max a b" |
|
29804
e15b74577368
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hoelzl
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changeset
|
216 |
|
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|
217 |
instance |
47600 | 218 |
by default |
219 |
(transfer, simp_all add: inf_float_def sup_float_def real_of_float_min real_of_float_max)+ |
|
29804
e15b74577368
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hoelzl
parents:
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changeset
|
220 |
end |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
221 |
|
47600 | 222 |
lemma float_numeral[simp]: "real (numeral x :: float) = numeral x" |
223 |
apply (induct x) |
|
224 |
apply simp |
|
225 |
apply (simp_all only: numeral_Bit0 numeral_Bit1 real_of_float_eq real_float |
|
47601
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hoelzl
parents:
47600
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changeset
|
226 |
plus_float.rep_eq one_float.rep_eq plus_float numeral_float one_float) |
47600 | 227 |
done |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
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changeset
|
228 |
|
53381 | 229 |
lemma transfer_numeral [transfer_rule]: |
55945 | 230 |
"rel_fun (op =) pcr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)" |
231 |
unfolding rel_fun_def float.pcr_cr_eq cr_float_def by simp |
|
47599
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|
232 |
|
54489
03ff4d1e6784
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haftmann
parents:
54230
diff
changeset
|
233 |
lemma float_neg_numeral[simp]: "real (- numeral x :: float) = - numeral x" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
234 |
by simp |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46670
diff
changeset
|
235 |
|
53381 | 236 |
lemma transfer_neg_numeral [transfer_rule]: |
55945 | 237 |
"rel_fun (op =) pcr_float (- numeral :: _ \<Rightarrow> real) (- numeral :: _ \<Rightarrow> float)" |
238 |
unfolding rel_fun_def float.pcr_cr_eq cr_float_def by simp |
|
47600 | 239 |
|
47599
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changeset
|
240 |
lemma |
47600 | 241 |
shows float_of_numeral[simp]: "numeral k = float_of (numeral k)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
242 |
and float_of_neg_numeral[simp]: "- numeral k = float_of (- numeral k)" |
47600 | 243 |
unfolding real_of_float_eq by simp_all |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46670
diff
changeset
|
244 |
|
47599
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changeset
|
245 |
subsection {* Represent floats as unique mantissa and exponent *} |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46670
diff
changeset
|
246 |
|
47599
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changeset
|
247 |
lemma int_induct_abs[case_names less]: |
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|
248 |
fixes j :: int |
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|
249 |
assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n" |
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diff
changeset
|
250 |
shows "P j" |
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parents:
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changeset
|
251 |
proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct) |
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|
252 |
case less show ?case by (rule H[OF less]) simp |
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changeset
|
253 |
qed |
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parents:
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diff
changeset
|
254 |
|
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hoelzl
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changeset
|
255 |
lemma int_cancel_factors: |
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hoelzl
parents:
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diff
changeset
|
256 |
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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changeset
|
257 |
proof (induct n rule: int_induct_abs) |
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parents:
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diff
changeset
|
258 |
case (less n) |
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changeset
|
259 |
{ fix m assume n: "n \<noteq> 0" "n = m * r" |
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hoelzl
parents:
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diff
changeset
|
260 |
then have "\<bar>m \<bar> < \<bar>n\<bar>" |
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hoelzl
parents:
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changeset
|
261 |
by (metis abs_dvd_iff abs_ge_self assms comm_semiring_1_class.normalizing_semiring_rules(7) |
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hoelzl
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diff
changeset
|
262 |
dvd_imp_le_int dvd_refl dvd_triv_right linorder_neq_iff linorder_not_le |
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hoelzl
parents:
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changeset
|
263 |
mult_eq_0_iff zdvd_mult_cancel1) |
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hoelzl
parents:
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diff
changeset
|
264 |
from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto } |
400b158f1589
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hoelzl
parents:
47230
diff
changeset
|
265 |
then show ?case |
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hoelzl
parents:
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changeset
|
266 |
by (metis comm_semiring_1_class.normalizing_semiring_rules(12,7) dvdE power_0) |
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parents:
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changeset
|
267 |
qed |
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parents:
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diff
changeset
|
268 |
|
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changeset
|
269 |
lemma mult_powr_eq_mult_powr_iff_asym: |
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changeset
|
270 |
fixes m1 m2 e1 e2 :: int |
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changeset
|
271 |
assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2" |
400b158f1589
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hoelzl
parents:
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changeset
|
272 |
shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2" |
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hoelzl
parents:
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diff
changeset
|
273 |
proof |
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hoelzl
parents:
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diff
changeset
|
274 |
have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto |
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hoelzl
parents:
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changeset
|
275 |
assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2" |
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hoelzl
parents:
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diff
changeset
|
276 |
with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2 - e1)" |
400b158f1589
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parents:
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diff
changeset
|
277 |
by (simp add: powr_divide2[symmetric] field_simps) |
400b158f1589
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hoelzl
parents:
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diff
changeset
|
278 |
also have "\<dots> = m2 * 2^nat (e2 - e1)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
279 |
by (simp add: powr_realpow) |
400b158f1589
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hoelzl
parents:
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diff
changeset
|
280 |
finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)" |
400b158f1589
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hoelzl
parents:
47230
diff
changeset
|
281 |
unfolding real_of_int_inject . |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset
|
282 |
with m1 have "m1 = m2" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
283 |
by (cases "nat (e2 - e1)") (auto simp add: dvd_def) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset
|
284 |
then show "m1 = m2 \<and> e1 = e2" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
285 |
using eq `m1 \<noteq> 0` by (simp add: powr_inj) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
286 |
qed simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
287 |
|
400b158f1589
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hoelzl
parents:
47230
diff
changeset
|
288 |
lemma mult_powr_eq_mult_powr_iff: |
400b158f1589
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hoelzl
parents:
47230
diff
changeset
|
289 |
fixes m1 m2 e1 e2 :: int |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
290 |
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" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
291 |
using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2] |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset
|
292 |
using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1] |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
293 |
by (cases e1 e2 rule: linorder_le_cases) auto |
400b158f1589
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hoelzl
parents:
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diff
changeset
|
294 |
|
400b158f1589
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parents:
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diff
changeset
|
295 |
lemma floatE_normed: |
400b158f1589
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parents:
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diff
changeset
|
296 |
assumes x: "x \<in> float" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
297 |
obtains (zero) "x = 0" |
400b158f1589
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parents:
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diff
changeset
|
298 |
| (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0" |
400b158f1589
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parents:
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diff
changeset
|
299 |
proof atomize_elim |
400b158f1589
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parents:
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diff
changeset
|
300 |
{ assume "x \<noteq> 0" |
400b158f1589
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parents:
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diff
changeset
|
301 |
from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
302 |
with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
303 |
by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
304 |
with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
305 |
by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"]) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
306 |
(simp add: powr_add powr_realpow) } |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
307 |
then show "x = 0 \<or> (\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m \<and> x \<noteq> 0)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
308 |
by blast |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
309 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
310 |
|
400b158f1589
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hoelzl
parents:
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diff
changeset
|
311 |
lemma float_normed_cases: |
400b158f1589
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hoelzl
parents:
47230
diff
changeset
|
312 |
fixes f :: float |
400b158f1589
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hoelzl
parents:
47230
diff
changeset
|
313 |
obtains (zero) "f = 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
314 |
| (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
315 |
proof (atomize_elim, induct f) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset
|
316 |
case (float_of y) then show ?case |
47600 | 317 |
by (cases rule: floatE_normed) (auto simp: zero_float_def) |
47599
400b158f1589
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hoelzl
parents:
47230
diff
changeset
|
318 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
319 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
320 |
definition mantissa :: "float \<Rightarrow> int" where |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
321 |
"mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) |
400b158f1589
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hoelzl
parents:
47230
diff
changeset
|
322 |
\<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
323 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
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diff
changeset
|
324 |
definition exponent :: "float \<Rightarrow> int" where |
400b158f1589
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hoelzl
parents:
47230
diff
changeset
|
325 |
"exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
326 |
\<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
327 |
|
53381 | 328 |
lemma |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
329 |
shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
330 |
and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
331 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
332 |
have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
333 |
then show ?E ?M |
47600 | 334 |
by (auto simp add: mantissa_def exponent_def zero_float_def) |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
335 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
336 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
337 |
lemma |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
338 |
shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
339 |
and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D") |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
340 |
proof cases |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
341 |
assume [simp]: "f \<noteq> (float_of 0)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
342 |
have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
343 |
proof (cases f rule: float_normed_cases) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
344 |
case (powr m e) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
345 |
then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
346 |
\<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
347 |
by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
348 |
then show ?thesis |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
349 |
unfolding exponent_def mantissa_def |
47600 | 350 |
by (rule someI2_ex) (simp add: zero_float_def) |
351 |
qed (simp add: zero_float_def) |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
352 |
then show ?E ?D by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
353 |
qed simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
354 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
355 |
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
356 |
using mantissa_not_dvd[of f] by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
357 |
|
53381 | 358 |
lemma |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
359 |
fixes m e :: int |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
360 |
defines "f \<equiv> float_of (m * 2 powr e)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
361 |
assumes dvd: "\<not> 2 dvd m" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
362 |
shows mantissa_float: "mantissa f = m" (is "?M") |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
363 |
and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E") |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
364 |
proof cases |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
365 |
assume "m = 0" with dvd show "mantissa f = m" by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
366 |
next |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
367 |
assume "m \<noteq> 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
368 |
then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
369 |
from mantissa_exponent[of f] |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
370 |
have "m * 2 powr e = mantissa f * 2 powr exponent f" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
371 |
by (auto simp add: f_def) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
372 |
then show "?M" "?E" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
373 |
using mantissa_not_dvd[OF f_not_0] dvd |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
374 |
by (auto simp: mult_powr_eq_mult_powr_iff) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
375 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
376 |
|
47600 | 377 |
subsection {* Compute arithmetic operations *} |
378 |
||
379 |
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f" |
|
380 |
unfolding real_of_float_eq mantissa_exponent[of f] by simp |
|
381 |
||
382 |
lemma Float_cases[case_names Float, cases type: float]: |
|
383 |
fixes f :: float |
|
384 |
obtains (Float) m e :: int where "f = Float m e" |
|
385 |
using Float_mantissa_exponent[symmetric] |
|
386 |
by (atomize_elim) auto |
|
387 |
||
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
388 |
lemma denormalize_shift: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
389 |
assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
390 |
obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
391 |
proof |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
392 |
from mantissa_exponent[of f] f_def |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
393 |
have "m * 2 powr e = mantissa f * 2 powr exponent f" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
394 |
by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
395 |
then have eq: "m = mantissa f * 2 powr (exponent f - e)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
396 |
by (simp add: powr_divide2[symmetric] field_simps) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
397 |
moreover |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
398 |
have "e \<le> exponent f" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
399 |
proof (rule ccontr) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
400 |
assume "\<not> e \<le> exponent f" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
401 |
then have pos: "exponent f < e" by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
402 |
then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
403 |
by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
404 |
also have "\<dots> = 1 / 2^nat (e - exponent f)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
405 |
using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric]) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
406 |
finally have "m * 2^nat (e - exponent f) = real (mantissa f)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
407 |
using eq by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
408 |
then have "mantissa f = m * 2^nat (e - exponent f)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
409 |
unfolding real_of_int_inject by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
410 |
with `exponent f < e` have "2 dvd mantissa f" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
411 |
apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"]) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
412 |
apply (cases "nat (e - exponent f)") |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
413 |
apply auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
414 |
done |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
415 |
then show False using mantissa_not_dvd[OF not_0] by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
416 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
417 |
ultimately have "real m = mantissa f * 2^nat (exponent f - e)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
418 |
by (simp add: powr_realpow[symmetric]) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
419 |
with `e \<le> exponent f` |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
420 |
show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
421 |
unfolding real_of_int_inject by auto |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
422 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
423 |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
424 |
lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0" |
47600 | 425 |
by transfer simp |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
426 |
hide_fact (open) compute_float_zero |
47600 | 427 |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
428 |
lemma compute_float_one[code_unfold, code]: "1 = Float 1 0" |
47600 | 429 |
by transfer simp |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
430 |
hide_fact (open) compute_float_one |
47600 | 431 |
|
432 |
definition normfloat :: "float \<Rightarrow> float" where |
|
433 |
[simp]: "normfloat x = x" |
|
434 |
||
435 |
lemma compute_normfloat[code]: "normfloat (Float m e) = |
|
436 |
(if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1)) |
|
437 |
else if m = 0 then 0 else Float m e)" |
|
438 |
unfolding normfloat_def |
|
439 |
by transfer (auto simp add: powr_add zmod_eq_0_iff) |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
440 |
hide_fact (open) compute_normfloat |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
441 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
442 |
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k" |
47600 | 443 |
by transfer simp |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
444 |
hide_fact (open) compute_float_numeral |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
445 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
446 |
lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k" |
47600 | 447 |
by transfer simp |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
448 |
hide_fact (open) compute_float_neg_numeral |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
449 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
450 |
lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1" |
47600 | 451 |
by transfer simp |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
452 |
hide_fact (open) compute_float_uminus |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
453 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
454 |
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)" |
47600 | 455 |
by transfer (simp add: field_simps powr_add) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
456 |
hide_fact (open) compute_float_times |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
457 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
458 |
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 = |
54783
25860d89a044
Float: prevent unnecessary large numbers when adding 0
immler
parents:
54782
diff
changeset
|
459 |
(if m1 = 0 then Float m2 e2 else if m2 = 0 then Float m1 e1 else |
25860d89a044
Float: prevent unnecessary large numbers when adding 0
immler
parents:
54782
diff
changeset
|
460 |
if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1 |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
461 |
else Float (m2 + m1 * 2^nat (e1 - e2)) e2)" |
47600 | 462 |
by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric]) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
463 |
hide_fact (open) compute_float_plus |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
464 |
|
47600 | 465 |
lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)" |
466 |
by simp |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
467 |
hide_fact (open) compute_float_minus |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
468 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
469 |
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)" |
47600 | 470 |
by transfer (simp add: sgn_times) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
471 |
hide_fact (open) compute_float_sgn |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
472 |
|
55565
f663fc1e653b
simplify proofs because of the stronger reflexivity prover
kuncar
parents:
54784
diff
changeset
|
473 |
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" . |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
474 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
475 |
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m" |
47600 | 476 |
by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0]) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
477 |
hide_fact (open) compute_is_float_pos |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
478 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
479 |
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)" |
47600 | 480 |
by transfer (simp add: field_simps) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
481 |
hide_fact (open) compute_float_less |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
482 |
|
55565
f663fc1e653b
simplify proofs because of the stronger reflexivity prover
kuncar
parents:
54784
diff
changeset
|
483 |
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" . |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
484 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
485 |
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m" |
47600 | 486 |
by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0]) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
487 |
hide_fact (open) compute_is_float_nonneg |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
488 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
489 |
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)" |
47600 | 490 |
by transfer (simp add: field_simps) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
491 |
hide_fact (open) compute_float_le |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
492 |
|
55565
f663fc1e653b
simplify proofs because of the stronger reflexivity prover
kuncar
parents:
54784
diff
changeset
|
493 |
lift_definition is_float_zero :: "float \<Rightarrow> bool" is "op = 0 :: real \<Rightarrow> bool" . |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
494 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
495 |
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m" |
47600 | 496 |
by transfer (auto simp add: is_float_zero_def) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
497 |
hide_fact (open) compute_is_float_zero |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
498 |
|
47600 | 499 |
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e" |
500 |
by transfer (simp add: abs_mult) |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
501 |
hide_fact (open) compute_float_abs |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
502 |
|
47600 | 503 |
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)" |
504 |
by transfer simp |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
505 |
hide_fact (open) compute_float_eq |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
506 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
507 |
subsection {* Rounding Real numbers *} |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
508 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
509 |
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
510 |
"round_down prec x = floor (x * 2 powr prec) * 2 powr -prec" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
511 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
512 |
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
513 |
"round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
514 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
515 |
lemma round_down_float[simp]: "round_down prec x \<in> float" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
516 |
unfolding round_down_def |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
517 |
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
518 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
519 |
lemma round_up_float[simp]: "round_up prec x \<in> float" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
520 |
unfolding round_up_def |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
521 |
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
522 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
523 |
lemma round_up: "x \<le> round_up prec x" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
524 |
by (simp add: powr_minus_divide le_divide_eq round_up_def) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
525 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
526 |
lemma round_down: "round_down prec x \<le> x" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
527 |
by (simp add: powr_minus_divide divide_le_eq round_down_def) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
528 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
529 |
lemma round_up_0[simp]: "round_up p 0 = 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
530 |
unfolding round_up_def by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
531 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
532 |
lemma round_down_0[simp]: "round_down p 0 = 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
533 |
unfolding round_down_def by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
534 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
535 |
lemma round_up_diff_round_down: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
536 |
"round_up prec x - round_down prec x \<le> 2 powr -prec" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
537 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
538 |
have "round_up prec x - round_down prec x = |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
539 |
(ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
540 |
by (simp add: round_up_def round_down_def field_simps) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
541 |
also have "\<dots> \<le> 1 * 2 powr -prec" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
542 |
by (rule mult_mono) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
543 |
(auto simp del: real_of_int_diff |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
544 |
simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
545 |
finally show ?thesis by simp |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
546 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
547 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
548 |
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
549 |
unfolding round_down_def |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
550 |
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric]) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
551 |
(simp add: powr_add[symmetric]) |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
552 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
553 |
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
554 |
unfolding round_up_def |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
555 |
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric]) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
556 |
(simp add: powr_add[symmetric]) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
557 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
558 |
subsection {* Rounding Floats *} |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
559 |
|
47600 | 560 |
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
561 |
declare float_up.rep_eq[simp] |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
562 |
|
54782 | 563 |
lemma round_up_correct: |
564 |
shows "round_up e f - f \<in> {0..2 powr -e}" |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
565 |
unfolding atLeastAtMost_iff |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
566 |
proof |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
567 |
have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
568 |
also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp |
54782 | 569 |
finally show "round_up e f - f \<le> 2 powr real (- e)" |
47600 | 570 |
by simp |
571 |
qed (simp add: algebra_simps round_up) |
|
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
572 |
|
54782 | 573 |
lemma float_up_correct: |
574 |
shows "real (float_up e f) - real f \<in> {0..2 powr -e}" |
|
575 |
by transfer (rule round_up_correct) |
|
576 |
||
47600 | 577 |
lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
578 |
declare float_down.rep_eq[simp] |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
579 |
|
54782 | 580 |
lemma round_down_correct: |
581 |
shows "f - (round_down e f) \<in> {0..2 powr -e}" |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
582 |
unfolding atLeastAtMost_iff |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
583 |
proof |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
584 |
have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
585 |
also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp |
54782 | 586 |
finally show "f - round_down e f \<le> 2 powr real (- e)" |
47600 | 587 |
by simp |
588 |
qed (simp add: algebra_simps round_down) |
|
24301 | 589 |
|
54782 | 590 |
lemma float_down_correct: |
591 |
shows "real f - real (float_down e f) \<in> {0..2 powr -e}" |
|
592 |
by transfer (rule round_down_correct) |
|
593 |
||
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
594 |
lemma compute_float_down[code]: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
595 |
"float_down p (Float m e) = |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
596 |
(if p + e < 0 then Float (m div 2^nat (-(p + e))) (-p) else Float m e)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
597 |
proof cases |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
598 |
assume "p + e < 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
599 |
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
|
600 |
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
|
601 |
also have "... = 1 / 2 powr p / 2 powr e" |
47600 | 602 |
unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add) |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
603 |
finally show ?thesis |
47600 | 604 |
using `p + e < 0` |
605 |
by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric]) |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
606 |
next |
47600 | 607 |
assume "\<not> p + e < 0" |
608 |
then have r: "real e + real p = real (nat (e + p))" by simp |
|
609 |
have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p" |
|
610 |
by (auto intro: exI[where x="m*2^nat (e+p)"] |
|
611 |
simp add: ac_simps powr_add[symmetric] r powr_realpow) |
|
612 |
with `\<not> p + e < 0` show ?thesis |
|
613 |
by transfer |
|
614 |
(auto simp add: round_down_def field_simps powr_add powr_minus inverse_eq_divide) |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
615 |
qed |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
616 |
hide_fact (open) compute_float_down |
24301 | 617 |
|
54782 | 618 |
lemma abs_round_down_le: "\<bar>f - (round_down e f)\<bar> \<le> 2 powr -e" |
619 |
using round_down_correct[of f e] by simp |
|
620 |
||
621 |
lemma abs_round_up_le: "\<bar>f - (round_up e f)\<bar> \<le> 2 powr -e" |
|
622 |
using round_up_correct[of e f] by simp |
|
623 |
||
624 |
lemma round_down_nonneg: "0 \<le> s \<Longrightarrow> 0 \<le> round_down p s" |
|
56536 | 625 |
by (auto simp: round_down_def) |
54782 | 626 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
627 |
lemma ceil_divide_floor_conv: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
628 |
assumes "b \<noteq> 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
629 |
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
|
630 |
proof cases |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
631 |
assume "\<not> b dvd a" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
632 |
hence "a mod b \<noteq> 0" by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
633 |
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
|
634 |
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
|
635 |
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
|
636 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
637 |
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
|
638 |
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
|
639 |
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
|
640 |
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
|
641 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
642 |
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
|
643 |
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric] |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
644 |
floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus) |
19765 | 645 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
646 |
lemma compute_float_up[code]: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
647 |
"float_up p (Float m e) = |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
648 |
(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
|
649 |
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
|
650 |
proof cases |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
651 |
assume "p + e < 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
652 |
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
|
653 |
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
|
654 |
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
|
655 |
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
|
656 |
finally have twopow_rewrite: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
657 |
"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
|
658 |
with `p + e < 0` have powr_rewrite: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
659 |
"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
|
660 |
unfolding powr_divide2 by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
661 |
show ?thesis |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
662 |
proof cases |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
663 |
assume "2^nat (-(p + e)) dvd m" |
47615 | 664 |
with `p + e < 0` twopow_rewrite show ?thesis |
47600 | 665 |
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
|
666 |
next |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
667 |
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
|
668 |
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
|
669 |
real m / real ((2::int) ^ nat (- (p + e)))" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
670 |
by (simp add: field_simps) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
671 |
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
|
672 |
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
|
673 |
using ndvd unfolding powr_rewrite one_div |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
674 |
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
|
675 |
thus ?thesis using `p + e < 0` twopow_rewrite |
47600 | 676 |
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
|
677 |
qed |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
678 |
next |
47600 | 679 |
assume "\<not> p + e < 0" |
680 |
then have r1: "real e + real p = real (nat (e + p))" by simp |
|
681 |
have r: "\<lceil>(m * 2 powr e) * 2 powr real p\<rceil> = (m * 2 powr e) * 2 powr real p" |
|
682 |
by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow |
|
683 |
intro: exI[where x="m*2^nat (e+p)"]) |
|
684 |
then show ?thesis using `\<not> p + e < 0` |
|
685 |
by transfer |
|
686 |
(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
|
687 |
qed |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
688 |
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
|
689 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
690 |
lemmas real_of_ints = |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
691 |
real_of_int_zero |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
692 |
real_of_one |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
693 |
real_of_int_add |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
694 |
real_of_int_minus |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
695 |
real_of_int_diff |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
696 |
real_of_int_mult |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
697 |
real_of_int_power |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
698 |
real_numeral |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
699 |
lemmas real_of_nats = |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
700 |
real_of_nat_zero |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
701 |
real_of_nat_one |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
702 |
real_of_nat_1 |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
703 |
real_of_nat_add |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
704 |
real_of_nat_mult |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
705 |
real_of_nat_power |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
706 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
707 |
lemmas int_of_reals = real_of_ints[symmetric] |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
708 |
lemmas nat_of_reals = real_of_nats[symmetric] |
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 |
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
|
711 |
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
|
712 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
713 |
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
|
714 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
715 |
subsection {* Compute bitlen of integers *} |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
716 |
|
47600 | 717 |
definition bitlen :: "int \<Rightarrow> int" where |
718 |
"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
|
719 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
720 |
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
|
721 |
proof - |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
722 |
{ |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
723 |
assume "0 > x" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
724 |
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
|
725 |
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
|
726 |
finally have "-1 < log 2 (-x)" . |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
727 |
} 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
|
728 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
729 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
730 |
lemma bitlen_bounds: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
731 |
assumes "x > 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
732 |
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
|
733 |
proof |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
734 |
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
|
735 |
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
|
736 |
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
|
737 |
by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
738 |
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
|
739 |
by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
740 |
also have "... = real x" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
741 |
using `0 < x` by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
742 |
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
|
743 |
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
|
744 |
by (simp add: bitlen_def) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
745 |
next |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
746 |
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
|
747 |
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
|
748 |
apply (simp add: powr_realpow[symmetric]) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
749 |
using `x > 0` by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
750 |
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
|
751 |
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
|
752 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
753 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
754 |
lemma bitlen_pow2[simp]: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
755 |
assumes "b > 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
756 |
shows "bitlen (b * 2 ^ c) = bitlen b + c" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
757 |
proof - |
56544 | 758 |
from assms have "b * 2 ^ c > 0" by auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
759 |
thus ?thesis |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
760 |
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
|
761 |
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
|
762 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
763 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
764 |
lemma bitlen_Float: |
53381 | 765 |
fixes m e |
766 |
defines "f \<equiv> Float m e" |
|
767 |
shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)" |
|
768 |
proof (cases "m = 0") |
|
769 |
case True |
|
770 |
then show ?thesis by (simp add: f_def bitlen_def Float_def) |
|
771 |
next |
|
772 |
case False |
|
47600 | 773 |
hence "f \<noteq> float_of 0" |
774 |
unfolding real_of_float_eq by (simp add: f_def) |
|
775 |
hence "mantissa f \<noteq> 0" |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
776 |
by (simp add: mantissa_noteq_0) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
777 |
moreover |
53381 | 778 |
obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i" |
779 |
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
|
780 |
ultimately show ?thesis by (simp add: abs_mult) |
53381 | 781 |
qed |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
782 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
783 |
lemma compute_bitlen[code]: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
784 |
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
|
785 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
786 |
{ assume "2 \<le> x" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
787 |
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
|
788 |
by (simp add: log_mult zmod_zdiv_equality') |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
789 |
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
|
790 |
proof cases |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
791 |
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
|
792 |
next |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
793 |
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
|
794 |
then have "0 \<le> n" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
795 |
using `2 \<le> x` by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
796 |
assume "x mod 2 \<noteq> 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
797 |
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
|
798 |
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
|
799 |
moreover |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
800 |
{ 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
|
801 |
by (simp add: powr_realpow) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
802 |
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
|
803 |
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
|
804 |
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
|
805 |
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
|
806 |
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
|
807 |
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
|
808 |
{ 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
|
809 |
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
|
810 |
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
|
811 |
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
|
812 |
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
|
813 |
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
|
814 |
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
|
815 |
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
|
816 |
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
|
817 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
818 |
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
|
819 |
moreover |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
820 |
{ assume "x < 2" "0 < x" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
821 |
then have "x = 1" by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
822 |
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
|
823 |
ultimately show ?thesis |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
824 |
unfolding bitlen_def |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
825 |
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
|
826 |
qed |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
827 |
hide_fact (open) compute_bitlen |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
828 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
829 |
lemma 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
|
830 |
shows "0 \<le> e + (bitlen m - 1)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
831 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
832 |
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
|
833 |
hence "0 < m" using powr_gt_zero[of 2 e] |
47600 | 834 |
by (auto simp: zero_less_mult_iff) |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
835 |
hence "m \<noteq> 0" by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
836 |
show ?thesis |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
837 |
proof (cases "0 \<le> e") |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
838 |
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
|
839 |
next |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
840 |
have "(1::int) < 2" by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
841 |
case False let ?S = "2^(nat (-e))" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
842 |
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
|
843 |
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
|
844 |
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
|
845 |
by (auto simp: powr_minus) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
846 |
hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
847 |
hence "?S \<le> real m" unfolding mult.assoc by auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
848 |
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
|
849 |
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
|
850 |
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
|
851 |
hence "-e < bitlen m" using False by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
852 |
thus ?thesis by auto |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
853 |
qed |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
854 |
qed |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
855 |
|
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
856 |
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
|
857 |
proof - |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
858 |
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
|
859 |
|
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
860 |
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
|
861 |
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
|
862 |
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
|
863 |
|
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
864 |
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
|
865 |
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
|
866 |
|
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
867 |
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
|
868 |
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
|
869 |
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
|
870 |
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
|
871 |
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
|
872 |
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
|
873 |
qed |
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 |
subsection {* Approximation of positive rationals *} |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
876 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
877 |
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
|
878 |
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
|
879 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
880 |
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
|
881 |
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
|
882 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
883 |
lemma real_div_nat_eq_floor_of_divide: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
884 |
fixes a b::nat |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
885 |
shows "a div b = real (floor (a/b))" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
886 |
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
|
887 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
888 |
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
|
889 |
|
47600 | 890 |
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" |
891 |
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
|
892 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
893 |
lemma compute_lapprox_posrat[code]: |
53381 | 894 |
fixes prec x y |
895 |
shows "lapprox_posrat prec x y = |
|
896 |
(let |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
897 |
l = rat_precision prec x y; |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
898 |
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
|
899 |
in normfloat (Float d (- l)))" |
47615 | 900 |
unfolding div_mult_twopow_eq normfloat_def |
47600 | 901 |
by transfer |
47615 | 902 |
(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
|
903 |
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
|
904 |
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
|
905 |
|
47600 | 906 |
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float" |
907 |
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
|
908 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
909 |
(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
910 |
lemma compute_rapprox_posrat[code]: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
911 |
fixes prec x y |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
912 |
defines "l \<equiv> rat_precision prec x y" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
913 |
shows "rapprox_posrat prec x y = (let |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
914 |
l = l ; |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
915 |
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
|
916 |
d = fst X div snd X ; |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
917 |
m = fst X mod snd X |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
918 |
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
|
919 |
proof (cases "y = 0") |
47615 | 920 |
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
|
921 |
next |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
922 |
assume "y \<noteq> 0" |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
923 |
show ?thesis |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
924 |
proof (cases "0 \<le> l") |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
925 |
assume "0 \<le> l" |
56777 | 926 |
def x' \<equiv> "x * 2 ^ nat l" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
927 |
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
|
928 |
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
|
929 |
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
|
930 |
ultimately show ?thesis |
47615 | 931 |
unfolding normfloat_def |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
932 |
using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0` |
47600 | 933 |
l_def[symmetric, THEN meta_eq_to_obj_eq] |
934 |
by transfer |
|
935 |
(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
|
936 |
next |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
937 |
assume "\<not> 0 \<le> l" |
56777 | 938 |
def y' \<equiv> "y * 2 ^ nat (- l)" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
939 |
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
|
940 |
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
|
941 |
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
|
942 |
using `\<not> 0 \<le> l` |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
943 |
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
|
944 |
ultimately show ?thesis |
47615 | 945 |
unfolding normfloat_def |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
946 |
using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0` |
47600 | 947 |
l_def[symmetric, THEN meta_eq_to_obj_eq] |
948 |
by transfer |
|
949 |
(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
|
950 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
951 |
qed |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
952 |
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
|
953 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
954 |
lemma rat_precision_pos: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
955 |
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
|
956 |
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
|
957 |
proof - |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
958 |
{ 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
|
959 |
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
|
960 |
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
|
961 |
(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
|
962 |
thus ?thesis |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
963 |
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
|
964 |
qed |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
965 |
|
56777 | 966 |
lemma power_aux: |
967 |
assumes "x > 0" |
|
968 |
shows "(2::int) ^ nat (x - 1) \<le> 2 ^ nat x - 1" |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
969 |
proof - |
56777 | 970 |
def y \<equiv> "nat (x - 1)" |
971 |
moreover |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
972 |
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
|
973 |
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
|
974 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
975 |
|
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
976 |
lemma rapprox_posrat_less1: |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
977 |
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
|
978 |
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
|
979 |
proof - |
53381 | 980 |
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
|
981 |
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
|
982 |
by (simp add: powr_realpow[symmetric]) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
983 |
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
|
984 |
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
|
985 |
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
|
986 |
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
|
987 |
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
|
988 |
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
|
989 |
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
|
990 |
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
|
991 |
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
|
992 |
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
|
993 |
using rat_precision_pos[OF assms] by (rule power_aux) |
47600 | 994 |
finally show ?thesis |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
995 |
apply (transfer fixing: n x y) |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
996 |
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
|
997 |
unfolding int_of_reals real_of_int_less_iff |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
998 |
apply (simp add: ceiling_less_eq) |
47600 | 999 |
done |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
1000 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
1001 |
|
47600 | 1002 |
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is |
1003 |
"\<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
|
1004 |
|
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
1005 |
lemma compute_lapprox_rat[code]: |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1006 |
"lapprox_rat prec x y = |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1007 |
(if y = 0 then 0 |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1008 |
else if 0 \<le> x then |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1009 |
(if 0 < y then lapprox_posrat prec (nat x) (nat y) |
53381 | 1010 |
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
|
1011 |
else (if 0 < y |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1012 |
then - (rapprox_posrat prec (nat (-x)) (nat y)) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1013 |
else lapprox_posrat prec (nat (-x)) (nat (-y))))" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
1014 |
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
|
1015 |
hide_fact (open) compute_lapprox_rat |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1016 |
|
47600 | 1017 |
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is |
1018 |
"\<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
|
1019 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1020 |
lemma compute_rapprox_rat[code]: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1021 |
"rapprox_rat prec x y = |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1022 |
(if y = 0 then 0 |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1023 |
else if 0 \<le> x then |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1024 |
(if 0 < y then rapprox_posrat prec (nat x) (nat y) |
53381 | 1025 |
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
|
1026 |
else (if 0 < y |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1027 |
then - (lapprox_posrat prec (nat (-x)) (nat y)) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1028 |
else rapprox_posrat prec (nat (-x)) (nat (-y))))" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
1029 |
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
|
1030 |
hide_fact (open) compute_rapprox_rat |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1031 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1032 |
subsection {* Division *} |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1033 |
|
54782 | 1034 |
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)" |
1035 |
||
1036 |
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)" |
|
1037 |
||
1038 |
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl |
|
1039 |
by (simp add: real_divl_def) |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1040 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1041 |
lemma compute_float_divl[code]: |
47600 | 1042 |
"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
|
1043 |
proof cases |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1044 |
let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2" |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1045 |
let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)" |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1046 |
assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0" |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1047 |
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 lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1048 |
by (simp add: abs_mult log_mult rat_precision_def bitlen_def) |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1049 |
have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s" |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1050 |
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
|
1051 |
|
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1052 |
show ?thesis |
53381 | 1053 |
using not_0 |
54782 | 1054 |
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def, |
1055 |
simp add: field_simps) |
|
1056 |
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
|
1057 |
hide_fact (open) compute_float_divl |
47600 | 1058 |
|
54782 | 1059 |
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr |
1060 |
by (simp add: real_divr_def) |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1061 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1062 |
lemma compute_float_divr[code]: |
47600 | 1063 |
"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
|
1064 |
proof cases |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1065 |
let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2" |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1066 |
let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)" |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1067 |
assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0" |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1068 |
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 lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1069 |
by (simp add: abs_mult log_mult rat_precision_def bitlen_def) |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1070 |
have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s" |
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1071 |
by (simp add: field_simps powr_divide2[symmetric]) |
47600 | 1072 |
|
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1073 |
show ?thesis |
53381 | 1074 |
using not_0 |
54782 | 1075 |
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift real_divr_def, |
1076 |
simp add: field_simps) |
|
1077 |
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
|
1078 |
hide_fact (open) compute_float_divr |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1079 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1080 |
subsection {* Lemmas needed by Approximate *} |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1081 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1082 |
lemma Float_num[simp]: shows |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1083 |
"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
|
1084 |
"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
|
1085 |
"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
|
1086 |
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
|
1087 |
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
|
1088 |
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
|
1089 |
by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1090 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1091 |
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
|
1092 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1093 |
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
|
1094 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1095 |
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
|
1096 |
by arith |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
1097 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1098 |
lemma lapprox_rat: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1099 |
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
|
1100 |
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
|
1101 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1102 |
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
|
1103 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1104 |
from zmod_zdiv_equality'[of a b] |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1105 |
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
|
1106 |
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
|
1107 |
using assms by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1108 |
finally show ?thesis by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1109 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1110 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1111 |
lemma lapprox_rat_nonneg: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1112 |
fixes n x y |
56777 | 1113 |
defines "p \<equiv> int n - ((bitlen \<bar>x\<bar>) - (bitlen \<bar>y\<bar>))" |
1114 |
assumes "0 \<le> x" and "0 < y" |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1115 |
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
|
1116 |
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
|
1117 |
powr_int[of 2, simplified] |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1118 |
by auto |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1119 |
|
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
|
1120 |
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
|
1121 |
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
|
1122 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1123 |
lemma rapprox_rat_le1: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1124 |
fixes n x y |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1125 |
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
|
1126 |
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
|
1127 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1128 |
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
|
1129 |
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
|
1130 |
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
|
1131 |
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
|
1132 |
\<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
|
1133 |
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
|
1134 |
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
|
1135 |
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
|
1136 |
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
|
1137 |
finally show ?thesis |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1138 |
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
|
1139 |
(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
|
1140 |
qed |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1141 |
|
53381 | 1142 |
lemma rapprox_rat_nonneg_neg: |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1143 |
"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
|
1144 |
unfolding rapprox_rat_def round_up_def |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1145 |
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
|
1146 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1147 |
lemma rapprox_rat_neg: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1148 |
"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
|
1149 |
unfolding rapprox_rat_def round_up_def |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1150 |
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
|
1151 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1152 |
lemma rapprox_rat_nonpos_pos: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1153 |
"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
|
1154 |
unfolding rapprox_rat_def round_up_def |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1155 |
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
|
1156 |
|
54782 | 1157 |
lemma real_divl: "real_divl prec x y \<le> x / y" |
1158 |
by (simp add: real_divl_def round_down) |
|
1159 |
||
1160 |
lemma real_divr: "x / y \<le> real_divr prec x y" |
|
1161 |
using round_up by (simp add: real_divr_def) |
|
1162 |
||
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
|
1163 |
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y" |
54782 | 1164 |
by transfer (rule real_divl) |
1165 |
||
1166 |
lemma real_divl_lower_bound: |
|
1167 |
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_divl prec x y" |
|
1168 |
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
|
1169 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1170 |
lemma float_divl_lower_bound: |
54782 | 1171 |
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real (float_divl prec x y)" |
1172 |
by transfer (rule real_divl_lower_bound) |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1173 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1174 |
lemma exponent_1: "exponent 1 = 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1175 |
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
|
1176 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1177 |
lemma mantissa_1: "mantissa 1 = 1" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1178 |
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
|
1179 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1180 |
lemma bitlen_1: "bitlen 1 = 1" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1181 |
by (simp add: bitlen_def) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1182 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1183 |
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
|
1184 |
proof |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1185 |
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
|
1186 |
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
|
1187 |
qed (simp add: zero_float_def) |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1188 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1189 |
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
|
1190 |
proof (cases "x = 0", simp) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1191 |
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
|
1192 |
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
|
1193 |
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
|
1194 |
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
|
1195 |
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
|
1196 |
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
|
1197 |
real_of_int_le_iff less_imp_le) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1198 |
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
|
1199 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
1200 |
|
54782 | 1201 |
lemma real_divl_pos_less1_bound: |
1202 |
"0 < x \<Longrightarrow> x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real_divl prec 1 x" |
|
1203 |
proof (unfold real_divl_def) |
|
1204 |
fix prec :: nat and x :: real assume x: "0 < x" "x < 1" and prec: "1 \<le> prec" |
|
53381 | 1205 |
def p \<equiv> "int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>" |
47600 | 1206 |
show "1 \<le> round_down (int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - \<lfloor>log 2 \<bar>1\<bar>\<rfloor>) (1 / x) " |
1207 |
proof cases |
|
1208 |
assume nonneg: "0 \<le> p" |
|
1209 |
hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)" |
|
1210 |
by (simp add: powr_int del: real_of_int_power) simp |
|
1211 |
also have "floor (1::real) \<le> floor (1 / x)" using x prec by simp |
|
1212 |
also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) \<le> |
|
1213 |
floor (real ((2::int) ^ nat p) * (1 / x))" |
|
1214 |
by (rule le_mult_floor) (auto simp: x prec less_imp_le) |
|
1215 |
finally have "2 powr real p \<le> floor (2 powr nat p / x)" by (simp add: powr_realpow) |
|
1216 |
thus ?thesis unfolding p_def[symmetric] |
|
1217 |
using x prec nonneg by (simp add: powr_minus inverse_eq_divide round_down_def) |
|
1218 |
next |
|
1219 |
assume neg: "\<not> 0 \<le> p" |
|
1220 |
||
1221 |
have "x = 2 powr (log 2 x)" |
|
1222 |
using x by simp |
|
1223 |
also have "2 powr (log 2 x) \<le> 2 powr p" |
|
1224 |
proof (rule powr_mono) |
|
1225 |
have "log 2 x \<le> \<lceil>log 2 x\<rceil>" |
|
1226 |
by simp |
|
1227 |
also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + 1" |
|
1228 |
using ceiling_diff_floor_le_1[of "log 2 x"] by simp |
|
1229 |
also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + prec" |
|
1230 |
using prec by simp |
|
1231 |
finally show "log 2 x \<le> real p" |
|
1232 |
using x by (simp add: p_def) |
|
1233 |
qed simp |
|
1234 |
finally have x_le: "x \<le> 2 powr p" . |
|
1235 |
||
1236 |
from neg have "2 powr real p \<le> 2 powr 0" |
|
1237 |
by (intro powr_mono) auto |
|
1238 |
also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp |
|
1239 |
also have "\<dots> \<le> \<lfloor>2 powr real p / x\<rfloor>" unfolding real_of_int_le_iff |
|
56544 | 1240 |
using x x_le by (intro floor_mono) (simp add: pos_le_divide_eq) |
47600 | 1241 |
finally show ?thesis |
1242 |
using prec x unfolding p_def[symmetric] |
|
56544 | 1243 |
by (simp add: round_down_def powr_minus_divide pos_le_divide_eq) |
47600 | 1244 |
qed |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
1245 |
qed |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1246 |
|
54782 | 1247 |
lemma float_divl_pos_less1_bound: |
1248 |
"0 < real x \<Longrightarrow> real x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)" |
|
1249 |
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
|
1250 |
|
54782 | 1251 |
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)" |
1252 |
by transfer (rule real_divr) |
|
1253 |
||
1254 |
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
|
1255 |
proof - |
54782 | 1256 |
have "1 \<le> 1 / x" using `0 < x` and `x < 1` by auto |
1257 |
also have "\<dots> \<le> real_divr prec 1 x" using real_divr[where x=1 and y=x] by auto |
|
47600 | 1258 |
finally show ?thesis by auto |
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
1259 |
qed |
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
1260 |
|
54782 | 1261 |
lemma float_divr_pos_less1_lower_bound: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> 1 \<le> float_divr prec 1 x" |
1262 |
by transfer (rule real_divr_pos_less1_lower_bound) |
|
1263 |
||
1264 |
lemma real_divr_nonpos_pos_upper_bound: |
|
1265 |
"x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real_divr prec x y \<le> 0" |
|
1266 |
by (auto simp: field_simps mult_le_0_iff divide_le_0_iff round_up_def real_divr_def) |
|
1267 |
||
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1268 |
lemma float_divr_nonpos_pos_upper_bound: |
47600 | 1269 |
"real x \<le> 0 \<Longrightarrow> 0 < real y \<Longrightarrow> real (float_divr prec x y) \<le> 0" |
54782 | 1270 |
by transfer (rule real_divr_nonpos_pos_upper_bound) |
1271 |
||
1272 |
lemma real_divr_nonneg_neg_upper_bound: |
|
1273 |
"0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real_divr prec x y \<le> 0" |
|
1274 |
by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff divide_le_0_iff round_up_def real_divr_def) |
|
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1275 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1276 |
lemma float_divr_nonneg_neg_upper_bound: |
47600 | 1277 |
"0 \<le> real x \<Longrightarrow> real y < 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0" |
54782 | 1278 |
by transfer (rule real_divr_nonneg_neg_upper_bound) |
1279 |
||
1280 |
definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real" where |
|
1281 |
"truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x" |
|
1282 |
||
1283 |
lemma truncate_down: "truncate_down prec x \<le> x" |
|
1284 |
using round_down by (simp add: truncate_down_def) |
|
1285 |
||
1286 |
lemma truncate_down_le: "x \<le> y \<Longrightarrow> truncate_down prec x \<le> y" |
|
1287 |
by (rule order_trans[OF truncate_down]) |
|
47600 | 1288 |
|
54782 | 1289 |
definition truncate_up::"nat \<Rightarrow> real \<Rightarrow> real" where |
1290 |
"truncate_up prec x = round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x" |
|
1291 |
||
1292 |
lemma truncate_up: "x \<le> truncate_up prec x" |
|
1293 |
using round_up by (simp add: truncate_up_def) |
|
1294 |
||
1295 |
lemma truncate_up_le: "x \<le> y \<Longrightarrow> x \<le> truncate_up prec y" |
|
1296 |
by (rule order_trans[OF _ truncate_up]) |
|
1297 |
||
1298 |
lemma truncate_up_zero[simp]: "truncate_up prec 0 = 0" |
|
1299 |
by (simp add: truncate_up_def) |
|
1300 |
||
1301 |
lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up |
|
1302 |
by (simp add: truncate_up_def) |
|
47600 | 1303 |
|
1304 |
lemma float_round_up: "real x \<le> real (float_round_up prec x)" |
|
54782 | 1305 |
using truncate_up by transfer simp |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1306 |
|
54782 | 1307 |
lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_down |
1308 |
by (simp add: truncate_down_def) |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1309 |
|
47600 | 1310 |
lemma float_round_down: "real (float_round_down prec x) \<le> real x" |
54782 | 1311 |
using truncate_down by transfer simp |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1312 |
|
47600 | 1313 |
lemma floor_add2[simp]: "\<lfloor> real i + x \<rfloor> = i + \<lfloor> x \<rfloor>" |
1314 |
using floor_add[of x i] by (simp del: floor_add add: ac_simps) |
|
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1315 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1316 |
lemma compute_float_round_down[code]: |
47600 | 1317 |
"float_round_down prec (Float m e) = (let d = bitlen (abs m) - int prec in |
1318 |
if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d) |
|
1319 |
else Float m e)" |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
1320 |
using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric] |
54782 | 1321 |
by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_down_def |
1322 |
cong del: if_weak_cong) |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
1323 |
hide_fact (open) compute_float_round_down |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1324 |
|
47600 | 1325 |
lemma compute_float_round_up[code]: |
1326 |
"float_round_up prec (Float m e) = (let d = (bitlen (abs m) - int prec) in |
|
1327 |
if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P |
|
1328 |
in Float (n + (if r = 0 then 0 else 1)) (e + d) |
|
1329 |
else Float m e)" |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
1330 |
using Float.compute_float_up[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric] |
47600 | 1331 |
unfolding Let_def |
54782 | 1332 |
by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_up_def |
1333 |
cong del: if_weak_cong) |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
1334 |
hide_fact (open) compute_float_round_up |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1335 |
|
54784 | 1336 |
lemma round_up_mono: "x \<le> y \<Longrightarrow> round_up p x \<le> round_up p y" |
1337 |
by (auto intro!: ceiling_mono simp: round_up_def) |
|
1338 |
||
1339 |
lemma truncate_up_nonneg_mono: |
|
1340 |
assumes "0 \<le> x" "x \<le> y" |
|
1341 |
shows "truncate_up prec x \<le> truncate_up prec y" |
|
1342 |
proof - |
|
1343 |
{ |
|
1344 |
assume "\<lfloor>log 2 x\<rfloor> = \<lfloor>log 2 y\<rfloor>" |
|
1345 |
hence ?thesis |
|
1346 |
using assms |
|
1347 |
by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono) |
|
1348 |
} moreover { |
|
1349 |
assume "0 < x" |
|
1350 |
hence "log 2 x \<le> log 2 y" using assms by auto |
|
1351 |
moreover |
|
1352 |
assume "\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>" |
|
1353 |
ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>" |
|
1354 |
unfolding atomize_conj |
|
1355 |
by (metis floor_less_cancel linorder_cases not_le) |
|
1356 |
have "truncate_up prec x = |
|
1357 |
real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> * 2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1)" |
|
1358 |
using assms by (simp add: truncate_up_def round_up_def) |
|
1359 |
also have "\<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> (2 ^ prec)" |
|
1360 |
proof (unfold ceiling_le_eq) |
|
1361 |
have "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> x * (2 powr real prec / (2 powr log 2 x))" |
|
1362 |
using real_of_int_floor_add_one_ge[of "log 2 x"] assms |
|
1363 |
by (auto simp add: algebra_simps powr_divide2 intro!: mult_left_mono) |
|
1364 |
thus "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)" |
|
1365 |
using `0 < x` by (simp add: powr_realpow) |
|
1366 |
qed |
|
1367 |
hence "real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec" |
|
1368 |
by (auto simp: powr_realpow) |
|
1369 |
also |
|
1370 |
have "2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)" |
|
1371 |
using logless flogless by (auto intro!: floor_mono) |
|
1372 |
also have "2 powr real (int prec) \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>))" |
|
1373 |
using assms `0 < x` |
|
1374 |
by (auto simp: algebra_simps) |
|
1375 |
finally have "truncate_up prec x \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>)) * 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)" |
|
1376 |
by simp |
|
1377 |
also have "\<dots> = 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>) - real (int prec - \<lfloor>log 2 y\<rfloor>))" |
|
1378 |
by (subst powr_add[symmetric]) simp |
|
1379 |
also have "\<dots> = y" |
|
1380 |
using `0 < x` assms |
|
1381 |
by (simp add: powr_add) |
|
1382 |
also have "\<dots> \<le> truncate_up prec y" |
|
1383 |
by (rule truncate_up) |
|
1384 |
finally have ?thesis . |
|
1385 |
} moreover { |
|
1386 |
assume "~ 0 < x" |
|
1387 |
hence ?thesis |
|
1388 |
using assms |
|
1389 |
by (auto intro!: truncate_up_le) |
|
1390 |
} ultimately show ?thesis |
|
1391 |
by blast |
|
1392 |
qed |
|
1393 |
||
1394 |
lemma truncate_up_nonpos: "x \<le> 0 \<Longrightarrow> truncate_up prec x \<le> 0" |
|
1395 |
by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg) |
|
1396 |
||
1397 |
lemma truncate_down_nonpos: "x \<le> 0 \<Longrightarrow> truncate_down prec x \<le> 0" |
|
1398 |
by (auto simp: truncate_down_def round_down_def intro!: mult_nonpos_nonneg |
|
1399 |
order_le_less_trans[of _ 0, OF mult_nonpos_nonneg]) |
|
1400 |
||
1401 |
lemma truncate_up_switch_sign_mono: |
|
1402 |
assumes "x \<le> 0" "0 \<le> y" |
|
1403 |
shows "truncate_up prec x \<le> truncate_up prec y" |
|
1404 |
proof - |
|
1405 |
note truncate_up_nonpos[OF `x \<le> 0`] |
|
1406 |
also note truncate_up_le[OF `0 \<le> y`] |
|
1407 |
finally show ?thesis . |
|
1408 |
qed |
|
1409 |
||
1410 |
lemma truncate_down_zeroprec_mono: |
|
1411 |
assumes "0 < x" "x \<le> y" |
|
1412 |
shows "truncate_down 0 x \<le> truncate_down 0 y" |
|
1413 |
proof - |
|
1414 |
have "x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1) = x * inverse (2 powr ((real \<lfloor>log 2 x\<rfloor> + 1)))" |
|
1415 |
by (simp add: powr_divide2[symmetric] powr_add powr_minus inverse_eq_divide) |
|
1416 |
also have "\<dots> = 2 powr (log 2 x - (real \<lfloor>log 2 x\<rfloor>) - 1)" |
|
1417 |
using `0 < x` |
|
1418 |
by (auto simp: inverse_eq_divide field_simps powr_add powr_divide2[symmetric]) |
|
1419 |
also have "\<dots> < 2 powr 0" |
|
1420 |
using real_of_int_floor_add_one_gt |
|
1421 |
unfolding neg_less_iff_less |
|
1422 |
by (intro powr_less_mono) (auto simp: algebra_simps) |
|
1423 |
finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> < 1" |
|
1424 |
unfolding less_ceiling_eq real_of_int_minus real_of_one |
|
1425 |
by simp |
|
1426 |
moreover |
|
1427 |
have "0 \<le> \<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor>" |
|
56536 | 1428 |
using `x > 0` by auto |
54784 | 1429 |
ultimately have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> \<in> {0 ..< 1}" |
1430 |
by simp |
|
1431 |
also have "\<dots> \<subseteq> {0}" by auto |
|
1432 |
finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0" by simp |
|
1433 |
with assms show ?thesis |
|
56536 | 1434 |
by (auto simp: truncate_down_def round_down_def) |
54784 | 1435 |
qed |
1436 |
||
1437 |
lemma truncate_down_nonneg: "0 \<le> y \<Longrightarrow> 0 \<le> truncate_down prec y" |
|
56536 | 1438 |
by (auto simp: truncate_down_def round_down_def) |
54784 | 1439 |
|
1440 |
lemma truncate_down_zero: "truncate_down prec 0 = 0" |
|
56536 | 1441 |
by (auto simp: truncate_down_def round_down_def) |
54784 | 1442 |
|
1443 |
lemma truncate_down_switch_sign_mono: |
|
1444 |
assumes "x \<le> 0" "0 \<le> y" |
|
1445 |
assumes "x \<le> y" |
|
1446 |
shows "truncate_down prec x \<le> truncate_down prec y" |
|
1447 |
proof - |
|
1448 |
note truncate_down_nonpos[OF `x \<le> 0`] |
|
1449 |
also note truncate_down_nonneg[OF `0 \<le> y`] |
|
1450 |
finally show ?thesis . |
|
1451 |
qed |
|
1452 |
||
1453 |
lemma truncate_up_uminus_truncate_down: |
|
1454 |
"truncate_up prec x = - truncate_down prec (- x)" |
|
1455 |
"truncate_up prec (-x) = - truncate_down prec x" |
|
1456 |
by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def) |
|
1457 |
||
1458 |
lemma truncate_down_uminus_truncate_up: |
|
1459 |
"truncate_down prec x = - truncate_up prec (- x)" |
|
1460 |
"truncate_down prec (-x) = - truncate_up prec x" |
|
1461 |
by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def) |
|
1462 |
||
1463 |
lemma truncate_down_nonneg_mono: |
|
1464 |
assumes "0 \<le> x" "x \<le> y" |
|
1465 |
shows "truncate_down prec x \<le> truncate_down prec y" |
|
1466 |
proof - |
|
1467 |
{ |
|
1468 |
assume "0 < x" "prec = 0" |
|
1469 |
with assms have ?thesis |
|
1470 |
by (simp add: truncate_down_zeroprec_mono) |
|
1471 |
} moreover { |
|
1472 |
assume "~ 0 < x" |
|
1473 |
with assms have "x = 0" "0 \<le> y" by simp_all |
|
1474 |
hence ?thesis |
|
1475 |
by (auto simp add: truncate_down_zero intro!: truncate_down_nonneg) |
|
1476 |
} moreover { |
|
1477 |
assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>" |
|
1478 |
hence ?thesis |
|
1479 |
using assms |
|
1480 |
by (auto simp: truncate_down_def round_down_def intro!: floor_mono) |
|
1481 |
} moreover { |
|
1482 |
assume "0 < x" |
|
1483 |
hence "log 2 x \<le> log 2 y" "0 < y" "0 \<le> y" using assms by auto |
|
1484 |
moreover |
|
1485 |
assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>" |
|
1486 |
ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>" |
|
1487 |
unfolding atomize_conj abs_of_pos[OF `0 < x`] abs_of_pos[OF `0 < y`] |
|
1488 |
by (metis floor_less_cancel linorder_cases not_le) |
|
1489 |
assume "prec \<noteq> 0" hence [simp]: "prec \<ge> Suc 0" by auto |
|
1490 |
have "2 powr (prec - 1) \<le> y * 2 powr real (prec - 1) / (2 powr log 2 y)" |
|
1491 |
using `0 < y` |
|
1492 |
by simp |
|
1493 |
also have "\<dots> \<le> y * 2 powr real prec / (2 powr (real \<lfloor>log 2 y\<rfloor> + 1))" |
|
1494 |
using `0 \<le> y` `0 \<le> x` assms(2) |
|
56544 | 1495 |
by (auto intro!: powr_mono divide_left_mono |
54784 | 1496 |
simp: real_of_nat_diff powr_add |
1497 |
powr_divide2[symmetric]) |
|
1498 |
also have "\<dots> = y * 2 powr real prec / (2 powr real \<lfloor>log 2 y\<rfloor> * 2)" |
|
1499 |
by (auto simp: powr_add) |
|
1500 |
finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>" |
|
1501 |
using `0 \<le> y` |
|
1502 |
by (auto simp: powr_divide2[symmetric] le_floor_eq powr_realpow) |
|
1503 |
hence "(2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y" |
|
1504 |
by (auto simp: truncate_down_def round_down_def) |
|
1505 |
moreover |
|
1506 |
{ |
|
1507 |
have "x = 2 powr (log 2 \<bar>x\<bar>)" using `0 < x` by simp |
|
1508 |
also have "\<dots> \<le> (2 ^ (prec )) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)" |
|
1509 |
using real_of_int_floor_add_one_ge[of "log 2 \<bar>x\<bar>"] |
|
1510 |
by (auto simp: powr_realpow[symmetric] powr_add[symmetric] algebra_simps) |
|
1511 |
also |
|
1512 |
have "2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor>)" |
|
1513 |
using logless flogless `x > 0` `y > 0` |
|
1514 |
by (auto intro!: floor_mono) |
|
1515 |
finally have "x \<le> (2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)" |
|
1516 |
by (auto simp: powr_realpow[symmetric] powr_divide2[symmetric] assms real_of_nat_diff) |
|
1517 |
} ultimately have ?thesis |
|
1518 |
by (metis dual_order.trans truncate_down) |
|
1519 |
} ultimately show ?thesis by blast |
|
1520 |
qed |
|
1521 |
||
1522 |
lemma truncate_down_mono: "x \<le> y \<Longrightarrow> truncate_down p x \<le> truncate_down p y" |
|
1523 |
apply (cases "0 \<le> x") |
|
1524 |
apply (rule truncate_down_nonneg_mono, assumption+) |
|
1525 |
apply (simp add: truncate_down_uminus_truncate_up) |
|
1526 |
apply (cases "0 \<le> y") |
|
1527 |
apply (auto intro: truncate_up_nonneg_mono truncate_up_switch_sign_mono) |
|
1528 |
done |
|
1529 |
||
1530 |
lemma truncate_up_mono: "x \<le> y \<Longrightarrow> truncate_up p x \<le> truncate_up p y" |
|
1531 |
by (simp add: truncate_up_uminus_truncate_down truncate_down_mono) |
|
1532 |
||
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1533 |
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1534 |
apply (auto simp: zero_float_def mult_le_0_iff) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1535 |
using powr_gt_zero[of 2 b] by simp |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1536 |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
1537 |
lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)" |
47600 | 1538 |
unfolding pprt_def sup_float_def max_def sup_real_def by auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1539 |
|
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
1540 |
lemma real_of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)" |
47600 | 1541 |
unfolding nprt_def inf_float_def min_def inf_real_def by auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1542 |
|
55565
f663fc1e653b
simplify proofs because of the stronger reflexivity prover
kuncar
parents:
54784
diff
changeset
|
1543 |
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor . |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1544 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1545 |
lemma compute_int_floor_fl[code]: |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1546 |
"int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))" |
47600 | 1547 |
by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
1548 |
hide_fact (open) compute_int_floor_fl |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1549 |
|
47600 | 1550 |
lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1551 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1552 |
lemma compute_floor_fl[code]: |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1553 |
"floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)" |
47600 | 1554 |
by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints) |
47621
4cf6011fb884
hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents:
47615
diff
changeset
|
1555 |
hide_fact (open) compute_floor_fl |
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1556 |
|
47600 | 1557 |
lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp |
1558 |
||
1559 |
lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp |
|
29804
e15b74577368
Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents:
29667
diff
changeset
|
1560 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1561 |
lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0" |
53381 | 1562 |
proof (cases "floor_fl x = float_of 0") |
1563 |
case True |
|
1564 |
then show ?thesis by (simp add: floor_fl_def) |
|
1565 |
next |
|
1566 |
case False |
|
1567 |
have eq: "floor_fl x = Float \<lfloor>real x\<rfloor> 0" by transfer simp |
|
1568 |
obtain i where "\<lfloor>real x\<rfloor> = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i" |
|
1569 |
by (rule denormalize_shift[OF eq[THEN eq_reflection] False]) |
|
1570 |
then show ?thesis by simp |
|
1571 |
qed |
|
16782
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1572 |
|
b214f21ae396
- use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset
|
1573 |
end |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47230
diff
changeset
|
1574 |