src/HOL/Library/Float.thy
author huffman
Sun, 25 Mar 2012 20:15:39 +0200
changeset 47108 2a1953f0d20d
parent 46670 e9aa6d151329
child 47165 9344891b504b
permissions -rw-r--r--
merged fork with new numeral representation (see NEWS)
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
30122
1c912a9d8200 standard headers;
wenzelm
parents: 30034
diff changeset
     1
(*  Title:      HOL/Library/Float.thy
1c912a9d8200 standard headers;
wenzelm
parents: 30034
diff changeset
     2
    Author:     Steven Obua 2008
1c912a9d8200 standard headers;
wenzelm
parents: 30034
diff changeset
     3
    Author:     Johannes Hoelzl, TU Muenchen <hoelzl@in.tum.de> 2008 / 2009
1c912a9d8200 standard headers;
wenzelm
parents: 30034
diff changeset
     4
*)
29988
747f0c519090 add header
huffman
parents: 29804
diff changeset
     5
747f0c519090 add header
huffman
parents: 29804
diff changeset
     6
header {* Floating-Point Numbers *}
747f0c519090 add header
huffman
parents: 29804
diff changeset
     7
20485
3078fd2eec7b got rid of Numeral.bin type
haftmann
parents: 20217
diff changeset
     8
theory Float
35032
7efe662e41b4 separate library theory for type classes combining lattices with various algebraic structures
haftmann
parents: 33555
diff changeset
     9
imports Complex_Main Lattice_Algebras
20485
3078fd2eec7b got rid of Numeral.bin type
haftmann
parents: 20217
diff changeset
    10
begin
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
    11
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
    12
definition pow2 :: "int \<Rightarrow> real" where
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    13
  [simp]: "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    14
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    15
datatype float = Float int int
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    16
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
    17
primrec of_float :: "float \<Rightarrow> real" where
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
    18
  "of_float (Float a b) = real a * pow2 b"
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
    19
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
    20
defs (overloaded)
31998
2c7a24f74db9 code attributes use common underscore convention
haftmann
parents: 31863
diff changeset
    21
  real_of_float_def [code_unfold]: "real == of_float"
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
    22
41024
ba961a606c67 move coercions to appropriate places
hoelzl
parents: 39161
diff changeset
    23
declare [[coercion "% x . Float x 0"]]
ba961a606c67 move coercions to appropriate places
hoelzl
parents: 39161
diff changeset
    24
declare [[coercion "real::float\<Rightarrow>real"]]
ba961a606c67 move coercions to appropriate places
hoelzl
parents: 39161
diff changeset
    25
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
    26
primrec mantissa :: "float \<Rightarrow> int" where
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
    27
  "mantissa (Float a b) = a"
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
    28
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
    29
primrec scale :: "float \<Rightarrow> int" where
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
    30
  "scale (Float a b) = b"
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21256
diff changeset
    31
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
    32
instantiation float :: zero
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
    33
begin
31467
f7d2aa438bee Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents: 31098
diff changeset
    34
definition zero_float where "0 = Float 0 0"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    35
instance ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    36
end
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    37
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
    38
instantiation float :: one
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
    39
begin
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    40
definition one_float where "1 = Float 1 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    41
instance ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    42
end
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    43
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
    44
lemma real_of_float_simp[simp]: "real (Float a b) = real a * pow2 b"
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
    45
  unfolding real_of_float_def using of_float.simps .
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    46
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
    47
lemma real_of_float_neg_exp: "e < 0 \<Longrightarrow> real (Float m e) = real m * inverse (2^nat (-e))" by auto
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
    48
lemma real_of_float_nge0_exp: "\<not> 0 \<le> e \<Longrightarrow> real (Float m e) = real m * inverse (2^nat (-e))" by auto
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
    49
lemma real_of_float_ge0_exp: "0 \<le> e \<Longrightarrow> real (Float m e) = real m * (2^nat e)" by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
    50
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    51
lemma Float_num[simp]: shows
31467
f7d2aa438bee Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents: 31098
diff changeset
    52
   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
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
    53
   "real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
    54
   "real (Float -1 0) = -1" and "real (Float (numeral n) 0) = numeral n"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    55
  by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
    56
31863
e391eee8bf14 Implemented taylor series expansion for approximation
hoelzl
parents: 31467
diff changeset
    57
lemma float_number_of_int[simp]: "real (Float n 0) = real n"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
    58
  by simp
31863
e391eee8bf14 Implemented taylor series expansion for approximation
hoelzl
parents: 31467
diff changeset
    59
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    60
lemma pow2_0[simp]: "pow2 0 = 1" by simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    61
lemma pow2_1[simp]: "pow2 1 = 2" by simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    62
lemma pow2_neg: "pow2 x = inverse (pow2 (-x))" by simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    63
45495
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    64
lemma pow2_powr: "pow2 a = 2 powr a"
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    65
  by (simp add: powr_realpow[symmetric] powr_minus)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
    66
45495
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    67
declare pow2_def[simp del]
19765
dfe940911617 misc cleanup;
wenzelm
parents: 16890
diff changeset
    68
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
    69
lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
45495
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    70
  by (simp add: pow2_powr powr_add)
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    71
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    72
lemma pow2_diff: "pow2 (a - b) = pow2 a / pow2 b"
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    73
  by (simp add: pow2_powr powr_divide2)
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    74
  
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    75
lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
    76
  by (simp add: pow2_add)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
    77
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
    78
lemma float_components[simp]: "Float (mantissa f) (scale f) = f" by (cases f) auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    79
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
    80
lemma float_split: "\<exists> a b. x = Float a b" by (cases x) auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
    81
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    82
lemma float_split2: "(\<forall> a b. x \<noteq> Float a b) = False" by (auto simp add: float_split)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    83
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
    84
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    85
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    86
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    87
by arith
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21256
diff changeset
    88
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    89
function normfloat :: "float \<Rightarrow> float" where
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
    90
  "normfloat (Float a b) =
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
    91
    (if a \<noteq> 0 \<and> even a then normfloat (Float (a div 2) (b+1))
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
    92
     else if a=0 then Float 0 0 else Float a b)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    93
by pat_completeness auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    94
termination by (relation "measure (nat o abs o mantissa)") (auto intro: abs_div_2_less)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    95
declare normfloat.simps[simp del]
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
    96
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
    97
theorem normfloat[symmetric, simp]: "real f = real (normfloat f)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    98
proof (induct f rule: normfloat.induct)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
    99
  case (1 a b)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   100
  have real2: "2 = real (2::int)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   101
    by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   102
  show ?case
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   103
    apply (subst normfloat.simps)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   104
    apply auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   105
    apply (subst 1[symmetric])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   106
    apply (auto simp add: pow2_add even_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   107
    done
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   108
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   109
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   110
lemma pow2_neq_zero[simp]: "pow2 x \<noteq> 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   111
  by (auto simp add: pow2_def)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   112
26313
8590bf5ef343 avoid rebinding of existing facts;
wenzelm
parents: 26086
diff changeset
   113
lemma pow2_int: "pow2 (int c) = 2^c"
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   114
  by (simp add: pow2_def)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   115
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   116
lemma zero_less_pow2[simp]: "0 < pow2 x"
45495
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   117
  by (simp add: pow2_powr)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   118
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   119
lemma normfloat_imp_odd_or_zero:
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   120
  "normfloat f = Float a b \<Longrightarrow> odd a \<or> (a = 0 \<and> b = 0)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   121
proof (induct f rule: normfloat.induct)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   122
  case (1 u v)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   123
  from 1 have ab: "normfloat (Float u v) = Float a b" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   124
  {
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   125
    assume eu: "even u"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   126
    assume z: "u \<noteq> 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   127
    have "normfloat (Float u v) = normfloat (Float (u div 2) (v + 1))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   128
      apply (subst normfloat.simps)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   129
      by (simp add: eu z)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   130
    with ab have "normfloat (Float (u div 2) (v + 1)) = Float a b" by simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   131
    with 1 eu z have ?case by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   132
  }
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   133
  note case1 = this
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   134
  {
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   135
    assume "odd u \<or> u = 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   136
    then have ou: "\<not> (u \<noteq> 0 \<and> even u)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   137
    have "normfloat (Float u v) = (if u = 0 then Float 0 0 else Float u v)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   138
      apply (subst normfloat.simps)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   139
      apply (simp add: ou)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   140
      done
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   141
    with ab have "Float a b = (if u = 0 then Float 0 0 else Float u v)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   142
    then have ?case
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   143
      apply (case_tac "u=0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   144
      apply (auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   145
      by (insert ou, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   146
  }
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   147
  note case2 = this
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   148
  show ?case
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   149
    apply (case_tac "odd u \<or> u = 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   150
    apply (rule case2)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   151
    apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   152
    apply (rule case1)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   153
    apply auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   154
    done
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   155
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   156
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   157
lemma float_eq_odd_helper: 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   158
  assumes odd: "odd a'"
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   159
    and floateq: "real (Float a b) = real (Float a' b')"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   160
  shows "b \<le> b'"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   161
proof - 
45495
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   162
  from odd have "a' \<noteq> 0" by auto
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   163
  with floateq have a': "real a' = real a * pow2 (b - b')"
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   164
    by (simp add: pow2_diff field_simps)
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   165
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   166
  {
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   167
    assume bcmp: "b > b'"
45495
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   168
    then have "\<exists>c::nat. b - b' = int c + 1"
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   169
      by arith
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   170
    then guess c ..
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   171
    with a' have "real a' = real (a * 2^c * 2)"
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   172
      by (simp add: pow2_def nat_add_distrib)
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   173
    with odd have False
c55a07526dbe cleaned up float theories; removed duplicate definitions and theorems
hoelzl
parents: 44766
diff changeset
   174
      unfolding real_of_int_inject by simp
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   175
  }
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   176
  then show ?thesis by arith
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   177
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   178
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   179
lemma float_eq_odd: 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   180
  assumes odd1: "odd a"
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   181
    and odd2: "odd a'"
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   182
    and floateq: "real (Float a b) = real (Float a' b')"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   183
  shows "a = a' \<and> b = b'"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   184
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   185
  from 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   186
     float_eq_odd_helper[OF odd2 floateq] 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   187
     float_eq_odd_helper[OF odd1 floateq[symmetric]]
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   188
  have beq: "b = b'" by arith
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   189
  with floateq show ?thesis by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   190
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   191
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   192
theorem normfloat_unique:
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
   193
  assumes real_of_float_eq: "real f = real g"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   194
  shows "normfloat f = normfloat g"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   195
proof - 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   196
  from float_split[of "normfloat f"] obtain a b where normf:"normfloat f = Float a b" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   197
  from float_split[of "normfloat g"] obtain a' b' where normg:"normfloat g = Float a' b'" by auto
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
   198
  have "real (normfloat f) = real (normfloat g)"
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
   199
    by (simp add: real_of_float_eq)
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
   200
  then have float_eq: "real (Float a b) = real (Float a' b')"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   201
    by (simp add: normf normg)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   202
  have ab: "odd a \<or> (a = 0 \<and> b = 0)" by (rule normfloat_imp_odd_or_zero[OF normf])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   203
  have ab': "odd a' \<or> (a' = 0 \<and> b' = 0)" by (rule normfloat_imp_odd_or_zero[OF normg])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   204
  {
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   205
    assume odd: "odd a"
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   206
    then have "a \<noteq> 0" by (simp add: even_def) arith
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   207
    with float_eq have "a' \<noteq> 0" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   208
    with ab' have "odd a'" by simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   209
    from odd this float_eq have "a = a' \<and> b = b'" by (rule float_eq_odd)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   210
  }
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   211
  note odd_case = this
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   212
  {
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   213
    assume even: "even a"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   214
    with ab have a0: "a = 0" by simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   215
    with float_eq have a0': "a' = 0" by auto 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   216
    from a0 a0' ab ab' have "a = a' \<and> b = b'" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   217
  }
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   218
  note even_case = this
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   219
  from odd_case even_case show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   220
    apply (simp add: normf normg)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   221
    apply (case_tac "even a")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   222
    apply auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   223
    done
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   224
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   225
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   226
instantiation float :: plus
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   227
begin
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   228
fun plus_float where
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   229
[simp del]: "(Float a_m a_e) + (Float b_m b_e) = 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   230
     (if a_e \<le> b_e then Float (a_m + b_m * 2^(nat(b_e - a_e))) a_e 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   231
                   else Float (a_m * 2^(nat (a_e - b_e)) + b_m) b_e)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   232
instance ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   233
end
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   234
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   235
instantiation float :: uminus
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   236
begin
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
   237
primrec uminus_float where [simp del]: "uminus_float (Float m e) = Float (-m) e"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   238
instance ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   239
end
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   240
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   241
instantiation float :: minus
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   242
begin
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   243
definition minus_float where "(z::float) - w = z + (- w)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   244
instance ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   245
end
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   246
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   247
instantiation float :: times
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   248
begin
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   249
fun times_float where [simp del]: "(Float a_m a_e) * (Float b_m b_e) = Float (a_m * b_m) (a_e + b_e)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   250
instance ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   251
end
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   252
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
   253
primrec float_pprt :: "float \<Rightarrow> float" where
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
   254
  "float_pprt (Float a e) = (if 0 <= a then (Float a e) else 0)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   255
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
   256
primrec float_nprt :: "float \<Rightarrow> float" where
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
   257
  "float_nprt (Float a e) = (if 0 <= a then 0 else (Float a e))" 
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   258
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   259
instantiation float :: ord
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   260
begin
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
   261
definition le_float_def: "z \<le> (w :: float) \<equiv> real z \<le> real w"
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
   262
definition less_float_def: "z < (w :: float) \<equiv> real z < real w"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   263
instance ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   264
end
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   265
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
   266
lemma real_of_float_add[simp]: "real (a + b) = real a + real (b :: float)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   267
  by (cases a, cases b) (simp add: algebra_simps plus_float.simps, 
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   268
      auto simp add: pow2_int[symmetric] pow2_add[symmetric])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   269
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
   270
lemma real_of_float_minus[simp]: "real (- a) = - real (a :: float)"
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   271
  by (cases a) simp
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   272
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
   273
lemma real_of_float_sub[simp]: "real (a - b) = real a - real (b :: float)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   274
  by (cases a, cases b) (simp add: minus_float_def)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   275
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
   276
lemma real_of_float_mult[simp]: "real (a*b) = real a * real (b :: float)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   277
  by (cases a, cases b) (simp add: times_float.simps pow2_add)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   278
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
   279
lemma real_of_float_0[simp]: "real (0 :: float) = 0"
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
   280
  by (auto simp add: zero_float_def)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   281
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
   282
lemma real_of_float_1[simp]: "real (1 :: float) = 1"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   283
  by (auto simp add: one_float_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   284
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   285
lemma zero_le_float:
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
   286
  "(0 <= real (Float a b)) = (0 <= a)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   287
  apply auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   288
  apply (auto simp add: zero_le_mult_iff)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   289
  apply (insert zero_less_pow2[of b])
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   290
  apply (simp_all)
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   291
  done
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   292
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   293
lemma float_le_zero:
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
   294
  "(real (Float a b) <= 0) = (a <= 0)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   295
  apply auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   296
  apply (auto simp add: mult_le_0_iff)
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   297
  apply (insert zero_less_pow2[of b])
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   298
  apply auto
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   299
  done
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   300
39161
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   301
lemma zero_less_float:
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   302
  "(0 < real (Float a b)) = (0 < a)"
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   303
  apply auto
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   304
  apply (auto simp add: zero_less_mult_iff)
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   305
  apply (insert zero_less_pow2[of b])
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   306
  apply (simp_all)
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   307
  done
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   308
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   309
lemma float_less_zero:
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   310
  "(real (Float a b) < 0) = (a < 0)"
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   311
  apply auto
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   312
  apply (auto simp add: mult_less_0_iff)
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   313
  apply (insert zero_less_pow2[of b])
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   314
  apply (simp_all)
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   315
  done
75849a560c09 When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
hoelzl
parents: 36778
diff changeset
   316
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
   317
declare real_of_float_simp[simp del]
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   318
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
   319
lemma real_of_float_pprt[simp]: "real (float_pprt a) = pprt (real a)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   320
  by (cases a) (auto simp add: zero_le_float float_le_zero)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   321
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
   322
lemma real_of_float_nprt[simp]: "real (float_nprt a) = nprt (real a)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   323
  by (cases a) (auto simp add: zero_le_float float_le_zero)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   324
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   325
instance float :: ab_semigroup_add
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   326
proof (intro_classes)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   327
  fix a b c :: float
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   328
  show "a + b + c = a + (b + c)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   329
    by (cases a, cases b, cases c)
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   330
      (auto simp add: algebra_simps power_add[symmetric] plus_float.simps)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   331
next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   332
  fix a b :: float
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   333
  show "a + b = b + a"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   334
    by (cases a, cases b) (simp add: plus_float.simps)
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
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   337
instance float :: numeral ..
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   338
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   339
lemma Float_add_same_scale: "Float x e + Float y e = Float (x + y) e"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   340
  by (simp add: plus_float.simps)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   341
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   342
(* FIXME: define other constant for code_unfold_post *)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   343
lemma numeral_float_Float (*[code_unfold_post]*):
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   344
  "numeral k = Float (numeral k) 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   345
  by (induct k, simp_all only: numeral.simps one_float_def
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   346
    Float_add_same_scale)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   347
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   348
lemma float_number_of[simp]: "real (numeral x :: float) = numeral x"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   349
  by (simp only: numeral_float_Float Float_num)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   350
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   351
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   352
instance float :: comm_monoid_mult
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   353
proof (intro_classes)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   354
  fix a b c :: float
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   355
  show "a * b * c = a * (b * c)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   356
    by (cases a, cases b, cases c) (simp add: times_float.simps)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   357
next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   358
  fix a b :: float
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   359
  show "a * b = b * a"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   360
    by (cases a, cases b) (simp add: times_float.simps)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   361
next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   362
  fix a :: float
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   363
  show "1 * a = a"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   364
    by (cases a) (simp add: times_float.simps one_float_def)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   365
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   366
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   367
(* Floats do NOT form a cancel_semigroup_add: *)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   368
lemma "0 + Float 0 1 = 0 + Float 0 2"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   369
  by (simp add: plus_float.simps zero_float_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   370
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   371
instance float :: comm_semiring
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   372
proof (intro_classes)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   373
  fix a b c :: float
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   374
  show "(a + b) * c = a * c + b * c"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   375
    by (cases a, cases b, cases c) (simp add: plus_float.simps times_float.simps algebra_simps)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   376
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   377
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   378
(* Floats do NOT form an order, because "(x < y) = (x <= y & x <> y)" does NOT hold *)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   379
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   380
instance float :: zero_neq_one
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   381
proof (intro_classes)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   382
  show "(0::float) \<noteq> 1"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   383
    by (simp add: zero_float_def one_float_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   384
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   385
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   386
lemma float_le_simp: "((x::float) \<le> y) = (0 \<le> y - x)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   387
  by (auto simp add: le_float_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   388
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   389
lemma float_less_simp: "((x::float) < y) = (0 < y - x)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   390
  by (auto simp add: less_float_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   391
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
   392
lemma real_of_float_min: "real (min x y :: float) = min (real x) (real y)" unfolding min_def le_float_def by auto
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
   393
lemma real_of_float_max: "real (max a b :: float) = max (real a) (real b)" unfolding max_def le_float_def by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   394
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
   395
lemma float_power: "real (x ^ n :: float) = real x ^ n"
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
   396
  by (induct n) simp_all
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   397
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   398
lemma zero_le_pow2[simp]: "0 \<le> pow2 s"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   399
  apply (subgoal_tac "0 < pow2 s")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   400
  apply (auto simp only:)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   401
  apply auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   402
  done
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   403
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   404
lemma pow2_less_0_eq_False[simp]: "(pow2 s < 0) = False"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   405
  apply auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   406
  apply (subgoal_tac "0 \<le> pow2 s")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   407
  apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   408
  apply simp
24301
6c7f226b24c3 changed floatarith lemmas
obua
parents: 24124
diff changeset
   409
  done
6c7f226b24c3 changed floatarith lemmas
obua
parents: 24124
diff changeset
   410
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   411
lemma pow2_le_0_eq_False[simp]: "(pow2 s \<le> 0) = False"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   412
  apply auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   413
  apply (subgoal_tac "0 < pow2 s")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   414
  apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   415
  apply simp
24301
6c7f226b24c3 changed floatarith lemmas
obua
parents: 24124
diff changeset
   416
  done
6c7f226b24c3 changed floatarith lemmas
obua
parents: 24124
diff changeset
   417
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   418
lemma float_pos_m_pos: "0 < Float m e \<Longrightarrow> 0 < m"
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
   419
  unfolding less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   420
  by auto
19765
dfe940911617 misc cleanup;
wenzelm
parents: 16890
diff changeset
   421
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   422
lemma float_pos_less1_e_neg: assumes "0 < Float m e" and "Float m e < 1" shows "e < 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   423
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   424
  have "0 < m" using float_pos_m_pos `0 < Float m e` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   425
  hence "0 \<le> real m" and "1 \<le> real m" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   426
  
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   427
  show "e < 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   428
  proof (rule ccontr)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   429
    assume "\<not> e < 0" hence "0 \<le> e" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   430
    hence "1 \<le> pow2 e" unfolding pow2_def by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   431
    from mult_mono[OF `1 \<le> real m` this `0 \<le> real m`]
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
   432
    have "1 \<le> Float m e" by (simp add: le_float_def real_of_float_simp)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   433
    thus False using `Float m e < 1` unfolding less_float_def le_float_def by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   434
  qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   435
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   436
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   437
lemma float_less1_mantissa_bound: assumes "0 < Float m e" "Float m e < 1" shows "m < 2^(nat (-e))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   438
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   439
  have "e < 0" using float_pos_less1_e_neg assms by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   440
  have "\<And>x. (0::real) < 2^x" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   441
  have "real m < 2^(nat (-e))" using `Float m e < 1`
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
   442
    unfolding less_float_def real_of_float_neg_exp[OF `e < 0`] real_of_float_1
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   443
          real_mult_less_iff1[of _ _ 1, OF `0 < 2^(nat (-e))`, symmetric] 
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 35344
diff changeset
   444
          mult_assoc by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   445
  thus ?thesis 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
   446
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   447
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   448
function bitlen :: "int \<Rightarrow> int" where
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   449
"bitlen 0 = 0" | 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   450
"bitlen -1 = 1" | 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   451
"0 < x \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))" | 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   452
"x < -1 \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   453
  apply (case_tac "x = 0 \<or> x = -1 \<or> x < -1 \<or> x > 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   454
  apply auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   455
  done
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   456
termination by (relation "measure (nat o abs)", auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   457
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   458
lemma bitlen_ge0: "0 \<le> bitlen x" by (induct x rule: bitlen.induct, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   459
lemma bitlen_ge1: "x \<noteq> 0 \<Longrightarrow> 1 \<le> bitlen x" by (induct x rule: bitlen.induct, auto simp add: bitlen_ge0)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   460
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   461
lemma bitlen_bounds': assumes "0 < x" shows "2^nat (bitlen x - 1) \<le> x \<and> x + 1 \<le> 2^nat (bitlen x)" (is "?P x")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   462
  using `0 < x`
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   463
proof (induct x rule: bitlen.induct)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   464
  fix x
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   465
  assume "0 < x" and hyp: "0 < x div 2 \<Longrightarrow> ?P (x div 2)" hence "0 \<le> x" and "x \<noteq> 0" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   466
  { fix x have "0 \<le> 1 + bitlen x" using bitlen_ge0[of x] by auto } note gt0_pls1 = this
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   467
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   468
  have "0 < (2::int)" by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   469
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   470
  show "?P x"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   471
  proof (cases "x = 1")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   472
    case True show "?P x" unfolding True by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   473
  next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   474
    case False hence "2 \<le> x" using `0 < x` `x \<noteq> 1` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   475
    hence "2 div 2 \<le> x div 2" by (rule zdiv_mono1, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   476
    hence "0 < x div 2" and "x div 2 \<noteq> 0" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   477
    hence bitlen_s1_ge0: "0 \<le> bitlen (x div 2) - 1" using bitlen_ge1[OF `x div 2 \<noteq> 0`] by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   478
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   479
    { from hyp[OF `0 < x div 2`]
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   480
      have "2 ^ nat (bitlen (x div 2) - 1) \<le> x div 2" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   481
      hence "2 ^ nat (bitlen (x div 2) - 1) * 2 \<le> x div 2 * 2" by (rule mult_right_mono, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   482
      also have "\<dots> \<le> x" using `0 < x` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   483
      finally have "2^nat (1 + bitlen (x div 2) - 1) \<le> x" unfolding power_Suc2[symmetric] Suc_nat_eq_nat_zadd1[OF bitlen_s1_ge0] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   484
    } moreover
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   485
    { have "x + 1 \<le> x - x mod 2 + 2"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   486
      proof -
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
   487
        have "x mod 2 < 2" using `0 < x` by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
   488
        hence "x < x - x mod 2 +  2" unfolding algebra_simps by auto
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
   489
        thus ?thesis by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   490
      qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   491
      also have "x - x mod 2 + 2 = (x div 2 + 1) * 2" unfolding algebra_simps using `0 < x` zdiv_zmod_equality2[of x 2 0] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   492
      also have "\<dots> \<le> 2^nat (bitlen (x div 2)) * 2" using hyp[OF `0 < x div 2`, THEN conjunct2] by (rule mult_right_mono, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   493
      also have "\<dots> = 2^(1 + nat (bitlen (x div 2)))" unfolding power_Suc2[symmetric] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   494
      finally have "x + 1 \<le> 2^(1 + nat (bitlen (x div 2)))" .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   495
    }
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   496
    ultimately show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   497
      unfolding bitlen.simps(3)[OF `0 < x`] nat_add_distrib[OF zero_le_one bitlen_ge0]
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   498
      unfolding add_commute nat_add_distrib[OF zero_le_one gt0_pls1]
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   499
      by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   500
  qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   501
next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   502
  fix x :: int assume "x < -1" and "0 < x" hence False by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   503
  thus "?P x" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   504
qed auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   505
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   506
lemma bitlen_bounds: assumes "0 < x" shows "2^nat (bitlen x - 1) \<le> x \<and> x < 2^nat (bitlen x)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   507
  using bitlen_bounds'[OF `0<x`] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   508
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   509
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
   510
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   511
  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
   512
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   513
  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
   514
  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
   515
  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
   516
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   517
  have "m \<noteq> 0" using assms by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   518
  have "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   519
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   520
  have "m < 2^nat(bitlen 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
   521
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   522
  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
   523
  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
   524
  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
   525
  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
   526
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   527
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   528
lemma float_gt1_scale: assumes "1 \<le> Float m e"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   529
  shows "0 \<le> e + (bitlen m - 1)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   530
proof (cases "0 \<le> e")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   531
  have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   532
  hence "0 < m" using float_pos_m_pos by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   533
  hence "m \<noteq> 0" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   534
  case True with bitlen_ge1[OF `m \<noteq> 0`] show ?thesis by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   535
next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   536
  have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   537
  hence "0 < m" using float_pos_m_pos by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   538
  hence "m \<noteq> 0" and "1 < (2::int)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   539
  case False let ?S = "2^(nat (-e))"
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
   540
  have "1 \<le> real m * inverse ?S" using assms unfolding le_float_def real_of_float_nge0_exp[OF False] by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   541
  hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   542
  hence "?S \<le> real m" unfolding mult_assoc by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   543
  hence "?S \<le> 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
   544
  from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   545
  have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   546
  hence "-e < bitlen m" using False bitlen_ge0 by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   547
  thus ?thesis by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   548
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   549
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
   550
lemma normalized_float: assumes "m \<noteq> 0" shows "real (Float m (- (bitlen m - 1))) = real m / 2^nat (bitlen m - 1)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   551
proof (cases "- (bitlen m - 1) = 0")
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
   552
  case True show ?thesis unfolding real_of_float_simp pow2_def using True by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   553
next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   554
  case False hence P: "\<not> 0 \<le> - (bitlen m - 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 35344
diff changeset
   555
  show ?thesis unfolding real_of_float_nge0_exp[OF P] divide_inverse by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   556
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   557
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   558
(* BROKEN
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   559
lemma bitlen_Pls: "bitlen (Int.Pls) = Int.Pls" by (subst Pls_def, subst Pls_def, simp)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   560
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   561
lemma bitlen_Min: "bitlen (Int.Min) = Int.Bit1 Int.Pls" by (subst Min_def, simp add: Bit1_def) 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   562
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   563
lemma bitlen_B0: "bitlen (Int.Bit0 b) = (if iszero b then Int.Pls else Int.succ (bitlen b))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   564
  apply (auto simp add: iszero_def succ_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   565
  apply (simp add: Bit0_def Pls_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   566
  apply (subst Bit0_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   567
  apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   568
  apply (subgoal_tac "0 < 2 * b \<or> 2 * b < -1")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   569
  apply auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   570
  done
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   571
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   572
lemma bitlen_B1: "bitlen (Int.Bit1 b) = (if iszero (Int.succ b) then Int.Bit1 Int.Pls else Int.succ (bitlen b))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   573
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   574
  have h: "! x. (2*x + 1) div 2 = (x::int)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   575
    by arith    
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   576
  show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   577
    apply (auto simp add: iszero_def succ_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   578
    apply (subst Bit1_def)+
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   579
    apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   580
    apply (subgoal_tac "2 * b + 1 = -1")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   581
    apply (simp only:)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   582
    apply simp_all
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   583
    apply (subst Bit1_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   584
    apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   585
    apply (subgoal_tac "0 < 2 * b + 1 \<or> 2 * b + 1 < -1")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   586
    apply (auto simp add: h)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   587
    done
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   588
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   589
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   590
lemma bitlen_number_of: "bitlen (number_of w) = number_of (bitlen w)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   591
  by (simp add: number_of_is_id)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   592
BH *)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   593
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   594
lemma [code]: "bitlen x = 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   595
     (if x = 0  then 0 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   596
 else if x = -1 then 1 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   597
                else (1 + (bitlen (x div 2))))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   598
  by (cases "x = 0 \<or> x = -1 \<or> 0 < x") auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   599
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   600
definition lapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   601
where
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   602
  "lapprox_posrat prec x y = 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   603
   (let 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   604
       l = nat (int prec + bitlen y - bitlen x) ;
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   605
       d = (x * 2^l) div y
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   606
    in normfloat (Float d (- (int l))))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   607
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   608
lemma pow2_minus: "pow2 (-x) = inverse (pow2 x)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   609
  unfolding pow2_neg[of "-x"] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   610
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   611
lemma lapprox_posrat: 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   612
  assumes x: "0 \<le> x"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   613
  and y: "0 < y"
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
   614
  shows "real (lapprox_posrat prec x y) \<le> real x / real y"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   615
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   616
  let ?l = "nat (int prec + bitlen y - bitlen x)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   617
  
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   618
  have "real (x * 2^?l div y) * inverse (2^?l) \<le> (real (x * 2^?l) / real y) * inverse (2^?l)" 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   619
    by (rule mult_right_mono, fact real_of_int_div4, simp)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   620
  also have "\<dots> \<le> (real x / real y) * 2^?l * inverse (2^?l)" by auto
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 35344
diff changeset
   621
  finally have "real (x * 2^?l div y) * inverse (2^?l) \<le> real x / real y" unfolding mult_assoc by auto
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
   622
  thus ?thesis unfolding lapprox_posrat_def Let_def normfloat real_of_float_simp
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   623
    unfolding pow2_minus pow2_int minus_minus .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   624
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   625
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   626
lemma real_of_int_div_mult: 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   627
  fixes x y c :: int assumes "0 < y" and "0 < c"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   628
  shows "real (x div y) \<le> real (x * c div y) * inverse (real c)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   629
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   630
  have "c * (x div y) + 0 \<le> c * x div y" unfolding zdiv_zmult1_eq[of c x y]
44766
d4d33a4d7548 avoid using legacy theorem names
huffman
parents: 42676
diff changeset
   631
    by (rule add_left_mono, 
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   632
        auto intro!: mult_nonneg_nonneg 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   633
             simp add: pos_imp_zdiv_nonneg_iff[OF `0 < y`] `0 < c`[THEN less_imp_le] pos_mod_sign[OF `0 < y`])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   634
  hence "real (x div y) * real c \<le> real (x * c div y)" 
44766
d4d33a4d7548 avoid using legacy theorem names
huffman
parents: 42676
diff changeset
   635
    unfolding real_of_int_mult[symmetric] real_of_int_le_iff mult_commute by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   636
  hence "real (x div y) * real c * inverse (real c) \<le> real (x * c div y) * inverse (real c)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   637
    using `0 < c` by auto
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 35344
diff changeset
   638
  thus ?thesis unfolding mult_assoc using `0 < c` by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   639
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   640
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   641
lemma lapprox_posrat_bottom: assumes "0 < y"
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
   642
  shows "real (x div y) \<le> real (lapprox_posrat n x y)" 
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   643
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   644
  have pow: "\<And>x. (0::int) < 2^x" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   645
  show ?thesis
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
   646
    unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   647
    using real_of_int_div_mult[OF `0 < y` pow] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   648
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   649
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   650
lemma lapprox_posrat_nonneg: assumes "0 \<le> x" and "0 < y"
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
   651
  shows "0 \<le> real (lapprox_posrat n x y)" 
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   652
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   653
  show ?thesis
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
   654
    unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   655
    using pos_imp_zdiv_nonneg_iff[OF `0 < y`] assms by (auto intro!: mult_nonneg_nonneg)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   656
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   657
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   658
definition rapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   659
where
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   660
  "rapprox_posrat prec x y = (let
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   661
     l = nat (int prec + bitlen y - bitlen x) ;
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   662
     X = x * 2^l ;
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   663
     d = X div y ;
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   664
     m = X mod y
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   665
   in normfloat (Float (d + (if m = 0 then 0 else 1)) (- (int l))))"
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   666
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   667
lemma rapprox_posrat:
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   668
  assumes x: "0 \<le> x"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   669
  and y: "0 < y"
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
   670
  shows "real x / real y \<le> real (rapprox_posrat prec x y)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   671
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   672
  let ?l = "nat (int prec + bitlen y - bitlen x)" let ?X = "x * 2^?l"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   673
  show ?thesis 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   674
  proof (cases "?X mod y = 0")
46670
e9aa6d151329 removing unnecessary assumptions in RealDef;
bulwahn
parents: 46573
diff changeset
   675
    case True hence "y dvd ?X" using `0 < y` by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   676
    from real_of_int_div[OF this]
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   677
    have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   678
    also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   679
    finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   680
    thus ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True] 
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
   681
      unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   682
  next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   683
    case False
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   684
    have "0 \<le> real y" and "real y \<noteq> 0" using `0 < y` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   685
    have "0 \<le> real y * 2^?l" by (rule mult_nonneg_nonneg, rule `0 \<le> real y`, auto)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   686
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   687
    have "?X = y * (?X div y) + ?X mod y" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   688
    also have "\<dots> \<le> y * (?X div y) + y" by (rule add_mono, auto simp add: pos_mod_bound[OF `0 < y`, THEN less_imp_le])
44766
d4d33a4d7548 avoid using legacy theorem names
huffman
parents: 42676
diff changeset
   689
    also have "\<dots> = y * (?X div y + 1)" unfolding right_distrib by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   690
    finally have "real ?X \<le> real y * real (?X div y + 1)" unfolding real_of_int_le_iff real_of_int_mult[symmetric] .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   691
    hence "real ?X / (real y * 2^?l) \<le> real y * real (?X div y + 1) / (real y * 2^?l)" 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   692
      by (rule divide_right_mono, simp only: `0 \<le> real y * 2^?l`)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   693
    also have "\<dots> = real y * real (?X div y + 1) / real y / 2^?l" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   694
    also have "\<dots> = real (?X div y + 1) * inverse (2^?l)" unfolding nonzero_mult_divide_cancel_left[OF `real y \<noteq> 0`] 
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 35344
diff changeset
   695
      unfolding divide_inverse ..
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
   696
    finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   697
      unfolding pow2_minus pow2_int minus_minus by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   698
  qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   699
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   700
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   701
lemma rapprox_posrat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
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
   702
  shows "real (rapprox_posrat n x y) \<le> 1"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   703
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   704
  let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   705
  show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   706
  proof (cases "?X mod y = 0")
46670
e9aa6d151329 removing unnecessary assumptions in RealDef;
bulwahn
parents: 46573
diff changeset
   707
    case True hence "y dvd ?X" using `0 < y` by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   708
    from real_of_int_div[OF this]
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   709
    have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   710
    also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   711
    finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   712
    also have "real x / real y \<le> 1" using `0 \<le> x` and `0 < y` and `x \<le> y` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   713
    finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True]
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
   714
      unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   715
  next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   716
    case False
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   717
    have "x \<noteq> y"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   718
    proof (rule ccontr)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   719
      assume "\<not> x \<noteq> y" hence "x = y" by auto
30034
60f64f112174 removed redundant thms
nipkow
parents: 29988
diff changeset
   720
      have "?X mod y = 0" unfolding `x = y` using mod_mult_self1_is_0 by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   721
      thus False using False by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   722
    qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   723
    hence "x < y" using `x \<le> y` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   724
    hence "real x / real y < 1" using `0 < y` and `0 \<le> x` by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   725
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   726
    from real_of_int_div4[of "?X" y]
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   727
    have "real (?X div y) \<le> (real x / real y) * 2^?l" unfolding real_of_int_mult times_divide_eq_left real_of_int_power real_numeral .
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   728
    also have "\<dots> < 1 * 2^?l" using `real x / real y < 1` by (rule mult_strict_right_mono, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   729
    finally have "?X div y < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   730
    hence "?X div y + 1 \<le> 2^?l" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   731
    hence "real (?X div y + 1) * inverse (2^?l) \<le> 2^?l * inverse (2^?l)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   732
      unfolding real_of_int_le_iff[of _ "2^?l", symmetric] real_of_int_power real_numeral
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   733
      by (rule mult_right_mono, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   734
    hence "real (?X div y + 1) * inverse (2^?l) \<le> 1" by auto
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
   735
    thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   736
      unfolding pow2_minus pow2_int minus_minus by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   737
  qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   738
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   739
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   740
lemma zdiv_greater_zero: fixes a b :: int assumes "0 < a" and "a \<le> b"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   741
  shows "0 < b div a"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   742
proof (rule ccontr)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   743
  have "0 \<le> b" using assms by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   744
  assume "\<not> 0 < b div a" hence "b div a = 0" using `0 \<le> b`[unfolded pos_imp_zdiv_nonneg_iff[OF `0<a`, of b, symmetric]] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   745
  have "b = a * (b div a) + b mod a" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   746
  hence "b = b mod a" unfolding `b div a = 0` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   747
  hence "b < a" using `0 < a`[THEN pos_mod_bound, of b] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   748
  thus False using `a \<le> b` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   749
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   750
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   751
lemma rapprox_posrat_less1: assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
   752
  shows "real (rapprox_posrat n x y) < 1"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   753
proof (cases "x = 0")
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
   754
  case True thus ?thesis unfolding rapprox_posrat_def True Let_def normfloat real_of_float_simp by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   755
next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   756
  case False hence "0 < x" using `0 \<le> x` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   757
  hence "x < y" using assms by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   758
  
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   759
  let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   760
  show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   761
  proof (cases "?X mod y = 0")
46670
e9aa6d151329 removing unnecessary assumptions in RealDef;
bulwahn
parents: 46573
diff changeset
   762
    case True hence "y dvd ?X" using `0 < y` by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   763
    from real_of_int_div[OF this]
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   764
    have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   765
    also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   766
    finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   767
    also have "real x / real y < 1" using `0 \<le> x` and `0 < y` and `x < y` by auto
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
   768
    finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_P[OF True]
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   769
      unfolding pow2_minus pow2_int minus_minus by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   770
  next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   771
    case False
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   772
    hence "(real x / real y) < 1 / 2" using `0 < y` and `0 \<le> x` `2 * x < y` by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   773
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   774
    have "0 < ?X div y"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   775
    proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   776
      have "2^nat (bitlen x - 1) \<le> y" and "y < 2^nat (bitlen y)"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
   777
        using bitlen_bounds[OF `0 < x`, THEN conjunct1] bitlen_bounds[OF `0 < y`, THEN conjunct2] `x < y` by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   778
      hence "(2::int)^nat (bitlen x - 1) < 2^nat (bitlen y)" by (rule order_le_less_trans)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   779
      hence "bitlen x \<le> bitlen y" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   780
      hence len_less: "nat (bitlen x - 1) \<le> nat (int (n - 1) + bitlen y)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   781
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   782
      have "x \<noteq> 0" and "y \<noteq> 0" using `0 < x` `0 < y` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   783
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   784
      have exp_eq: "nat (int (n - 1) + bitlen y) - nat (bitlen x - 1) = ?l"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
   785
        using `bitlen x \<le> bitlen y` bitlen_ge1[OF `x \<noteq> 0`] bitlen_ge1[OF `y \<noteq> 0`] `0 < n` by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   786
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   787
      have "y * 2^nat (bitlen x - 1) \<le> y * x" 
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
   788
        using bitlen_bounds[OF `0 < x`, THEN conjunct1] `0 < y`[THEN less_imp_le] by (rule mult_left_mono)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   789
      also have "\<dots> \<le> 2^nat (bitlen y) * x" using bitlen_bounds[OF `0 < y`, THEN conjunct2, THEN less_imp_le] `0 \<le> x` by (rule mult_right_mono)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   790
      also have "\<dots> \<le> x * 2^nat (int (n - 1) + bitlen y)" unfolding mult_commute[of x] by (rule mult_right_mono, auto simp add: `0 \<le> x`)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   791
      finally have "real y * 2^nat (bitlen x - 1) * inverse (2^nat (bitlen x - 1)) \<le> real x * 2^nat (int (n - 1) + bitlen y) * inverse (2^nat (bitlen x - 1))"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
   792
        unfolding real_of_int_le_iff[symmetric] by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   793
      hence "real y \<le> real x * (2^nat (int (n - 1) + bitlen y) / (2^nat (bitlen x - 1)))" 
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 35344
diff changeset
   794
        unfolding mult_assoc divide_inverse by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   795
      also have "\<dots> = real x * (2^(nat (int (n - 1) + bitlen y) - nat (bitlen x - 1)))" using power_diff[of "2::real", OF _ len_less] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   796
      finally have "y \<le> x * 2^?l" unfolding exp_eq 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
   797
      thus ?thesis using zdiv_greater_zero[OF `0 < y`] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   798
    qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   799
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   800
    from real_of_int_div4[of "?X" y]
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   801
    have "real (?X div y) \<le> (real x / real y) * 2^?l" unfolding real_of_int_mult times_divide_eq_left real_of_int_power real_numeral .
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   802
    also have "\<dots> < 1/2 * 2^?l" using `real x / real y < 1/2` by (rule mult_strict_right_mono, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   803
    finally have "?X div y * 2 < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   804
    hence "?X div y + 1 < 2^?l" using `0 < ?X div y` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   805
    hence "real (?X div y + 1) * inverse (2^?l) < 2^?l * inverse (2^?l)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
   806
      unfolding real_of_int_less_iff[of _ "2^?l", symmetric] real_of_int_power real_numeral
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   807
      by (rule mult_strict_right_mono, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   808
    hence "real (?X div y + 1) * inverse (2^?l) < 1" by auto
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
   809
    thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   810
      unfolding pow2_minus pow2_int minus_minus by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   811
  qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   812
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   813
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   814
lemma approx_rat_pattern: fixes P and ps :: "nat * int * int"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   815
  assumes Y: "\<And>y prec x. \<lbrakk>y = 0; ps = (prec, x, 0)\<rbrakk> \<Longrightarrow> P" 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   816
  and A: "\<And>x y prec. \<lbrakk>0 \<le> x; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   817
  and B: "\<And>x y prec. \<lbrakk>x < 0; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   818
  and C: "\<And>x y prec. \<lbrakk>x < 0; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   819
  and D: "\<And>x y prec. \<lbrakk>0 \<le> x; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   820
  shows P
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   821
proof -
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   822
  obtain prec x y where [simp]: "ps = (prec, x, y)" by (cases ps) auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   823
  from Y have "y = 0 \<Longrightarrow> P" by auto
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   824
  moreover {
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   825
    assume "0 < y"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   826
    have P
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   827
    proof (cases "0 \<le> x")
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   828
      case True
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   829
      with A and `0 < y` show P by auto
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   830
    next
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   831
      case False
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   832
      with B and `0 < y` show P by auto
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   833
    qed
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   834
  } 
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   835
  moreover {
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   836
    assume "y < 0"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   837
    have P
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   838
    proof (cases "0 \<le> x")
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   839
      case True
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   840
      with D and `y < 0` show P by auto
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   841
    next
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   842
      case False
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   843
      with C and `y < 0` show P by auto
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   844
    qed
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   845
  }
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   846
  ultimately show P by (cases "y = 0 \<or> 0 < y \<or> y < 0") auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   847
qed
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   848
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   849
function lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   850
where
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   851
  "y = 0 \<Longrightarrow> lapprox_rat prec x y = 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   852
| "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec x y"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   853
| "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec (-x) y)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   854
| "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec (-x) (-y)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   855
| "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec x (-y))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   856
apply simp_all by (rule approx_rat_pattern)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   857
termination by lexicographic_order
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   858
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   859
lemma compute_lapprox_rat[code]:
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   860
      "lapprox_rat prec x y = (if y = 0 then 0 else if 0 \<le> x then (if 0 < y then lapprox_posrat prec x y else - (rapprox_posrat prec x (-y))) 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   861
                                                             else (if 0 < y then - (rapprox_posrat prec (-x) y) else lapprox_posrat prec (-x) (-y)))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   862
  by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   863
            
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
   864
lemma lapprox_rat: "real (lapprox_rat prec x y) \<le> real x / real y"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   865
proof -      
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   866
  have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   867
  show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   868
    apply (case_tac "y = 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   869
    apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   870
    apply (case_tac "0 \<le> x \<and> 0 < y")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   871
    apply (simp add: lapprox_posrat)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   872
    apply (case_tac "x < 0 \<and> 0 < y")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   873
    apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   874
    apply (subst minus_le_iff)   
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   875
    apply (rule h[OF rapprox_posrat])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   876
    apply (simp_all)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   877
    apply (case_tac "x < 0 \<and> y < 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   878
    apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   879
    apply (rule h[OF _ lapprox_posrat])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   880
    apply (simp_all)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   881
    apply (case_tac "0 \<le> x \<and> y < 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   882
    apply (simp)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   883
    apply (subst minus_le_iff)   
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   884
    apply (rule h[OF rapprox_posrat])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   885
    apply simp_all
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   886
    apply arith
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   887
    done
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   888
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   889
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   890
lemma lapprox_rat_bottom: assumes "0 \<le> x" and "0 < y"
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
   891
  shows "real (x div y) \<le> real (lapprox_rat n x y)" 
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   892
  unfolding lapprox_rat.simps(2)[OF assms]  using lapprox_posrat_bottom[OF `0<y`] .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   893
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   894
function rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   895
where
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   896
  "y = 0 \<Longrightarrow> rapprox_rat prec x y = 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   897
| "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec x y"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   898
| "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec (-x) y)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   899
| "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec (-x) (-y)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   900
| "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec x (-y))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   901
apply simp_all by (rule approx_rat_pattern)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   902
termination by lexicographic_order
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   903
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   904
lemma compute_rapprox_rat[code]:
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   905
      "rapprox_rat prec x y = (if y = 0 then 0 else if 0 \<le> x then (if 0 < y then rapprox_posrat prec x y else - (lapprox_posrat prec x (-y))) else 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   906
                                                                  (if 0 < y then - (lapprox_posrat prec (-x) y) else rapprox_posrat prec (-x) (-y)))"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   907
  by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   908
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
   909
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   910
proof -      
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   911
  have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   912
  show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   913
    apply (case_tac "y = 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   914
    apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   915
    apply (case_tac "0 \<le> x \<and> 0 < y")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   916
    apply (simp add: rapprox_posrat)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   917
    apply (case_tac "x < 0 \<and> 0 < y")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   918
    apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   919
    apply (subst le_minus_iff)   
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   920
    apply (rule h[OF _ lapprox_posrat])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   921
    apply (simp_all)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   922
    apply (case_tac "x < 0 \<and> y < 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   923
    apply simp
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   924
    apply (rule h[OF rapprox_posrat])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   925
    apply (simp_all)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   926
    apply (case_tac "0 \<le> x \<and> y < 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   927
    apply (simp)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   928
    apply (subst le_minus_iff)   
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   929
    apply (rule h[OF _ lapprox_posrat])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   930
    apply simp_all
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   931
    apply arith
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   932
    done
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   933
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   934
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   935
lemma rapprox_rat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
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
   936
  shows "real (rapprox_rat n x y) \<le> 1"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   937
  unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`] using rapprox_posrat_le1[OF assms] .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   938
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   939
lemma rapprox_rat_neg: assumes "x < 0" and "0 < y"
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
   940
  shows "real (rapprox_rat n x y) \<le> 0"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   941
  unfolding rapprox_rat.simps(3)[OF assms] using lapprox_posrat_nonneg[of "-x" y n] assms by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   942
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   943
lemma rapprox_rat_nonneg_neg: assumes "0 \<le> x" and "y < 0"
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
   944
  shows "real (rapprox_rat n x y) \<le> 0"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   945
  unfolding rapprox_rat.simps(5)[OF assms] using lapprox_posrat_nonneg[of x "-y" n] assms by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   946
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   947
lemma rapprox_rat_nonpos_pos: assumes "x \<le> 0" and "0 < y"
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
   948
  shows "real (rapprox_rat n x y) \<le> 0"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   949
proof (cases "x = 0") 
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   950
  case True
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   951
  hence "0 \<le> x" by auto show ?thesis
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   952
    unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`]
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   953
    unfolding True rapprox_posrat_def Let_def
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   954
    by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   955
next
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   956
  case False
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   957
  hence "x < 0" using assms by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   958
  show ?thesis using rapprox_rat_neg[OF `x < 0` `0 < y`] .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   959
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   960
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   961
fun float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   962
where
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   963
  "float_divl prec (Float m1 s1) (Float m2 s2) = 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   964
    (let
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   965
       l = lapprox_rat prec m1 m2;
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   966
       f = Float 1 (s1 - s2)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   967
     in
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   968
       f * l)"     
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   969
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
   970
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
45772
8a8f78ce0dcf tuned proofs
hoelzl
parents: 45495
diff changeset
   971
  using lapprox_rat[of prec "mantissa x" "mantissa y"]
8a8f78ce0dcf tuned proofs
hoelzl
parents: 45495
diff changeset
   972
  by (cases x y rule: float.exhaust[case_product float.exhaust])
8a8f78ce0dcf tuned proofs
hoelzl
parents: 45495
diff changeset
   973
     (simp split: split_if_asm
8a8f78ce0dcf tuned proofs
hoelzl
parents: 45495
diff changeset
   974
           add: real_of_float_simp pow2_diff field_simps le_divide_eq mult_less_0_iff zero_less_mult_iff)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   975
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   976
lemma float_divl_lower_bound: assumes "0 \<le> x" and "0 < y" shows "0 \<le> float_divl prec x y"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   977
proof (cases x, cases y)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   978
  fix xm xe ym ye :: int
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   979
  assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   980
  have "0 \<le> xm"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   981
    using `0 \<le> x`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 zero_le_mult_iff]
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   982
    by auto
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   983
  have "0 < ym"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   984
    using `0 < y`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff]
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   985
    by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   986
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   987
  have "\<And>n. 0 \<le> real (Float 1 n)"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   988
    unfolding real_of_float_simp using zero_le_pow2 by auto
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   989
  moreover have "0 \<le> real (lapprox_rat prec xm ym)"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   990
    apply (rule order_trans[OF _ lapprox_rat_bottom[OF `0 \<le> xm` `0 < ym`]])
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   991
    apply (auto simp add: `0 \<le> xm` pos_imp_zdiv_nonneg_iff[OF `0 < ym`])
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   992
    done
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   993
  ultimately show "0 \<le> float_divl prec x y"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   994
    unfolding x_eq y_eq float_divl.simps Let_def le_float_def real_of_float_0
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   995
    by (auto intro!: mult_nonneg_nonneg)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   996
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   997
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   998
lemma float_divl_pos_less1_bound:
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
   999
  assumes "0 < x" and "x < 1" and "0 < prec"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1000
  shows "1 \<le> float_divl prec 1 x"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1001
proof (cases x)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1002
  case (Float m e)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1003
  from `0 < x` `x < 1` have "0 < m" "e < 0"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1004
    using float_pos_m_pos float_pos_less1_e_neg unfolding Float by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1005
  let ?b = "nat (bitlen m)" and ?e = "nat (-e)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1006
  have "1 \<le> m" and "m \<noteq> 0" using `0 < m` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1007
  with bitlen_bounds[OF `0 < m`] have "m < 2^?b" and "(2::int) \<le> 2^?b" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1008
  hence "1 \<le> bitlen m" using power_le_imp_le_exp[of "2::int" 1 ?b] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1009
  hence pow_split: "nat (int prec + bitlen m - 1) = (prec - 1) + ?b" using `0 < prec` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1010
  
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1011
  have pow_not0: "\<And>x. (2::real)^x \<noteq> 0" by auto
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1012
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1013
  from float_less1_mantissa_bound `0 < x` `x < 1` Float 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1014
  have "m < 2^?e" by auto
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1015
  with bitlen_bounds[OF `0 < m`, THEN conjunct1] have "(2::int)^nat (bitlen m - 1) < 2^?e"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1016
    by (rule order_le_less_trans)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1017
  from power_less_imp_less_exp[OF _ this]
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1018
  have "bitlen m \<le> - e" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1019
  hence "(2::real)^?b \<le> 2^?e" by auto
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1020
  hence "(2::real)^?b * inverse (2^?b) \<le> 2^?e * inverse (2^?b)"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1021
    by (rule mult_right_mono) auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1022
  hence "(1::real) \<le> 2^?e * inverse (2^?b)" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1023
  also
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1024
  let ?d = "real (2 ^ nat (int prec + bitlen m - 1) div m) * inverse (2 ^ nat (int prec + bitlen m - 1))"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1025
  {
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1026
    have "2^(prec - 1) * m \<le> 2^(prec - 1) * 2^?b"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1027
      using `m < 2^?b`[THEN less_imp_le] by (rule mult_left_mono) auto
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1028
    also have "\<dots> = 2 ^ nat (int prec + bitlen m - 1)"
44766
d4d33a4d7548 avoid using legacy theorem names
huffman
parents: 42676
diff changeset
  1029
      unfolding pow_split power_add by auto
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1030
    finally have "2^(prec - 1) * m div m \<le> 2 ^ nat (int prec + bitlen m - 1) div m"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1031
      using `0 < m` by (rule zdiv_mono1)
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1032
    hence "2^(prec - 1) \<le> 2 ^ nat (int prec + bitlen m - 1) div m"
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1033
      unfolding div_mult_self2_is_id[OF `m \<noteq> 0`] .
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1034
    hence "2^(prec - 1) * inverse (2 ^ nat (int prec + bitlen m - 1)) \<le> ?d"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1035
      unfolding real_of_int_le_iff[of "2^(prec - 1)", symmetric] by auto
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1036
  }
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1037
  from mult_left_mono[OF this [unfolded pow_split power_add inverse_mult_distrib mult_assoc[symmetric] right_inverse[OF pow_not0] mult_1_left], of "2^?e"]
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1038
  have "2^?e * inverse (2^?b) \<le> 2^?e * ?d" unfolding pow_split power_add by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1039
  finally have "1 \<le> 2^?e * ?d" .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1040
  
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1041
  have e_nat: "0 - e = int (nat (-e))" using `e < 0` by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1042
  have "bitlen 1 = 1" using bitlen.simps by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1043
  
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1044
  show ?thesis 
41528
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1045
    unfolding one_float_def Float float_divl.simps Let_def
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1046
      lapprox_rat.simps(2)[OF zero_le_one `0 < m`]
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1047
      lapprox_posrat_def `bitlen 1 = 1`
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1048
    unfolding le_float_def real_of_float_mult normfloat real_of_float_simp
276078f01ada eliminated global prems;
wenzelm
parents: 41024
diff changeset
  1049
      pow2_minus pow2_int e_nat
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1050
    using `1 \<le> 2^?e * ?d` by (auto simp add: pow2_def)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1051
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1052
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1053
fun float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1054
where
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1055
  "float_divr prec (Float m1 s1) (Float m2 s2) = 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1056
    (let
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1057
       r = rapprox_rat prec m1 m2;
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1058
       f = Float 1 (s1 - s2)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1059
     in
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1060
       f * r)"  
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1061
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
  1062
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
45772
8a8f78ce0dcf tuned proofs
hoelzl
parents: 45495
diff changeset
  1063
  using rapprox_rat[of "mantissa x" "mantissa y" prec]
8a8f78ce0dcf tuned proofs
hoelzl
parents: 45495
diff changeset
  1064
  by (cases x y rule: float.exhaust[case_product float.exhaust])
8a8f78ce0dcf tuned proofs
hoelzl
parents: 45495
diff changeset
  1065
     (simp split: split_if_asm
8a8f78ce0dcf tuned proofs
hoelzl
parents: 45495
diff changeset
  1066
           add: real_of_float_simp pow2_diff field_simps divide_le_eq mult_less_0_iff zero_less_mult_iff)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1067
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1068
lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1069
proof -
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
  1070
  have "1 \<le> 1 / real x" using `0 < x` and `x < 1` unfolding less_float_def by auto
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
  1071
  also have "\<dots> \<le> real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1072
  finally show ?thesis unfolding le_float_def by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1073
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1074
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1075
lemma float_divr_nonpos_pos_upper_bound: assumes "x \<le> 0" and "0 < y" shows "float_divr prec x y \<le> 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1076
proof (cases x, cases y)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1077
  fix xm xe ym ye :: int
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1078
  assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
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
  1079
  have "xm \<le> 0" using `x \<le> 0`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 mult_le_0_iff] by auto
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
  1080
  have "0 < ym" using `0 < y`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff] by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1081
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
  1082
  have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
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
  1083
  moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonpos_pos[OF `xm \<le> 0` `0 < ym`] .
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1084
  ultimately show "float_divr prec x y \<le> 0"
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
  1085
    unfolding x_eq y_eq float_divr.simps Let_def le_float_def real_of_float_0 real_of_float_mult by (auto intro!: mult_nonneg_nonpos)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1086
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1087
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1088
lemma float_divr_nonneg_neg_upper_bound: assumes "0 \<le> x" and "y < 0" shows "float_divr prec x y \<le> 0"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1089
proof (cases x, cases y)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1090
  fix xm xe ym ye :: int
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1091
  assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
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
  1092
  have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 zero_le_mult_iff] by auto
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
  1093
  have "ym < 0" using `y < 0`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 mult_less_0_iff] by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1094
  hence "0 < - ym" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1095
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
  1096
  have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
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
  1097
  moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonneg_neg[OF `0 \<le> xm` `ym < 0`] .
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1098
  ultimately show "float_divr prec x y \<le> 0"
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
  1099
    unfolding x_eq y_eq float_divr.simps Let_def le_float_def real_of_float_0 real_of_float_mult by (auto intro!: mult_nonneg_nonpos)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1100
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1101
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
  1102
primrec round_down :: "nat \<Rightarrow> float \<Rightarrow> float" where
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1103
"round_down prec (Float m e) = (let d = bitlen m - int prec in
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1104
     if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1105
              else Float m e)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1106
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
  1107
primrec round_up :: "nat \<Rightarrow> float \<Rightarrow> float" where
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1108
"round_up prec (Float m e) = (let d = bitlen m - int prec in
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1109
  if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P in Float (n + (if r = 0 then 0 else 1)) (e + d) 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1110
           else Float m e)"
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1111
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
  1112
lemma round_up: "real x \<le> real (round_up prec x)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1113
proof (cases x)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1114
  case (Float m e)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1115
  let ?d = "bitlen m - int prec"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1116
  let ?p = "(2::int)^nat ?d"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1117
  have "0 < ?p" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1118
  show "?thesis"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1119
  proof (cases "0 < ?d")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1120
    case True
35344
e0b46cd72414 moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents: 35032
diff changeset
  1121
    hence pow_d: "pow2 ?d = real ?p" using pow2_int[symmetric] by simp
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1122
    show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1123
    proof (cases "m mod ?p = 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1124
      case True
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1125
      have m: "m = m div ?p * ?p + 0" unfolding True[symmetric] using zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right, symmetric] .
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
  1126
      have "real (Float m e) = real (Float (m div ?p) (e + ?d))" unfolding real_of_float_simp arg_cong[OF m, of real]
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
  1127
        by (auto simp add: pow2_add `0 < ?d` pow_d)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1128
      thus ?thesis
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
  1129
        unfolding Float round_up.simps Let_def if_P[OF `m mod ?p = 0`] if_P[OF `0 < ?d`]
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
  1130
        by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1131
    next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1132
      case False
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1133
      have "m = m div ?p * ?p + m mod ?p" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
44766
d4d33a4d7548 avoid using legacy theorem names
huffman
parents: 42676
diff changeset
  1134
      also have "\<dots> \<le> (m div ?p + 1) * ?p" unfolding left_distrib mult_1 by (rule add_left_mono, rule pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
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
  1135
      finally have "real (Float m e) \<le> real (Float (m div ?p + 1) (e + ?d))" unfolding real_of_float_simp add_commute[of e]
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
  1136
        unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of m, symmetric]
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
  1137
        by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1138
      thus ?thesis
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
  1139
        unfolding Float round_up.simps Let_def if_not_P[OF `\<not> m mod ?p = 0`] if_P[OF `0 < ?d`] .
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1140
    qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1141
  next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1142
    case False
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1143
    show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1144
      unfolding Float round_up.simps Let_def if_not_P[OF False] .. 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1145
  qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1146
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1147
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
  1148
lemma round_down: "real (round_down prec x) \<le> real x"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1149
proof (cases x)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1150
  case (Float m e)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1151
  let ?d = "bitlen m - int prec"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1152
  let ?p = "(2::int)^nat ?d"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1153
  have "0 < ?p" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1154
  show "?thesis"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1155
  proof (cases "0 < ?d")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1156
    case True
35344
e0b46cd72414 moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents: 35032
diff changeset
  1157
    hence pow_d: "pow2 ?d = real ?p" using pow2_int[symmetric] by simp
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1158
    have "m div ?p * ?p \<le> m div ?p * ?p + m mod ?p" by (auto simp add: pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1159
    also have "\<dots> \<le> m" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
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
  1160
    finally have "real (Float (m div ?p) (e + ?d)) \<le> real (Float m e)" unfolding real_of_float_simp add_commute[of e]
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1161
      unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of _ m, symmetric]
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1162
      by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1163
    thus ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1164
      unfolding Float round_down.simps Let_def if_P[OF `0 < ?d`] .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1165
  next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1166
    case False
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1167
    show ?thesis
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1168
      unfolding Float round_down.simps Let_def if_not_P[OF False] .. 
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1169
  qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1170
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1171
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1172
definition lb_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1173
  "lb_mult prec x y =
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1174
    (case normfloat (x * y) of Float m e \<Rightarrow>
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1175
      let
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1176
        l = bitlen m - int prec
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1177
      in if l > 0 then Float (m div (2^nat l)) (e + l)
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1178
                  else Float m e)"
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1179
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1180
definition ub_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
46573
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1181
  "ub_mult prec x y =
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1182
    (case normfloat (x * y) of Float m e \<Rightarrow>
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1183
      let
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1184
        l = bitlen m - int prec
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1185
      in if l > 0 then Float (m div (2^nat l) + 1) (e + l)
8c4c5c8dcf7a misc tuning;
wenzelm
parents: 46028
diff changeset
  1186
                  else Float m e)"
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1187
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
  1188
lemma lb_mult: "real (lb_mult prec x y) \<le> real (x * y)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1189
proof (cases "normfloat (x * y)")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1190
  case (Float m e)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1191
  hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1192
  let ?l = "bitlen m - int prec"
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
  1193
  have "real (lb_mult prec x y) \<le> real (normfloat (x * y))"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1194
  proof (cases "?l > 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1195
    case False thus ?thesis unfolding lb_mult_def Float Let_def float.cases by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1196
  next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1197
    case True
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1198
    have "real (m div 2^(nat ?l)) * pow2 ?l \<le> real m"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1199
    proof -
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46670
diff changeset
  1200
      have "real (m div 2^(nat ?l)) * pow2 ?l = real (2^(nat ?l) * (m div 2^(nat ?l)))" unfolding real_of_int_mult real_of_int_power real_numeral unfolding pow2_int[symmetric] 
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32069
diff changeset
  1201
        using `?l > 0` by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1202
      also have "\<dots> \<le> real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding real_of_int_add by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1203
      also have "\<dots> = real m" unfolding zmod_zdiv_equality[symmetric] ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1204
      finally show ?thesis by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1205
    qed
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 35344
diff changeset
  1206
    thus ?thesis unfolding lb_mult_def Float Let_def float.cases if_P[OF True] real_of_float_simp pow2_add mult_commute mult_assoc by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1207
  qed
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
  1208
  also have "\<dots> = real (x * y)" unfolding normfloat ..
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1209
  finally show ?thesis .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1210
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1211
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
  1212
lemma ub_mult: "real (x * y) \<le> real (ub_mult prec x y)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1213
proof (cases "normfloat (x * y)")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1214
  case (Float m e)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1215
  hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1216
  let ?l = "bitlen m - int prec"
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
  1217
  have "real (x * y) = real (normfloat (x * y))" unfolding normfloat ..
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
  1218
  also have "\<dots> \<le> real (ub_mult prec x y)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1219
  proof (cases "?l > 0")
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1220
    case False thus ?thesis unfolding ub_mult_def Float Let_def float.cases by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1221
  next
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1222
    case True
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1223
    have "real m \<le> real (m div 2^(nat ?l) + 1) * pow2 ?l"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1224
    proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1225
      have "m mod 2^(nat ?l) < 2^(nat ?l)" by (rule pos_mod_bound) auto
44766
d4d33a4d7548 avoid using legacy theorem names
huffman
parents: 42676
diff changeset
  1226
      hence mod_uneq: "real (m mod 2^(nat ?l)) \<le> 1 * 2^(nat ?l)" unfolding mult_1 real_of_int_less_iff[symmetric] by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1227
      
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1228
      have "real m = real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding zmod_zdiv_equality[symmetric] ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1229
      also have "\<dots> = real (m div 2^(nat ?l)) * 2^(nat ?l) + real (m mod 2^(nat ?l))" unfolding real_of_int_add by auto
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 35344
diff changeset
  1230
      also have "\<dots> \<le> (real (m div 2^(nat ?l)) + 1) * 2^(nat ?l)" unfolding left_distrib using mod_uneq by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1231
      finally show ?thesis unfolding pow2_int[symmetric] using True by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1232
    qed
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 35344
diff changeset
  1233
    thus ?thesis unfolding ub_mult_def Float Let_def float.cases if_P[OF True] real_of_float_simp pow2_add mult_commute mult_assoc by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1234
  qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1235
  finally show ?thesis .
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1236
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1237
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
  1238
primrec float_abs :: "float \<Rightarrow> float" where
fec1a04b7220 power operation defined generic
haftmann
parents: 30242
diff changeset
  1239
  "float_abs (Float m e) = Float \<bar>m\<bar> e"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset