author | nipkow |
Mon, 17 Oct 2016 17:33:07 +0200 | |
changeset 64272 | f76b6dda2e56 |
parent 64267 | b9a1486e79be |
child 65109 | a79c1080f1e9 |
permissions | -rw-r--r-- |
58834 | 1 |
(* Author: Johannes Hoelzl, TU Muenchen |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2 |
Coercions removed by Dmitriy Traytel *) |
30122 | 3 |
|
60533 | 4 |
section \<open>Prove Real Valued Inequalities by Computation\<close> |
30122 | 5 |
|
40892 | 6 |
theory Approximation |
41413
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41024
diff
changeset
|
7 |
imports |
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41024
diff
changeset
|
8 |
Complex_Main |
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41024
diff
changeset
|
9 |
"~~/src/HOL/Library/Float" |
51544 | 10 |
Dense_Linear_Order |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
49962
diff
changeset
|
11 |
"~~/src/HOL/Library/Code_Target_Numeral" |
56923 | 12 |
keywords "approximate" :: diag |
29805 | 13 |
begin |
14 |
||
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54269
diff
changeset
|
15 |
declare powr_numeral [simp] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54269
diff
changeset
|
16 |
declare powr_neg_one [simp] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54269
diff
changeset
|
17 |
declare powr_neg_numeral [simp] |
47600 | 18 |
|
29805 | 19 |
section "Horner Scheme" |
20 |
||
61586 | 21 |
subsection \<open>Define auxiliary helper \<open>horner\<close> function\<close> |
29805 | 22 |
|
31098
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
23 |
primrec horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where |
29805 | 24 |
"horner F G 0 i k x = 0" | |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
25 |
"horner F G (Suc n) i k x = 1 / k - x * horner F G n (F i) (G i k) x" |
29805 | 26 |
|
49351 | 27 |
lemma horner_schema': |
28 |
fixes x :: real and a :: "nat \<Rightarrow> real" |
|
29805 | 29 |
shows "a 0 - x * (\<Sum> i=0..<n. (-1)^i * a (Suc i) * x^i) = (\<Sum> i=0..<Suc n. (-1)^i * a i * x^i)" |
30 |
proof - |
|
49351 | 31 |
have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)" |
32 |
by auto |
|
33 |
show ?thesis |
|
64267 | 34 |
unfolding sum_distrib_left shift_pow uminus_add_conv_diff [symmetric] sum_negf[symmetric] |
35 |
sum_head_upt_Suc[OF zero_less_Suc] |
|
36 |
sum.reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n *a n * x^n"] by auto |
|
29805 | 37 |
qed |
38 |
||
49351 | 39 |
lemma horner_schema: |
40 |
fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat" |
|
30971 | 41 |
assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)" |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
42 |
shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum> j = 0..< n. (- 1) ^ j * (1 / (f (j' + j))) * x ^ j)" |
45129
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
44821
diff
changeset
|
43 |
proof (induct n arbitrary: j') |
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
44821
diff
changeset
|
44 |
case 0 |
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
44821
diff
changeset
|
45 |
then show ?case by auto |
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
44821
diff
changeset
|
46 |
next |
29805 | 47 |
case (Suc n) |
48 |
show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc] |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
49 |
using horner_schema'[of "\<lambda> j. 1 / (f (j' + j))"] by auto |
45129
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
44821
diff
changeset
|
50 |
qed |
29805 | 51 |
|
52 |
lemma horner_bounds': |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
53 |
fixes lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
54 |
assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)" |
49351 | 55 |
and lb_0: "\<And> i k x. lb 0 i k x = 0" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
56 |
and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
57 |
(lapprox_rat prec 1 k) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
58 |
(- float_round_up prec (x * (ub n (F i) (G i k) x)))" |
49351 | 59 |
and ub_0: "\<And> i k x. ub 0 i k x = 0" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
60 |
and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
61 |
(rapprox_rat prec 1 k) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
62 |
(- float_round_down prec (x * (lb n (F i) (G i k) x)))" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
63 |
shows "(lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') x \<and> |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
64 |
horner F G n ((F ^^ j') s) (f j') x \<le> (ub n ((F ^^ j') s) (f j') x)" |
29805 | 65 |
(is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'") |
66 |
proof (induct n arbitrary: j') |
|
49351 | 67 |
case 0 |
68 |
thus ?case unfolding lb_0 ub_0 horner.simps by auto |
|
29805 | 69 |
next |
70 |
case (Suc n) |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
71 |
thus ?case using lapprox_rat[of prec 1 "f j'"] using rapprox_rat[of 1 "f j'" prec] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
72 |
Suc[where j'="Suc j'"] \<open>0 \<le> real_of_float x\<close> |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
73 |
by (auto intro!: add_mono mult_left_mono float_round_down_le float_round_up_le |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
74 |
order_trans[OF add_mono[OF _ float_plus_down_le]] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
75 |
order_trans[OF _ add_mono[OF _ float_plus_up_le]] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
76 |
simp add: lb_Suc ub_Suc field_simps f_Suc) |
29805 | 77 |
qed |
78 |
||
79 |
subsection "Theorems for floating point functions implementing the horner scheme" |
|
80 |
||
60533 | 81 |
text \<open> |
29805 | 82 |
|
83 |
Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are |
|
84 |
all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}. |
|
85 |
||
60533 | 86 |
\<close> |
29805 | 87 |
|
49351 | 88 |
lemma horner_bounds: |
89 |
fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
90 |
assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)" |
49351 | 91 |
and lb_0: "\<And> i k x. lb 0 i k x = 0" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
92 |
and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
93 |
(lapprox_rat prec 1 k) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
94 |
(- float_round_up prec (x * (ub n (F i) (G i k) x)))" |
49351 | 95 |
and ub_0: "\<And> i k x. ub 0 i k x = 0" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
96 |
and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
97 |
(rapprox_rat prec 1 k) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
98 |
(- float_round_down prec (x * (lb n (F i) (G i k) x)))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
99 |
shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
100 |
(is "?lb") |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
101 |
and "(\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j)) \<le> (ub n ((F ^^ j') s) (f j') x)" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
102 |
(is "?ub") |
29805 | 103 |
proof - |
31809 | 104 |
have "?lb \<and> ?ub" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
105 |
using horner_bounds'[where lb=lb, OF \<open>0 \<le> real_of_float x\<close> f_Suc lb_0 lb_Suc ub_0 ub_Suc] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
106 |
unfolding horner_schema[where f=f, OF f_Suc] by simp |
29805 | 107 |
thus "?lb" and "?ub" by auto |
108 |
qed |
|
109 |
||
49351 | 110 |
lemma horner_bounds_nonpos: |
111 |
fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
112 |
assumes "real_of_float x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)" |
49351 | 113 |
and lb_0: "\<And> i k x. lb 0 i k x = 0" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
114 |
and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
115 |
(lapprox_rat prec 1 k) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
116 |
(float_round_down prec (x * (ub n (F i) (G i k) x)))" |
49351 | 117 |
and ub_0: "\<And> i k x. ub 0 i k x = 0" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
118 |
and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
119 |
(rapprox_rat prec 1 k) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
120 |
(float_round_up prec (x * (lb n (F i) (G i k) x)))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
121 |
shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j)" (is "?lb") |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
122 |
and "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub") |
29805 | 123 |
proof - |
60680 | 124 |
have diff_mult_minus: "x - y * z = x + - y * z" for x y z :: float by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
125 |
have sum_eq: "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
126 |
(\<Sum>j = 0..<n. (- 1) ^ j * (1 / (f (j' + j))) * real_of_float (- x) ^ j)" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
127 |
by (auto simp add: field_simps power_mult_distrib[symmetric]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
128 |
have "0 \<le> real_of_float (-x)" using assms by auto |
29805 | 129 |
from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
130 |
and lb="\<lambda> n i k x. lb n i k (-x)" and ub="\<lambda> n i k x. ub n i k (-x)", |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
131 |
unfolded lb_Suc ub_Suc diff_mult_minus, |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
132 |
OF this f_Suc lb_0 _ ub_0 _] |
29805 | 133 |
show "?lb" and "?ub" unfolding minus_minus sum_eq |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
134 |
by (auto simp: minus_float_round_up_eq minus_float_round_down_eq) |
29805 | 135 |
qed |
136 |
||
60680 | 137 |
|
60533 | 138 |
subsection \<open>Selectors for next even or odd number\<close> |
139 |
||
140 |
text \<open> |
|
29805 | 141 |
The horner scheme computes alternating series. To get the upper and lower bounds we need to |
142 |
guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}. |
|
60533 | 143 |
\<close> |
29805 | 144 |
|
145 |
definition get_odd :: "nat \<Rightarrow> nat" where |
|
146 |
"get_odd n = (if odd n then n else (Suc n))" |
|
147 |
||
148 |
definition get_even :: "nat \<Rightarrow> nat" where |
|
149 |
"get_even n = (if even n then n else (Suc n))" |
|
150 |
||
60680 | 151 |
lemma get_odd[simp]: "odd (get_odd n)" |
152 |
unfolding get_odd_def by (cases "odd n") auto |
|
153 |
||
154 |
lemma get_even[simp]: "even (get_even n)" |
|
155 |
unfolding get_even_def by (cases "even n") auto |
|
156 |
||
29805 | 157 |
lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)" |
54269 | 158 |
by (auto simp: get_odd_def odd_pos intro!: exI[of _ "n - 1"]) |
29805 | 159 |
|
60680 | 160 |
lemma get_even_double: "\<exists>i. get_even n = 2 * i" |
161 |
using get_even by (blast elim: evenE) |
|
162 |
||
163 |
lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1" |
|
164 |
using get_odd by (blast elim: oddE) |
|
165 |
||
29805 | 166 |
|
167 |
section "Power function" |
|
168 |
||
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
169 |
definition float_power_bnds :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
170 |
"float_power_bnds prec n l u = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
171 |
(if 0 < l then (power_down_fl prec l n, power_up_fl prec u n) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
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172 |
else if odd n then |
61945 | 173 |
(- power_up_fl prec \<bar>l\<bar> n, |
174 |
if u < 0 then - power_down_fl prec \<bar>u\<bar> n else power_up_fl prec u n) |
|
175 |
else if u < 0 then (power_down_fl prec \<bar>u\<bar> n, power_up_fl prec \<bar>l\<bar> n) |
|
176 |
else (0, power_up_fl prec (max \<bar>l\<bar> \<bar>u\<bar>) n))" |
|
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177 |
|
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178 |
lemma le_minus_power_downI: "0 \<le> x \<Longrightarrow> x ^ n \<le> - a \<Longrightarrow> a \<le> - power_down prec x n" |
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179 |
by (subst le_minus_iff) (auto intro: power_down_le power_mono_odd) |
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180 |
|
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181 |
lemma float_power_bnds: |
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182 |
"(l1, u1) = float_power_bnds prec n l u \<Longrightarrow> x \<in> {l .. u} \<Longrightarrow> (x::real) ^ n \<in> {l1..u1}" |
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183 |
by (auto |
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184 |
simp: float_power_bnds_def max_def real_power_up_fl real_power_down_fl minus_le_iff |
62390 | 185 |
split: if_split_asm |
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186 |
intro!: power_up_le power_down_le le_minus_power_downI |
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187 |
intro: power_mono_odd power_mono power_mono_even zero_le_even_power) |
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188 |
|
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189 |
lemma bnds_power: |
60680 | 190 |
"\<forall>(x::real) l u. (l1, u1) = float_power_bnds prec n l u \<and> x \<in> {l .. u} \<longrightarrow> |
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191 |
l1 \<le> x ^ n \<and> x ^ n \<le> u1" |
29805 | 192 |
using float_power_bnds by auto |
193 |
||
62200
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194 |
section \<open>Approximation utility functions\<close> |
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|
195 |
|
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|
196 |
definition bnds_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<times> float" where |
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197 |
"bnds_mult prec a1 a2 b1 b2 = |
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|
198 |
(float_plus_down prec (nprt a1 * pprt b2) |
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|
199 |
(float_plus_down prec (nprt a2 * nprt b2) |
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changeset
|
200 |
(float_plus_down prec (pprt a1 * pprt b1) (pprt a2 * nprt b1))), |
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diff
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|
201 |
float_plus_up prec (pprt a2 * pprt b2) |
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|
202 |
(float_plus_up prec (pprt a1 * nprt b2) |
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|
203 |
(float_plus_up prec (nprt a2 * pprt b1) (nprt a1 * nprt b1))))" |
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changeset
|
204 |
|
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|
205 |
lemma bnds_mult: |
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|
206 |
fixes prec :: nat and a1 aa2 b1 b2 :: float |
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changeset
|
207 |
assumes "(l, u) = bnds_mult prec a1 a2 b1 b2" |
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changeset
|
208 |
assumes "a \<in> {real_of_float a1..real_of_float a2}" |
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changeset
|
209 |
assumes "b \<in> {real_of_float b1..real_of_float b2}" |
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changeset
|
210 |
shows "a * b \<in> {real_of_float l..real_of_float u}" |
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diff
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|
211 |
proof - |
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changeset
|
212 |
from assms have "real_of_float l \<le> a * b" |
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changeset
|
213 |
by (intro order.trans[OF _ mult_ge_prts[of a1 a a2 b1 b b2]]) |
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changeset
|
214 |
(auto simp: bnds_mult_def intro!: float_plus_down_le) |
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|
215 |
moreover from assms have "real_of_float u \<ge> a * b" |
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changeset
|
216 |
by (intro order.trans[OF mult_le_prts[of a1 a a2 b1 b b2]]) |
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changeset
|
217 |
(auto simp: bnds_mult_def intro!: float_plus_up_le) |
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|
218 |
ultimately show ?thesis by simp |
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|
219 |
qed |
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changeset
|
220 |
|
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|
221 |
definition map_bnds :: "(nat \<Rightarrow> float \<Rightarrow> float) \<Rightarrow> (nat \<Rightarrow> float \<Rightarrow> float) \<Rightarrow> |
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|
222 |
nat \<Rightarrow> (float \<times> float) \<Rightarrow> (float \<times> float)" where |
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changeset
|
223 |
"map_bnds lb ub prec = (\<lambda>(l,u). (lb prec l, ub prec u))" |
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diff
changeset
|
224 |
|
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|
225 |
lemma map_bnds: |
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|
226 |
assumes "(lf, uf) = map_bnds lb ub prec (l, u)" |
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|
227 |
assumes "mono f" |
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changeset
|
228 |
assumes "x \<in> {real_of_float l..real_of_float u}" |
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changeset
|
229 |
assumes "real_of_float (lb prec l) \<le> f (real_of_float l)" |
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diff
changeset
|
230 |
assumes "real_of_float (ub prec u) \<ge> f (real_of_float u)" |
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changeset
|
231 |
shows "f x \<in> {real_of_float lf..real_of_float uf}" |
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|
232 |
proof - |
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changeset
|
233 |
from assms have "real_of_float lf = real_of_float (lb prec l)" |
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changeset
|
234 |
by (simp add: map_bnds_def) |
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diff
changeset
|
235 |
also have "real_of_float (lb prec l) \<le> f (real_of_float l)" by fact |
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changeset
|
236 |
also from assms have "\<dots> \<le> f x" |
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|
237 |
by (intro monoD[OF \<open>mono f\<close>]) auto |
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changeset
|
238 |
finally have lf: "real_of_float lf \<le> f x" . |
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diff
changeset
|
239 |
|
67792e4a5486
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changeset
|
240 |
from assms have "f x \<le> f (real_of_float u)" |
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|
241 |
by (intro monoD[OF \<open>mono f\<close>]) auto |
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parents:
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diff
changeset
|
242 |
also have "\<dots> \<le> real_of_float (ub prec u)" by fact |
67792e4a5486
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parents:
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diff
changeset
|
243 |
also from assms have "\<dots> = real_of_float uf" |
67792e4a5486
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parents:
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|
244 |
by (simp add: map_bnds_def) |
67792e4a5486
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parents:
61969
diff
changeset
|
245 |
finally have uf: "f x \<le> real_of_float uf" . |
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|
246 |
|
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|
247 |
from lf uf show ?thesis by simp |
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|
248 |
qed |
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|
249 |
|
60680 | 250 |
|
29805 | 251 |
section "Square root" |
252 |
||
60533 | 253 |
text \<open> |
29805 | 254 |
The square root computation is implemented as newton iteration. As first first step we use the |
255 |
nearest power of two greater than the square root. |
|
60533 | 256 |
\<close> |
29805 | 257 |
|
258 |
fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
47599
400b158f1589
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diff
changeset
|
259 |
"sqrt_iteration prec 0 x = Float 1 ((bitlen \<bar>mantissa x\<bar> + exponent x) div 2 + 1)" | |
31467
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
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31148
diff
changeset
|
260 |
"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x |
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|
261 |
in Float 1 (- 1) * float_plus_up prec y (float_divr prec x y))" |
29805 | 262 |
|
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|
263 |
lemma compute_sqrt_iteration_base[code]: |
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|
264 |
shows "sqrt_iteration prec n (Float m e) = |
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|
265 |
(if n = 0 then Float 1 ((if m = 0 then 0 else bitlen \<bar>m\<bar> + e) div 2 + 1) |
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|
266 |
else (let y = sqrt_iteration prec (n - 1) (Float m e) in |
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immler
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|
267 |
Float 1 (- 1) * float_plus_up prec y (float_divr prec (Float m e) y)))" |
47599
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|
268 |
using bitlen_Float by (cases n) simp_all |
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|
269 |
|
31467
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
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31148
diff
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|
270 |
function ub_sqrt lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float" where |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
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diff
changeset
|
271 |
"ub_sqrt prec x = (if 0 < x then (sqrt_iteration prec prec x) |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
272 |
else if x < 0 then - lb_sqrt prec (- x) |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
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diff
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|
273 |
else 0)" | |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
274 |
"lb_sqrt prec x = (if 0 < x then (float_divl prec x (sqrt_iteration prec prec x)) |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
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31148
diff
changeset
|
275 |
else if x < 0 then - ub_sqrt prec (- x) |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
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|
276 |
else 0)" |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
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31148
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|
277 |
by pat_completeness auto |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55413
diff
changeset
|
278 |
termination by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto) |
29805 | 279 |
|
31467
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
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|
280 |
declare lb_sqrt.simps[simp del] |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
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|
281 |
declare ub_sqrt.simps[simp del] |
29805 | 282 |
|
283 |
lemma sqrt_ub_pos_pos_1: |
|
284 |
assumes "sqrt x < b" and "0 < b" and "0 < x" |
|
285 |
shows "sqrt x < (b + x / b)/2" |
|
286 |
proof - |
|
53077 | 287 |
from assms have "0 < (b - sqrt x)\<^sup>2 " by simp |
288 |
also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + (sqrt x)\<^sup>2" by algebra |
|
289 |
also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + x" using assms by simp |
|
290 |
finally have "0 < b\<^sup>2 - 2 * b * sqrt x + x" . |
|
29805 | 291 |
hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms |
292 |
by (simp add: field_simps power2_eq_square) |
|
293 |
thus ?thesis by (simp add: field_simps) |
|
294 |
qed |
|
295 |
||
60680 | 296 |
lemma sqrt_iteration_bound: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
297 |
assumes "0 < real_of_float x" |
54269 | 298 |
shows "sqrt x < sqrt_iteration prec n x" |
29805 | 299 |
proof (induct n) |
300 |
case 0 |
|
301 |
show ?case |
|
302 |
proof (cases x) |
|
303 |
case (Float m e) |
|
60680 | 304 |
hence "0 < m" |
305 |
using assms |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59850
diff
changeset
|
306 |
apply (auto simp: sign_simps) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59850
diff
changeset
|
307 |
by (meson not_less powr_ge_pzero) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
308 |
hence "0 < sqrt m" by auto |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
309 |
|
60680 | 310 |
have int_nat_bl: "(nat (bitlen m)) = bitlen m" |
311 |
using bitlen_nonneg by auto |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
312 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
313 |
have "x = (m / 2^nat (bitlen m)) * 2 powr (e + (nat (bitlen m)))" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
314 |
unfolding Float by (auto simp: powr_realpow[symmetric] field_simps powr_add) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
315 |
also have "\<dots> < 1 * 2 powr (e + nat (bitlen m))" |
29805 | 316 |
proof (rule mult_strict_right_mono, auto) |
60680 | 317 |
show "m < 2^nat (bitlen m)" |
318 |
using bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
319 |
unfolding of_int_less_iff[of m, symmetric] by auto |
29805 | 320 |
qed |
60680 | 321 |
finally have "sqrt x < sqrt (2 powr (e + bitlen m))" |
322 |
unfolding int_nat_bl by auto |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
323 |
also have "\<dots> \<le> 2 powr ((e + bitlen m) div 2 + 1)" |
29805 | 324 |
proof - |
325 |
let ?E = "e + bitlen m" |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
326 |
have E_mod_pow: "2 powr (?E mod 2) < 4" |
29805 | 327 |
proof (cases "?E mod 2 = 1") |
60680 | 328 |
case True |
329 |
thus ?thesis by auto |
|
29805 | 330 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
331 |
case False |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
332 |
have "0 \<le> ?E mod 2" by auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
333 |
have "?E mod 2 < 2" by auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
334 |
from this[THEN zless_imp_add1_zle] |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
335 |
have "?E mod 2 \<le> 0" using False by auto |
60533 | 336 |
from xt1(5)[OF \<open>0 \<le> ?E mod 2\<close> this] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
337 |
show ?thesis by auto |
29805 | 338 |
qed |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56813
diff
changeset
|
339 |
hence "sqrt (2 powr (?E mod 2)) < sqrt (2 * 2)" |
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56813
diff
changeset
|
340 |
by (auto simp del: real_sqrt_four) |
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56813
diff
changeset
|
341 |
hence E_mod_pow: "sqrt (2 powr (?E mod 2)) < 2" by auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
342 |
|
60680 | 343 |
have E_eq: "2 powr ?E = 2 powr (?E div 2 + ?E div 2 + ?E mod 2)" |
344 |
by auto |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
345 |
have "sqrt (2 powr ?E) = sqrt (2 powr (?E div 2) * 2 powr (?E div 2) * 2 powr (?E mod 2))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
346 |
unfolding E_eq unfolding powr_add[symmetric] by (metis of_int_add) |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
347 |
also have "\<dots> = 2 powr (?E div 2) * sqrt (2 powr (?E mod 2))" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
348 |
unfolding real_sqrt_mult[of _ "2 powr (?E mod 2)"] real_sqrt_abs2 by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
349 |
also have "\<dots> < 2 powr (?E div 2) * 2 powr 1" |
60680 | 350 |
by (rule mult_strict_left_mono) (auto intro: E_mod_pow) |
351 |
also have "\<dots> = 2 powr (?E div 2 + 1)" |
|
352 |
unfolding add.commute[of _ 1] powr_add[symmetric] by simp |
|
29805 | 353 |
finally show ?thesis by auto |
354 |
qed |
|
60533 | 355 |
finally show ?thesis using \<open>0 < m\<close> |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
356 |
unfolding Float |
47600 | 357 |
by (subst compute_sqrt_iteration_base) (simp add: ac_simps) |
29805 | 358 |
qed |
359 |
next |
|
360 |
case (Suc n) |
|
361 |
let ?b = "sqrt_iteration prec n x" |
|
60680 | 362 |
have "0 < sqrt x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
363 |
using \<open>0 < real_of_float x\<close> by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
364 |
also have "\<dots> < real_of_float ?b" |
60680 | 365 |
using Suc . |
366 |
finally have "sqrt x < (?b + x / ?b)/2" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
367 |
using sqrt_ub_pos_pos_1[OF Suc _ \<open>0 < real_of_float x\<close>] by auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
368 |
also have "\<dots> \<le> (?b + (float_divr prec x ?b))/2" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
369 |
by (rule divide_right_mono, auto simp add: float_divr) |
60680 | 370 |
also have "\<dots> = (Float 1 (- 1)) * (?b + (float_divr prec x ?b))" |
371 |
by simp |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
372 |
also have "\<dots> \<le> (Float 1 (- 1)) * (float_plus_up prec ?b (float_divr prec x ?b))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
373 |
by (auto simp add: algebra_simps float_plus_up_le) |
60680 | 374 |
finally show ?case |
375 |
unfolding sqrt_iteration.simps Let_def distrib_left . |
|
29805 | 376 |
qed |
377 |
||
60680 | 378 |
lemma sqrt_iteration_lower_bound: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
379 |
assumes "0 < real_of_float x" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
380 |
shows "0 < real_of_float (sqrt_iteration prec n x)" (is "0 < ?sqrt") |
29805 | 381 |
proof - |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
382 |
have "0 < sqrt x" using assms by auto |
29805 | 383 |
also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] . |
384 |
finally show ?thesis . |
|
385 |
qed |
|
386 |
||
60680 | 387 |
lemma lb_sqrt_lower_bound: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
388 |
assumes "0 \<le> real_of_float x" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
389 |
shows "0 \<le> real_of_float (lb_sqrt prec x)" |
29805 | 390 |
proof (cases "0 < x") |
60680 | 391 |
case True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
392 |
hence "0 < real_of_float x" and "0 \<le> x" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
393 |
using \<open>0 \<le> real_of_float x\<close> by auto |
60680 | 394 |
hence "0 < sqrt_iteration prec prec x" |
395 |
using sqrt_iteration_lower_bound by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
396 |
hence "0 \<le> real_of_float (float_divl prec x (sqrt_iteration prec prec x))" |
60680 | 397 |
using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] unfolding less_eq_float_def by auto |
398 |
thus ?thesis |
|
399 |
unfolding lb_sqrt.simps using True by auto |
|
29805 | 400 |
next |
60680 | 401 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
402 |
with \<open>0 \<le> real_of_float x\<close> have "real_of_float x = 0" by auto |
60680 | 403 |
thus ?thesis |
404 |
unfolding lb_sqrt.simps by auto |
|
29805 | 405 |
qed |
406 |
||
49351 | 407 |
lemma bnds_sqrt': "sqrt x \<in> {(lb_sqrt prec x) .. (ub_sqrt prec x)}" |
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:
31148
diff
changeset
|
408 |
proof - |
60680 | 409 |
have lb: "lb_sqrt prec x \<le> sqrt x" if "0 < x" for x :: float |
410 |
proof - |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
411 |
from that have "0 < real_of_float x" and "0 \<le> real_of_float x" by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
412 |
hence sqrt_gt0: "0 < sqrt x" by auto |
60680 | 413 |
hence sqrt_ub: "sqrt x < sqrt_iteration prec prec x" |
414 |
using sqrt_iteration_bound by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
415 |
have "(float_divl prec x (sqrt_iteration prec prec x)) \<le> |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
416 |
x / (sqrt_iteration prec prec x)" by (rule float_divl) |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
417 |
also have "\<dots> < x / sqrt x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
418 |
by (rule divide_strict_left_mono[OF sqrt_ub \<open>0 < real_of_float x\<close> |
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:
31148
diff
changeset
|
419 |
mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]]) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
420 |
also have "\<dots> = sqrt x" |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
421 |
unfolding inverse_eq_iff_eq[of _ "sqrt x", symmetric] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
422 |
sqrt_divide_self_eq[OF \<open>0 \<le> real_of_float x\<close>, symmetric] by auto |
60680 | 423 |
finally show ?thesis |
424 |
unfolding lb_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto |
|
425 |
qed |
|
426 |
have ub: "sqrt x \<le> ub_sqrt prec x" if "0 < x" for x :: float |
|
427 |
proof - |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
428 |
from that have "0 < real_of_float x" by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
429 |
hence "0 < sqrt x" by auto |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
430 |
hence "sqrt x < sqrt_iteration prec prec x" |
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:
31148
diff
changeset
|
431 |
using sqrt_iteration_bound by auto |
60680 | 432 |
then show ?thesis |
433 |
unfolding ub_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto |
|
434 |
qed |
|
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:
31148
diff
changeset
|
435 |
show ?thesis |
54269 | 436 |
using lb[of "-x"] ub[of "-x"] lb[of x] ub[of x] |
437 |
by (auto simp add: lb_sqrt.simps ub_sqrt.simps real_sqrt_minus) |
|
29805 | 438 |
qed |
439 |
||
60680 | 440 |
lemma bnds_sqrt: "\<forall>(x::real) lx ux. |
441 |
(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> sqrt x \<and> sqrt x \<le> u" |
|
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:
31148
diff
changeset
|
442 |
proof ((rule allI) +, rule impI, erule conjE, rule conjI) |
60680 | 443 |
fix x :: real |
444 |
fix lx ux |
|
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:
31148
diff
changeset
|
445 |
assume "(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux)" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
446 |
and x: "x \<in> {lx .. ux}" |
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:
31148
diff
changeset
|
447 |
hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto |
29805 | 448 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
449 |
have "sqrt lx \<le> sqrt x" using x by auto |
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:
31148
diff
changeset
|
450 |
from order_trans[OF _ this] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
451 |
show "l \<le> sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
452 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
453 |
have "sqrt x \<le> sqrt ux" using x by auto |
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:
31148
diff
changeset
|
454 |
from order_trans[OF this] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
455 |
show "sqrt x \<le> u" unfolding u using bnds_sqrt'[of ux prec] by auto |
29805 | 456 |
qed |
457 |
||
60680 | 458 |
|
29805 | 459 |
section "Arcus tangens and \<pi>" |
460 |
||
461 |
subsection "Compute arcus tangens series" |
|
462 |
||
60533 | 463 |
text \<open> |
29805 | 464 |
As first step we implement the computation of the arcus tangens series. This is only valid in the range |
465 |
@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens. |
|
60533 | 466 |
\<close> |
29805 | 467 |
|
468 |
fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
|
469 |
and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
470 |
"ub_arctan_horner prec 0 k x = 0" |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
471 |
| "ub_arctan_horner prec (Suc n) k x = float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
472 |
(rapprox_rat prec 1 k) (- float_round_down prec (x * (lb_arctan_horner prec n (k + 2) x)))" |
29805 | 473 |
| "lb_arctan_horner prec 0 k x = 0" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
474 |
| "lb_arctan_horner prec (Suc n) k x = float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
475 |
(lapprox_rat prec 1 k) (- float_round_up prec (x * (ub_arctan_horner prec n (k + 2) x)))" |
29805 | 476 |
|
49351 | 477 |
lemma arctan_0_1_bounds': |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
478 |
assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1" |
60680 | 479 |
and "even n" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
480 |
shows "arctan (sqrt y) \<in> |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
481 |
{(sqrt y * lb_arctan_horner prec n 1 y) .. (sqrt y * ub_arctan_horner prec (Suc n) 1 y)}" |
29805 | 482 |
proof - |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
483 |
let ?c = "\<lambda>i. (- 1) ^ i * (1 / (i * 2 + (1::nat)) * sqrt y ^ (i * 2 + 1))" |
54269 | 484 |
let ?S = "\<lambda>n. \<Sum> i=0..<n. ?c i" |
29805 | 485 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
486 |
have "0 \<le> sqrt y" using assms by auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
487 |
have "sqrt y \<le> 1" using assms by auto |
60533 | 488 |
from \<open>even n\<close> obtain m where "2 * m = n" by (blast elim: evenE) |
31809 | 489 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
490 |
have "arctan (sqrt y) \<in> { ?S n .. ?S (Suc n) }" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
491 |
proof (cases "sqrt y = 0") |
60680 | 492 |
case True |
493 |
then show ?thesis by simp |
|
494 |
next |
|
29805 | 495 |
case False |
60533 | 496 |
hence "0 < sqrt y" using \<open>0 \<le> sqrt y\<close> by auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
497 |
hence prem: "0 < 1 / (0 * 2 + (1::nat)) * sqrt y ^ (0 * 2 + 1)" by auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
498 |
|
60533 | 499 |
have "\<bar> sqrt y \<bar> \<le> 1" using \<open>0 \<le> sqrt y\<close> \<open>sqrt y \<le> 1\<close> by auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
500 |
from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this] |
60533 | 501 |
monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded \<open>2 * m = n\<close>] |
502 |
show ?thesis unfolding arctan_series[OF \<open>\<bar> sqrt y \<bar> \<le> 1\<close>] Suc_eq_plus1 atLeast0LessThan . |
|
60680 | 503 |
qed |
29805 | 504 |
note arctan_bounds = this[unfolded atLeastAtMost_iff] |
505 |
||
506 |
have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto |
|
507 |
||
31809 | 508 |
note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0 |
29805 | 509 |
and lb="\<lambda>n i k x. lb_arctan_horner prec n k x" |
31809 | 510 |
and ub="\<lambda>n i k x. ub_arctan_horner prec n k x", |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
511 |
OF \<open>0 \<le> real_of_float y\<close> F lb_arctan_horner.simps ub_arctan_horner.simps] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
512 |
|
60680 | 513 |
have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> arctan (sqrt y)" |
514 |
proof - |
|
515 |
have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> ?S n" |
|
60533 | 516 |
using bounds(1) \<open>0 \<le> sqrt y\<close> |
64267 | 517 |
apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric]) |
63170 | 518 |
apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult) |
519 |
apply (auto intro!: mult_left_mono) |
|
520 |
done |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
521 |
also have "\<dots> \<le> arctan (sqrt y)" using arctan_bounds .. |
60680 | 522 |
finally show ?thesis . |
523 |
qed |
|
29805 | 524 |
moreover |
60680 | 525 |
have "arctan (sqrt y) \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)" |
526 |
proof - |
|
527 |
have "arctan (sqrt y) \<le> ?S (Suc n)" using arctan_bounds .. |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
528 |
also have "\<dots> \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)" |
60533 | 529 |
using bounds(2)[of "Suc n"] \<open>0 \<le> sqrt y\<close> |
64267 | 530 |
apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric]) |
63170 | 531 |
apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult) |
532 |
apply (auto intro!: mult_left_mono) |
|
533 |
done |
|
60680 | 534 |
finally show ?thesis . |
535 |
qed |
|
29805 | 536 |
ultimately show ?thesis by auto |
537 |
qed |
|
538 |
||
60680 | 539 |
lemma arctan_0_1_bounds: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
540 |
assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
541 |
shows "arctan (sqrt y) \<in> |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
542 |
{(sqrt y * lb_arctan_horner prec (get_even n) 1 y) .. |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
543 |
(sqrt y * ub_arctan_horner prec (get_odd n) 1 y)}" |
54269 | 544 |
using |
545 |
arctan_0_1_bounds'[OF assms, of n prec] |
|
546 |
arctan_0_1_bounds'[OF assms, of "n + 1" prec] |
|
547 |
arctan_0_1_bounds'[OF assms, of "n - 1" prec] |
|
60680 | 548 |
by (auto simp: get_even_def get_odd_def odd_pos |
549 |
simp del: ub_arctan_horner.simps lb_arctan_horner.simps) |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
550 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
551 |
lemma arctan_lower_bound: |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
552 |
assumes "0 \<le> x" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
553 |
shows "x / (1 + x\<^sup>2) \<le> arctan x" (is "?l x \<le> _") |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
554 |
proof - |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
555 |
have "?l x - arctan x \<le> ?l 0 - arctan 0" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
556 |
using assms |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
557 |
by (intro DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. ?l x - arctan x"]) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
558 |
(auto intro!: derivative_eq_intros simp: add_nonneg_eq_0_iff field_simps) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
559 |
thus ?thesis by simp |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
560 |
qed |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
561 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
562 |
lemma arctan_divide_mono: "0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> arctan y / y \<le> arctan x / x" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
563 |
by (rule DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. arctan x / x"]) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
564 |
(auto intro!: derivative_eq_intros divide_nonpos_nonneg |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
565 |
simp: inverse_eq_divide arctan_lower_bound) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
566 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
567 |
lemma arctan_mult_mono: "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> x * arctan y \<le> y * arctan x" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
568 |
using arctan_divide_mono[of x y] by (cases "x = 0") (simp_all add: field_simps) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
569 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
570 |
lemma arctan_mult_le: |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
571 |
assumes "0 \<le> x" "x \<le> y" "y * z \<le> arctan y" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
572 |
shows "x * z \<le> arctan x" |
60680 | 573 |
proof (cases "x = 0") |
574 |
case True |
|
575 |
then show ?thesis by simp |
|
576 |
next |
|
577 |
case False |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
578 |
with assms have "z \<le> arctan y / y" by (simp add: field_simps) |
60533 | 579 |
also have "\<dots> \<le> arctan x / x" using assms \<open>x \<noteq> 0\<close> by (auto intro!: arctan_divide_mono) |
580 |
finally show ?thesis using assms \<open>x \<noteq> 0\<close> by (simp add: field_simps) |
|
60680 | 581 |
qed |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
582 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
583 |
lemma arctan_le_mult: |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
584 |
assumes "0 < x" "x \<le> y" "arctan x \<le> x * z" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
585 |
shows "arctan y \<le> y * z" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
586 |
proof - |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
587 |
from assms have "arctan y / y \<le> arctan x / x" by (auto intro!: arctan_divide_mono) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
588 |
also have "\<dots> \<le> z" using assms by (auto simp: field_simps) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
589 |
finally show ?thesis using assms by (simp add: field_simps) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
590 |
qed |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
591 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
592 |
lemma arctan_0_1_bounds_le: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
593 |
assumes "0 \<le> x" "x \<le> 1" "0 < real_of_float xl" "real_of_float xl \<le> x * x" "x * x \<le> real_of_float xu" "real_of_float xu \<le> 1" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
594 |
shows "arctan x \<in> |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
595 |
{x * lb_arctan_horner p1 (get_even n) 1 xu .. x * ub_arctan_horner p2 (get_odd n) 1 xl}" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
596 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
597 |
from assms have "real_of_float xl \<le> 1" "sqrt (real_of_float xl) \<le> x" "x \<le> sqrt (real_of_float xu)" "0 \<le> real_of_float xu" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
598 |
"0 \<le> real_of_float xl" "0 < sqrt (real_of_float xl)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
599 |
by (auto intro!: real_le_rsqrt real_le_lsqrt simp: power2_eq_square) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
600 |
from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xu\<close> \<open>real_of_float xu \<le> 1\<close>] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
601 |
have "sqrt (real_of_float xu) * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan (sqrt (real_of_float xu))" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
602 |
by simp |
60533 | 603 |
from arctan_mult_le[OF \<open>0 \<le> x\<close> \<open>x \<le> sqrt _\<close> this] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
604 |
have "x * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan x" . |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
605 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
606 |
from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xl\<close> \<open>real_of_float xl \<le> 1\<close>] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
607 |
have "arctan (sqrt (real_of_float xl)) \<le> sqrt (real_of_float xl) * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
608 |
by simp |
60533 | 609 |
from arctan_le_mult[OF \<open>0 < sqrt xl\<close> \<open>sqrt xl \<le> x\<close> this] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
610 |
have "arctan x \<le> x * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)" . |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
611 |
ultimately show ?thesis by simp |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
612 |
qed |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
613 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
614 |
lemma arctan_0_1_bounds_round: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
615 |
assumes "0 \<le> real_of_float x" "real_of_float x \<le> 1" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
616 |
shows "arctan x \<in> |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
617 |
{real_of_float x * lb_arctan_horner p1 (get_even n) 1 (float_round_up (Suc p2) (x * x)) .. |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
618 |
real_of_float x * ub_arctan_horner p3 (get_odd n) 1 (float_round_down (Suc p4) (x * x))}" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
619 |
using assms |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
620 |
apply (cases "x > 0") |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
621 |
apply (intro arctan_0_1_bounds_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
622 |
apply (auto simp: float_round_down.rep_eq float_round_up.rep_eq |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
623 |
intro!: truncate_up_le1 mult_le_one truncate_down_le truncate_up_le truncate_down_pos |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
624 |
mult_pos_pos) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
625 |
done |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
626 |
|
29805 | 627 |
|
628 |
subsection "Compute \<pi>" |
|
629 |
||
630 |
definition ub_pi :: "nat \<Rightarrow> float" where |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
631 |
"ub_pi prec = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
632 |
(let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
633 |
A = rapprox_rat prec 1 5 ; |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
634 |
B = lapprox_rat prec 1 239 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
635 |
in ((Float 1 2) * float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
636 |
((Float 1 2) * float_round_up prec (A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
637 |
(float_round_down (Suc prec) (A * A))))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
638 |
(- float_round_down prec (B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
639 |
(float_round_up (Suc prec) (B * B)))))))" |
29805 | 640 |
|
641 |
definition lb_pi :: "nat \<Rightarrow> float" where |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
642 |
"lb_pi prec = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
643 |
(let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
644 |
A = lapprox_rat prec 1 5 ; |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
645 |
B = rapprox_rat prec 1 239 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
646 |
in ((Float 1 2) * float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
647 |
((Float 1 2) * float_round_down prec (A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
648 |
(float_round_up (Suc prec) (A * A))))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
649 |
(- float_round_up prec (B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
650 |
(float_round_down (Suc prec) (B * B)))))))" |
29805 | 651 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
652 |
lemma pi_boundaries: "pi \<in> {(lb_pi n) .. (ub_pi n)}" |
29805 | 653 |
proof - |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
654 |
have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
655 |
unfolding machin[symmetric] by auto |
29805 | 656 |
|
60680 | 657 |
{ |
658 |
fix prec n :: nat |
|
659 |
fix k :: int |
|
660 |
assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto |
|
29805 | 661 |
let ?k = "rapprox_rat prec 1 k" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
662 |
let ?kl = "float_round_down (Suc prec) (?k * ?k)" |
60533 | 663 |
have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto |
664 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
665 |
have "0 \<le> real_of_float ?k" by (rule order_trans[OF _ rapprox_rat]) (auto simp add: \<open>0 \<le> k\<close>) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
666 |
have "real_of_float ?k \<le> 1" |
60533 | 667 |
by (auto simp add: \<open>0 < k\<close> \<open>1 \<le> k\<close> less_imp_le |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
668 |
intro!: mult_le_one order_trans[OF _ rapprox_rat] rapprox_rat_le1) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
669 |
have "1 / k \<le> ?k" using rapprox_rat[where x=1 and y=k] by auto |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
670 |
hence "arctan (1 / k) \<le> arctan ?k" by (rule arctan_monotone') |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
671 |
also have "\<dots> \<le> (?k * ub_arctan_horner prec (get_odd n) 1 ?kl)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
672 |
using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
673 |
by auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
674 |
finally have "arctan (1 / k) \<le> ?k * ub_arctan_horner prec (get_odd n) 1 ?kl" . |
29805 | 675 |
} note ub_arctan = this |
676 |
||
60680 | 677 |
{ |
678 |
fix prec n :: nat |
|
679 |
fix k :: int |
|
680 |
assume "1 < k" hence "0 \<le> k" and "0 < k" by auto |
|
29805 | 681 |
let ?k = "lapprox_rat prec 1 k" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
682 |
let ?ku = "float_round_up (Suc prec) (?k * ?k)" |
60533 | 683 |
have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto |
684 |
have "1 / k \<le> 1" using \<open>1 < k\<close> by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
685 |
have "0 \<le> real_of_float ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one \<open>0 \<le> k\<close>] |
60533 | 686 |
by (auto simp add: \<open>1 div k = 0\<close>) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
687 |
have "0 \<le> real_of_float (?k * ?k)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
688 |
have "real_of_float ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: \<open>1 / k \<le> 1\<close>) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
689 |
hence "real_of_float (?k * ?k) \<le> 1" using \<open>0 \<le> real_of_float ?k\<close> by (auto intro!: mult_le_one) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
690 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
691 |
have "?k \<le> 1 / k" using lapprox_rat[where x=1 and y=k] by auto |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
692 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
693 |
have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan ?k" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
694 |
using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
695 |
by auto |
60533 | 696 |
also have "\<dots> \<le> arctan (1 / k)" using \<open>?k \<le> 1 / k\<close> by (rule arctan_monotone') |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
697 |
finally have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan (1 / k)" . |
29805 | 698 |
} note lb_arctan = this |
699 |
||
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
700 |
have "pi \<le> ub_pi n " |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
701 |
unfolding ub_pi_def machin_pi Let_def times_float.rep_eq Float_num |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
702 |
using lb_arctan[of 239] ub_arctan[of 5] powr_realpow[of 2 2] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
703 |
by (intro mult_left_mono float_plus_up_le float_plus_down_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
704 |
(auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
705 |
moreover have "lb_pi n \<le> pi" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
706 |
unfolding lb_pi_def machin_pi Let_def times_float.rep_eq Float_num |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
707 |
using lb_arctan[of 5] ub_arctan[of 239] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
708 |
by (intro mult_left_mono float_plus_up_le float_plus_down_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
709 |
(auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
710 |
ultimately show ?thesis by auto |
29805 | 711 |
qed |
712 |
||
60680 | 713 |
|
29805 | 714 |
subsection "Compute arcus tangens in the entire domain" |
715 |
||
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:
31148
diff
changeset
|
716 |
function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
717 |
"lb_arctan prec x = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
718 |
(let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
719 |
ub_horner = \<lambda> x. float_round_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
720 |
(x * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
721 |
ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))); |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
722 |
lb_horner = \<lambda> x. float_round_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
723 |
(x * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
724 |
lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
725 |
in |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
726 |
if x < 0 then - ub_arctan prec (-x) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
727 |
else if x \<le> Float 1 (- 1) then lb_horner x |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
728 |
else if x \<le> Float 1 1 then |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
729 |
Float 1 1 * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
730 |
lb_horner |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
731 |
(float_divl prec x |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
732 |
(float_plus_up prec 1 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
733 |
(ub_sqrt prec (float_plus_up prec 1 (float_round_up prec (x * x)))))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
734 |
else let inv = float_divr prec 1 x in |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
735 |
if inv > 1 then 0 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
736 |
else float_plus_down prec (lb_pi prec * Float 1 (- 1)) ( - ub_horner inv))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
737 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
738 |
| "ub_arctan prec x = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
739 |
(let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
740 |
lb_horner = \<lambda> x. float_round_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
741 |
(x * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
742 |
lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))) ; |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
743 |
ub_horner = \<lambda> x. float_round_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
744 |
(x * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
745 |
ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
746 |
in if x < 0 then - lb_arctan prec (-x) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
747 |
else if x \<le> Float 1 (- 1) then ub_horner x |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
748 |
else if x \<le> Float 1 1 then |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
749 |
let y = float_divr prec x |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
750 |
(float_plus_down |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
751 |
(Suc prec) 1 (lb_sqrt prec (float_plus_down prec 1 (float_round_down prec (x * x))))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
752 |
in if y > 1 then ub_pi prec * Float 1 (- 1) else Float 1 1 * ub_horner y |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
753 |
else float_plus_up prec (ub_pi prec * Float 1 (- 1)) ( - lb_horner (float_divl prec 1 x)))" |
29805 | 754 |
by pat_completeness auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
755 |
termination |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
756 |
by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto) |
29805 | 757 |
|
758 |
declare ub_arctan_horner.simps[simp del] |
|
759 |
declare lb_arctan_horner.simps[simp del] |
|
760 |
||
60680 | 761 |
lemma lb_arctan_bound': |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
762 |
assumes "0 \<le> real_of_float x" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
763 |
shows "lb_arctan prec x \<le> arctan x" |
29805 | 764 |
proof - |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
765 |
have "\<not> x < 0" and "0 \<le> x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
766 |
using \<open>0 \<le> real_of_float x\<close> by (auto intro!: truncate_up_le ) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
767 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
768 |
let "?ub_horner x" = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
769 |
"x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
770 |
and "?lb_horner x" = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
771 |
"x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))" |
29805 | 772 |
|
773 |
show ?thesis |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
774 |
proof (cases "x \<le> Float 1 (- 1)") |
60680 | 775 |
case True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
776 |
hence "real_of_float x \<le> 1" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
777 |
from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
778 |
show ?thesis |
60533 | 779 |
unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True] using \<open>0 \<le> x\<close> |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
780 |
by (auto intro!: float_round_down_le) |
29805 | 781 |
next |
60680 | 782 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
783 |
hence "0 < real_of_float x" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
784 |
let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
785 |
let ?sxx = "float_plus_up prec 1 (float_round_up prec (x * x))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
786 |
let ?fR = "float_plus_up prec 1 (ub_sqrt prec ?sxx)" |
29805 | 787 |
let ?DIV = "float_divl prec x ?fR" |
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:
31148
diff
changeset
|
788 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
789 |
have divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
790 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
791 |
have "sqrt (1 + x*x) \<le> sqrt ?sxx" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
792 |
by (auto simp: float_plus_up.rep_eq plus_up_def float_round_up.rep_eq intro!: truncate_up_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
793 |
also have "\<dots> \<le> ub_sqrt prec ?sxx" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
794 |
using bnds_sqrt'[of ?sxx prec] by auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
795 |
finally |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
796 |
have "sqrt (1 + x*x) \<le> ub_sqrt prec ?sxx" . |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
797 |
hence "?R \<le> ?fR" by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
798 |
hence "0 < ?fR" and "0 < real_of_float ?fR" using \<open>0 < ?R\<close> by auto |
29805 | 799 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
800 |
have monotone: "?DIV \<le> x / ?R" |
29805 | 801 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
802 |
have "?DIV \<le> real_of_float x / ?fR" by (rule float_divl) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
803 |
also have "\<dots> \<le> x / ?R" by (rule divide_left_mono[OF \<open>?R \<le> ?fR\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 \<open>?R \<le> real_of_float ?fR\<close>] divisor_gt0]]) |
29805 | 804 |
finally show ?thesis . |
805 |
qed |
|
806 |
||
807 |
show ?thesis |
|
808 |
proof (cases "x \<le> Float 1 1") |
|
809 |
case True |
|
60680 | 810 |
have "x \<le> sqrt (1 + x * x)" |
811 |
using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto |
|
60533 | 812 |
also note \<open>\<dots> \<le> (ub_sqrt prec ?sxx)\<close> |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
813 |
finally have "real_of_float x \<le> ?fR" |
60680 | 814 |
by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
815 |
moreover have "?DIV \<le> real_of_float x / ?fR" |
60680 | 816 |
by (rule float_divl) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
817 |
ultimately have "real_of_float ?DIV \<le> 1" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
818 |
unfolding divide_le_eq_1_pos[OF \<open>0 < real_of_float ?fR\<close>, symmetric] by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
819 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
820 |
have "0 \<le> real_of_float ?DIV" |
60680 | 821 |
using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] \<open>0 < ?fR\<close> |
822 |
unfolding less_eq_float_def by auto |
|
60533 | 823 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
824 |
from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float (?DIV)\<close> \<open>real_of_float (?DIV) \<le> 1\<close>] |
60680 | 825 |
have "Float 1 1 * ?lb_horner ?DIV \<le> 2 * arctan ?DIV" |
826 |
by simp |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
827 |
also have "\<dots> \<le> 2 * arctan (x / ?R)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
828 |
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono arctan_monotone') |
60680 | 829 |
also have "2 * arctan (x / ?R) = arctan x" |
830 |
using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left . |
|
831 |
finally show ?thesis |
|
832 |
unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
833 |
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF True] |
|
834 |
by (auto simp: float_round_down.rep_eq |
|
835 |
intro!: order_trans[OF mult_left_mono[OF truncate_down]]) |
|
29805 | 836 |
next |
837 |
case False |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
838 |
hence "2 < real_of_float x" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
839 |
hence "1 \<le> real_of_float x" by auto |
29805 | 840 |
|
841 |
let "?invx" = "float_divr prec 1 x" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
842 |
have "0 \<le> arctan x" using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>] |
60680 | 843 |
using arctan_tan[of 0, unfolded tan_zero] by auto |
29805 | 844 |
|
845 |
show ?thesis |
|
846 |
proof (cases "1 < ?invx") |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
847 |
case True |
60680 | 848 |
show ?thesis |
849 |
unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
850 |
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF False] if_P[OF True] |
|
60533 | 851 |
using \<open>0 \<le> arctan x\<close> by auto |
29805 | 852 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
853 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
854 |
hence "real_of_float ?invx \<le> 1" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
855 |
have "0 \<le> real_of_float ?invx" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
856 |
by (rule order_trans[OF _ float_divr]) (auto simp add: \<open>0 \<le> real_of_float x\<close>) |
60680 | 857 |
|
858 |
have "1 / x \<noteq> 0" and "0 < 1 / x" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
859 |
using \<open>0 < real_of_float x\<close> by auto |
60680 | 860 |
|
861 |
have "arctan (1 / x) \<le> arctan ?invx" |
|
862 |
unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divr) |
|
863 |
also have "\<dots> \<le> ?ub_horner ?invx" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
864 |
using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
865 |
by (auto intro!: float_round_up_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
866 |
also note float_round_up |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
867 |
finally have "pi / 2 - float_round_up prec (?ub_horner ?invx) \<le> arctan x" |
60533 | 868 |
using \<open>0 \<le> arctan x\<close> arctan_inverse[OF \<open>1 / x \<noteq> 0\<close>] |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61610
diff
changeset
|
869 |
unfolding sgn_pos[OF \<open>0 < 1 / real_of_float x\<close>] le_diff_eq by auto |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
870 |
moreover |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
871 |
have "lb_pi prec * Float 1 (- 1) \<le> pi / 2" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
872 |
unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by simp |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
873 |
ultimately |
60680 | 874 |
show ?thesis |
875 |
unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
876 |
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x \<le> Float 1 1\<close>] if_not_P[OF False] |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
877 |
by (auto intro!: float_plus_down_le) |
29805 | 878 |
qed |
879 |
qed |
|
880 |
qed |
|
881 |
qed |
|
882 |
||
60680 | 883 |
lemma ub_arctan_bound': |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
884 |
assumes "0 \<le> real_of_float x" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
885 |
shows "arctan x \<le> ub_arctan prec x" |
29805 | 886 |
proof - |
60680 | 887 |
have "\<not> x < 0" and "0 \<le> x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
888 |
using \<open>0 \<le> real_of_float x\<close> by auto |
60680 | 889 |
|
890 |
let "?ub_horner x" = |
|
891 |
"float_round_up prec (x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))" |
|
892 |
let "?lb_horner x" = |
|
893 |
"float_round_down prec (x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))" |
|
29805 | 894 |
|
895 |
show ?thesis |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
896 |
proof (cases "x \<le> Float 1 (- 1)") |
60680 | 897 |
case True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
898 |
hence "real_of_float x \<le> 1" by auto |
60680 | 899 |
show ?thesis |
900 |
unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
901 |
using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
902 |
by (auto intro!: float_round_up_le) |
29805 | 903 |
next |
60680 | 904 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
905 |
hence "0 < real_of_float x" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
906 |
let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
907 |
let ?sxx = "float_plus_down prec 1 (float_round_down prec (x * x))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
908 |
let ?fR = "float_plus_down (Suc prec) 1 (lb_sqrt prec ?sxx)" |
29805 | 909 |
let ?DIV = "float_divr prec x ?fR" |
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:
31148
diff
changeset
|
910 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
911 |
have sqr_ge0: "0 \<le> 1 + real_of_float x * real_of_float x" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
912 |
using sum_power2_ge_zero[of 1 "real_of_float x", unfolded numeral_2_eq_2] by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
913 |
hence "0 \<le> real_of_float (1 + x*x)" by auto |
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:
31148
diff
changeset
|
914 |
|
29805 | 915 |
hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg) |
916 |
||
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
917 |
have "lb_sqrt prec ?sxx \<le> sqrt ?sxx" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
918 |
using bnds_sqrt'[of ?sxx] by auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
919 |
also have "\<dots> \<le> sqrt (1 + x*x)" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
920 |
by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq truncate_down_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
921 |
finally have "lb_sqrt prec ?sxx \<le> sqrt (1 + x*x)" . |
60680 | 922 |
hence "?fR \<le> ?R" |
923 |
by (auto simp: float_plus_down.rep_eq plus_down_def truncate_down_le) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
924 |
have "0 < real_of_float ?fR" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
925 |
by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
926 |
intro!: truncate_down_ge1 lb_sqrt_lower_bound order_less_le_trans[OF zero_less_one] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
927 |
truncate_down_nonneg add_nonneg_nonneg) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
928 |
have monotone: "x / ?R \<le> (float_divr prec x ?fR)" |
29805 | 929 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
930 |
from divide_left_mono[OF \<open>?fR \<le> ?R\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF divisor_gt0 \<open>0 < real_of_float ?fR\<close>]] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
931 |
have "x / ?R \<le> x / ?fR" . |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
932 |
also have "\<dots> \<le> ?DIV" by (rule float_divr) |
29805 | 933 |
finally show ?thesis . |
934 |
qed |
|
935 |
||
936 |
show ?thesis |
|
937 |
proof (cases "x \<le> Float 1 1") |
|
938 |
case True |
|
939 |
show ?thesis |
|
940 |
proof (cases "?DIV > 1") |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
941 |
case True |
60680 | 942 |
have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)" |
943 |
unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
944 |
from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le] |
60680 | 945 |
show ?thesis |
946 |
unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
947 |
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_P[OF True] . |
|
29805 | 948 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
949 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
950 |
hence "real_of_float ?DIV \<le> 1" by auto |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
951 |
|
60680 | 952 |
have "0 \<le> x / ?R" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
953 |
using \<open>0 \<le> real_of_float x\<close> \<open>0 < ?R\<close> unfolding zero_le_divide_iff by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
954 |
hence "0 \<le> real_of_float ?DIV" |
60680 | 955 |
using monotone by (rule order_trans) |
956 |
||
957 |
have "arctan x = 2 * arctan (x / ?R)" |
|
958 |
using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left . |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
959 |
also have "\<dots> \<le> 2 * arctan (?DIV)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
960 |
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono) |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
961 |
also have "\<dots> \<le> (Float 1 1 * ?ub_horner ?DIV)" unfolding Float_num |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
962 |
using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?DIV\<close> \<open>real_of_float ?DIV \<le> 1\<close>] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
963 |
by (auto intro!: float_round_up_le) |
60680 | 964 |
finally show ?thesis |
965 |
unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
966 |
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_not_P[OF False] . |
|
29805 | 967 |
qed |
968 |
next |
|
969 |
case False |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
970 |
hence "2 < real_of_float x" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
971 |
hence "1 \<le> real_of_float x" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
972 |
hence "0 < real_of_float x" by auto |
47600 | 973 |
hence "0 < x" by auto |
29805 | 974 |
|
975 |
let "?invx" = "float_divl prec 1 x" |
|
60680 | 976 |
have "0 \<le> arctan x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
977 |
using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>] and arctan_tan[of 0, unfolded tan_zero] by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
978 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
979 |
have "real_of_float ?invx \<le> 1" |
60680 | 980 |
unfolding less_float_def |
981 |
by (rule order_trans[OF float_divl]) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
982 |
(auto simp add: \<open>1 \<le> real_of_float x\<close> divide_le_eq_1_pos[OF \<open>0 < real_of_float x\<close>]) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
983 |
have "0 \<le> real_of_float ?invx" |
60680 | 984 |
using \<open>0 < x\<close> by (intro float_divl_lower_bound) auto |
985 |
||
986 |
have "1 / x \<noteq> 0" and "0 < 1 / x" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
987 |
using \<open>0 < real_of_float x\<close> by auto |
60680 | 988 |
|
989 |
have "(?lb_horner ?invx) \<le> arctan (?invx)" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
990 |
using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
991 |
by (auto intro!: float_round_down_le) |
60680 | 992 |
also have "\<dots> \<le> arctan (1 / x)" |
993 |
unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone') (rule float_divl) |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
994 |
finally have "arctan x \<le> pi / 2 - (?lb_horner ?invx)" |
60533 | 995 |
using \<open>0 \<le> arctan x\<close> arctan_inverse[OF \<open>1 / x \<noteq> 0\<close>] |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61610
diff
changeset
|
996 |
unfolding sgn_pos[OF \<open>0 < 1 / x\<close>] le_diff_eq by auto |
29805 | 997 |
moreover |
60680 | 998 |
have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)" |
999 |
unfolding Float_num times_divide_eq_right mult_1_right |
|
1000 |
using pi_boundaries by auto |
|
29805 | 1001 |
ultimately |
60680 | 1002 |
show ?thesis |
1003 |
unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
1004 |
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF False] |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1005 |
by (auto intro!: float_round_up_le float_plus_up_le) |
29805 | 1006 |
qed |
1007 |
qed |
|
1008 |
qed |
|
1009 |
||
60680 | 1010 |
lemma arctan_boundaries: "arctan x \<in> {(lb_arctan prec x) .. (ub_arctan prec x)}" |
29805 | 1011 |
proof (cases "0 \<le> x") |
60680 | 1012 |
case True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1013 |
hence "0 \<le> real_of_float x" by auto |
60680 | 1014 |
show ?thesis |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1015 |
using ub_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>] |
60680 | 1016 |
unfolding atLeastAtMost_iff by auto |
29805 | 1017 |
next |
60680 | 1018 |
case False |
29805 | 1019 |
let ?mx = "-x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1020 |
from False have "x < 0" and "0 \<le> real_of_float ?mx" |
60680 | 1021 |
by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1022 |
hence bounds: "lb_arctan prec ?mx \<le> arctan ?mx \<and> arctan ?mx \<le> ub_arctan prec ?mx" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1023 |
using ub_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] by auto |
60680 | 1024 |
show ?thesis |
1025 |
unfolding minus_float.rep_eq arctan_minus lb_arctan.simps[where x=x] |
|
1026 |
ub_arctan.simps[where x=x] Let_def if_P[OF \<open>x < 0\<close>] |
|
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1027 |
unfolding atLeastAtMost_iff using bounds[unfolded minus_float.rep_eq arctan_minus] |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1028 |
by (simp add: arctan_minus) |
29805 | 1029 |
qed |
1030 |
||
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1031 |
lemma bnds_arctan: "\<forall> (x::real) lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> arctan x \<and> arctan x \<le> u" |
29805 | 1032 |
proof (rule allI, rule allI, rule allI, rule impI) |
60680 | 1033 |
fix x :: real |
1034 |
fix lx ux |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1035 |
assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux}" |
60680 | 1036 |
hence l: "lb_arctan prec lx = l " |
1037 |
and u: "ub_arctan prec ux = u" |
|
1038 |
and x: "x \<in> {lx .. ux}" |
|
1039 |
by auto |
|
1040 |
show "l \<le> arctan x \<and> arctan x \<le> u" |
|
1041 |
proof |
|
1042 |
show "l \<le> arctan x" |
|
1043 |
proof - |
|
1044 |
from arctan_boundaries[of lx prec, unfolded l] |
|
1045 |
have "l \<le> arctan lx" by (auto simp del: lb_arctan.simps) |
|
1046 |
also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone') |
|
1047 |
finally show ?thesis . |
|
1048 |
qed |
|
1049 |
show "arctan x \<le> u" |
|
1050 |
proof - |
|
1051 |
have "arctan x \<le> arctan ux" using x by (auto intro: arctan_monotone') |
|
1052 |
also have "\<dots> \<le> u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps) |
|
1053 |
finally show ?thesis . |
|
1054 |
qed |
|
1055 |
qed |
|
29805 | 1056 |
qed |
1057 |
||
60680 | 1058 |
|
29805 | 1059 |
section "Sinus and Cosinus" |
1060 |
||
1061 |
subsection "Compute the cosinus and sinus series" |
|
1062 |
||
1063 |
fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
|
1064 |
and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
1065 |
"ub_sin_cos_aux prec 0 i k x = 0" |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1066 |
| "ub_sin_cos_aux prec (Suc n) i k x = float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1067 |
(rapprox_rat prec 1 k) (- |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1068 |
float_round_down prec (x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))" |
29805 | 1069 |
| "lb_sin_cos_aux prec 0 i k x = 0" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1070 |
| "lb_sin_cos_aux prec (Suc n) i k x = float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1071 |
(lapprox_rat prec 1 k) (- |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1072 |
float_round_up prec (x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))" |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1073 |
|
29805 | 1074 |
lemma cos_aux: |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1075 |
shows "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x ^(2 * i))" (is "?lb") |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1076 |
and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x^(2 * i)) \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub") |
29805 | 1077 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1078 |
have "0 \<le> real_of_float (x * x)" by auto |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1079 |
let "?f n" = "fact (2 * n) :: nat" |
60680 | 1080 |
have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)" for n |
1081 |
proof - |
|
1082 |
have "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto |
|
1083 |
then show ?thesis by auto |
|
1084 |
qed |
|
31809 | 1085 |
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0, |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1086 |
OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps] |
60680 | 1087 |
show ?lb and ?ub |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1088 |
by (auto simp add: power_mult power2_eq_square[of "real_of_float x"]) |
29805 | 1089 |
qed |
1090 |
||
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1091 |
lemma lb_sin_cos_aux_zero_le_one: "lb_sin_cos_aux prec n i j 0 \<le> 1" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1092 |
by (cases j n rule: nat.exhaust[case_product nat.exhaust]) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1093 |
(auto intro!: float_plus_down_le order_trans[OF lapprox_rat]) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1094 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1095 |
lemma one_le_ub_sin_cos_aux: "odd n \<Longrightarrow> 1 \<le> ub_sin_cos_aux prec n i (Suc 0) 0" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1096 |
by (cases n) (auto intro!: float_plus_up_le order_trans[OF _ rapprox_rat]) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1097 |
|
60680 | 1098 |
lemma cos_boundaries: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1099 |
assumes "0 \<le> real_of_float x" and "x \<le> pi / 2" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1100 |
shows "cos x \<in> {(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1101 |
proof (cases "real_of_float x = 0") |
60680 | 1102 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1103 |
hence "real_of_float x \<noteq> 0" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1104 |
hence "0 < x" and "0 < real_of_float x" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1105 |
using \<open>0 \<le> real_of_float x\<close> by auto |
60680 | 1106 |
have "0 < x * x" |
1107 |
using \<open>0 < x\<close> by simp |
|
1108 |
||
1109 |
have morph_to_if_power: "(\<Sum> i=0..<n. (-1::real) ^ i * (1/(fact (2 * i))) * x ^ (2 * i)) = |
|
1110 |
(\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * x ^ i)" |
|
1111 |
(is "?sum = ?ifsum") for x n |
|
29805 | 1112 |
proof - |
1113 |
have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto |
|
31809 | 1114 |
also have "\<dots> = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1115 |
(\<Sum> j = 0 ..< n. (- 1) ^ ((2 * j) div 2) / ((fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1116 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then (- 1) ^ (i div 2) / ((fact i)) * x ^ i else 0)" |
56195 | 1117 |
unfolding sum_split_even_odd atLeast0LessThan .. |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1118 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then (- 1) ^ (i div 2) / ((fact i)) else 0) * x ^ i)" |
64267 | 1119 |
by (rule sum.cong) auto |
60680 | 1120 |
finally show ?thesis . |
1121 |
qed |
|
29805 | 1122 |
|
1123 |
{ fix n :: nat assume "0 < n" |
|
1124 |
hence "0 < 2 * n" by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1125 |
obtain t where "0 < t" and "t < real_of_float x" and |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1126 |
cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * (real_of_float x) ^ i) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1127 |
+ (cos (t + 1/2 * (2 * n) * pi) / (fact (2*n))) * (real_of_float x)^(2*n)" |
29805 | 1128 |
(is "_ = ?SUM + ?rest / ?fact * ?pow") |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1129 |
using Maclaurin_cos_expansion2[OF \<open>0 < real_of_float x\<close> \<open>0 < 2 * n\<close>] |
56195 | 1130 |
unfolding cos_coeff_def atLeast0LessThan by auto |
29805 | 1131 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1132 |
have "cos t * (- 1) ^ n = cos t * cos (n * pi) + sin t * sin (n * pi)" by auto |
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
1133 |
also have "\<dots> = cos (t + n * pi)" by (simp add: cos_add) |
29805 | 1134 |
also have "\<dots> = ?rest" by auto |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1135 |
finally have "cos t * (- 1) ^ n = ?rest" . |
29805 | 1136 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1137 |
have "t \<le> pi / 2" using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto |
60533 | 1138 |
hence "0 \<le> cos t" using \<open>0 < t\<close> and cos_ge_zero by auto |
29805 | 1139 |
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto |
1140 |
||
1141 |
have "0 < ?fact" by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1142 |
have "0 < ?pow" using \<open>0 < real_of_float x\<close> by auto |
29805 | 1143 |
|
1144 |
{ |
|
1145 |
assume "even n" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1146 |
have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1147 |
unfolding morph_to_if_power[symmetric] using cos_aux by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1148 |
also have "\<dots> \<le> cos x" |
29805 | 1149 |
proof - |
60533 | 1150 |
from even[OF \<open>even n\<close>] \<open>0 < ?fact\<close> \<open>0 < ?pow\<close> |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1151 |
have "0 \<le> (?rest / ?fact) * ?pow" by simp |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1152 |
thus ?thesis unfolding cos_eq by auto |
29805 | 1153 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1154 |
finally have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos x" . |
29805 | 1155 |
} note lb = this |
1156 |
||
1157 |
{ |
|
1158 |
assume "odd n" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1159 |
have "cos x \<le> ?SUM" |
29805 | 1160 |
proof - |
60533 | 1161 |
from \<open>0 < ?fact\<close> and \<open>0 < ?pow\<close> and odd[OF \<open>odd n\<close>] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1162 |
have "0 \<le> (- ?rest) / ?fact * ?pow" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1163 |
by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1164 |
thus ?thesis unfolding cos_eq by auto |
29805 | 1165 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1166 |
also have "\<dots> \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1167 |
unfolding morph_to_if_power[symmetric] using cos_aux by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1168 |
finally have "cos x \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" . |
29805 | 1169 |
} note ub = this and lb |
1170 |
} note ub = this(1) and lb = this(2) |
|
1171 |
||
60680 | 1172 |
have "cos x \<le> (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" |
1173 |
using ub[OF odd_pos[OF get_odd] get_odd] . |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1174 |
moreover have "(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos x" |
29805 | 1175 |
proof (cases "0 < get_even n") |
60680 | 1176 |
case True |
1177 |
show ?thesis using lb[OF True get_even] . |
|
29805 | 1178 |
next |
1179 |
case False |
|
1180 |
hence "get_even n = 0" by auto |
|
60680 | 1181 |
have "- (pi / 2) \<le> x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1182 |
by (rule order_trans[OF _ \<open>0 < real_of_float x\<close>[THEN less_imp_le]]) auto |
60680 | 1183 |
with \<open>x \<le> pi / 2\<close> show ?thesis |
1184 |
unfolding \<open>get_even n = 0\<close> lb_sin_cos_aux.simps minus_float.rep_eq zero_float.rep_eq |
|
1185 |
using cos_ge_zero by auto |
|
29805 | 1186 |
qed |
1187 |
ultimately show ?thesis by auto |
|
1188 |
next |
|
1189 |
case True |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1190 |
hence "x = 0" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1191 |
by transfer |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1192 |
thus ?thesis |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1193 |
using lb_sin_cos_aux_zero_le_one one_le_ub_sin_cos_aux |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1194 |
by simp |
29805 | 1195 |
qed |
1196 |
||
60680 | 1197 |
lemma sin_aux: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1198 |
assumes "0 \<le> real_of_float x" |
60680 | 1199 |
shows "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> |
1200 |
(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb") |
|
1201 |
and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1)) \<le> |
|
1202 |
(x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub") |
|
29805 | 1203 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1204 |
have "0 \<le> real_of_float (x * x)" by auto |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1205 |
let "?f n" = "fact (2 * n + 1) :: nat" |
60680 | 1206 |
have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)" for n |
1207 |
proof - |
|
45129
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
44821
diff
changeset
|
1208 |
have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto |
60680 | 1209 |
show ?thesis |
1210 |
unfolding F by auto |
|
1211 |
qed |
|
29805 | 1212 |
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0, |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1213 |
OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1214 |
show "?lb" and "?ub" using \<open>0 \<le> real_of_float x\<close> |
64267 | 1215 |
apply (simp_all only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric]) |
63170 | 1216 |
apply (simp_all only: mult.commute[where 'a=real] of_nat_fact) |
1217 |
apply (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real_of_float x"]) |
|
1218 |
done |
|
29805 | 1219 |
qed |
1220 |
||
60680 | 1221 |
lemma sin_boundaries: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1222 |
assumes "0 \<le> real_of_float x" |
60680 | 1223 |
and "x \<le> pi / 2" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1224 |
shows "sin x \<in> {(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1225 |
proof (cases "real_of_float x = 0") |
60680 | 1226 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1227 |
hence "real_of_float x \<noteq> 0" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1228 |
hence "0 < x" and "0 < real_of_float x" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1229 |
using \<open>0 \<le> real_of_float x\<close> by auto |
60680 | 1230 |
have "0 < x * x" |
1231 |
using \<open>0 < x\<close> by simp |
|
1232 |
||
64267 | 1233 |
have sum_morph: "(\<Sum>j = 0 ..< n. (- 1) ^ (((2 * j + 1) - Suc 0) div 2) / ((fact (2 * j + 1))) * x ^(2 * j + 1)) = |
60680 | 1234 |
(\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * x ^ i)" |
1235 |
(is "?SUM = _") for x :: real and n |
|
1236 |
proof - |
|
1237 |
have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" |
|
1238 |
by auto |
|
1239 |
have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" |
|
1240 |
by auto |
|
1241 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i)) * x ^ i)" |
|
1242 |
unfolding sum_split_even_odd atLeast0LessThan .. |
|
1243 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i))) * x ^ i)" |
|
64267 | 1244 |
by (rule sum.cong) auto |
60680 | 1245 |
finally show ?thesis . |
1246 |
qed |
|
29805 | 1247 |
|
1248 |
{ fix n :: nat assume "0 < n" |
|
1249 |
hence "0 < 2 * n + 1" by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1250 |
obtain t where "0 < t" and "t < real_of_float x" and |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1251 |
sin_eq: "sin x = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1252 |
+ (sin (t + 1/2 * (2 * n + 1) * pi) / (fact (2*n + 1))) * (real_of_float x)^(2*n + 1)" |
29805 | 1253 |
(is "_ = ?SUM + ?rest / ?fact * ?pow") |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1254 |
using Maclaurin_sin_expansion3[OF \<open>0 < 2 * n + 1\<close> \<open>0 < real_of_float x\<close>] |
56195 | 1255 |
unfolding sin_coeff_def atLeast0LessThan by auto |
29805 | 1256 |
|
60680 | 1257 |
have "?rest = cos t * (- 1) ^ n" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1258 |
unfolding sin_add cos_add of_nat_add distrib_right distrib_left by auto |
29805 | 1259 |
moreover |
60680 | 1260 |
have "t \<le> pi / 2" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1261 |
using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto |
60680 | 1262 |
hence "0 \<le> cos t" |
1263 |
using \<open>0 < t\<close> and cos_ge_zero by auto |
|
1264 |
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest" |
|
1265 |
by auto |
|
1266 |
||
1267 |
have "0 < ?fact" |
|
1268 |
by (simp del: fact_Suc) |
|
1269 |
have "0 < ?pow" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1270 |
using \<open>0 < real_of_float x\<close> by (rule zero_less_power) |
29805 | 1271 |
|
1272 |
{ |
|
1273 |
assume "even n" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1274 |
have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1275 |
(\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)" |
64267 | 1276 |
using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding sum_morph[symmetric] by auto |
29805 | 1277 |
also have "\<dots> \<le> ?SUM" by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1278 |
also have "\<dots> \<le> sin x" |
29805 | 1279 |
proof - |
60533 | 1280 |
from even[OF \<open>even n\<close>] \<open>0 < ?fact\<close> \<open>0 < ?pow\<close> |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1281 |
have "0 \<le> (?rest / ?fact) * ?pow" by simp |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1282 |
thus ?thesis unfolding sin_eq by auto |
29805 | 1283 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1284 |
finally have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin x" . |
29805 | 1285 |
} note lb = this |
1286 |
||
1287 |
{ |
|
1288 |
assume "odd n" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1289 |
have "sin x \<le> ?SUM" |
29805 | 1290 |
proof - |
60533 | 1291 |
from \<open>0 < ?fact\<close> and \<open>0 < ?pow\<close> and odd[OF \<open>odd n\<close>] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1292 |
have "0 \<le> (- ?rest) / ?fact * ?pow" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1293 |
by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1294 |
thus ?thesis unfolding sin_eq by auto |
29805 | 1295 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1296 |
also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1297 |
by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1298 |
also have "\<dots> \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" |
64267 | 1299 |
using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding sum_morph[symmetric] by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1300 |
finally have "sin x \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" . |
29805 | 1301 |
} note ub = this and lb |
1302 |
} note ub = this(1) and lb = this(2) |
|
1303 |
||
60680 | 1304 |
have "sin x \<le> (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" |
1305 |
using ub[OF odd_pos[OF get_odd] get_odd] . |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1306 |
moreover have "(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin x" |
29805 | 1307 |
proof (cases "0 < get_even n") |
60680 | 1308 |
case True |
1309 |
show ?thesis |
|
1310 |
using lb[OF True get_even] . |
|
29805 | 1311 |
next |
1312 |
case False |
|
1313 |
hence "get_even n = 0" by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1314 |
with \<open>x \<le> pi / 2\<close> \<open>0 \<le> real_of_float x\<close> |
60680 | 1315 |
show ?thesis |
1316 |
unfolding \<open>get_even n = 0\<close> ub_sin_cos_aux.simps minus_float.rep_eq |
|
1317 |
using sin_ge_zero by auto |
|
29805 | 1318 |
qed |
1319 |
ultimately show ?thesis by auto |
|
1320 |
next |
|
1321 |
case True |
|
1322 |
show ?thesis |
|
1323 |
proof (cases "n = 0") |
|
31809 | 1324 |
case True |
60680 | 1325 |
thus ?thesis |
1326 |
unfolding \<open>n = 0\<close> get_even_def get_odd_def |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1327 |
using \<open>real_of_float x = 0\<close> lapprox_rat[where x="-1" and y=1] by auto |
29805 | 1328 |
next |
60680 | 1329 |
case False |
1330 |
with not0_implies_Suc obtain m where "n = Suc m" by blast |
|
1331 |
thus ?thesis |
|
1332 |
unfolding \<open>n = Suc m\<close> get_even_def get_odd_def |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1333 |
using \<open>real_of_float x = 0\<close> rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] |
60680 | 1334 |
by (cases "even (Suc m)") auto |
29805 | 1335 |
qed |
1336 |
qed |
|
1337 |
||
60680 | 1338 |
|
29805 | 1339 |
subsection "Compute the cosinus in the entire domain" |
1340 |
||
1341 |
definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
1342 |
"lb_cos prec x = (let |
|
1343 |
horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ; |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1344 |
half = \<lambda> x. if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1345 |
in if x < Float 1 (- 1) then horner x |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1346 |
else if x < 1 then half (horner (x * Float 1 (- 1))) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1347 |
else half (half (horner (x * Float 1 (- 2)))))" |
29805 | 1348 |
|
1349 |
definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
1350 |
"ub_cos prec x = (let |
|
1351 |
horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ; |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1352 |
half = \<lambda> x. float_plus_up prec (Float 1 1 * x * x) (- 1) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1353 |
in if x < Float 1 (- 1) then horner x |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1354 |
else if x < 1 then half (horner (x * Float 1 (- 1))) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1355 |
else half (half (horner (x * Float 1 (- 2)))))" |
29805 | 1356 |
|
60680 | 1357 |
lemma lb_cos: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1358 |
assumes "0 \<le> real_of_float x" and "x \<le> pi" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1359 |
shows "cos x \<in> {(lb_cos prec x) .. (ub_cos prec x)}" (is "?cos x \<in> {(?lb x) .. (?ub x) }") |
29805 | 1360 |
proof - |
60680 | 1361 |
have x_half[symmetric]: "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" for x :: real |
1362 |
proof - |
|
1363 |
have "cos x = cos (x / 2 + x / 2)" |
|
1364 |
by auto |
|
29805 | 1365 |
also have "\<dots> = cos (x / 2) * cos (x / 2) + sin (x / 2) * sin (x / 2) - sin (x / 2) * sin (x / 2) + cos (x / 2) * cos (x / 2) - 1" |
1366 |
unfolding cos_add by auto |
|
60680 | 1367 |
also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" |
1368 |
by algebra |
|
1369 |
finally show ?thesis . |
|
1370 |
qed |
|
29805 | 1371 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1372 |
have "\<not> x < 0" using \<open>0 \<le> real_of_float x\<close> by auto |
29805 | 1373 |
let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)" |
1374 |
let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)" |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1375 |
let "?ub_half x" = "float_plus_up prec (Float 1 1 * x * x) (- 1)" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1376 |
let "?lb_half x" = "if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)" |
29805 | 1377 |
|
1378 |
show ?thesis |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1379 |
proof (cases "x < Float 1 (- 1)") |
60680 | 1380 |
case True |
1381 |
hence "x \<le> pi / 2" |
|
1382 |
using pi_ge_two by auto |
|
1383 |
show ?thesis |
|
1384 |
unfolding lb_cos_def[where x=x] ub_cos_def[where x=x] |
|
1385 |
if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF \<open>x < Float 1 (- 1)\<close>] Let_def |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1386 |
using cos_boundaries[OF \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi / 2\<close>] . |
29805 | 1387 |
next |
1388 |
case False |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1389 |
{ fix y x :: float let ?x2 = "(x * Float 1 (- 1))" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1390 |
assume "y \<le> cos ?x2" and "-pi \<le> x" and "x \<le> pi" |
60680 | 1391 |
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" |
1392 |
using pi_ge_two unfolding Float_num by auto |
|
1393 |
hence "0 \<le> cos ?x2" |
|
1394 |
by (rule cos_ge_zero) |
|
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:
31148
diff
changeset
|
1395 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1396 |
have "(?lb_half y) \<le> cos x" |
29805 | 1397 |
proof (cases "y < 0") |
60680 | 1398 |
case True |
1399 |
show ?thesis |
|
1400 |
using cos_ge_minus_one unfolding if_P[OF True] by auto |
|
29805 | 1401 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1402 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1403 |
hence "0 \<le> real_of_float y" by auto |
60533 | 1404 |
from mult_mono[OF \<open>y \<le> cos ?x2\<close> \<open>y \<le> cos ?x2\<close> \<open>0 \<le> cos ?x2\<close> this] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1405 |
have "real_of_float y * real_of_float y \<le> cos ?x2 * cos ?x2" . |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1406 |
hence "2 * real_of_float y * real_of_float y \<le> 2 * cos ?x2 * cos ?x2" |
60680 | 1407 |
by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1408 |
hence "2 * real_of_float y * real_of_float y - 1 \<le> 2 * cos (x / 2) * cos (x / 2) - 1" |
60680 | 1409 |
unfolding Float_num by auto |
1410 |
thus ?thesis |
|
1411 |
unfolding if_not_P[OF False] x_half Float_num |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1412 |
by (auto intro!: float_plus_down_le) |
29805 | 1413 |
qed |
1414 |
} note lb_half = this |
|
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:
31148
diff
changeset
|
1415 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1416 |
{ fix y x :: float let ?x2 = "(x * Float 1 (- 1))" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1417 |
assume ub: "cos ?x2 \<le> y" and "- pi \<le> x" and "x \<le> pi" |
60680 | 1418 |
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" |
1419 |
using pi_ge_two unfolding Float_num by auto |
|
29805 | 1420 |
hence "0 \<le> cos ?x2" by (rule cos_ge_zero) |
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:
31148
diff
changeset
|
1421 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1422 |
have "cos x \<le> (?ub_half y)" |
29805 | 1423 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1424 |
have "0 \<le> real_of_float y" |
60680 | 1425 |
using \<open>0 \<le> cos ?x2\<close> ub by (rule order_trans) |
60533 | 1426 |
from mult_mono[OF ub ub this \<open>0 \<le> cos ?x2\<close>] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1427 |
have "cos ?x2 * cos ?x2 \<le> real_of_float y * real_of_float y" . |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1428 |
hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real_of_float y * real_of_float y" |
60680 | 1429 |
by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1430 |
hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real_of_float y * real_of_float y - 1" |
60680 | 1431 |
unfolding Float_num by auto |
1432 |
thus ?thesis |
|
1433 |
unfolding x_half Float_num |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1434 |
by (auto intro!: float_plus_up_le) |
29805 | 1435 |
qed |
1436 |
} note ub_half = this |
|
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:
31148
diff
changeset
|
1437 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1438 |
let ?x2 = "x * Float 1 (- 1)" |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1439 |
let ?x4 = "x * Float 1 (- 1) * Float 1 (- 1)" |
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:
31148
diff
changeset
|
1440 |
|
60680 | 1441 |
have "-pi \<le> x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1442 |
using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] \<open>0 \<le> real_of_float x\<close> |
60680 | 1443 |
by (rule order_trans) |
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:
31148
diff
changeset
|
1444 |
|
29805 | 1445 |
show ?thesis |
1446 |
proof (cases "x < 1") |
|
60680 | 1447 |
case True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1448 |
hence "real_of_float x \<le> 1" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1449 |
have "0 \<le> real_of_float ?x2" and "?x2 \<le> pi / 2" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1450 |
using pi_ge_two \<open>0 \<le> real_of_float x\<close> using assms by auto |
29805 | 1451 |
from cos_boundaries[OF this] |
60680 | 1452 |
have lb: "(?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> (?ub_horner ?x2)" |
1453 |
by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1454 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1455 |
have "(?lb x) \<le> ?cos x" |
29805 | 1456 |
proof - |
60533 | 1457 |
from lb_half[OF lb \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>] |
60680 | 1458 |
show ?thesis |
1459 |
unfolding lb_cos_def[where x=x] Let_def |
|
1460 |
using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto |
|
29805 | 1461 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1462 |
moreover have "?cos x \<le> (?ub x)" |
29805 | 1463 |
proof - |
60533 | 1464 |
from ub_half[OF ub \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>] |
60680 | 1465 |
show ?thesis |
1466 |
unfolding ub_cos_def[where x=x] Let_def |
|
1467 |
using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto |
|
29805 | 1468 |
qed |
1469 |
ultimately show ?thesis by auto |
|
1470 |
next |
|
1471 |
case False |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1472 |
have "0 \<le> real_of_float ?x4" and "?x4 \<le> pi / 2" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1473 |
using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> unfolding Float_num by auto |
29805 | 1474 |
from cos_boundaries[OF this] |
60680 | 1475 |
have lb: "(?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> (?ub_horner ?x4)" |
1476 |
by auto |
|
1477 |
||
1478 |
have eq_4: "?x2 * Float 1 (- 1) = x * Float 1 (- 2)" |
|
1479 |
by transfer simp |
|
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:
31148
diff
changeset
|
1480 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1481 |
have "(?lb x) \<le> ?cos x" |
29805 | 1482 |
proof - |
60680 | 1483 |
have "-pi \<le> ?x2" and "?x2 \<le> pi" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1484 |
using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> by auto |
60533 | 1485 |
from lb_half[OF lb_half[OF lb this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4] |
60680 | 1486 |
show ?thesis |
1487 |
unfolding lb_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>] |
|
1488 |
if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def . |
|
29805 | 1489 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1490 |
moreover have "?cos x \<le> (?ub x)" |
29805 | 1491 |
proof - |
60680 | 1492 |
have "-pi \<le> ?x2" and "?x2 \<le> pi" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1493 |
using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open> x \<le> pi\<close> by auto |
60533 | 1494 |
from ub_half[OF ub_half[OF ub this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4] |
60680 | 1495 |
show ?thesis |
1496 |
unfolding ub_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>] |
|
1497 |
if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def . |
|
29805 | 1498 |
qed |
1499 |
ultimately show ?thesis by auto |
|
1500 |
qed |
|
1501 |
qed |
|
1502 |
qed |
|
1503 |
||
60680 | 1504 |
lemma lb_cos_minus: |
1505 |
assumes "-pi \<le> x" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1506 |
and "real_of_float x \<le> 0" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1507 |
shows "cos (real_of_float(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}" |
29805 | 1508 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1509 |
have "0 \<le> real_of_float (-x)" and "(-x) \<le> pi" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1510 |
using \<open>-pi \<le> x\<close> \<open>real_of_float x \<le> 0\<close> by auto |
29805 | 1511 |
from lb_cos[OF this] show ?thesis . |
1512 |
qed |
|
1513 |
||
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:
31148
diff
changeset
|
1514 |
definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1515 |
"bnds_cos prec lx ux = (let |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1516 |
lpi = float_round_down prec (lb_pi prec) ; |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1517 |
upi = float_round_up prec (ub_pi prec) ; |
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:
31148
diff
changeset
|
1518 |
k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)) ; |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1519 |
lx = float_plus_down prec lx (- k * 2 * (if k < 0 then lpi else upi)) ; |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1520 |
ux = float_plus_up prec ux (- k * 2 * (if k < 0 then upi else lpi)) |
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:
31148
diff
changeset
|
1521 |
in if - lpi \<le> lx \<and> ux \<le> 0 then (lb_cos prec (-lx), ub_cos prec (-ux)) |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1522 |
else if 0 \<le> lx \<and> ux \<le> lpi then (lb_cos prec ux, ub_cos prec lx) |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1523 |
else if - lpi \<le> lx \<and> ux \<le> lpi then (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1524 |
else if 0 \<le> lx \<and> ux \<le> 2 * lpi then (Float (- 1) 0, max (ub_cos prec lx) (ub_cos prec (- (ux - 2 * lpi)))) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1525 |
else if -2 * lpi \<le> lx \<and> ux \<le> 0 then (Float (- 1) 0, max (ub_cos prec (lx + 2 * lpi)) (ub_cos prec (-ux))) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1526 |
else (Float (- 1) 0, Float 1 0))" |
29805 | 1527 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1528 |
lemma floor_int: obtains k :: int where "real_of_int k = (floor_fl f)" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1529 |
by (simp add: floor_fl_def) |
29805 | 1530 |
|
60680 | 1531 |
lemma cos_periodic_nat[simp]: |
1532 |
fixes n :: nat |
|
1533 |
shows "cos (x + n * (2 * pi)) = cos x" |
|
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:
31148
diff
changeset
|
1534 |
proof (induct n arbitrary: x) |
60680 | 1535 |
case 0 |
1536 |
then show ?case by simp |
|
1537 |
next |
|
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:
31148
diff
changeset
|
1538 |
case (Suc n) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1539 |
have split_pi_off: "x + (Suc n) * (2 * pi) = (x + n * (2 * pi)) + 2 * pi" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1540 |
unfolding Suc_eq_plus1 of_nat_add of_int_1 distrib_right by auto |
60680 | 1541 |
show ?case |
1542 |
unfolding split_pi_off using Suc by auto |
|
1543 |
qed |
|
1544 |
||
1545 |
lemma cos_periodic_int[simp]: |
|
1546 |
fixes i :: int |
|
1547 |
shows "cos (x + i * (2 * pi)) = cos x" |
|
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:
31148
diff
changeset
|
1548 |
proof (cases "0 \<le> i") |
60680 | 1549 |
case True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1550 |
hence i_nat: "real_of_int i = nat i" by auto |
60680 | 1551 |
show ?thesis |
1552 |
unfolding i_nat by auto |
|
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:
31148
diff
changeset
|
1553 |
next |
60680 | 1554 |
case False |
1555 |
hence i_nat: "i = - real (nat (-i))" by auto |
|
1556 |
have "cos x = cos (x + i * (2 * pi) - i * (2 * pi))" |
|
1557 |
by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1558 |
also have "\<dots> = cos (x + i * (2 * pi))" |
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:
31148
diff
changeset
|
1559 |
unfolding i_nat mult_minus_left diff_minus_eq_add by (rule cos_periodic_nat) |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1560 |
finally show ?thesis by auto |
29805 | 1561 |
qed |
1562 |
||
60680 | 1563 |
lemma bnds_cos: "\<forall>(x::real) lx ux. (l, u) = |
1564 |
bnds_cos prec lx ux \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> cos x \<and> cos x \<le> u" |
|
1565 |
proof (rule allI | rule impI | erule conjE)+ |
|
1566 |
fix x :: real |
|
1567 |
fix lx ux |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1568 |
assume bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {lx .. ux}" |
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:
31148
diff
changeset
|
1569 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1570 |
let ?lpi = "float_round_down prec (lb_pi prec)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1571 |
let ?upi = "float_round_up prec (ub_pi prec)" |
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:
31148
diff
changeset
|
1572 |
let ?k = "floor_fl (float_divr prec (lx + ?lpi) (2 * ?lpi))" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1573 |
let ?lx2 = "(- ?k * 2 * (if ?k < 0 then ?lpi else ?upi))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1574 |
let ?ux2 = "(- ?k * 2 * (if ?k < 0 then ?upi else ?lpi))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1575 |
let ?lx = "float_plus_down prec lx ?lx2" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1576 |
let ?ux = "float_plus_up prec ux ?ux2" |
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:
31148
diff
changeset
|
1577 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1578 |
obtain k :: int where k: "k = real_of_float ?k" |
60680 | 1579 |
by (rule floor_int) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1580 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1581 |
have upi: "pi \<le> ?upi" and lpi: "?lpi \<le> pi" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1582 |
using float_round_up[of "ub_pi prec" prec] pi_boundaries[of prec] |
60680 | 1583 |
float_round_down[of prec "lb_pi prec"] |
1584 |
by auto |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1585 |
hence "lx + ?lx2 \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ux + ?ux2" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1586 |
using x |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1587 |
by (cases "k = 0") |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1588 |
(auto intro!: add_mono |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1589 |
simp add: k [symmetric] uminus_add_conv_diff [symmetric] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1590 |
simp del: float_of_numeral uminus_add_conv_diff) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1591 |
hence "?lx \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ?ux" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1592 |
by (auto intro!: float_plus_down_le float_plus_up_le) |
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:
31148
diff
changeset
|
1593 |
note lx = this[THEN conjunct1] and ux = this[THEN conjunct2] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1594 |
hence lx_less_ux: "?lx \<le> real_of_float ?ux" by (rule order_trans) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1595 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1596 |
{ assume "- ?lpi \<le> ?lx" and x_le_0: "x - k * (2 * pi) \<le> 0" |
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:
31148
diff
changeset
|
1597 |
with lpi[THEN le_imp_neg_le] lx |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1598 |
have pi_lx: "- pi \<le> ?lx" and lx_0: "real_of_float ?lx \<le> 0" |
47600 | 1599 |
by simp_all |
29805 | 1600 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1601 |
have "(lb_cos prec (- ?lx)) \<le> cos (real_of_float (- ?lx))" |
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:
31148
diff
changeset
|
1602 |
using lb_cos_minus[OF pi_lx lx_0] by simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1603 |
also have "\<dots> \<le> cos (x + (-k) * (2 * pi))" |
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:
31148
diff
changeset
|
1604 |
using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1605 |
by (simp only: uminus_float.rep_eq of_int_minus |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
1606 |
cos_minus mult_minus_left) simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1607 |
finally have "(lb_cos prec (- ?lx)) \<le> cos x" |
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:
31148
diff
changeset
|
1608 |
unfolding cos_periodic_int . } |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1609 |
note negative_lx = this |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1610 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1611 |
{ assume "0 \<le> ?lx" and pi_x: "x - k * (2 * pi) \<le> pi" |
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:
31148
diff
changeset
|
1612 |
with lx |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1613 |
have pi_lx: "?lx \<le> pi" and lx_0: "0 \<le> real_of_float ?lx" |
47600 | 1614 |
by auto |
29805 | 1615 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1616 |
have "cos (x + (-k) * (2 * pi)) \<le> cos ?lx" |
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
1617 |
using cos_monotone_0_pi_le[OF lx_0 lx pi_x] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1618 |
by (simp only: of_int_minus |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
1619 |
cos_minus mult_minus_left) simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1620 |
also have "\<dots> \<le> (ub_cos prec ?lx)" |
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:
31148
diff
changeset
|
1621 |
using lb_cos[OF lx_0 pi_lx] by simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1622 |
finally have "cos x \<le> (ub_cos prec ?lx)" |
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:
31148
diff
changeset
|
1623 |
unfolding cos_periodic_int . } |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1624 |
note positive_lx = this |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1625 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1626 |
{ assume pi_x: "- pi \<le> x - k * (2 * pi)" and "?ux \<le> 0" |
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:
31148
diff
changeset
|
1627 |
with ux |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1628 |
have pi_ux: "- pi \<le> ?ux" and ux_0: "real_of_float ?ux \<le> 0" |
47600 | 1629 |
by simp_all |
29805 | 1630 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1631 |
have "cos (x + (-k) * (2 * pi)) \<le> cos (real_of_float (- ?ux))" |
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:
31148
diff
changeset
|
1632 |
using cos_monotone_minus_pi_0'[OF pi_x ux ux_0] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1633 |
by (simp only: uminus_float.rep_eq of_int_minus |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
1634 |
cos_minus mult_minus_left) simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1635 |
also have "\<dots> \<le> (ub_cos prec (- ?ux))" |
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:
31148
diff
changeset
|
1636 |
using lb_cos_minus[OF pi_ux ux_0, of prec] by simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1637 |
finally have "cos x \<le> (ub_cos prec (- ?ux))" |
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:
31148
diff
changeset
|
1638 |
unfolding cos_periodic_int . } |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1639 |
note negative_ux = this |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1640 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1641 |
{ assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - k * (2 * pi)" |
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:
31148
diff
changeset
|
1642 |
with lpi ux |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1643 |
have pi_ux: "?ux \<le> pi" and ux_0: "0 \<le> real_of_float ?ux" |
47600 | 1644 |
by simp_all |
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:
31148
diff
changeset
|
1645 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1646 |
have "(lb_cos prec ?ux) \<le> cos ?ux" |
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:
31148
diff
changeset
|
1647 |
using lb_cos[OF ux_0 pi_ux] by simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1648 |
also have "\<dots> \<le> cos (x + (-k) * (2 * pi))" |
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
1649 |
using cos_monotone_0_pi_le[OF x_ge_0 ux pi_ux] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1650 |
by (simp only: of_int_minus |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
1651 |
cos_minus mult_minus_left) simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1652 |
finally have "(lb_cos prec ?ux) \<le> cos x" |
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:
31148
diff
changeset
|
1653 |
unfolding cos_periodic_int . } |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1654 |
note positive_ux = this |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1655 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1656 |
show "l \<le> cos x \<and> cos x \<le> u" |
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:
31148
diff
changeset
|
1657 |
proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0") |
60680 | 1658 |
case True |
1659 |
with bnds have l: "l = lb_cos prec (-?lx)" and u: "u = ub_cos prec (-?ux)" |
|
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:
31148
diff
changeset
|
1660 |
by (auto simp add: bnds_cos_def Let_def) |
f7d2aa438bee
Approximation: Implemented argument reduction for cosine. Sinus is now implemented in terms of cosine. Sqrt computes on the entire real numbers
hoelzl
parents:
31148
diff
changeset
|
1661 |
from True lpi[THEN le_imp_neg_le] lx ux |
60680 | 1662 |
have "- pi \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> 0" |
47600 | 1663 |
by auto |
60680 | 1664 |
with True negative_ux negative_lx show ?thesis |
1665 |
unfolding l u by simp |
|
1666 |
next |
|
1667 |
case 1: False |
|
1668 |
show ?thesis |
|
1669 |
proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi") |
|
1670 |
case True with bnds 1 |
|
1671 |
have l: "l = lb_cos prec ?ux" |
|
1672 |
and u: "u = ub_cos prec ?lx" |
|
1673 |
by (auto simp add: bnds_cos_def Let_def) |
|
1674 |
from True lpi lx ux |
|
1675 |
have "0 \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> pi" |
|
1676 |
by auto |
|
1677 |
with True positive_ux positive_lx show ?thesis |
|
1678 |
unfolding l u by simp |
|
29805 | 1679 |
next |
60680 | 1680 |
case 2: False |
1681 |
show ?thesis |
|
1682 |
proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi") |
|
1683 |
case Cond: True |
|
1684 |
with bnds 1 2 have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)" |
|
1685 |
and u: "u = Float 1 0" |
|
1686 |
by (auto simp add: bnds_cos_def Let_def) |
|
1687 |
show ?thesis |
|
1688 |
unfolding u l using negative_lx positive_ux Cond |
|
1689 |
by (cases "x - k * (2 * pi) < 0") (auto simp add: real_of_float_min) |
|
1690 |
next |
|
1691 |
case 3: False |
|
1692 |
show ?thesis |
|
1693 |
proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi") |
|
1694 |
case Cond: True |
|
1695 |
with bnds 1 2 3 |
|
1696 |
have l: "l = Float (- 1) 0" |
|
1697 |
and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))" |
|
1698 |
by (auto simp add: bnds_cos_def Let_def) |
|
1699 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1700 |
have "cos x \<le> real_of_float u" |
60680 | 1701 |
proof (cases "x - k * (2 * pi) < pi") |
1702 |
case True |
|
1703 |
hence "x - k * (2 * pi) \<le> pi" by simp |
|
1704 |
from positive_lx[OF Cond[THEN conjunct1] this] show ?thesis |
|
1705 |
unfolding u by (simp add: real_of_float_max) |
|
1706 |
next |
|
1707 |
case False |
|
1708 |
hence "pi \<le> x - k * (2 * pi)" by simp |
|
1709 |
hence pi_x: "- pi \<le> x - k * (2 * pi) - 2 * pi" by simp |
|
1710 |
||
1711 |
have "?ux \<le> 2 * pi" |
|
1712 |
using Cond lpi by auto |
|
1713 |
hence "x - k * (2 * pi) - 2 * pi \<le> 0" |
|
1714 |
using ux by simp |
|
1715 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1716 |
have ux_0: "real_of_float (?ux - 2 * ?lpi) \<le> 0" |
60680 | 1717 |
using Cond by auto |
1718 |
||
1719 |
from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto |
|
1720 |
hence "- ?lpi \<le> ?ux - 2 * ?lpi" by auto |
|
1721 |
hence pi_ux: "- pi \<le> (?ux - 2 * ?lpi)" |
|
1722 |
using lpi[THEN le_imp_neg_le] by auto |
|
1723 |
||
1724 |
have x_le_ux: "x - k * (2 * pi) - 2 * pi \<le> (?ux - 2 * ?lpi)" |
|
1725 |
using ux lpi by auto |
|
1726 |
have "cos x = cos (x + (-k) * (2 * pi) + (-1::int) * (2 * pi))" |
|
1727 |
unfolding cos_periodic_int .. |
|
1728 |
also have "\<dots> \<le> cos ((?ux - 2 * ?lpi))" |
|
1729 |
using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1730 |
by (simp only: minus_float.rep_eq of_int_minus of_int_1 |
60680 | 1731 |
mult_minus_left mult_1_left) simp |
1732 |
also have "\<dots> = cos ((- (?ux - 2 * ?lpi)))" |
|
1733 |
unfolding uminus_float.rep_eq cos_minus .. |
|
1734 |
also have "\<dots> \<le> (ub_cos prec (- (?ux - 2 * ?lpi)))" |
|
1735 |
using lb_cos_minus[OF pi_ux ux_0] by simp |
|
1736 |
finally show ?thesis unfolding u by (simp add: real_of_float_max) |
|
1737 |
qed |
|
1738 |
thus ?thesis unfolding l by auto |
|
1739 |
next |
|
1740 |
case 4: False |
|
1741 |
show ?thesis |
|
1742 |
proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0") |
|
1743 |
case Cond: True |
|
1744 |
with bnds 1 2 3 4 have l: "l = Float (- 1) 0" |
|
1745 |
and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))" |
|
1746 |
by (auto simp add: bnds_cos_def Let_def) |
|
1747 |
||
1748 |
have "cos x \<le> u" |
|
1749 |
proof (cases "-pi < x - k * (2 * pi)") |
|
1750 |
case True |
|
1751 |
hence "-pi \<le> x - k * (2 * pi)" by simp |
|
1752 |
from negative_ux[OF this Cond[THEN conjunct2]] show ?thesis |
|
1753 |
unfolding u by (simp add: real_of_float_max) |
|
1754 |
next |
|
1755 |
case False |
|
1756 |
hence "x - k * (2 * pi) \<le> -pi" by simp |
|
1757 |
hence pi_x: "x - k * (2 * pi) + 2 * pi \<le> pi" by simp |
|
1758 |
||
1759 |
have "-2 * pi \<le> ?lx" using Cond lpi by auto |
|
1760 |
||
1761 |
hence "0 \<le> x - k * (2 * pi) + 2 * pi" using lx by simp |
|
1762 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1763 |
have lx_0: "0 \<le> real_of_float (?lx + 2 * ?lpi)" |
60680 | 1764 |
using Cond lpi by auto |
1765 |
||
1766 |
from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto |
|
1767 |
hence "?lx + 2 * ?lpi \<le> ?lpi" by auto |
|
1768 |
hence pi_lx: "(?lx + 2 * ?lpi) \<le> pi" |
|
1769 |
using lpi[THEN le_imp_neg_le] by auto |
|
1770 |
||
1771 |
have lx_le_x: "(?lx + 2 * ?lpi) \<le> x - k * (2 * pi) + 2 * pi" |
|
1772 |
using lx lpi by auto |
|
1773 |
||
1774 |
have "cos x = cos (x + (-k) * (2 * pi) + (1 :: int) * (2 * pi))" |
|
1775 |
unfolding cos_periodic_int .. |
|
1776 |
also have "\<dots> \<le> cos ((?lx + 2 * ?lpi))" |
|
1777 |
using cos_monotone_0_pi_le[OF lx_0 lx_le_x pi_x] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1778 |
by (simp only: minus_float.rep_eq of_int_minus of_int_1 |
60680 | 1779 |
mult_minus_left mult_1_left) simp |
1780 |
also have "\<dots> \<le> (ub_cos prec (?lx + 2 * ?lpi))" |
|
1781 |
using lb_cos[OF lx_0 pi_lx] by simp |
|
1782 |
finally show ?thesis unfolding u by (simp add: real_of_float_max) |
|
1783 |
qed |
|
1784 |
thus ?thesis unfolding l by auto |
|
1785 |
next |
|
1786 |
case False |
|
1787 |
with bnds 1 2 3 4 show ?thesis |
|
1788 |
by (auto simp add: bnds_cos_def Let_def) |
|
1789 |
qed |
|
1790 |
qed |
|
1791 |
qed |
|
29805 | 1792 |
qed |
60680 | 1793 |
qed |
29805 | 1794 |
qed |
1795 |
||
60680 | 1796 |
|
29805 | 1797 |
section "Exponential function" |
1798 |
||
1799 |
subsection "Compute the series of the exponential function" |
|
1800 |
||
60680 | 1801 |
fun ub_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
1802 |
and lb_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
|
1803 |
where |
|
29805 | 1804 |
"ub_exp_horner prec 0 i k x = 0" | |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1805 |
"ub_exp_horner prec (Suc n) i k x = float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1806 |
(rapprox_rat prec 1 (int k)) (float_round_up prec (x * lb_exp_horner prec n (i + 1) (k * i) x))" | |
29805 | 1807 |
"lb_exp_horner prec 0 i k x = 0" | |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1808 |
"lb_exp_horner prec (Suc n) i k x = float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1809 |
(lapprox_rat prec 1 (int k)) (float_round_down prec (x * ub_exp_horner prec n (i + 1) (k * i) x))" |
29805 | 1810 |
|
60680 | 1811 |
lemma bnds_exp_horner: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1812 |
assumes "real_of_float x \<le> 0" |
60680 | 1813 |
shows "exp x \<in> {lb_exp_horner prec (get_even n) 1 1 x .. ub_exp_horner prec (get_odd n) 1 1 x}" |
29805 | 1814 |
proof - |
60680 | 1815 |
have f_eq: "fact (Suc n) = fact n * ((\<lambda>i::nat. i + 1) ^^ n) 1" for n |
1816 |
proof - |
|
1817 |
have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m" |
|
1818 |
by (induct n) auto |
|
1819 |
show ?thesis |
|
1820 |
unfolding F by auto |
|
1821 |
qed |
|
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:
31148
diff
changeset
|
1822 |
|
29805 | 1823 |
note bounds = horner_bounds_nonpos[where f="fact" and lb="lb_exp_horner prec" and ub="ub_exp_horner prec" and j'=0 and s=1, |
1824 |
OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps] |
|
1825 |
||
60680 | 1826 |
have "lb_exp_horner prec (get_even n) 1 1 x \<le> exp x" |
1827 |
proof - |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1828 |
have "lb_exp_horner prec (get_even n) 1 1 x \<le> (\<Sum>j = 0..<get_even n. 1 / (fact j) * real_of_float x ^ j)" |
29805 | 1829 |
using bounds(1) by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1830 |
also have "\<dots> \<le> exp x" |
29805 | 1831 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1832 |
obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>" and "exp x = (\<Sum>m = 0..<get_even n. real_of_float x ^ m / (fact m)) + exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)" |
56195 | 1833 |
using Maclaurin_exp_le unfolding atLeast0LessThan by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1834 |
moreover have "0 \<le> exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)" |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1835 |
by (auto simp: zero_le_even_power) |
56536 | 1836 |
ultimately show ?thesis using get_odd exp_gt_zero by auto |
29805 | 1837 |
qed |
60680 | 1838 |
finally show ?thesis . |
1839 |
qed |
|
1840 |
moreover |
|
1841 |
have "exp x \<le> ub_exp_horner prec (get_odd n) 1 1 x" |
|
1842 |
proof - |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1843 |
have x_less_zero: "real_of_float x ^ get_odd n \<le> 0" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1844 |
proof (cases "real_of_float x = 0") |
29805 | 1845 |
case True |
1846 |
have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto |
|
1847 |
thus ?thesis unfolding True power_0_left by auto |
|
1848 |
next |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1849 |
case False hence "real_of_float x < 0" using \<open>real_of_float x \<le> 0\<close> by auto |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
1850 |
show ?thesis by (rule less_imp_le, auto simp add: \<open>real_of_float x < 0\<close>) |
29805 | 1851 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1852 |
obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1853 |
and "exp x = (\<Sum>m = 0..<get_odd n. (real_of_float x) ^ m / (fact m)) + exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n)" |
56195 | 1854 |
using Maclaurin_exp_le unfolding atLeast0LessThan by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1855 |
moreover have "exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n) \<le> 0" |
46545 | 1856 |
by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1857 |
ultimately have "exp x \<le> (\<Sum>j = 0..<get_odd n. 1 / (fact j) * real_of_float x ^ j)" |
56536 | 1858 |
using get_odd exp_gt_zero by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1859 |
also have "\<dots> \<le> ub_exp_horner prec (get_odd n) 1 1 x" |
29805 | 1860 |
using bounds(2) by auto |
60680 | 1861 |
finally show ?thesis . |
1862 |
qed |
|
1863 |
ultimately show ?thesis by auto |
|
29805 | 1864 |
qed |
1865 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1866 |
lemma ub_exp_horner_nonneg: "real_of_float x \<le> 0 \<Longrightarrow> |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1867 |
0 \<le> real_of_float (ub_exp_horner prec (get_odd n) (Suc 0) (Suc 0) x)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1868 |
using bnds_exp_horner[of x prec n] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1869 |
by (intro order_trans[OF exp_ge_zero]) auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1870 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1871 |
|
29805 | 1872 |
subsection "Compute the exponential function on the entire domain" |
1873 |
||
1874 |
function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1875 |
"lb_exp prec x = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1876 |
(if 0 < x then float_divl prec 1 (ub_exp prec (-x)) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1877 |
else |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1878 |
let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1879 |
horner = (\<lambda> x. let y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x in |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1880 |
if y \<le> 0 then Float 1 (- 2) else y) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1881 |
in |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1882 |
if x < - 1 then |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1883 |
power_down_fl prec (horner (float_divl prec x (- floor_fl x))) (nat (- int_floor_fl x)) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1884 |
else horner x)" | |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1885 |
"ub_exp prec x = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1886 |
(if 0 < x then float_divr prec 1 (lb_exp prec (-x)) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1887 |
else if x < - 1 then |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1888 |
power_up_fl prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1889 |
(ub_exp_horner prec (get_odd (prec + 2)) 1 1 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1890 |
(float_divr prec x (- floor_fl x))) (nat (- int_floor_fl x)) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1891 |
else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)" |
60680 | 1892 |
by pat_completeness auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1893 |
termination |
60680 | 1894 |
by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if 0 < x then 1 else 0))") auto |
29805 | 1895 |
|
1896 |
lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)" |
|
1897 |
proof - |
|
1898 |
have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto |
|
60680 | 1899 |
have "1 / 4 = (Float 1 (- 2))" |
1900 |
unfolding Float_num by auto |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1901 |
also have "\<dots> \<le> lb_exp_horner 3 (get_even 3) 1 1 (- 1)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1902 |
by (subst less_eq_float.rep_eq [symmetric]) code_simp |
60680 | 1903 |
also have "\<dots> \<le> exp (- 1 :: float)" |
1904 |
using bnds_exp_horner[where x="- 1"] by auto |
|
1905 |
finally show ?thesis |
|
1906 |
by simp |
|
29805 | 1907 |
qed |
1908 |
||
60680 | 1909 |
lemma lb_exp_pos: |
1910 |
assumes "\<not> 0 < x" |
|
1911 |
shows "0 < lb_exp prec x" |
|
29805 | 1912 |
proof - |
1913 |
let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x" |
|
60680 | 1914 |
let "?horner x" = "let y = ?lb_horner x in if y \<le> 0 then Float 1 (- 2) else y" |
1915 |
have pos_horner: "0 < ?horner x" for x |
|
1916 |
unfolding Let_def by (cases "?lb_horner x \<le> 0") auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1917 |
moreover have "0 < real_of_float ((?horner x) ^ num)" for x :: float and num :: nat |
60680 | 1918 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1919 |
have "0 < real_of_float (?horner x) ^ num" using \<open>0 < ?horner x\<close> by simp |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1920 |
also have "\<dots> = (?horner x) ^ num" by auto |
60680 | 1921 |
finally show ?thesis . |
1922 |
qed |
|
29805 | 1923 |
ultimately show ?thesis |
60533 | 1924 |
unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] Let_def |
60680 | 1925 |
by (cases "floor_fl x", cases "x < - 1") |
1926 |
(auto simp: real_power_up_fl real_power_down_fl intro!: power_up_less power_down_pos) |
|
29805 | 1927 |
qed |
1928 |
||
60680 | 1929 |
lemma exp_boundaries': |
1930 |
assumes "x \<le> 0" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1931 |
shows "exp x \<in> { (lb_exp prec x) .. (ub_exp prec x)}" |
29805 | 1932 |
proof - |
1933 |
let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x" |
|
1934 |
let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x" |
|
1935 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1936 |
have "real_of_float x \<le> 0" and "\<not> x > 0" |
60680 | 1937 |
using \<open>x \<le> 0\<close> by auto |
29805 | 1938 |
show ?thesis |
1939 |
proof (cases "x < - 1") |
|
60680 | 1940 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1941 |
hence "- 1 \<le> real_of_float x" by auto |
29805 | 1942 |
show ?thesis |
1943 |
proof (cases "?lb_exp_horner x \<le> 0") |
|
60680 | 1944 |
case True |
1945 |
from \<open>\<not> x < - 1\<close> |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1946 |
have "- 1 \<le> real_of_float x" by auto |
60680 | 1947 |
hence "exp (- 1) \<le> exp x" |
1948 |
unfolding exp_le_cancel_iff . |
|
1949 |
from order_trans[OF exp_m1_ge_quarter this] have "Float 1 (- 2) \<le> exp x" |
|
1950 |
unfolding Float_num . |
|
1951 |
with True show ?thesis |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1952 |
using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by auto |
29805 | 1953 |
next |
60680 | 1954 |
case False |
1955 |
thus ?thesis |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1956 |
using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by (auto simp add: Let_def) |
29805 | 1957 |
qed |
1958 |
next |
|
1959 |
case True |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1960 |
let ?num = "nat (- int_floor_fl x)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1961 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1962 |
have "real_of_int (int_floor_fl x) < - 1" |
60680 | 1963 |
using int_floor_fl[of x] \<open>x < - 1\<close> by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1964 |
hence "real_of_int (int_floor_fl x) < 0" by simp |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1965 |
hence "int_floor_fl x < 0" by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1966 |
hence "1 \<le> - int_floor_fl x" by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1967 |
hence "0 < nat (- int_floor_fl x)" by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1968 |
hence "0 < ?num" by auto |
29805 | 1969 |
hence "real ?num \<noteq> 0" by auto |
60680 | 1970 |
have num_eq: "real ?num = - int_floor_fl x" |
1971 |
using \<open>0 < nat (- int_floor_fl x)\<close> by auto |
|
1972 |
have "0 < - int_floor_fl x" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1973 |
using \<open>0 < ?num\<close>[unfolded of_nat_less_iff[symmetric]] by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1974 |
hence "real_of_int (int_floor_fl x) < 0" |
60680 | 1975 |
unfolding less_float_def by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1976 |
have fl_eq: "real_of_int (- int_floor_fl x) = real_of_float (- floor_fl x)" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1977 |
by (simp add: floor_fl_def int_floor_fl_def) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1978 |
from \<open>0 < - int_floor_fl x\<close> have "0 \<le> real_of_float (- floor_fl x)" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1979 |
by (simp add: floor_fl_def int_floor_fl_def) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1980 |
from \<open>real_of_int (int_floor_fl x) < 0\<close> have "real_of_float (floor_fl x) < 0" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1981 |
by (simp add: floor_fl_def int_floor_fl_def) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1982 |
have "exp x \<le> ub_exp prec x" |
29805 | 1983 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1984 |
have div_less_zero: "real_of_float (float_divr prec x (- floor_fl x)) \<le> 0" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1985 |
using float_divr_nonpos_pos_upper_bound[OF \<open>real_of_float x \<le> 0\<close> \<open>0 \<le> real_of_float (- floor_fl x)\<close>] |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
1986 |
unfolding less_eq_float_def zero_float.rep_eq . |
31809 | 1987 |
|
60680 | 1988 |
have "exp x = exp (?num * (x / ?num))" |
1989 |
using \<open>real ?num \<noteq> 0\<close> by auto |
|
1990 |
also have "\<dots> = exp (x / ?num) ^ ?num" |
|
1991 |
unfolding exp_real_of_nat_mult .. |
|
1992 |
also have "\<dots> \<le> exp (float_divr prec x (- floor_fl x)) ^ ?num" |
|
1993 |
unfolding num_eq fl_eq |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1994 |
by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1995 |
also have "\<dots> \<le> (?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1996 |
unfolding real_of_float_power |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1997 |
by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1998 |
also have "\<dots> \<le> real_of_float (power_up_fl prec (?ub_exp_horner (float_divr prec x (- floor_fl x))) ?num)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1999 |
by (auto simp add: real_power_up_fl intro!: power_up ub_exp_horner_nonneg div_less_zero) |
60680 | 2000 |
finally show ?thesis |
2001 |
unfolding ub_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>] floor_fl_def Let_def . |
|
29805 | 2002 |
qed |
31809 | 2003 |
moreover |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2004 |
have "lb_exp prec x \<le> exp x" |
29805 | 2005 |
proof - |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2006 |
let ?divl = "float_divl prec x (- floor_fl x)" |
29805 | 2007 |
let ?horner = "?lb_exp_horner ?divl" |
31809 | 2008 |
|
29805 | 2009 |
show ?thesis |
2010 |
proof (cases "?horner \<le> 0") |
|
60680 | 2011 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2012 |
hence "0 \<le> real_of_float ?horner" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2013 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2014 |
have div_less_zero: "real_of_float (float_divl prec x (- floor_fl x)) \<le> 0" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2015 |
using \<open>real_of_float (floor_fl x) < 0\<close> \<open>real_of_float x \<le> 0\<close> |
60680 | 2016 |
by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg) |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56410
diff
changeset
|
2017 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2018 |
have "(?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num \<le> |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2019 |
exp (float_divl prec x (- floor_fl x)) ^ ?num" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2020 |
using \<open>0 \<le> real_of_float ?horner\<close>[unfolded floor_fl_def[symmetric]] |
60680 | 2021 |
bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] |
2022 |
by (auto intro!: power_mono) |
|
2023 |
also have "\<dots> \<le> exp (x / ?num) ^ ?num" |
|
2024 |
unfolding num_eq fl_eq |
|
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
2025 |
using float_divl by (auto intro!: power_mono simp del: uminus_float.rep_eq) |
60680 | 2026 |
also have "\<dots> = exp (?num * (x / ?num))" |
2027 |
unfolding exp_real_of_nat_mult .. |
|
2028 |
also have "\<dots> = exp x" |
|
2029 |
using \<open>real ?num \<noteq> 0\<close> by auto |
|
2030 |
finally show ?thesis |
|
2031 |
using False |
|
2032 |
unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>] |
|
2033 |
int_floor_fl_def Let_def if_not_P[OF False] |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2034 |
by (auto simp: real_power_down_fl intro!: power_down_le) |
29805 | 2035 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2036 |
case True |
59741
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
2037 |
have "power_down_fl prec (Float 1 (- 2)) ?num \<le> (Float 1 (- 2)) ^ ?num" |
60680 | 2038 |
by (metis Float_le_zero_iff less_imp_le linorder_not_less |
2039 |
not_numeral_le_zero numeral_One power_down_fl) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2040 |
then have "power_down_fl prec (Float 1 (- 2)) ?num \<le> real_of_float (Float 1 (- 2)) ^ ?num" |
59741
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
2041 |
by simp |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2042 |
also |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2043 |
have "real_of_float (floor_fl x) \<noteq> 0" and "real_of_float (floor_fl x) \<le> 0" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2044 |
using \<open>real_of_float (floor_fl x) < 0\<close> by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2045 |
from divide_right_mono_neg[OF floor_fl[of x] \<open>real_of_float (floor_fl x) \<le> 0\<close>, unfolded divide_self[OF \<open>real_of_float (floor_fl x) \<noteq> 0\<close>]] |
60680 | 2046 |
have "- 1 \<le> x / (- floor_fl x)" |
2047 |
unfolding minus_float.rep_eq by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2048 |
from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]] |
60680 | 2049 |
have "Float 1 (- 2) \<le> exp (x / (- floor_fl x))" |
2050 |
unfolding Float_num . |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2051 |
hence "real_of_float (Float 1 (- 2)) ^ ?num \<le> exp (x / (- floor_fl x)) ^ ?num" |
59741
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
2052 |
by (metis Float_num(5) power_mono zero_le_divide_1_iff zero_le_numeral) |
60680 | 2053 |
also have "\<dots> = exp x" |
2054 |
unfolding num_eq fl_eq exp_real_of_nat_mult[symmetric] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2055 |
using \<open>real_of_float (floor_fl x) \<noteq> 0\<close> by auto |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2056 |
finally show ?thesis |
60680 | 2057 |
unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>] |
2058 |
int_floor_fl_def Let_def if_P[OF True] real_of_float_power . |
|
29805 | 2059 |
qed |
2060 |
qed |
|
2061 |
ultimately show ?thesis by auto |
|
2062 |
qed |
|
2063 |
qed |
|
2064 |
||
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2065 |
lemma exp_boundaries: "exp x \<in> { lb_exp prec x .. ub_exp prec x }" |
29805 | 2066 |
proof - |
2067 |
show ?thesis |
|
2068 |
proof (cases "0 < x") |
|
60680 | 2069 |
case False |
2070 |
hence "x \<le> 0" by auto |
|
29805 | 2071 |
from exp_boundaries'[OF this] show ?thesis . |
2072 |
next |
|
60680 | 2073 |
case True |
2074 |
hence "-x \<le> 0" by auto |
|
31809 | 2075 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2076 |
have "lb_exp prec x \<le> exp x" |
29805 | 2077 |
proof - |
60533 | 2078 |
from exp_boundaries'[OF \<open>-x \<le> 0\<close>] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2079 |
have ub_exp: "exp (- real_of_float x) \<le> ub_exp prec (-x)" |
60680 | 2080 |
unfolding atLeastAtMost_iff minus_float.rep_eq by auto |
2081 |
||
2082 |
have "float_divl prec 1 (ub_exp prec (-x)) \<le> 1 / ub_exp prec (-x)" |
|
2083 |
using float_divl[where x=1] by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2084 |
also have "\<dots> \<le> exp x" |
60680 | 2085 |
using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] |
2086 |
exp_gt_zero, symmetric]] |
|
2087 |
unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide |
|
2088 |
by auto |
|
2089 |
finally show ?thesis |
|
2090 |
unfolding lb_exp.simps if_P[OF True] . |
|
29805 | 2091 |
qed |
2092 |
moreover |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2093 |
have "exp x \<le> ub_exp prec x" |
29805 | 2094 |
proof - |
60533 | 2095 |
have "\<not> 0 < -x" using \<open>0 < x\<close> by auto |
2096 |
||
2097 |
from exp_boundaries'[OF \<open>-x \<le> 0\<close>] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2098 |
have lb_exp: "lb_exp prec (-x) \<le> exp (- real_of_float x)" |
60680 | 2099 |
unfolding atLeastAtMost_iff minus_float.rep_eq by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2100 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2101 |
have "exp x \<le> (1 :: float) / lb_exp prec (-x)" |
60533 | 2102 |
using lb_exp lb_exp_pos[OF \<open>\<not> 0 < -x\<close>, of prec] |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
2103 |
by (simp del: lb_exp.simps add: exp_minus field_simps) |
60680 | 2104 |
also have "\<dots> \<le> float_divr prec 1 (lb_exp prec (-x))" |
2105 |
using float_divr . |
|
2106 |
finally show ?thesis |
|
2107 |
unfolding ub_exp.simps if_P[OF True] . |
|
29805 | 2108 |
qed |
60680 | 2109 |
ultimately show ?thesis |
2110 |
by auto |
|
29805 | 2111 |
qed |
2112 |
qed |
|
2113 |
||
60680 | 2114 |
lemma bnds_exp: "\<forall>(x::real) lx ux. (l, u) = |
2115 |
(lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> exp x \<and> exp x \<le> u" |
|
29805 | 2116 |
proof (rule allI, rule allI, rule allI, rule impI) |
60680 | 2117 |
fix x :: real and lx ux |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2118 |
assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux}" |
60680 | 2119 |
hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {lx .. ux}" |
2120 |
by auto |
|
2121 |
show "l \<le> exp x \<and> exp x \<le> u" |
|
2122 |
proof |
|
2123 |
show "l \<le> exp x" |
|
2124 |
proof - |
|
2125 |
from exp_boundaries[of lx prec, unfolded l] |
|
2126 |
have "l \<le> exp lx" by (auto simp del: lb_exp.simps) |
|
2127 |
also have "\<dots> \<le> exp x" using x by auto |
|
2128 |
finally show ?thesis . |
|
2129 |
qed |
|
2130 |
show "exp x \<le> u" |
|
2131 |
proof - |
|
2132 |
have "exp x \<le> exp ux" using x by auto |
|
2133 |
also have "\<dots> \<le> u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps) |
|
2134 |
finally show ?thesis . |
|
2135 |
qed |
|
2136 |
qed |
|
29805 | 2137 |
qed |
2138 |
||
60680 | 2139 |
|
29805 | 2140 |
section "Logarithm" |
2141 |
||
2142 |
subsection "Compute the logarithm series" |
|
2143 |
||
31809 | 2144 |
fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
29805 | 2145 |
and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
2146 |
"ub_ln_horner prec 0 i x = 0" | |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2147 |
"ub_ln_horner prec (Suc n) i x = float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2148 |
(rapprox_rat prec 1 (int i)) (- float_round_down prec (x * lb_ln_horner prec n (Suc i) x))" | |
29805 | 2149 |
"lb_ln_horner prec 0 i x = 0" | |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2150 |
"lb_ln_horner prec (Suc n) i x = float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2151 |
(lapprox_rat prec 1 (int i)) (- float_round_up prec (x * ub_ln_horner prec n (Suc i) x))" |
29805 | 2152 |
|
2153 |
lemma ln_bounds: |
|
60680 | 2154 |
assumes "0 \<le> x" |
2155 |
and "x < 1" |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2156 |
shows "(\<Sum>i=0..<2*n. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb") |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2157 |
and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub") |
29805 | 2158 |
proof - |
30952
7ab2716dd93b
power operation on functions with syntax o^; power operation on relations with syntax ^^
haftmann
parents:
30886
diff
changeset
|
2159 |
let "?a n" = "(1/real (n +1)) * x ^ (Suc n)" |
29805 | 2160 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2161 |
have ln_eq: "(\<Sum> i. (- 1) ^ i * ?a i) = ln (x + 1)" |
60533 | 2162 |
using ln_series[of "x + 1"] \<open>0 \<le> x\<close> \<open>x < 1\<close> by auto |
29805 | 2163 |
|
2164 |
have "norm x < 1" using assms by auto |
|
61969 | 2165 |
have "?a \<longlonglongrightarrow> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric] |
60533 | 2166 |
using tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF \<open>norm x < 1\<close>]]] by auto |
60680 | 2167 |
have "0 \<le> ?a n" for n |
2168 |
by (rule mult_nonneg_nonneg) (auto simp: \<open>0 \<le> x\<close>) |
|
2169 |
have "?a (Suc n) \<le> ?a n" for n |
|
2170 |
unfolding inverse_eq_divide[symmetric] |
|
2171 |
proof (rule mult_mono) |
|
2172 |
show "0 \<le> x ^ Suc (Suc n)" |
|
2173 |
by (auto simp add: \<open>0 \<le> x\<close>) |
|
2174 |
have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" |
|
2175 |
unfolding power_Suc2 mult.assoc[symmetric] |
|
2176 |
by (rule mult_left_mono, fact less_imp_le[OF \<open>x < 1\<close>]) (auto simp: \<open>0 \<le> x\<close>) |
|
2177 |
thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto |
|
2178 |
qed auto |
|
61969 | 2179 |
from summable_Leibniz'(2,4)[OF \<open>?a \<longlonglongrightarrow> 0\<close> \<open>\<And>n. 0 \<le> ?a n\<close>, OF \<open>\<And>n. ?a (Suc n) \<le> ?a n\<close>, unfolded ln_eq] |
60680 | 2180 |
show ?lb and ?ub |
2181 |
unfolding atLeast0LessThan by auto |
|
29805 | 2182 |
qed |
2183 |
||
31809 | 2184 |
lemma ln_float_bounds: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2185 |
assumes "0 \<le> real_of_float x" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2186 |
and "real_of_float x < 1" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2187 |
shows "x * lb_ln_horner prec (get_even n) 1 x \<le> ln (x + 1)" (is "?lb \<le> ?ln") |
60680 | 2188 |
and "ln (x + 1) \<le> x * ub_ln_horner prec (get_odd n) 1 x" (is "?ln \<le> ?ub") |
29805 | 2189 |
proof - |
2190 |
obtain ev where ev: "get_even n = 2 * ev" using get_even_double .. |
|
2191 |
obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double .. |
|
2192 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2193 |
let "?s n" = "(- 1) ^ n * (1 / real (1 + n)) * (real_of_float x)^(Suc n)" |
29805 | 2194 |
|
64267 | 2195 |
have "?lb \<le> sum ?s {0 ..< 2 * ev}" |
2196 |
unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq sum_distrib_right[symmetric] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2197 |
unfolding mult.commute[of "real_of_float x"] ev |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2198 |
using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2199 |
and lb="\<lambda>n i k x. lb_ln_horner prec n k x" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2200 |
and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev", |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2201 |
OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close> |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2202 |
unfolding real_of_float_power |
29805 | 2203 |
by (rule mult_right_mono) |
60680 | 2204 |
also have "\<dots> \<le> ?ln" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2205 |
using ln_bounds(1)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto |
31809 | 2206 |
finally show "?lb \<le> ?ln" . |
29805 | 2207 |
|
64267 | 2208 |
have "?ln \<le> sum ?s {0 ..< 2 * od + 1}" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2209 |
using ln_bounds(2)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto |
60680 | 2210 |
also have "\<dots> \<le> ?ub" |
64267 | 2211 |
unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq sum_distrib_right[symmetric] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2212 |
unfolding mult.commute[of "real_of_float x"] od |
29805 | 2213 |
using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1", |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2214 |
OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close> |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2215 |
unfolding real_of_float_power |
29805 | 2216 |
by (rule mult_right_mono) |
31809 | 2217 |
finally show "?ln \<le> ?ub" . |
29805 | 2218 |
qed |
2219 |
||
60680 | 2220 |
lemma ln_add: |
2221 |
fixes x :: real |
|
2222 |
assumes "0 < x" and "0 < y" |
|
2223 |
shows "ln (x + y) = ln x + ln (1 + y / x)" |
|
29805 | 2224 |
proof - |
2225 |
have "x \<noteq> 0" using assms by auto |
|
60680 | 2226 |
have "x + y = x * (1 + y / x)" |
64240 | 2227 |
unfolding distrib_left times_divide_eq_right nonzero_mult_div_cancel_left[OF \<open>x \<noteq> 0\<close>] |
60680 | 2228 |
by auto |
31809 | 2229 |
moreover |
56541 | 2230 |
have "0 < y / x" using assms by auto |
29805 | 2231 |
hence "0 < 1 + y / x" by auto |
60680 | 2232 |
ultimately show ?thesis |
2233 |
using ln_mult assms by auto |
|
29805 | 2234 |
qed |
2235 |
||
60680 | 2236 |
|
29805 | 2237 |
subsection "Compute the logarithm of 2" |
2238 |
||
31809 | 2239 |
definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3 |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2240 |
in float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2241 |
((Float 1 (- 1) * ub_ln_horner prec (get_odd prec) 1 (Float 1 (- 1)))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2242 |
(float_round_up prec (third * ub_ln_horner prec (get_odd prec) 1 third)))" |
31809 | 2243 |
definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3 |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2244 |
in float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2245 |
((Float 1 (- 1) * lb_ln_horner prec (get_even prec) 1 (Float 1 (- 1)))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2246 |
(float_round_down prec (third * lb_ln_horner prec (get_even prec) 1 third)))" |
29805 | 2247 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2248 |
lemma ub_ln2: "ln 2 \<le> ub_ln2 prec" (is "?ub_ln2") |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2249 |
and lb_ln2: "lb_ln2 prec \<le> ln 2" (is "?lb_ln2") |
29805 | 2250 |
proof - |
2251 |
let ?uthird = "rapprox_rat (max prec 1) 1 3" |
|
2252 |
let ?lthird = "lapprox_rat prec 1 3" |
|
2253 |
||
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59850
diff
changeset
|
2254 |
have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1::real)" |
29805 | 2255 |
using ln_add[of "3 / 2" "1 / 2"] by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2256 |
have lb3: "?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2257 |
hence lb3_ub: "real_of_float ?lthird < 1" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2258 |
have lb3_lb: "0 \<le> real_of_float ?lthird" using lapprox_rat_nonneg[of 1 3] by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2259 |
have ub3: "1 / 3 \<le> ?uthird" using rapprox_rat[of 1 3] by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2260 |
hence ub3_lb: "0 \<le> real_of_float ?uthird" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2261 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2262 |
have lb2: "0 \<le> real_of_float (Float 1 (- 1))" and ub2: "real_of_float (Float 1 (- 1)) < 1" |
60680 | 2263 |
unfolding Float_num by auto |
29805 | 2264 |
|
2265 |
have "0 \<le> (1::int)" and "0 < (3::int)" by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2266 |
have ub3_ub: "real_of_float ?uthird < 1" |
58982
27e7e3f9e665
simplified computations based on round_up by reducing to round_down;
immler
parents:
58889
diff
changeset
|
2267 |
by (simp add: Float.compute_rapprox_rat Float.compute_lapprox_rat rapprox_posrat_less1) |
29805 | 2268 |
|
2269 |
have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2270 |
have uthird_gt0: "0 < real_of_float ?uthird + 1" using ub3_lb by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2271 |
have lthird_gt0: "0 < real_of_float ?lthird + 1" using lb3_lb by auto |
29805 | 2272 |
|
60680 | 2273 |
show ?ub_ln2 |
2274 |
unfolding ub_ln2_def Let_def ln2_sum Float_num(4)[symmetric] |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2275 |
proof (rule float_plus_up_le, rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2276 |
have "ln (1 / 3 + 1) \<le> ln (real_of_float ?uthird + 1)" |
60680 | 2277 |
unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2278 |
also have "\<dots> \<le> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird" |
29805 | 2279 |
using ln_float_bounds(2)[OF ub3_lb ub3_ub] . |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2280 |
also note float_round_up |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2281 |
finally show "ln (1 / 3 + 1) \<le> float_round_up prec (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" . |
29805 | 2282 |
qed |
60680 | 2283 |
show ?lb_ln2 |
2284 |
unfolding lb_ln2_def Let_def ln2_sum Float_num(4)[symmetric] |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2285 |
proof (rule float_plus_down_le, rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2286 |
have "?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird \<le> ln (real_of_float ?lthird + 1)" |
29805 | 2287 |
using ln_float_bounds(1)[OF lb3_lb lb3_ub] . |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2288 |
note float_round_down_le[OF this] |
60680 | 2289 |
also have "\<dots> \<le> ln (1 / 3 + 1)" |
2290 |
unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] |
|
2291 |
using lb3 by auto |
|
2292 |
finally show "float_round_down prec (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> |
|
2293 |
ln (1 / 3 + 1)" . |
|
29805 | 2294 |
qed |
2295 |
qed |
|
2296 |
||
60680 | 2297 |
|
29805 | 2298 |
subsection "Compute the logarithm in the entire domain" |
2299 |
||
2300 |
function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where |
|
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2301 |
"ub_ln prec x = (if x \<le> 0 then None |
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2302 |
else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x))) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2303 |
else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2304 |
if x \<le> Float 3 (- 1) then Some (horner (x - 1)) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2305 |
else if x < Float 1 1 then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1))) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2306 |
else let l = bitlen (mantissa x) - 1 in |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2307 |
Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))" | |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2308 |
"lb_ln prec x = (if x \<le> 0 then None |
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2309 |
else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x))) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2310 |
else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2311 |
if x \<le> Float 3 (- 1) then Some (horner (x - 1)) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2312 |
else if x < Float 1 1 then Some (float_round_down prec (horner (Float 1 (- 1)) + |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2313 |
horner (max (x * lapprox_rat prec 2 3 - 1) 0))) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2314 |
else let l = bitlen (mantissa x) - 1 in |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2315 |
Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))" |
60680 | 2316 |
by pat_completeness auto |
2317 |
||
2318 |
termination |
|
2319 |
proof (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 1 then 1 else 0))", auto) |
|
2320 |
fix prec and x :: float |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2321 |
assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2322 |
hence "0 < real_of_float x" "1 \<le> max prec (Suc 0)" "real_of_float x < 1" |
60680 | 2323 |
by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2324 |
from float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x < 1\<close>[THEN less_imp_le] \<open>1 \<le> max prec (Suc 0)\<close>] |
60680 | 2325 |
show False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2326 |
using \<open>real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1\<close> by auto |
29805 | 2327 |
next |
60680 | 2328 |
fix prec x |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2329 |
assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divr prec 1 x) < 1" |
47600 | 2330 |
hence "0 < x" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2331 |
from float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close>, of prec] \<open>real_of_float x < 1\<close> show False |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2332 |
using \<open>real_of_float (float_divr prec 1 x) < 1\<close> by auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2333 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2334 |
|
60680 | 2335 |
lemma float_pos_eq_mantissa_pos: "x > 0 \<longleftrightarrow> mantissa x > 0" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2336 |
apply (subst Float_mantissa_exponent[of x, symmetric]) |
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59850
diff
changeset
|
2337 |
apply (auto simp add: zero_less_mult_iff zero_float_def dest: less_zeroE) |
60680 | 2338 |
apply (metis not_le powr_ge_pzero) |
2339 |
done |
|
2340 |
||
2341 |
lemma Float_pos_eq_mantissa_pos: "Float m e > 0 \<longleftrightarrow> m > 0" |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2342 |
using powr_gt_zero[of 2 "e"] |
54269 | 2343 |
by (auto simp add: zero_less_mult_iff zero_float_def simp del: powr_gt_zero dest: less_zeroE) |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2344 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2345 |
lemma Float_representation_aux: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2346 |
fixes m e |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2347 |
defines "x \<equiv> Float m e" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2348 |
assumes "x > 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2349 |
shows "Float (exponent x + (bitlen (mantissa x) - 1)) 0 = Float (e + (bitlen m - 1)) 0" (is ?th1) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2350 |
and "Float (mantissa x) (- (bitlen (mantissa x) - 1)) = Float m ( - (bitlen m - 1))" (is ?th2) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2351 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2352 |
from assms have mantissa_pos: "m > 0" "mantissa x > 0" |
47600 | 2353 |
using Float_pos_eq_mantissa_pos[of m e] float_pos_eq_mantissa_pos[of x] by simp_all |
60680 | 2354 |
thus ?th1 |
2355 |
using bitlen_Float[of m e] assms |
|
2356 |
by (auto simp add: zero_less_mult_iff intro!: arg_cong2[where f=Float]) |
|
47600 | 2357 |
have "x \<noteq> float_of 0" |
60533 | 2358 |
unfolding zero_float_def[symmetric] using \<open>0 < x\<close> by auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2359 |
from denormalize_shift[OF assms(1) this] guess i . note i = this |
47600 | 2360 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2361 |
have "2 powr (1 - (real_of_int (bitlen (mantissa x)) + real_of_int i)) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2362 |
2 powr (1 - (real_of_int (bitlen (mantissa x)))) * inverse (2 powr (real i))" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2363 |
by (simp add: powr_minus[symmetric] powr_add[symmetric] field_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2364 |
hence "real_of_int (mantissa x) * 2 powr (1 - real_of_int (bitlen (mantissa x))) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2365 |
(real_of_int (mantissa x) * 2 ^ i) * 2 powr (1 - real_of_int (bitlen (mantissa x * 2 ^ i)))" |
60533 | 2366 |
using \<open>mantissa x > 0\<close> by (simp add: powr_realpow) |
47600 | 2367 |
then show ?th2 |
2368 |
unfolding i by transfer auto |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2369 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2370 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2371 |
lemma compute_ln[code]: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2372 |
fixes m e |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2373 |
defines "x \<equiv> Float m e" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2374 |
shows "ub_ln prec x = (if x \<le> 0 then None |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2375 |
else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x))) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2376 |
else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2377 |
if x \<le> Float 3 (- 1) then Some (horner (x - 1)) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2378 |
else if x < Float 1 1 then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1))) |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2379 |
else let l = bitlen m - 1 in |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2380 |
Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2381 |
(is ?th1) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2382 |
and "lb_ln prec x = (if x \<le> 0 then None |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2383 |
else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x))) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2384 |
else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2385 |
if x \<le> Float 3 (- 1) then Some (horner (x - 1)) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2386 |
else if x < Float 1 1 then Some (float_round_down prec (horner (Float 1 (- 1)) + |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2387 |
horner (max (x * lapprox_rat prec 2 3 - 1) 0))) |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2388 |
else let l = bitlen m - 1 in |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2389 |
Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2390 |
(is ?th2) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2391 |
proof - |
60680 | 2392 |
from assms Float_pos_eq_mantissa_pos have "x > 0 \<Longrightarrow> m > 0" |
2393 |
by simp |
|
2394 |
thus ?th1 ?th2 |
|
2395 |
using Float_representation_aux[of m e] |
|
2396 |
unfolding x_def[symmetric] |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
2397 |
by (auto dest: not_le_imp_less) |
29805 | 2398 |
qed |
2399 |
||
60680 | 2400 |
lemma ln_shifted_float: |
2401 |
assumes "0 < m" |
|
2402 |
shows "ln (Float m e) = ln 2 * (e + (bitlen m - 1)) + ln (Float m (- (bitlen m - 1)))" |
|
29805 | 2403 |
proof - |
2404 |
let ?B = "2^nat (bitlen m - 1)" |
|
63040 | 2405 |
define bl where "bl = bitlen m - 1" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2406 |
have "0 < real_of_int m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" |
60680 | 2407 |
using assms by auto |
63248 | 2408 |
hence "0 \<le> bl" by (simp add: bitlen_alt_def bl_def) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2409 |
show ?thesis |
29805 | 2410 |
proof (cases "0 \<le> e") |
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
2411 |
case True |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2412 |
thus ?thesis |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2413 |
unfolding bl_def[symmetric] using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close> |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2414 |
apply (simp add: ln_mult) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2415 |
apply (cases "e=0") |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2416 |
apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2417 |
apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr field_simps) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2418 |
done |
29805 | 2419 |
next |
60680 | 2420 |
case False |
2421 |
hence "0 < -e" by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2422 |
have lne: "ln (2 powr real_of_int e) = ln (inverse (2 powr - e))" |
60680 | 2423 |
by (simp add: powr_minus) |
2424 |
hence pow_gt0: "(0::real) < 2^nat (-e)" |
|
2425 |
by auto |
|
2426 |
hence inv_gt0: "(0::real) < inverse (2^nat (-e))" |
|
2427 |
by auto |
|
2428 |
show ?thesis |
|
2429 |
using False unfolding bl_def[symmetric] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2430 |
using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close> |
56483 | 2431 |
by (auto simp add: lne ln_mult ln_powr ln_div field_simps) |
29805 | 2432 |
qed |
2433 |
qed |
|
2434 |
||
60680 | 2435 |
lemma ub_ln_lb_ln_bounds': |
2436 |
assumes "1 \<le> x" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2437 |
shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)" |
60680 | 2438 |
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub") |
29805 | 2439 |
proof (cases "x < Float 1 1") |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2440 |
case True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2441 |
hence "real_of_float (x - 1) < 1" and "real_of_float x < 2" by auto |
60533 | 2442 |
have "\<not> x \<le> 0" and "\<not> x < 1" using \<open>1 \<le> x\<close> by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2443 |
hence "0 \<le> real_of_float (x - 1)" using \<open>1 \<le> x\<close> by auto |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2444 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2445 |
have [simp]: "(Float 3 (- 1)) = 3 / 2" by simp |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2446 |
|
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2447 |
show ?thesis |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2448 |
proof (cases "x \<le> Float 3 (- 1)") |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2449 |
case True |
60680 | 2450 |
show ?thesis |
2451 |
unfolding lb_ln.simps |
|
2452 |
unfolding ub_ln.simps Let_def |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2453 |
using ln_float_bounds[OF \<open>0 \<le> real_of_float (x - 1)\<close> \<open>real_of_float (x - 1) < 1\<close>, of prec] |
60680 | 2454 |
\<open>\<not> x \<le> 0\<close> \<open>\<not> x < 1\<close> True |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2455 |
by (auto intro!: float_round_down_le float_round_up_le) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2456 |
next |
60680 | 2457 |
case False |
2458 |
hence *: "3 / 2 < x" by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2459 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2460 |
with ln_add[of "3 / 2" "x - 3 / 2"] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2461 |
have add: "ln x = ln (3 / 2) + ln (real_of_float x * 2 / 3)" |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2462 |
by (auto simp add: algebra_simps diff_divide_distrib) |
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2463 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2464 |
let "?ub_horner x" = "float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x)" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2465 |
let "?lb_horner x" = "float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x)" |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2466 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2467 |
{ have up: "real_of_float (rapprox_rat prec 2 3) \<le> 1" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2468 |
by (rule rapprox_rat_le1) simp_all |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2469 |
have low: "2 / 3 \<le> rapprox_rat prec 2 3" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2470 |
by (rule order_trans[OF _ rapprox_rat]) simp |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2471 |
from mult_less_le_imp_less[OF * low] * |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2472 |
have pos: "0 < real_of_float (x * rapprox_rat prec 2 3 - 1)" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2473 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2474 |
have "ln (real_of_float x * 2/3) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2475 |
\<le> ln (real_of_float (x * rapprox_rat prec 2 3 - 1) + 1)" |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2476 |
proof (rule ln_le_cancel_iff[symmetric, THEN iffD1]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2477 |
show "real_of_float x * 2 / 3 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1) + 1" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2478 |
using * low by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2479 |
show "0 < real_of_float x * 2 / 3" using * by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2480 |
show "0 < real_of_float (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2481 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2482 |
also have "\<dots> \<le> ?ub_horner (x * rapprox_rat prec 2 3 - 1)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2483 |
proof (rule float_round_up_le, rule ln_float_bounds(2)) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2484 |
from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] low * |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2485 |
show "real_of_float (x * rapprox_rat prec 2 3 - 1) < 1" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2486 |
show "0 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1)" using pos by auto |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2487 |
qed |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2488 |
finally have "ln x \<le> ?ub_horner (Float 1 (-1)) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2489 |
+ ?ub_horner ((x * rapprox_rat prec 2 3 - 1))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2490 |
using ln_float_bounds(2)[of "Float 1 (- 1)" prec prec] add |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2491 |
by (auto intro!: add_mono float_round_up_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2492 |
note float_round_up_le[OF this, of prec] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2493 |
} |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2494 |
moreover |
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2495 |
{ let ?max = "max (x * lapprox_rat prec 2 3 - 1) 0" |
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2496 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2497 |
have up: "lapprox_rat prec 2 3 \<le> 2/3" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2498 |
by (rule order_trans[OF lapprox_rat], simp) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2499 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2500 |
have low: "0 \<le> real_of_float (lapprox_rat prec 2 3)" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2501 |
using lapprox_rat_nonneg[of 2 3 prec] by simp |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2502 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2503 |
have "?lb_horner ?max |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2504 |
\<le> ln (real_of_float ?max + 1)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2505 |
proof (rule float_round_down_le, rule ln_float_bounds(1)) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2506 |
from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] * low |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2507 |
show "real_of_float ?max < 1" by (cases "real_of_float (lapprox_rat prec 2 3) = 0", |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2508 |
auto simp add: real_of_float_max) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2509 |
show "0 \<le> real_of_float ?max" by (auto simp add: real_of_float_max) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2510 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2511 |
also have "\<dots> \<le> ln (real_of_float x * 2/3)" |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2512 |
proof (rule ln_le_cancel_iff[symmetric, THEN iffD1]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2513 |
show "0 < real_of_float ?max + 1" by (auto simp add: real_of_float_max) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2514 |
show "0 < real_of_float x * 2/3" using * by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2515 |
show "real_of_float ?max + 1 \<le> real_of_float x * 2/3" using * up |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2516 |
by (cases "0 < real_of_float x * real_of_float (lapprox_posrat prec 2 3) - 1", |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2517 |
auto simp add: max_def) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2518 |
qed |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2519 |
finally have "?lb_horner (Float 1 (- 1)) + ?lb_horner ?max \<le> ln x" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2520 |
using ln_float_bounds(1)[of "Float 1 (- 1)" prec prec] add |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2521 |
by (auto intro!: add_mono float_round_down_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2522 |
note float_round_down_le[OF this, of prec] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2523 |
} |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2524 |
ultimately |
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2525 |
show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def |
60533 | 2526 |
using \<open>\<not> x \<le> 0\<close> \<open>\<not> x < 1\<close> True False by auto |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2527 |
qed |
29805 | 2528 |
next |
2529 |
case False |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2530 |
hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 (- 1)" |
60533 | 2531 |
using \<open>1 \<le> x\<close> by auto |
29805 | 2532 |
show ?thesis |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2533 |
proof - |
63040 | 2534 |
define m where "m = mantissa x" |
2535 |
define e where "e = exponent x" |
|
60680 | 2536 |
from Float_mantissa_exponent[of x] have Float: "x = Float m e" |
2537 |
by (simp add: m_def e_def) |
|
29805 | 2538 |
let ?s = "Float (e + (bitlen m - 1)) 0" |
2539 |
let ?x = "Float m (- (bitlen m - 1))" |
|
2540 |
||
60680 | 2541 |
have "0 < m" and "m \<noteq> 0" using \<open>0 < x\<close> Float powr_gt_zero[of 2 e] |
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59850
diff
changeset
|
2542 |
apply (auto simp add: zero_less_mult_iff) |
60680 | 2543 |
using not_le powr_ge_pzero apply blast |
2544 |
done |
|
63040 | 2545 |
define bl where "bl = bitlen m - 1" |
60680 | 2546 |
hence "bl \<ge> 0" |
63248 | 2547 |
using \<open>m > 0\<close> by (simp add: bitlen_alt_def) |
60680 | 2548 |
have "1 \<le> Float m e" |
2549 |
using \<open>1 \<le> x\<close> Float unfolding less_eq_float_def by auto |
|
60533 | 2550 |
from bitlen_div[OF \<open>0 < m\<close>] float_gt1_scale[OF \<open>1 \<le> Float m e\<close>] \<open>bl \<ge> 0\<close> |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2551 |
have x_bnds: "0 \<le> real_of_float (?x - 1)" "real_of_float (?x - 1) < 1" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2552 |
unfolding bl_def[symmetric] |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
2553 |
by (auto simp: powr_realpow[symmetric] field_simps) |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
2554 |
(auto simp : powr_minus field_simps) |
29805 | 2555 |
|
2556 |
{ |
|
60680 | 2557 |
have "float_round_down prec (lb_ln2 prec * ?s) \<le> ln 2 * (e + (bitlen m - 1))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2558 |
(is "real_of_float ?lb2 \<le> _") |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2559 |
apply (rule float_round_down_le) |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
2560 |
unfolding nat_0 power_0 mult_1_right times_float.rep_eq |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2561 |
using lb_ln2[of prec] |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2562 |
proof (rule mult_mono) |
60533 | 2563 |
from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2564 |
show "0 \<le> real_of_float (Float (e + (bitlen m - 1)) 0)" by simp |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2565 |
qed auto |
29805 | 2566 |
moreover |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2567 |
from ln_float_bounds(1)[OF x_bnds] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2568 |
have "float_round_down prec ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln ?x" (is "real_of_float ?lb_horner \<le> _") |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2569 |
by (auto intro!: float_round_down_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2570 |
ultimately have "float_plus_down prec ?lb2 ?lb_horner \<le> ln x" |
60533 | 2571 |
unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e] by (auto intro!: float_plus_down_le) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2572 |
} |
29805 | 2573 |
moreover |
2574 |
{ |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2575 |
from ln_float_bounds(2)[OF x_bnds] |
60680 | 2576 |
have "ln ?x \<le> float_round_up prec ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2577 |
(is "_ \<le> real_of_float ?ub_horner") |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2578 |
by (auto intro!: float_round_up_le) |
29805 | 2579 |
moreover |
60680 | 2580 |
have "ln 2 * (e + (bitlen m - 1)) \<le> float_round_up prec (ub_ln2 prec * ?s)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2581 |
(is "_ \<le> real_of_float ?ub2") |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2582 |
apply (rule float_round_up_le) |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
2583 |
unfolding nat_0 power_0 mult_1_right times_float.rep_eq |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
2584 |
using ub_ln2[of prec] |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2585 |
proof (rule mult_mono) |
60533 | 2586 |
from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2587 |
show "0 \<le> real_of_int (e + (bitlen m - 1))" by auto |
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59850
diff
changeset
|
2588 |
have "0 \<le> ln (2 :: real)" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2589 |
thus "0 \<le> real_of_float (ub_ln2 prec)" using ub_ln2[of prec] by arith |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2590 |
qed auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2591 |
ultimately have "ln x \<le> float_plus_up prec ?ub2 ?ub_horner" |
60533 | 2592 |
unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2593 |
by (auto intro!: float_plus_up_le) |
29805 | 2594 |
} |
60680 | 2595 |
ultimately show ?thesis |
2596 |
unfolding lb_ln.simps |
|
2597 |
unfolding ub_ln.simps |
|
2598 |
unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] |
|
2599 |
if_not_P[OF False] if_not_P[OF \<open>\<not> x \<le> Float 3 (- 1)\<close>] Let_def |
|
2600 |
unfolding plus_float.rep_eq e_def[symmetric] m_def[symmetric] |
|
2601 |
by simp |
|
29805 | 2602 |
qed |
2603 |
qed |
|
2604 |
||
49351 | 2605 |
lemma ub_ln_lb_ln_bounds: |
2606 |
assumes "0 < x" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2607 |
shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)" |
60680 | 2608 |
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub") |
29805 | 2609 |
proof (cases "x < 1") |
60680 | 2610 |
case False |
2611 |
hence "1 \<le> x" |
|
2612 |
unfolding less_float_def less_eq_float_def by auto |
|
2613 |
show ?thesis |
|
2614 |
using ub_ln_lb_ln_bounds'[OF \<open>1 \<le> x\<close>] . |
|
29805 | 2615 |
next |
60680 | 2616 |
case True |
2617 |
have "\<not> x \<le> 0" using \<open>0 < x\<close> by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2618 |
from True have "real_of_float x \<le> 1" "x \<le> 1" |
60680 | 2619 |
by simp_all |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2620 |
have "0 < real_of_float x" and "real_of_float x \<noteq> 0" |
60680 | 2621 |
using \<open>0 < x\<close> by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2622 |
hence A: "0 < 1 / real_of_float x" by auto |
29805 | 2623 |
|
2624 |
{ |
|
2625 |
let ?divl = "float_divl (max prec 1) 1 x" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2626 |
have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>] by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2627 |
hence B: "0 < real_of_float ?divl" by auto |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2628 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2629 |
have "ln ?divl \<le> ln (1 / x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2630 |
hence "ln x \<le> - ln ?divl" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2631 |
from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le] |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
2632 |
have "?ln \<le> - the (lb_ln prec ?divl)" unfolding uminus_float.rep_eq by (rule order_trans) |
29805 | 2633 |
} moreover |
2634 |
{ |
|
2635 |
let ?divr = "float_divr prec 1 x" |
|
60533 | 2636 |
have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close> \<open>x \<le> 1\<close>] unfolding less_eq_float_def less_float_def by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2637 |
hence B: "0 < real_of_float ?divr" by auto |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2638 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2639 |
have "ln (1 / x) \<le> ln ?divr" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2640 |
hence "- ln ?divr \<le> ln x" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto |
29805 | 2641 |
from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this |
47601
050718fe6eee
use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents:
47600
diff
changeset
|
2642 |
have "- the (ub_ln prec ?divr) \<le> ?ln" unfolding uminus_float.rep_eq by (rule order_trans) |
29805 | 2643 |
} |
2644 |
ultimately show ?thesis unfolding lb_ln.simps[where x=x] ub_ln.simps[where x=x] |
|
60533 | 2645 |
unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_P[OF True] by auto |
29805 | 2646 |
qed |
2647 |
||
49351 | 2648 |
lemma lb_ln: |
2649 |
assumes "Some y = lb_ln prec x" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2650 |
shows "y \<le> ln x" and "0 < real_of_float x" |
29805 | 2651 |
proof - |
2652 |
have "0 < x" |
|
2653 |
proof (rule ccontr) |
|
60680 | 2654 |
assume "\<not> 0 < x" |
2655 |
hence "x \<le> 0" |
|
2656 |
unfolding less_eq_float_def less_float_def by auto |
|
2657 |
thus False |
|
2658 |
using assms by auto |
|
29805 | 2659 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2660 |
thus "0 < real_of_float x" by auto |
60680 | 2661 |
have "the (lb_ln prec x) \<le> ln x" |
2662 |
using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] .. |
|
2663 |
thus "y \<le> ln x" |
|
2664 |
unfolding assms[symmetric] by auto |
|
29805 | 2665 |
qed |
2666 |
||
49351 | 2667 |
lemma ub_ln: |
2668 |
assumes "Some y = ub_ln prec x" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2669 |
shows "ln x \<le> y" and "0 < real_of_float x" |
29805 | 2670 |
proof - |
2671 |
have "0 < x" |
|
2672 |
proof (rule ccontr) |
|
60680 | 2673 |
assume "\<not> 0 < x" |
2674 |
hence "x \<le> 0" by auto |
|
2675 |
thus False |
|
2676 |
using assms by auto |
|
29805 | 2677 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2678 |
thus "0 < real_of_float x" by auto |
60680 | 2679 |
have "ln x \<le> the (ub_ln prec x)" |
2680 |
using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] .. |
|
2681 |
thus "ln x \<le> y" |
|
2682 |
unfolding assms[symmetric] by auto |
|
29805 | 2683 |
qed |
2684 |
||
60680 | 2685 |
lemma bnds_ln: "\<forall>(x::real) lx ux. (Some l, Some u) = |
2686 |
(lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> ln x \<and> ln x \<le> u" |
|
29805 | 2687 |
proof (rule allI, rule allI, rule allI, rule impI) |
60680 | 2688 |
fix x :: real |
2689 |
fix lx ux |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2690 |
assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux}" |
60680 | 2691 |
hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {lx .. ux}" |
2692 |
by auto |
|
2693 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2694 |
have "ln ux \<le> u" and "0 < real_of_float ux" |
60680 | 2695 |
using ub_ln u by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2696 |
have "l \<le> ln lx" and "0 < real_of_float lx" and "0 < x" |
60680 | 2697 |
using lb_ln[OF l] x by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2698 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2699 |
from ln_le_cancel_iff[OF \<open>0 < real_of_float lx\<close> \<open>0 < x\<close>] \<open>l \<le> ln lx\<close> |
60680 | 2700 |
have "l \<le> ln x" |
2701 |
using x unfolding atLeastAtMost_iff by auto |
|
29805 | 2702 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2703 |
from ln_le_cancel_iff[OF \<open>0 < x\<close> \<open>0 < real_of_float ux\<close>] \<open>ln ux \<le> real_of_float u\<close> |
60680 | 2704 |
have "ln x \<le> u" |
2705 |
using x unfolding atLeastAtMost_iff by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2706 |
ultimately show "l \<le> ln x \<and> ln x \<le> u" .. |
29805 | 2707 |
qed |
2708 |
||
60680 | 2709 |
|
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2710 |
section \<open>Real power function\<close> |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2711 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2712 |
definition bnds_powr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float \<times> float) option" where |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2713 |
"bnds_powr prec l1 u1 l2 u2 = ( |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2714 |
if l1 = 0 \<and> u1 = 0 then |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2715 |
Some (0, 0) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2716 |
else if l1 = 0 \<and> l2 \<ge> 1 then |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2717 |
let uln = the (ub_ln prec u1) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2718 |
in Some (0, ub_exp prec (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2)))) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2719 |
else if l1 \<le> 0 then |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2720 |
None |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2721 |
else |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2722 |
Some (map_bnds lb_exp ub_exp prec |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2723 |
(bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2)))" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2724 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2725 |
lemmas [simp del] = lb_exp.simps ub_exp.simps |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2726 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2727 |
lemma mono_exp_real: "mono (exp :: real \<Rightarrow> real)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2728 |
by (auto simp: mono_def) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2729 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2730 |
lemma ub_exp_nonneg: "real_of_float (ub_exp prec x) \<ge> 0" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2731 |
proof - |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2732 |
have "0 \<le> exp (real_of_float x)" by simp |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2733 |
also from exp_boundaries[of x prec] |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2734 |
have "\<dots> \<le> real_of_float (ub_exp prec x)" by simp |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2735 |
finally show ?thesis . |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2736 |
qed |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2737 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2738 |
lemma bnds_powr: |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2739 |
assumes lu: "Some (l, u) = bnds_powr prec l1 u1 l2 u2" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2740 |
assumes x: "x \<in> {real_of_float l1..real_of_float u1}" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2741 |
assumes y: "y \<in> {real_of_float l2..real_of_float u2}" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2742 |
shows "x powr y \<in> {real_of_float l..real_of_float u}" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2743 |
proof - |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2744 |
consider "l1 = 0" "u1 = 0" | "l1 = 0" "u1 \<noteq> 0" "l2 \<ge> 1" | |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2745 |
"l1 \<le> 0" "\<not>(l1 = 0 \<and> (u1 = 0 \<or> l2 \<ge> 1))" | "l1 > 0" by force |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2746 |
thus ?thesis |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2747 |
proof cases |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2748 |
assume "l1 = 0" "u1 = 0" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2749 |
with x lu show ?thesis by (auto simp: bnds_powr_def) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2750 |
next |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2751 |
assume A: "l1 = 0" "u1 \<noteq> 0" "l2 \<ge> 1" |
63040 | 2752 |
define uln where "uln = the (ub_ln prec u1)" |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2753 |
show ?thesis |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2754 |
proof (cases "x = 0") |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2755 |
case False |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2756 |
with A x y have "x powr y = exp (ln x * y)" by (simp add: powr_def) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2757 |
also { |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2758 |
from A x False have "ln x \<le> ln (real_of_float u1)" by simp |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2759 |
also from ub_ln_lb_ln_bounds[of u1 prec] A y x False |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2760 |
have "ln (real_of_float u1) \<le> real_of_float uln" by (simp add: uln_def del: lb_ln.simps) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2761 |
also from A x y have "\<dots> * y \<le> real_of_float uln * (if uln \<ge> 0 then u2 else l2)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2762 |
by (auto intro: mult_left_mono mult_left_mono_neg) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2763 |
also have "\<dots> \<le> real_of_float (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2)))" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2764 |
by (simp add: float_round_up_le) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2765 |
finally have "ln x * y \<le> \<dots>" using A y by - simp |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2766 |
} |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2767 |
also have "exp (real_of_float (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2)))) \<le> |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2768 |
real_of_float (ub_exp prec (float_round_up prec |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2769 |
(uln * (if uln \<ge> 0 then u2 else l2))))" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2770 |
using exp_boundaries by simp |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2771 |
finally show ?thesis using A x y lu |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2772 |
by (simp add: bnds_powr_def uln_def Let_def del: lb_ln.simps ub_ln.simps) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2773 |
qed (insert x y lu A, simp_all add: bnds_powr_def Let_def ub_exp_nonneg |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2774 |
del: lb_ln.simps ub_ln.simps) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2775 |
next |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2776 |
assume "l1 \<le> 0" "\<not>(l1 = 0 \<and> (u1 = 0 \<or> l2 \<ge> 1))" |
62390 | 2777 |
with lu show ?thesis by (simp add: bnds_powr_def split: if_split_asm) |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2778 |
next |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2779 |
assume l1: "l1 > 0" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2780 |
obtain lm um where lmum: |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2781 |
"(lm, um) = bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2782 |
by (cases "bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2") simp |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2783 |
with l1 have "(l, u) = map_bnds lb_exp ub_exp prec (lm, um)" |
62390 | 2784 |
using lu by (simp add: bnds_powr_def del: lb_ln.simps ub_ln.simps split: if_split_asm) |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2785 |
hence "exp (ln x * y) \<in> {real_of_float l..real_of_float u}" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2786 |
proof (rule map_bnds[OF _ mono_exp_real], goal_cases) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2787 |
case 1 |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2788 |
let ?lln = "the (lb_ln prec l1)" and ?uln = "the (ub_ln prec u1)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2789 |
from ub_ln_lb_ln_bounds[of l1 prec] ub_ln_lb_ln_bounds[of u1 prec] x l1 |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2790 |
have "real_of_float ?lln \<le> ln (real_of_float l1) \<and> |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2791 |
ln (real_of_float u1) \<le> real_of_float ?uln" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2792 |
by (auto simp del: lb_ln.simps ub_ln.simps) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2793 |
moreover from l1 x have "ln (real_of_float l1) \<le> ln x \<and> ln x \<le> ln (real_of_float u1)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2794 |
by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2795 |
ultimately have ln: "real_of_float ?lln \<le> ln x \<and> ln x \<le> real_of_float ?uln" by simp |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2796 |
from lmum show ?case |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2797 |
by (rule bnds_mult) (insert y ln, simp_all) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2798 |
qed (insert exp_boundaries[of lm prec] exp_boundaries[of um prec], simp_all) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2799 |
with x l1 show ?thesis |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2800 |
by (simp add: powr_def mult_ac) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2801 |
qed |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2802 |
qed |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2803 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2804 |
|
29805 | 2805 |
section "Implement floatarith" |
2806 |
||
2807 |
subsection "Define syntax and semantics" |
|
2808 |
||
58310 | 2809 |
datatype floatarith |
29805 | 2810 |
= Add floatarith floatarith |
2811 |
| Minus floatarith |
|
2812 |
| Mult floatarith floatarith |
|
2813 |
| Inverse floatarith |
|
2814 |
| Cos floatarith |
|
2815 |
| Arctan floatarith |
|
2816 |
| Abs floatarith |
|
2817 |
| Max floatarith floatarith |
|
2818 |
| Min floatarith floatarith |
|
2819 |
| Pi |
|
2820 |
| Sqrt floatarith |
|
2821 |
| Exp floatarith |
|
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2822 |
| Powr floatarith floatarith |
29805 | 2823 |
| Ln floatarith |
2824 |
| Power floatarith nat |
|
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
2825 |
| Floor floatarith |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
2826 |
| Var nat |
29805 | 2827 |
| Num float |
2828 |
||
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
2829 |
fun interpret_floatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real" where |
31098
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2830 |
"interpret_floatarith (Add a b) vs = (interpret_floatarith a vs) + (interpret_floatarith b vs)" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2831 |
"interpret_floatarith (Minus a) vs = - (interpret_floatarith a vs)" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2832 |
"interpret_floatarith (Mult a b) vs = (interpret_floatarith a vs) * (interpret_floatarith b vs)" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2833 |
"interpret_floatarith (Inverse a) vs = inverse (interpret_floatarith a vs)" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2834 |
"interpret_floatarith (Cos a) vs = cos (interpret_floatarith a vs)" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2835 |
"interpret_floatarith (Arctan a) vs = arctan (interpret_floatarith a vs)" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2836 |
"interpret_floatarith (Min a b) vs = min (interpret_floatarith a vs) (interpret_floatarith b vs)" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2837 |
"interpret_floatarith (Max a b) vs = max (interpret_floatarith a vs) (interpret_floatarith b vs)" | |
61945 | 2838 |
"interpret_floatarith (Abs a) vs = \<bar>interpret_floatarith a vs\<bar>" | |
31098
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2839 |
"interpret_floatarith Pi vs = pi" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2840 |
"interpret_floatarith (Sqrt a) vs = sqrt (interpret_floatarith a vs)" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2841 |
"interpret_floatarith (Exp a) vs = exp (interpret_floatarith a vs)" | |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2842 |
"interpret_floatarith (Powr a b) vs = interpret_floatarith a vs powr interpret_floatarith b vs" | |
31098
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2843 |
"interpret_floatarith (Ln a) vs = ln (interpret_floatarith a vs)" | |
73dd67adf90a
replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents:
30971
diff
changeset
|
2844 |
"interpret_floatarith (Power a n) vs = (interpret_floatarith a vs)^n" | |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
2845 |
"interpret_floatarith (Floor a) vs = floor (interpret_floatarith a vs)" | |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2846 |
"interpret_floatarith (Num f) vs = f" | |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
2847 |
"interpret_floatarith (Var n) vs = vs ! n" |
29805 | 2848 |
|
60680 | 2849 |
lemma interpret_floatarith_divide: |
2850 |
"interpret_floatarith (Mult a (Inverse b)) vs = |
|
2851 |
(interpret_floatarith a vs) / (interpret_floatarith b vs)" |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2852 |
unfolding divide_inverse interpret_floatarith.simps .. |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2853 |
|
60680 | 2854 |
lemma interpret_floatarith_diff: |
2855 |
"interpret_floatarith (Add a (Minus b)) vs = |
|
2856 |
(interpret_floatarith a vs) - (interpret_floatarith b vs)" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
2857 |
unfolding interpret_floatarith.simps by simp |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2858 |
|
60680 | 2859 |
lemma interpret_floatarith_sin: |
2860 |
"interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) vs = |
|
2861 |
sin (interpret_floatarith a vs)" |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2862 |
unfolding sin_cos_eq interpret_floatarith.simps |
60680 | 2863 |
interpret_floatarith_divide interpret_floatarith_diff |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2864 |
by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2865 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2866 |
|
29805 | 2867 |
subsection "Implement approximation function" |
2868 |
||
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2869 |
fun lift_bin :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float) option) \<Rightarrow> (float * float) option" where |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2870 |
"lift_bin (Some (l1, u1)) (Some (l2, u2)) f = f l1 u1 l2 u2" | |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2871 |
"lift_bin a b f = None" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2872 |
|
29805 | 2873 |
fun lift_bin' :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where |
2874 |
"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" | |
|
2875 |
"lift_bin' a b f = None" |
|
2876 |
||
2877 |
fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where |
|
2878 |
"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u) |
|
2879 |
| t \<Rightarrow> None)" | |
|
2880 |
"lift_un b f = None" |
|
2881 |
||
2882 |
fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where |
|
2883 |
"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" | |
|
2884 |
"lift_un' b f = None" |
|
2885 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2886 |
definition bounded_by :: "real list \<Rightarrow> (float \<times> float) option list \<Rightarrow> bool" where |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2887 |
"bounded_by xs vs \<longleftrightarrow> |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2888 |
(\<forall> i < length vs. case vs ! i of None \<Rightarrow> True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2889 |
| Some (l, u) \<Rightarrow> xs ! i \<in> { real_of_float l .. real_of_float u })" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2890 |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2891 |
lemma bounded_byE: |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2892 |
assumes "bounded_by xs vs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2893 |
shows "\<And> i. i < length vs \<Longrightarrow> case vs ! i of None \<Rightarrow> True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2894 |
| Some (l, u) \<Rightarrow> xs ! i \<in> { real_of_float l .. real_of_float u }" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2895 |
using assms bounded_by_def by blast |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2896 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2897 |
lemma bounded_by_update: |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2898 |
assumes "bounded_by xs vs" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2899 |
and bnd: "xs ! i \<in> { real_of_float l .. real_of_float u }" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2900 |
shows "bounded_by xs (vs[i := Some (l,u)])" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2901 |
proof - |
60680 | 2902 |
{ |
2903 |
fix j |
|
2904 |
let ?vs = "vs[i := Some (l,u)]" |
|
2905 |
assume "j < length ?vs" |
|
2906 |
hence [simp]: "j < length vs" by simp |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2907 |
have "case ?vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> xs ! j \<in> { real_of_float l .. real_of_float u }" |
60680 | 2908 |
proof (cases "?vs ! j") |
2909 |
case (Some b) |
|
2910 |
thus ?thesis |
|
2911 |
proof (cases "i = j") |
|
2912 |
case True |
|
2913 |
thus ?thesis using \<open>?vs ! j = Some b\<close> and bnd by auto |
|
2914 |
next |
|
2915 |
case False |
|
2916 |
thus ?thesis using \<open>bounded_by xs vs\<close> unfolding bounded_by_def by auto |
|
2917 |
qed |
|
2918 |
qed auto |
|
2919 |
} |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2920 |
thus ?thesis unfolding bounded_by_def by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2921 |
qed |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2922 |
|
60680 | 2923 |
lemma bounded_by_None: "bounded_by xs (replicate (length xs) None)" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2924 |
unfolding bounded_by_def by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2925 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2926 |
fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option" where |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2927 |
"approx' prec a bs = (case (approx prec a bs) of Some (l, u) \<Rightarrow> Some (float_round_down prec l, float_round_up prec u) | None \<Rightarrow> None)" | |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2928 |
"approx prec (Add a b) bs = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2929 |
lift_bin' (approx' prec a bs) (approx' prec b bs) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2930 |
(\<lambda> l1 u1 l2 u2. (float_plus_down prec l1 l2, float_plus_up prec u1 u2))" | |
29805 | 2931 |
"approx prec (Minus a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" | |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2932 |
"approx prec (Mult a b) bs = |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2933 |
lift_bin' (approx' prec a bs) (approx' prec b bs) (bnds_mult prec)" | |
29805 | 2934 |
"approx prec (Inverse a) bs = lift_un (approx' prec a bs) (\<lambda> l u. if (0 < l \<or> u < 0) then (Some (float_divl prec 1 u), Some (float_divr prec 1 l)) else (None, None))" | |
2935 |
"approx prec (Cos a) bs = lift_un' (approx' prec a bs) (bnds_cos prec)" | |
|
2936 |
"approx prec Pi bs = Some (lb_pi prec, ub_pi prec)" | |
|
2937 |
"approx prec (Min a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (min l1 l2, min u1 u2))" | |
|
2938 |
"approx prec (Max a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (max l1 l2, max u1 u2))" | |
|
2939 |
"approx prec (Abs a) bs = lift_un' (approx' prec a bs) (\<lambda>l u. (if l < 0 \<and> 0 < u then 0 else min \<bar>l\<bar> \<bar>u\<bar>, max \<bar>l\<bar> \<bar>u\<bar>))" | |
|
2940 |
"approx prec (Arctan a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" | |
|
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:
31148
diff
changeset
|
2941 |
"approx prec (Sqrt a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" | |
29805 | 2942 |
"approx prec (Exp a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" | |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2943 |
"approx prec (Powr a b) bs = lift_bin (approx' prec a bs) (approx' prec b bs) (bnds_powr prec)" | |
29805 | 2944 |
"approx prec (Ln a) bs = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" | |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2945 |
"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds prec n)" | |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
2946 |
"approx prec (Floor a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (floor_fl l, floor_fl u))" | |
29805 | 2947 |
"approx prec (Num f) bs = Some (f, f)" | |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
2948 |
"approx prec (Var i) bs = (if i < length bs then bs ! i else None)" |
29805 | 2949 |
|
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2950 |
lemma approx_approx': |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2951 |
assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2952 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2953 |
and approx': "Some (l, u) = approx' prec a vs" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2954 |
shows "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2955 |
proof - |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2956 |
obtain l' u' where S: "Some (l', u') = approx prec a vs" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2957 |
using approx' unfolding approx'.simps by (cases "approx prec a vs") auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2958 |
have l': "l = float_round_down prec l'" and u': "u = float_round_up prec u'" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2959 |
using approx' unfolding approx'.simps S[symmetric] by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2960 |
show ?thesis unfolding l' u' |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2961 |
using order_trans[OF Pa[OF S, THEN conjunct2] float_round_up[of u']] |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2962 |
using order_trans[OF float_round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2963 |
qed |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2964 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2965 |
lemma lift_bin_ex: |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2966 |
assumes lift_bin_Some: "Some (l, u) = lift_bin a b f" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2967 |
shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2968 |
proof (cases a) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2969 |
case None |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2970 |
hence "None = lift_bin a b f" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2971 |
unfolding None lift_bin.simps .. |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2972 |
thus ?thesis |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2973 |
using lift_bin_Some by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2974 |
next |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2975 |
case (Some a') |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2976 |
show ?thesis |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2977 |
proof (cases b) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2978 |
case None |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2979 |
hence "None = lift_bin a b f" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2980 |
unfolding None lift_bin.simps .. |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2981 |
thus ?thesis using lift_bin_Some by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2982 |
next |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2983 |
case (Some b') |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2984 |
obtain la ua where a': "a' = (la, ua)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2985 |
by (cases a') auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2986 |
obtain lb ub where b': "b' = (lb, ub)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2987 |
by (cases b') auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2988 |
thus ?thesis |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2989 |
unfolding \<open>a = Some a'\<close> \<open>b = Some b'\<close> a' b' by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2990 |
qed |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2991 |
qed |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2992 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2993 |
lemma lift_bin_f: |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2994 |
assumes lift_bin_Some: "Some (l, u) = lift_bin (g a) (g b) f" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2995 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2996 |
and Pb: "\<And>l u. Some (l, u) = g b \<Longrightarrow> P l u b" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2997 |
shows "\<exists> l1 u1 l2 u2. P l1 u1 a \<and> P l2 u2 b \<and> Some (l, u) = f l1 u1 l2 u2" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2998 |
proof - |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
2999 |
obtain l1 u1 l2 u2 |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3000 |
where Sa: "Some (l1, u1) = g a" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3001 |
and Sb: "Some (l2, u2) = g b" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3002 |
using lift_bin_ex[OF assms(1)] by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3003 |
have lu: "Some (l, u) = f l1 u1 l2 u2" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3004 |
using lift_bin_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin.simps] by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3005 |
thus ?thesis |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3006 |
using Pa[OF Sa] Pb[OF Sb] by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3007 |
qed |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3008 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3009 |
lemma lift_bin: |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3010 |
assumes lift_bin_Some: "Some (l, u) = lift_bin (approx' prec a bs) (approx' prec b bs) f" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3011 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3012 |
real_of_float l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real_of_float u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3013 |
and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3014 |
real_of_float l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real_of_float u" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3015 |
shows "\<exists>l1 u1 l2 u2. (real_of_float l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real_of_float u1) \<and> |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3016 |
(real_of_float l2 \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real_of_float u2) \<and> |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3017 |
Some (l, u) = (f l1 u1 l2 u2)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3018 |
proof - |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3019 |
{ fix l u assume "Some (l, u) = approx' prec a bs" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3020 |
with approx_approx'[of prec a bs, OF _ this] Pa |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3021 |
have "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" by auto } note Pa = this |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3022 |
{ fix l u assume "Some (l, u) = approx' prec b bs" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3023 |
with approx_approx'[of prec b bs, OF _ this] Pb |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3024 |
have "l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u" by auto } note Pb = this |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3025 |
|
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3026 |
from lift_bin_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin_Some, OF Pa Pb] |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3027 |
show ?thesis by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3028 |
qed |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3029 |
|
29805 | 3030 |
lemma lift_bin'_ex: |
3031 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f" |
|
3032 |
shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b" |
|
3033 |
proof (cases a) |
|
60680 | 3034 |
case None |
3035 |
hence "None = lift_bin' a b f" |
|
3036 |
unfolding None lift_bin'.simps .. |
|
3037 |
thus ?thesis |
|
3038 |
using lift_bin'_Some by auto |
|
29805 | 3039 |
next |
3040 |
case (Some a') |
|
3041 |
show ?thesis |
|
3042 |
proof (cases b) |
|
60680 | 3043 |
case None |
3044 |
hence "None = lift_bin' a b f" |
|
3045 |
unfolding None lift_bin'.simps .. |
|
29805 | 3046 |
thus ?thesis using lift_bin'_Some by auto |
3047 |
next |
|
3048 |
case (Some b') |
|
60680 | 3049 |
obtain la ua where a': "a' = (la, ua)" |
3050 |
by (cases a') auto |
|
3051 |
obtain lb ub where b': "b' = (lb, ub)" |
|
3052 |
by (cases b') auto |
|
3053 |
thus ?thesis |
|
3054 |
unfolding \<open>a = Some a'\<close> \<open>b = Some b'\<close> a' b' by auto |
|
29805 | 3055 |
qed |
3056 |
qed |
|
3057 |
||
3058 |
lemma lift_bin'_f: |
|
3059 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f" |
|
60680 | 3060 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" |
3061 |
and Pb: "\<And>l u. Some (l, u) = g b \<Longrightarrow> P l u b" |
|
29805 | 3062 |
shows "\<exists> l1 u1 l2 u2. P l1 u1 a \<and> P l2 u2 b \<and> l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)" |
3063 |
proof - |
|
3064 |
obtain l1 u1 l2 u2 |
|
60680 | 3065 |
where Sa: "Some (l1, u1) = g a" |
3066 |
and Sb: "Some (l2, u2) = g b" |
|
3067 |
using lift_bin'_ex[OF assms(1)] by auto |
|
3068 |
have lu: "(l, u) = f l1 u1 l2 u2" |
|
3069 |
using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto |
|
3070 |
have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" |
|
3071 |
unfolding lu[symmetric] by auto |
|
3072 |
thus ?thesis |
|
3073 |
using Pa[OF Sa] Pb[OF Sb] by auto |
|
29805 | 3074 |
qed |
3075 |
||
3076 |
lemma lift_bin': |
|
3077 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f" |
|
60680 | 3078 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
3079 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
|
3080 |
and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> |
|
3081 |
l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u" |
|
3082 |
shows "\<exists>l1 u1 l2 u2. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and> |
|
3083 |
(l2 \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u2) \<and> |
|
3084 |
l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)" |
|
29805 | 3085 |
proof - |
3086 |
{ fix l u assume "Some (l, u) = approx' prec a bs" |
|
3087 |
with approx_approx'[of prec a bs, OF _ this] Pa |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3088 |
have "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" by auto } note Pa = this |
29805 | 3089 |
{ fix l u assume "Some (l, u) = approx' prec b bs" |
3090 |
with approx_approx'[of prec b bs, OF _ this] Pb |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3091 |
have "l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u" by auto } note Pb = this |
29805 | 3092 |
|
3093 |
from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb] |
|
3094 |
show ?thesis by auto |
|
3095 |
qed |
|
3096 |
||
3097 |
lemma lift_un'_ex: |
|
3098 |
assumes lift_un'_Some: "Some (l, u) = lift_un' a f" |
|
3099 |
shows "\<exists> l u. Some (l, u) = a" |
|
3100 |
proof (cases a) |
|
60680 | 3101 |
case None |
3102 |
hence "None = lift_un' a f" |
|
3103 |
unfolding None lift_un'.simps .. |
|
3104 |
thus ?thesis |
|
3105 |
using lift_un'_Some by auto |
|
29805 | 3106 |
next |
3107 |
case (Some a') |
|
60680 | 3108 |
obtain la ua where a': "a' = (la, ua)" |
3109 |
by (cases a') auto |
|
3110 |
thus ?thesis |
|
3111 |
unfolding \<open>a = Some a'\<close> a' by auto |
|
29805 | 3112 |
qed |
3113 |
||
3114 |
lemma lift_un'_f: |
|
3115 |
assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f" |
|
60680 | 3116 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" |
29805 | 3117 |
shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)" |
3118 |
proof - |
|
60680 | 3119 |
obtain l1 u1 where Sa: "Some (l1, u1) = g a" |
3120 |
using lift_un'_ex[OF assms(1)] by auto |
|
3121 |
have lu: "(l, u) = f l1 u1" |
|
3122 |
using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto |
|
3123 |
have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" |
|
3124 |
unfolding lu[symmetric] by auto |
|
3125 |
thus ?thesis |
|
3126 |
using Pa[OF Sa] by auto |
|
29805 | 3127 |
qed |
3128 |
||
3129 |
lemma lift_un': |
|
3130 |
assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f" |
|
60680 | 3131 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
3132 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
|
3133 |
(is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
|
3134 |
shows "\<exists>l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and> |
|
3135 |
l = fst (f l1 u1) \<and> u = snd (f l1 u1)" |
|
29805 | 3136 |
proof - |
60680 | 3137 |
have Pa: "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
3138 |
if "Some (l, u) = approx' prec a bs" for l u |
|
3139 |
using approx_approx'[of prec a bs, OF _ that] Pa |
|
3140 |
by auto |
|
29805 | 3141 |
from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa] |
3142 |
show ?thesis by auto |
|
3143 |
qed |
|
3144 |
||
3145 |
lemma lift_un'_bnds: |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3146 |
assumes bnds: "\<forall> (x::real) lx ux. (l, u) = f lx ux \<and> x \<in> { lx .. ux } \<longrightarrow> l \<le> f' x \<and> f' x \<le> u" |
60680 | 3147 |
and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f" |
3148 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
|
3149 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3150 |
shows "real_of_float l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real_of_float u" |
29805 | 3151 |
proof - |
3152 |
from lift_un'[OF lift_un'_Some Pa] |
|
60680 | 3153 |
obtain l1 u1 where "l1 \<le> interpret_floatarith a xs" |
3154 |
and "interpret_floatarith a xs \<le> u1" |
|
3155 |
and "l = fst (f l1 u1)" |
|
3156 |
and "u = snd (f l1 u1)" |
|
3157 |
by blast |
|
3158 |
hence "(l, u) = f l1 u1" and "interpret_floatarith a xs \<in> {l1 .. u1}" |
|
3159 |
by auto |
|
3160 |
thus ?thesis |
|
3161 |
using bnds by auto |
|
29805 | 3162 |
qed |
3163 |
||
3164 |
lemma lift_un_ex: |
|
3165 |
assumes lift_un_Some: "Some (l, u) = lift_un a f" |
|
60680 | 3166 |
shows "\<exists>l u. Some (l, u) = a" |
29805 | 3167 |
proof (cases a) |
60680 | 3168 |
case None |
3169 |
hence "None = lift_un a f" |
|
3170 |
unfolding None lift_un.simps .. |
|
3171 |
thus ?thesis |
|
3172 |
using lift_un_Some by auto |
|
29805 | 3173 |
next |
3174 |
case (Some a') |
|
60680 | 3175 |
obtain la ua where a': "a' = (la, ua)" |
3176 |
by (cases a') auto |
|
3177 |
thus ?thesis |
|
3178 |
unfolding \<open>a = Some a'\<close> a' by auto |
|
29805 | 3179 |
qed |
3180 |
||
3181 |
lemma lift_un_f: |
|
3182 |
assumes lift_un_Some: "Some (l, u) = lift_un (g a) f" |
|
60680 | 3183 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" |
29805 | 3184 |
shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)" |
3185 |
proof - |
|
60680 | 3186 |
obtain l1 u1 where Sa: "Some (l1, u1) = g a" |
3187 |
using lift_un_ex[OF assms(1)] by auto |
|
29805 | 3188 |
have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None" |
3189 |
proof (rule ccontr) |
|
3190 |
assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)" |
|
3191 |
hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto |
|
31809 | 3192 |
hence "lift_un (g a) f = None" |
29805 | 3193 |
proof (cases "fst (f l1 u1) = None") |
3194 |
case True |
|
60680 | 3195 |
then obtain b where b: "f l1 u1 = (None, b)" |
3196 |
by (cases "f l1 u1") auto |
|
3197 |
thus ?thesis |
|
3198 |
unfolding Sa[symmetric] lift_un.simps b by auto |
|
29805 | 3199 |
next |
60680 | 3200 |
case False |
3201 |
hence "snd (f l1 u1) = None" |
|
3202 |
using or by auto |
|
3203 |
with False obtain b where b: "f l1 u1 = (Some b, None)" |
|
3204 |
by (cases "f l1 u1") auto |
|
3205 |
thus ?thesis |
|
3206 |
unfolding Sa[symmetric] lift_un.simps b by auto |
|
29805 | 3207 |
qed |
60680 | 3208 |
thus False |
3209 |
using lift_un_Some by auto |
|
29805 | 3210 |
qed |
60680 | 3211 |
then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" |
3212 |
by (cases "f l1 u1") auto |
|
29805 | 3213 |
from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f] |
60680 | 3214 |
have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" |
3215 |
unfolding f by auto |
|
3216 |
thus ?thesis |
|
3217 |
unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto |
|
29805 | 3218 |
qed |
3219 |
||
3220 |
lemma lift_un: |
|
3221 |
assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f" |
|
60680 | 3222 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
3223 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
|
3224 |
(is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
|
3225 |
shows "\<exists>l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and> |
|
29805 | 3226 |
Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)" |
3227 |
proof - |
|
60680 | 3228 |
have Pa: "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
3229 |
if "Some (l, u) = approx' prec a bs" for l u |
|
3230 |
using approx_approx'[of prec a bs, OF _ that] Pa by auto |
|
29805 | 3231 |
from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa] |
3232 |
show ?thesis by auto |
|
3233 |
qed |
|
3234 |
||
3235 |
lemma lift_un_bnds: |
|
60680 | 3236 |
assumes bnds: "\<forall>(x::real) lx ux. (Some l, Some u) = f lx ux \<and> x \<in> { lx .. ux } \<longrightarrow> l \<le> f' x \<and> f' x \<le> u" |
3237 |
and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f" |
|
3238 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
|
3239 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3240 |
shows "real_of_float l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real_of_float u" |
29805 | 3241 |
proof - |
3242 |
from lift_un[OF lift_un_Some Pa] |
|
60680 | 3243 |
obtain l1 u1 where "l1 \<le> interpret_floatarith a xs" |
3244 |
and "interpret_floatarith a xs \<le> u1" |
|
3245 |
and "Some l = fst (f l1 u1)" |
|
3246 |
and "Some u = snd (f l1 u1)" |
|
3247 |
by blast |
|
3248 |
hence "(Some l, Some u) = f l1 u1" and "interpret_floatarith a xs \<in> {l1 .. u1}" |
|
3249 |
by auto |
|
3250 |
thus ?thesis |
|
3251 |
using bnds by auto |
|
29805 | 3252 |
qed |
3253 |
||
3254 |
lemma approx: |
|
3255 |
assumes "bounded_by xs vs" |
|
60680 | 3256 |
and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith") |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3257 |
shows "l \<le> interpret_floatarith arith xs \<and> interpret_floatarith arith xs \<le> u" (is "?P l u arith") |
60533 | 3258 |
using \<open>Some (l, u) = approx prec arith vs\<close> |
45129
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
44821
diff
changeset
|
3259 |
proof (induct arith arbitrary: l u) |
29805 | 3260 |
case (Add a b) |
3261 |
from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps |
|
60680 | 3262 |
obtain l1 u1 l2 u2 where "l = float_plus_down prec l1 l2" |
3263 |
and "u = float_plus_up prec u1 u2" "l1 \<le> interpret_floatarith a xs" |
|
3264 |
and "interpret_floatarith a xs \<le> u1" "l2 \<le> interpret_floatarith b xs" |
|
3265 |
and "interpret_floatarith b xs \<le> u2" |
|
3266 |
unfolding fst_conv snd_conv by blast |
|
3267 |
thus ?case |
|
3268 |
unfolding interpret_floatarith.simps by (auto intro!: float_plus_up_le float_plus_down_le) |
|
29805 | 3269 |
next |
3270 |
case (Minus a) |
|
3271 |
from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps |
|
60680 | 3272 |
obtain l1 u1 where "l = -u1" "u = -l1" |
3273 |
and "l1 \<le> interpret_floatarith a xs" "interpret_floatarith a xs \<le> u1" |
|
3274 |
unfolding fst_conv snd_conv by blast |
|
3275 |
thus ?case |
|
3276 |
unfolding interpret_floatarith.simps using minus_float.rep_eq by auto |
|
29805 | 3277 |
next |
3278 |
case (Mult a b) |
|
3279 |
from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps |
|
31809 | 3280 |
obtain l1 u1 l2 u2 |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3281 |
where l: "l = fst (bnds_mult prec l1 u1 l2 u2)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3282 |
and u: "u = snd (bnds_mult prec l1 u1 l2 u2)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3283 |
and a: "l1 \<le> interpret_floatarith a xs" "interpret_floatarith a xs \<le> u1" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3284 |
and b: "l2 \<le> interpret_floatarith b xs" "interpret_floatarith b xs \<le> u2" unfolding fst_conv snd_conv by blast |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3285 |
from l u have lu: "(l, u) = bnds_mult prec l1 u1 l2 u2" by simp |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3286 |
from bnds_mult[OF lu] a b show ?case by simp |
29805 | 3287 |
next |
3288 |
case (Inverse a) |
|
3289 |
from lift_un[OF Inverse.prems[unfolded approx.simps], unfolded if_distrib[of fst] if_distrib[of snd] fst_conv snd_conv] Inverse.hyps |
|
31809 | 3290 |
obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)" |
29805 | 3291 |
and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)" |
60680 | 3292 |
and l1: "l1 \<le> interpret_floatarith a xs" |
3293 |
and u1: "interpret_floatarith a xs \<le> u1" |
|
3294 |
by blast |
|
3295 |
have either: "0 < l1 \<or> u1 < 0" |
|
3296 |
proof (rule ccontr) |
|
3297 |
assume P: "\<not> (0 < l1 \<or> u1 < 0)" |
|
3298 |
show False |
|
3299 |
using l' unfolding if_not_P[OF P] by auto |
|
3300 |
qed |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3301 |
moreover have l1_le_u1: "real_of_float l1 \<le> real_of_float u1" |
60680 | 3302 |
using l1 u1 by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3303 |
ultimately have "real_of_float l1 \<noteq> 0" and "real_of_float u1 \<noteq> 0" |
60680 | 3304 |
by auto |
29805 | 3305 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3306 |
have inv: "inverse u1 \<le> inverse (interpret_floatarith a xs) |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3307 |
\<and> inverse (interpret_floatarith a xs) \<le> inverse l1" |
29805 | 3308 |
proof (cases "0 < l1") |
60680 | 3309 |
case True |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3310 |
hence "0 < real_of_float u1" and "0 < real_of_float l1" "0 < interpret_floatarith a xs" |
47600 | 3311 |
using l1_le_u1 l1 by auto |
29805 | 3312 |
show ?thesis |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3313 |
unfolding inverse_le_iff_le[OF \<open>0 < real_of_float u1\<close> \<open>0 < interpret_floatarith a xs\<close>] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3314 |
inverse_le_iff_le[OF \<open>0 < interpret_floatarith a xs\<close> \<open>0 < real_of_float l1\<close>] |
29805 | 3315 |
using l1 u1 by auto |
3316 |
next |
|
60680 | 3317 |
case False |
3318 |
hence "u1 < 0" |
|
3319 |
using either by blast |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3320 |
hence "real_of_float u1 < 0" and "real_of_float l1 < 0" "interpret_floatarith a xs < 0" |
47600 | 3321 |
using l1_le_u1 u1 by auto |
29805 | 3322 |
show ?thesis |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3323 |
unfolding inverse_le_iff_le_neg[OF \<open>real_of_float u1 < 0\<close> \<open>interpret_floatarith a xs < 0\<close>] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3324 |
inverse_le_iff_le_neg[OF \<open>interpret_floatarith a xs < 0\<close> \<open>real_of_float l1 < 0\<close>] |
29805 | 3325 |
using l1 u1 by auto |
3326 |
qed |
|
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
3327 |
|
60680 | 3328 |
from l' have "l = float_divl prec 1 u1" |
3329 |
by (cases "0 < l1 \<or> u1 < 0") auto |
|
3330 |
hence "l \<le> inverse u1" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3331 |
unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float u1 \<noteq> 0\<close>] |
60680 | 3332 |
using float_divl[of prec 1 u1] by auto |
3333 |
also have "\<dots> \<le> inverse (interpret_floatarith a xs)" |
|
3334 |
using inv by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3335 |
finally have "l \<le> inverse (interpret_floatarith a xs)" . |
29805 | 3336 |
moreover |
60680 | 3337 |
from u' have "u = float_divr prec 1 l1" |
3338 |
by (cases "0 < l1 \<or> u1 < 0") auto |
|
3339 |
hence "inverse l1 \<le> u" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3340 |
unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float l1 \<noteq> 0\<close>] |
60680 | 3341 |
using float_divr[of 1 l1 prec] by auto |
3342 |
hence "inverse (interpret_floatarith a xs) \<le> u" |
|
3343 |
by (rule order_trans[OF inv[THEN conjunct2]]) |
|
3344 |
ultimately show ?case |
|
3345 |
unfolding interpret_floatarith.simps using l1 u1 by auto |
|
29805 | 3346 |
next |
3347 |
case (Abs x) |
|
3348 |
from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps |
|
60680 | 3349 |
obtain l1 u1 where l': "l = (if l1 < 0 \<and> 0 < u1 then 0 else min \<bar>l1\<bar> \<bar>u1\<bar>)" |
3350 |
and u': "u = max \<bar>l1\<bar> \<bar>u1\<bar>" |
|
3351 |
and l1: "l1 \<le> interpret_floatarith x xs" |
|
3352 |
and u1: "interpret_floatarith x xs \<le> u1" |
|
3353 |
by blast |
|
3354 |
thus ?case |
|
3355 |
unfolding l' u' |
|
3356 |
by (cases "l1 < 0 \<and> 0 < u1") (auto simp add: real_of_float_min real_of_float_max) |
|
29805 | 3357 |
next |
3358 |
case (Min a b) |
|
3359 |
from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps |
|
3360 |
obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3361 |
and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1" |
60680 | 3362 |
and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> u2" |
3363 |
by blast |
|
3364 |
thus ?case |
|
3365 |
unfolding l' u' by (auto simp add: real_of_float_min) |
|
29805 | 3366 |
next |
3367 |
case (Max a b) |
|
3368 |
from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps |
|
3369 |
obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3370 |
and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1" |
60680 | 3371 |
and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> u2" |
3372 |
by blast |
|
3373 |
thus ?case |
|
3374 |
unfolding l' u' by (auto simp add: real_of_float_max) |
|
3375 |
next |
|
3376 |
case (Cos a) |
|
3377 |
with lift_un'_bnds[OF bnds_cos] show ?case by auto |
|
3378 |
next |
|
3379 |
case (Arctan a) |
|
3380 |
with lift_un'_bnds[OF bnds_arctan] show ?case by auto |
|
3381 |
next |
|
3382 |
case Pi |
|
3383 |
with pi_boundaries show ?case by auto |
|
3384 |
next |
|
3385 |
case (Sqrt a) |
|
3386 |
with lift_un'_bnds[OF bnds_sqrt] show ?case by auto |
|
3387 |
next |
|
3388 |
case (Exp a) |
|
3389 |
with lift_un'_bnds[OF bnds_exp] show ?case by auto |
|
3390 |
next |
|
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3391 |
case (Powr a b) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3392 |
from lift_bin[OF Powr.prems[unfolded approx.simps]] Powr.hyps |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3393 |
obtain l1 u1 l2 u2 where lu: "Some (l, u) = bnds_powr prec l1 u1 l2 u2" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3394 |
and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3395 |
and l2: "l2 \<le> interpret_floatarith b xs" and u2: "interpret_floatarith b xs \<le> u2" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3396 |
by blast |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3397 |
from bnds_powr[OF lu] l1 u1 l2 u2 |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3398 |
show ?case by simp |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3399 |
next |
60680 | 3400 |
case (Ln a) |
3401 |
with lift_un_bnds[OF bnds_ln] show ?case by auto |
|
3402 |
next |
|
3403 |
case (Power a n) |
|
3404 |
with lift_un'_bnds[OF bnds_power] show ?case by auto |
|
3405 |
next |
|
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3406 |
case (Floor a) |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3407 |
from lift_un'[OF Floor.prems[unfolded approx.simps] Floor.hyps] |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3408 |
show ?case by (auto simp: floor_fl.rep_eq floor_mono) |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3409 |
next |
60680 | 3410 |
case (Num f) |
3411 |
thus ?case by auto |
|
29805 | 3412 |
next |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3413 |
case (Var n) |
60533 | 3414 |
from this[symmetric] \<open>bounded_by xs vs\<close>[THEN bounded_byE, of n] |
60680 | 3415 |
show ?case by (cases "n < length vs") auto |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3416 |
qed |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3417 |
|
58310 | 3418 |
datatype form = Bound floatarith floatarith floatarith form |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3419 |
| Assign floatarith floatarith form |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3420 |
| Less floatarith floatarith |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3421 |
| LessEqual floatarith floatarith |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3422 |
| AtLeastAtMost floatarith floatarith floatarith |
58986 | 3423 |
| Conj form form |
3424 |
| Disj form form |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3425 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3426 |
fun interpret_form :: "form \<Rightarrow> real list \<Rightarrow> bool" where |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3427 |
"interpret_form (Bound x a b f) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs } \<longrightarrow> interpret_form f vs)" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3428 |
"interpret_form (Assign x a f) vs = (interpret_floatarith x vs = interpret_floatarith a vs \<longrightarrow> interpret_form f vs)" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3429 |
"interpret_form (Less a b) vs = (interpret_floatarith a vs < interpret_floatarith b vs)" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3430 |
"interpret_form (LessEqual a b) vs = (interpret_floatarith a vs \<le> interpret_floatarith b vs)" | |
58986 | 3431 |
"interpret_form (AtLeastAtMost x a b) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs })" | |
3432 |
"interpret_form (Conj f g) vs \<longleftrightarrow> interpret_form f vs \<and> interpret_form g vs" | |
|
3433 |
"interpret_form (Disj f g) vs \<longleftrightarrow> interpret_form f vs \<or> interpret_form g vs" |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3434 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3435 |
fun approx_form' and approx_form :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> nat list \<Rightarrow> bool" where |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3436 |
"approx_form' prec f 0 n l u bs ss = approx_form prec f (bs[n := Some (l, u)]) ss" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3437 |
"approx_form' prec f (Suc s) n l u bs ss = |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
3438 |
(let m = (l + u) * Float 1 (- 1) |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3439 |
in (if approx_form' prec f s n l m bs ss then approx_form' prec f s n m u bs ss else False))" | |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3440 |
"approx_form prec (Bound (Var n) a b f) bs ss = |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3441 |
(case (approx prec a bs, approx prec b bs) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3442 |
of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3443 |
| _ \<Rightarrow> False)" | |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3444 |
"approx_form prec (Assign (Var n) a f) bs ss = |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3445 |
(case (approx prec a bs) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3446 |
of (Some (l, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3447 |
| _ \<Rightarrow> False)" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3448 |
"approx_form prec (Less a b) bs ss = |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3449 |
(case (approx prec a bs, approx prec b bs) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3450 |
of (Some (l, u), Some (l', u')) \<Rightarrow> float_plus_up prec u (-l') < 0 |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3451 |
| _ \<Rightarrow> False)" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3452 |
"approx_form prec (LessEqual a b) bs ss = |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3453 |
(case (approx prec a bs, approx prec b bs) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3454 |
of (Some (l, u), Some (l', u')) \<Rightarrow> float_plus_up prec u (-l') \<le> 0 |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3455 |
| _ \<Rightarrow> False)" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3456 |
"approx_form prec (AtLeastAtMost x a b) bs ss = |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3457 |
(case (approx prec x bs, approx prec a bs, approx prec b bs) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3458 |
of (Some (lx, ux), Some (l, u), Some (l', u')) \<Rightarrow> float_plus_up prec u (-lx) \<le> 0 \<and> float_plus_up prec ux (-l') \<le> 0 |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3459 |
| _ \<Rightarrow> False)" | |
58986 | 3460 |
"approx_form prec (Conj a b) bs ss \<longleftrightarrow> approx_form prec a bs ss \<and> approx_form prec b bs ss" | |
3461 |
"approx_form prec (Disj a b) bs ss \<longleftrightarrow> approx_form prec a bs ss \<or> approx_form prec b bs ss" | |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3462 |
"approx_form _ _ _ _ = False" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3463 |
|
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3464 |
lemma lazy_conj: "(if A then B else False) = (A \<and> B)" by simp |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3465 |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3466 |
lemma approx_form_approx_form': |
60680 | 3467 |
assumes "approx_form' prec f s n l u bs ss" |
3468 |
and "(x::real) \<in> { l .. u }" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3469 |
obtains l' u' where "x \<in> { l' .. u' }" |
49351 | 3470 |
and "approx_form prec f (bs[n := Some (l', u')]) ss" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3471 |
using assms proof (induct s arbitrary: l u) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3472 |
case 0 |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3473 |
from this(1)[of l u] this(2,3) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3474 |
show thesis by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3475 |
next |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3476 |
case (Suc s) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3477 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
3478 |
let ?m = "(l + u) * Float 1 (- 1)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3479 |
have "real_of_float l \<le> ?m" and "?m \<le> real_of_float u" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
3480 |
unfolding less_eq_float_def using Suc.prems by auto |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3481 |
|
60533 | 3482 |
with \<open>x \<in> { l .. u }\<close> |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3483 |
have "x \<in> { l .. ?m} \<or> x \<in> { ?m .. u }" by auto |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3484 |
thus thesis |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3485 |
proof (rule disjE) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3486 |
assume *: "x \<in> { l .. ?m }" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3487 |
with Suc.hyps[OF _ _ *] Suc.prems |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3488 |
show thesis by (simp add: Let_def lazy_conj) |
29805 | 3489 |
next |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3490 |
assume *: "x \<in> { ?m .. u }" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3491 |
with Suc.hyps[OF _ _ *] Suc.prems |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3492 |
show thesis by (simp add: Let_def lazy_conj) |
29805 | 3493 |
qed |
3494 |
qed |
|
3495 |
||
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3496 |
lemma approx_form_aux: |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3497 |
assumes "approx_form prec f vs ss" |
49351 | 3498 |
and "bounded_by xs vs" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3499 |
shows "interpret_form f xs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3500 |
using assms proof (induct f arbitrary: vs) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3501 |
case (Bound x a b f) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3502 |
then obtain n |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3503 |
where x_eq: "x = Var n" by (cases x) auto |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3504 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3505 |
with Bound.prems obtain l u' l' u |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3506 |
where l_eq: "Some (l, u') = approx prec a vs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3507 |
and u_eq: "Some (l', u) = approx prec b vs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3508 |
and approx_form': "approx_form' prec f (ss ! n) n l u vs ss" |
37411
c88c44156083
removed simplifier congruence rule of "prod_case"
haftmann
parents:
37391
diff
changeset
|
3509 |
by (cases "approx prec a vs", simp) (cases "approx prec b vs", auto) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3510 |
|
60680 | 3511 |
have "interpret_form f xs" |
3512 |
if "xs ! n \<in> { interpret_floatarith a xs .. interpret_floatarith b xs }" |
|
3513 |
proof - |
|
3514 |
from approx[OF Bound.prems(2) l_eq] and approx[OF Bound.prems(2) u_eq] that |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3515 |
have "xs ! n \<in> { l .. u}" by auto |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3516 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3517 |
from approx_form_approx_form'[OF approx_form' this] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3518 |
obtain lx ux where bnds: "xs ! n \<in> { lx .. ux }" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3519 |
and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" . |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3520 |
|
60680 | 3521 |
from \<open>bounded_by xs vs\<close> bnds have "bounded_by xs (vs[n := Some (lx, ux)])" |
3522 |
by (rule bounded_by_update) |
|
3523 |
with Bound.hyps[OF approx_form] show ?thesis |
|
3524 |
by blast |
|
3525 |
qed |
|
3526 |
thus ?case |
|
3527 |
using interpret_form.simps x_eq and interpret_floatarith.simps by simp |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3528 |
next |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3529 |
case (Assign x a f) |
60680 | 3530 |
then obtain n where x_eq: "x = Var n" |
3531 |
by (cases x) auto |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3532 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
3533 |
with Assign.prems obtain l u |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3534 |
where bnd_eq: "Some (l, u) = approx prec a vs" |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3535 |
and x_eq: "x = Var n" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3536 |
and approx_form': "approx_form' prec f (ss ! n) n l u vs ss" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3537 |
by (cases "approx prec a vs") auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3538 |
|
60680 | 3539 |
have "interpret_form f xs" |
3540 |
if bnds: "xs ! n = interpret_floatarith a xs" |
|
3541 |
proof - |
|
3542 |
from approx[OF Assign.prems(2) bnd_eq] bnds |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3543 |
have "xs ! n \<in> { l .. u}" by auto |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3544 |
from approx_form_approx_form'[OF approx_form' this] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3545 |
obtain lx ux where bnds: "xs ! n \<in> { lx .. ux }" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3546 |
and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" . |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3547 |
|
60680 | 3548 |
from \<open>bounded_by xs vs\<close> bnds have "bounded_by xs (vs[n := Some (lx, ux)])" |
3549 |
by (rule bounded_by_update) |
|
3550 |
with Assign.hyps[OF approx_form] show ?thesis |
|
3551 |
by blast |
|
3552 |
qed |
|
3553 |
thus ?case |
|
3554 |
using interpret_form.simps x_eq and interpret_floatarith.simps by simp |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3555 |
next |
29805 | 3556 |
case (Less a b) |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3557 |
then obtain l u l' u' |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3558 |
where l_eq: "Some (l, u) = approx prec a vs" |
49351 | 3559 |
and u_eq: "Some (l', u') = approx prec b vs" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3560 |
and inequality: "real_of_float (float_plus_up prec u (-l')) < 0" |
60680 | 3561 |
by (cases "approx prec a vs", auto, cases "approx prec b vs", auto) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3562 |
from le_less_trans[OF float_plus_up inequality] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3563 |
approx[OF Less.prems(2) l_eq] approx[OF Less.prems(2) u_eq] |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3564 |
show ?case by auto |
29805 | 3565 |
next |
3566 |
case (LessEqual a b) |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3567 |
then obtain l u l' u' |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3568 |
where l_eq: "Some (l, u) = approx prec a vs" |
49351 | 3569 |
and u_eq: "Some (l', u') = approx prec b vs" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3570 |
and inequality: "real_of_float (float_plus_up prec u (-l')) \<le> 0" |
60680 | 3571 |
by (cases "approx prec a vs", auto, cases "approx prec b vs", auto) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3572 |
from order_trans[OF float_plus_up inequality] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3573 |
approx[OF LessEqual.prems(2) l_eq] approx[OF LessEqual.prems(2) u_eq] |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3574 |
show ?case by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3575 |
next |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3576 |
case (AtLeastAtMost x a b) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3577 |
then obtain lx ux l u l' u' |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3578 |
where x_eq: "Some (lx, ux) = approx prec x vs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3579 |
and l_eq: "Some (l, u) = approx prec a vs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3580 |
and u_eq: "Some (l', u') = approx prec b vs" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3581 |
and inequality: "real_of_float (float_plus_up prec u (-lx)) \<le> 0" "real_of_float (float_plus_up prec ux (-l')) \<le> 0" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3582 |
by (cases "approx prec x vs", auto, |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3583 |
cases "approx prec a vs", auto, |
56073
29e308b56d23
enhanced simplifier solver for preconditions of rewrite rule, can now deal with conjunctions
nipkow
parents:
55506
diff
changeset
|
3584 |
cases "approx prec b vs", auto) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3585 |
from order_trans[OF float_plus_up inequality(1)] order_trans[OF float_plus_up inequality(2)] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3586 |
approx[OF AtLeastAtMost.prems(2) l_eq] approx[OF AtLeastAtMost.prems(2) u_eq] approx[OF AtLeastAtMost.prems(2) x_eq] |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3587 |
show ?case by auto |
58986 | 3588 |
qed auto |
29805 | 3589 |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3590 |
lemma approx_form: |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3591 |
assumes "n = length xs" |
60680 | 3592 |
and "approx_form prec f (replicate n None) ss" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3593 |
shows "interpret_form f xs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3594 |
using approx_form_aux[OF _ bounded_by_None] assms by auto |
29805 | 3595 |
|
60680 | 3596 |
|
60533 | 3597 |
subsection \<open>Implementing Taylor series expansion\<close> |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3598 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3599 |
fun isDERIV :: "nat \<Rightarrow> floatarith \<Rightarrow> real list \<Rightarrow> bool" where |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3600 |
"isDERIV x (Add a b) vs = (isDERIV x a vs \<and> isDERIV x b vs)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3601 |
"isDERIV x (Mult a b) vs = (isDERIV x a vs \<and> isDERIV x b vs)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3602 |
"isDERIV x (Minus a) vs = isDERIV x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3603 |
"isDERIV x (Inverse a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs \<noteq> 0)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3604 |
"isDERIV x (Cos a) vs = isDERIV x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3605 |
"isDERIV x (Arctan a) vs = isDERIV x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3606 |
"isDERIV x (Min a b) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3607 |
"isDERIV x (Max a b) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3608 |
"isDERIV x (Abs a) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3609 |
"isDERIV x Pi vs = True" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3610 |
"isDERIV x (Sqrt a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3611 |
"isDERIV x (Exp a) vs = isDERIV x a vs" | |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3612 |
"isDERIV x (Powr a b) vs = |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3613 |
(isDERIV x a vs \<and> isDERIV x b vs \<and> interpret_floatarith a vs > 0)" | |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3614 |
"isDERIV x (Ln a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" | |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3615 |
"isDERIV x (Floor a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs \<notin> \<int>)" | |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3616 |
"isDERIV x (Power a 0) vs = True" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3617 |
"isDERIV x (Power a (Suc n)) vs = isDERIV x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3618 |
"isDERIV x (Num f) vs = True" | |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3619 |
"isDERIV x (Var n) vs = True" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3620 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3621 |
fun DERIV_floatarith :: "nat \<Rightarrow> floatarith \<Rightarrow> floatarith" where |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3622 |
"DERIV_floatarith x (Add a b) = Add (DERIV_floatarith x a) (DERIV_floatarith x b)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3623 |
"DERIV_floatarith x (Mult a b) = Add (Mult a (DERIV_floatarith x b)) (Mult (DERIV_floatarith x a) b)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3624 |
"DERIV_floatarith x (Minus a) = Minus (DERIV_floatarith x a)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3625 |
"DERIV_floatarith x (Inverse a) = Minus (Mult (DERIV_floatarith x a) (Inverse (Power a 2)))" | |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
3626 |
"DERIV_floatarith x (Cos a) = Minus (Mult (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) (DERIV_floatarith x a))" | |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3627 |
"DERIV_floatarith x (Arctan a) = Mult (Inverse (Add (Num 1) (Power a 2))) (DERIV_floatarith x a)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3628 |
"DERIV_floatarith x (Min a b) = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3629 |
"DERIV_floatarith x (Max a b) = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3630 |
"DERIV_floatarith x (Abs a) = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3631 |
"DERIV_floatarith x Pi = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3632 |
"DERIV_floatarith x (Sqrt a) = (Mult (Inverse (Mult (Sqrt a) (Num 2))) (DERIV_floatarith x a))" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3633 |
"DERIV_floatarith x (Exp a) = Mult (Exp a) (DERIV_floatarith x a)" | |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3634 |
"DERIV_floatarith x (Powr a b) = |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3635 |
Mult (Powr a b) (Add (Mult (DERIV_floatarith x b) (Ln a)) (Mult (Mult (DERIV_floatarith x a) b) (Inverse a)))" | |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3636 |
"DERIV_floatarith x (Ln a) = Mult (Inverse a) (DERIV_floatarith x a)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3637 |
"DERIV_floatarith x (Power a 0) = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3638 |
"DERIV_floatarith x (Power a (Suc n)) = Mult (Num (Float (int (Suc n)) 0)) (Mult (Power a n) (DERIV_floatarith x a))" | |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3639 |
"DERIV_floatarith x (Floor a) = Num 0" | |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3640 |
"DERIV_floatarith x (Num f) = Num 0" | |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3641 |
"DERIV_floatarith x (Var n) = (if x = n then Num 1 else Num 0)" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3642 |
|
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3643 |
lemma has_real_derivative_powr': |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3644 |
fixes f g :: "real \<Rightarrow> real" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3645 |
assumes "(f has_real_derivative f') (at x)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3646 |
assumes "(g has_real_derivative g') (at x)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3647 |
assumes "f x > 0" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3648 |
defines "h \<equiv> \<lambda>x. f x powr g x * (g' * ln (f x) + f' * g x / f x)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3649 |
shows "((\<lambda>x. f x powr g x) has_real_derivative h x) (at x)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3650 |
proof (subst DERIV_cong_ev[OF refl _ refl]) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3651 |
from assms have "isCont f x" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3652 |
by (simp add: DERIV_continuous) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3653 |
hence "f \<midarrow>x\<rightarrow> f x" by (simp add: continuous_at) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3654 |
with \<open>f x > 0\<close> have "eventually (\<lambda>x. f x > 0) (nhds x)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3655 |
by (auto simp: tendsto_at_iff_tendsto_nhds dest: order_tendstoD) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3656 |
thus "eventually (\<lambda>x. f x powr g x = exp (g x * ln (f x))) (nhds x)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3657 |
by eventually_elim (simp add: powr_def) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3658 |
next |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3659 |
from assms show "((\<lambda>x. exp (g x * ln (f x))) has_real_derivative h x) (at x)" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3660 |
by (auto intro!: derivative_eq_intros simp: h_def powr_def) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3661 |
qed |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3662 |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3663 |
lemma DERIV_floatarith: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3664 |
assumes "n < length vs" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3665 |
assumes isDERIV: "isDERIV n f (vs[n := x])" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3666 |
shows "DERIV (\<lambda> x'. interpret_floatarith f (vs[n := x'])) x :> |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3667 |
interpret_floatarith (DERIV_floatarith n f) (vs[n := x])" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3668 |
(is "DERIV (?i f) x :> _") |
49351 | 3669 |
using isDERIV |
3670 |
proof (induct f arbitrary: x) |
|
3671 |
case (Inverse a) |
|
3672 |
thus ?case |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56195
diff
changeset
|
3673 |
by (auto intro!: derivative_eq_intros simp add: algebra_simps power2_eq_square) |
49351 | 3674 |
next |
3675 |
case (Cos a) |
|
3676 |
thus ?case |
|
56382 | 3677 |
by (auto intro!: derivative_eq_intros |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3678 |
simp del: interpret_floatarith.simps(5) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3679 |
simp add: interpret_floatarith_sin interpret_floatarith.simps(5)[of a]) |
49351 | 3680 |
next |
3681 |
case (Power a n) |
|
3682 |
thus ?case |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3683 |
by (cases n) (auto intro!: derivative_eq_intros simp del: power_Suc) |
49351 | 3684 |
next |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3685 |
case (Floor a) |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3686 |
thus ?case |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3687 |
by (auto intro!: derivative_eq_intros DERIV_isCont floor_has_real_derivative) |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3688 |
next |
49351 | 3689 |
case (Ln a) |
56382 | 3690 |
thus ?case by (auto intro!: derivative_eq_intros simp add: divide_inverse) |
49351 | 3691 |
next |
3692 |
case (Var i) |
|
60533 | 3693 |
thus ?case using \<open>n < length vs\<close> by auto |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3694 |
next |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3695 |
case (Powr a b) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3696 |
note [derivative_intros] = has_real_derivative_powr' |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3697 |
from Powr show ?case |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3698 |
by (auto intro!: derivative_eq_intros simp: field_simps) |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56195
diff
changeset
|
3699 |
qed (auto intro!: derivative_eq_intros) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3700 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3701 |
declare approx.simps[simp del] |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3702 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3703 |
fun isDERIV_approx :: "nat \<Rightarrow> nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> bool" where |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3704 |
"isDERIV_approx prec x (Add a b) vs = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3705 |
"isDERIV_approx prec x (Mult a b) vs = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3706 |
"isDERIV_approx prec x (Minus a) vs = isDERIV_approx prec x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3707 |
"isDERIV_approx prec x (Inverse a) vs = |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3708 |
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l \<or> u < 0 | None \<Rightarrow> False))" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3709 |
"isDERIV_approx prec x (Cos a) vs = isDERIV_approx prec x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3710 |
"isDERIV_approx prec x (Arctan a) vs = isDERIV_approx prec x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3711 |
"isDERIV_approx prec x (Min a b) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3712 |
"isDERIV_approx prec x (Max a b) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3713 |
"isDERIV_approx prec x (Abs a) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3714 |
"isDERIV_approx prec x Pi vs = True" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3715 |
"isDERIV_approx prec x (Sqrt a) vs = |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3716 |
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3717 |
"isDERIV_approx prec x (Exp a) vs = isDERIV_approx prec x a vs" | |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3718 |
"isDERIV_approx prec x (Powr a b) vs = |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3719 |
(isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" | |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3720 |
"isDERIV_approx prec x (Ln a) vs = |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3721 |
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3722 |
"isDERIV_approx prec x (Power a 0) vs = True" | |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3723 |
"isDERIV_approx prec x (Floor a) vs = |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3724 |
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> l > floor u \<and> u < ceiling l | None \<Rightarrow> False))" | |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3725 |
"isDERIV_approx prec x (Power a (Suc n)) vs = isDERIV_approx prec x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3726 |
"isDERIV_approx prec x (Num f) vs = True" | |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3727 |
"isDERIV_approx prec x (Var n) vs = True" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3728 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3729 |
lemma isDERIV_approx: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3730 |
assumes "bounded_by xs vs" |
49351 | 3731 |
and isDERIV_approx: "isDERIV_approx prec x f vs" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3732 |
shows "isDERIV x f xs" |
49351 | 3733 |
using isDERIV_approx |
3734 |
proof (induct f) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3735 |
case (Inverse a) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3736 |
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3737 |
and *: "0 < l \<or> u < 0" |
49351 | 3738 |
by (cases "approx prec a vs") auto |
60533 | 3739 |
with approx[OF \<open>bounded_by xs vs\<close> approx_Some] |
47600 | 3740 |
have "interpret_floatarith a xs \<noteq> 0" by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3741 |
thus ?case using Inverse by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3742 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3743 |
case (Ln a) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3744 |
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3745 |
and *: "0 < l" |
49351 | 3746 |
by (cases "approx prec a vs") auto |
60533 | 3747 |
with approx[OF \<open>bounded_by xs vs\<close> approx_Some] |
47600 | 3748 |
have "0 < interpret_floatarith a xs" by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3749 |
thus ?case using Ln by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3750 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3751 |
case (Sqrt a) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3752 |
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3753 |
and *: "0 < l" |
49351 | 3754 |
by (cases "approx prec a vs") auto |
60533 | 3755 |
with approx[OF \<open>bounded_by xs vs\<close> approx_Some] |
47600 | 3756 |
have "0 < interpret_floatarith a xs" by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3757 |
thus ?case using Sqrt by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3758 |
next |
60680 | 3759 |
case (Power a n) |
3760 |
thus ?case by (cases n) auto |
|
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3761 |
next |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3762 |
case (Powr a b) |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3763 |
from Powr obtain l1 u1 where a: "Some (l1, u1) = approx prec a vs" and pos: "0 < l1" |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3764 |
by (cases "approx prec a vs") auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3765 |
with approx[OF \<open>bounded_by xs vs\<close> a] |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3766 |
have "0 < interpret_floatarith a xs" by auto |
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3767 |
with Powr show ?case by auto |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3768 |
next |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3769 |
case (Floor a) |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3770 |
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs" |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3771 |
and "real_of_int \<lfloor>real_of_float u\<rfloor> < real_of_float l" "real_of_float u < real_of_int \<lceil>real_of_float l\<rceil>" |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3772 |
and "isDERIV x a xs" |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3773 |
by (cases "approx prec a vs") auto |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3774 |
with approx[OF \<open>bounded_by xs vs\<close> approx_Some] le_floor_iff |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3775 |
show ?case |
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
3776 |
by (force elim!: Ints_cases) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3777 |
qed auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3778 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3779 |
lemma bounded_by_update_var: |
60680 | 3780 |
assumes "bounded_by xs vs" |
3781 |
and "vs ! i = Some (l, u)" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3782 |
and bnd: "x \<in> { real_of_float l .. real_of_float u }" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3783 |
shows "bounded_by (xs[i := x]) vs" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3784 |
proof (cases "i < length xs") |
49351 | 3785 |
case False |
60680 | 3786 |
thus ?thesis |
3787 |
using \<open>bounded_by xs vs\<close> by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3788 |
next |
60680 | 3789 |
case True |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3790 |
let ?xs = "xs[i := x]" |
60680 | 3791 |
from True have "i < length ?xs" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3792 |
have "case vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> ?xs ! j \<in> {real_of_float l .. real_of_float u}" |
60680 | 3793 |
if "j < length vs" for j |
3794 |
proof (cases "vs ! j") |
|
3795 |
case None |
|
3796 |
then show ?thesis by simp |
|
3797 |
next |
|
3798 |
case (Some b) |
|
3799 |
thus ?thesis |
|
3800 |
proof (cases "i = j") |
|
3801 |
case True |
|
3802 |
thus ?thesis using \<open>vs ! i = Some (l, u)\<close> Some and bnd \<open>i < length ?xs\<close> |
|
3803 |
by auto |
|
3804 |
next |
|
3805 |
case False |
|
49351 | 3806 |
thus ?thesis |
60680 | 3807 |
using \<open>bounded_by xs vs\<close>[THEN bounded_byE, OF \<open>j < length vs\<close>] Some by auto |
3808 |
qed |
|
3809 |
qed |
|
3810 |
thus ?thesis |
|
3811 |
unfolding bounded_by_def by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3812 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3813 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3814 |
lemma isDERIV_approx': |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3815 |
assumes "bounded_by xs vs" |
60680 | 3816 |
and vs_x: "vs ! x = Some (l, u)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3817 |
and X_in: "X \<in> {real_of_float l .. real_of_float u}" |
49351 | 3818 |
and approx: "isDERIV_approx prec x f vs" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3819 |
shows "isDERIV x f (xs[x := X])" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3820 |
proof - |
60680 | 3821 |
from bounded_by_update_var[OF \<open>bounded_by xs vs\<close> vs_x X_in] approx |
3822 |
show ?thesis by (rule isDERIV_approx) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3823 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3824 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3825 |
lemma DERIV_approx: |
60680 | 3826 |
assumes "n < length xs" |
3827 |
and bnd: "bounded_by xs vs" |
|
49351 | 3828 |
and isD: "isDERIV_approx prec n f vs" |
3829 |
and app: "Some (l, u) = approx prec (DERIV_floatarith n f) vs" (is "_ = approx _ ?D _") |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3830 |
shows "\<exists>(x::real). l \<le> x \<and> x \<le> u \<and> |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3831 |
DERIV (\<lambda> x. interpret_floatarith f (xs[n := x])) (xs!n) :> x" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3832 |
(is "\<exists> x. _ \<and> _ \<and> DERIV (?i f) _ :> _") |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3833 |
proof (rule exI[of _ "?i ?D (xs!n)"], rule conjI[OF _ conjI]) |
60680 | 3834 |
let "?i f" = "\<lambda>x. interpret_floatarith f (xs[n := x])" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3835 |
from approx[OF bnd app] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3836 |
show "l \<le> ?i ?D (xs!n)" and "?i ?D (xs!n) \<le> u" |
60533 | 3837 |
using \<open>n < length xs\<close> by auto |
3838 |
from DERIV_floatarith[OF \<open>n < length xs\<close>, of f "xs!n"] isDERIV_approx[OF bnd isD] |
|
60680 | 3839 |
show "DERIV (?i f) (xs!n) :> (?i ?D (xs!n))" |
3840 |
by simp |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3841 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3842 |
|
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3843 |
lemma lift_bin_aux: |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3844 |
assumes lift_bin_Some: "Some (l, u) = lift_bin a b f" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3845 |
obtains l1 u1 l2 u2 |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3846 |
where "a = Some (l1, u1)" |
49351 | 3847 |
and "b = Some (l2, u2)" |
3848 |
and "f l1 u1 l2 u2 = Some (l, u)" |
|
3849 |
using assms by (cases a, simp, cases b, simp, auto) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3850 |
|
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3851 |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3852 |
fun approx_tse where |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3853 |
"approx_tse prec n 0 c k f bs = approx prec f bs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3854 |
"approx_tse prec n (Suc s) c k f bs = |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3855 |
(if isDERIV_approx prec n f bs then |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3856 |
lift_bin (approx prec f (bs[n := Some (c,c)])) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3857 |
(approx_tse prec n s c (Suc k) (DERIV_floatarith n f) bs) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3858 |
(\<lambda> l1 u1 l2 u2. approx prec |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3859 |
(Add (Var 0) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3860 |
(Mult (Inverse (Num (Float (int k) 0))) |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3861 |
(Mult (Add (Var (Suc (Suc 0))) (Minus (Num c))) |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
3862 |
(Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), bs!n]) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3863 |
else approx prec f bs)" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3864 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3865 |
lemma bounded_by_Cons: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3866 |
assumes bnd: "bounded_by xs vs" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3867 |
and x: "x \<in> { real_of_float l .. real_of_float u }" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3868 |
shows "bounded_by (x#xs) ((Some (l, u))#vs)" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3869 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3870 |
have "case ((Some (l,u))#vs) ! i of Some (l, u) \<Rightarrow> (x#xs)!i \<in> { real_of_float l .. real_of_float u } | None \<Rightarrow> True" |
60680 | 3871 |
if *: "i < length ((Some (l, u))#vs)" for i |
3872 |
proof (cases i) |
|
3873 |
case 0 |
|
3874 |
with x show ?thesis by auto |
|
3875 |
next |
|
3876 |
case (Suc i) |
|
3877 |
with * have "i < length vs" by auto |
|
3878 |
from bnd[THEN bounded_byE, OF this] |
|
3879 |
show ?thesis unfolding Suc nth_Cons_Suc . |
|
3880 |
qed |
|
3881 |
thus ?thesis |
|
3882 |
by (auto simp add: bounded_by_def) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3883 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3884 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3885 |
lemma approx_tse_generic: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3886 |
assumes "bounded_by xs vs" |
60680 | 3887 |
and bnd_c: "bounded_by (xs[x := c]) vs" |
3888 |
and "x < length vs" and "x < length xs" |
|
49351 | 3889 |
and bnd_x: "vs ! x = Some (lx, ux)" |
3890 |
and ate: "Some (l, u) = approx_tse prec x s c k f vs" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3891 |
shows "\<exists> n. (\<forall> m < n. \<forall> (z::real) \<in> {lx .. ux}. |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3892 |
DERIV (\<lambda> y. interpret_floatarith ((DERIV_floatarith x ^^ m) f) (xs[x := y])) z :> |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3893 |
(interpret_floatarith ((DERIV_floatarith x ^^ (Suc m)) f) (xs[x := z]))) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3894 |
\<and> (\<forall> (t::real) \<in> {lx .. ux}. (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) * |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3895 |
interpret_floatarith ((DERIV_floatarith x ^^ i) f) (xs[x := c]) * |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3896 |
(xs!x - c)^i) + |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3897 |
inverse (real (\<Prod> j \<in> {k..<k+n}. j)) * |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3898 |
interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := t]) * |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3899 |
(xs!x - c)^n \<in> {l .. u})" (is "\<exists> n. ?taylor f k l u n") |
60680 | 3900 |
using ate |
3901 |
proof (induct s arbitrary: k f l u) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3902 |
case 0 |
49351 | 3903 |
{ |
3904 |
fix t::real assume "t \<in> {lx .. ux}" |
|
60533 | 3905 |
note bounded_by_update_var[OF \<open>bounded_by xs vs\<close> bnd_x this] |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3906 |
from approx[OF this 0[unfolded approx_tse.simps]] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3907 |
have "(interpret_floatarith f (xs[x := t])) \<in> {l .. u}" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3908 |
by (auto simp add: algebra_simps) |
49351 | 3909 |
} |
3910 |
thus ?case by (auto intro!: exI[of _ 0]) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3911 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3912 |
case (Suc s) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3913 |
show ?case |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3914 |
proof (cases "isDERIV_approx prec x f vs") |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3915 |
case False |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3916 |
note ap = Suc.prems[unfolded approx_tse.simps if_not_P[OF False]] |
49351 | 3917 |
{ |
3918 |
fix t::real assume "t \<in> {lx .. ux}" |
|
60533 | 3919 |
note bounded_by_update_var[OF \<open>bounded_by xs vs\<close> bnd_x this] |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3920 |
from approx[OF this ap] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3921 |
have "(interpret_floatarith f (xs[x := t])) \<in> {l .. u}" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
3922 |
by (auto simp add: algebra_simps) |
49351 | 3923 |
} |
3924 |
thus ?thesis by (auto intro!: exI[of _ 0]) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3925 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3926 |
case True |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3927 |
with Suc.prems |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3928 |
obtain l1 u1 l2 u2 |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3929 |
where a: "Some (l1, u1) = approx prec f (vs[x := Some (c,c)])" |
49351 | 3930 |
and ate: "Some (l2, u2) = approx_tse prec x s c (Suc k) (DERIV_floatarith x f) vs" |
3931 |
and final: "Some (l, u) = approx prec |
|
3932 |
(Add (Var 0) |
|
3933 |
(Mult (Inverse (Num (Float (int k) 0))) |
|
3934 |
(Mult (Add (Var (Suc (Suc 0))) (Minus (Num c))) |
|
3935 |
(Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), vs!x]" |
|
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
3936 |
by (auto elim!: lift_bin_aux) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3937 |
|
60533 | 3938 |
from bnd_c \<open>x < length xs\<close> |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3939 |
have bnd: "bounded_by (xs[x:=c]) (vs[x:= Some (c,c)])" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3940 |
by (auto intro!: bounded_by_update) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3941 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3942 |
from approx[OF this a] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3943 |
have f_c: "interpret_floatarith ((DERIV_floatarith x ^^ 0) f) (xs[x := c]) \<in> { l1 .. u1 }" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3944 |
(is "?f 0 (real_of_float c) \<in> _") |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3945 |
by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3946 |
|
60680 | 3947 |
have funpow_Suc[symmetric]: "(f ^^ Suc n) x = (f ^^ n) (f x)" |
3948 |
for f :: "'a \<Rightarrow> 'a" and n :: nat and x :: 'a |
|
3949 |
by (induct n) auto |
|
3950 |
from Suc.hyps[OF ate, unfolded this] obtain n |
|
3951 |
where DERIV_hyp: "\<And>m z. \<lbrakk> m < n ; (z::real) \<in> { lx .. ux } \<rbrakk> \<Longrightarrow> |
|
3952 |
DERIV (?f (Suc m)) z :> ?f (Suc (Suc m)) z" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3953 |
and hyp: "\<forall>t \<in> {real_of_float lx .. real_of_float ux}. |
60680 | 3954 |
(\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {Suc k..<Suc k + i}. j)) * ?f (Suc i) c * (xs!x - c)^i) + |
3955 |
inverse (real (\<Prod> j \<in> {Suc k..<Suc k + n}. j)) * ?f (Suc n) t * (xs!x - c)^n \<in> {l2 .. u2}" |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3956 |
(is "\<forall> t \<in> _. ?X (Suc k) f n t \<in> _") |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3957 |
by blast |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3958 |
|
60680 | 3959 |
have DERIV: "DERIV (?f m) z :> ?f (Suc m) z" |
3960 |
if "m < Suc n" and bnd_z: "z \<in> { lx .. ux }" for m and z::real |
|
3961 |
proof (cases m) |
|
3962 |
case 0 |
|
3963 |
with DERIV_floatarith[OF \<open>x < length xs\<close> |
|
3964 |
isDERIV_approx'[OF \<open>bounded_by xs vs\<close> bnd_x bnd_z True]] |
|
3965 |
show ?thesis by simp |
|
3966 |
next |
|
3967 |
case (Suc m') |
|
3968 |
hence "m' < n" |
|
3969 |
using \<open>m < Suc n\<close> by auto |
|
3970 |
from DERIV_hyp[OF this bnd_z] show ?thesis |
|
3971 |
using Suc by simp |
|
3972 |
qed |
|
3973 |
||
3974 |
have "\<And>k i. k < i \<Longrightarrow> {k ..< i} = insert k {Suc k ..< i}" by auto |
|
64272 | 3975 |
hence prod_head_Suc: "\<And>k i. \<Prod>{k ..< k + Suc i} = k * \<Prod>{Suc k ..< Suc k + i}" |
60680 | 3976 |
by auto |
64267 | 3977 |
have sum_move0: "\<And>k F. sum F {0..<Suc k} = F 0 + sum (\<lambda> k. F (Suc k)) {0..<k}" |
3978 |
unfolding sum_shift_bounds_Suc_ivl[symmetric] |
|
3979 |
unfolding sum_head_upt_Suc[OF zero_less_Suc] .. |
|
63040 | 3980 |
define C where "C = xs!x - c" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3981 |
|
49351 | 3982 |
{ |
3983 |
fix t::real assume t: "t \<in> {lx .. ux}" |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3984 |
hence "bounded_by [xs!x] [vs!x]" |
60533 | 3985 |
using \<open>bounded_by xs vs\<close>[THEN bounded_byE, OF \<open>x < length vs\<close>] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
3986 |
by (cases "vs!x", auto simp add: bounded_by_def) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3987 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3988 |
with hyp[THEN bspec, OF t] f_c |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3989 |
have "bounded_by [?f 0 c, ?X (Suc k) f n t, xs!x] [Some (l1, u1), Some (l2, u2), vs!x]" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
3990 |
by (auto intro!: bounded_by_Cons) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3991 |
from approx[OF this final, unfolded atLeastAtMost_iff[symmetric]] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3992 |
have "?X (Suc k) f n t * (xs!x - real_of_float c) * inverse k + ?f 0 c \<in> {l .. u}" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
3993 |
by (auto simp add: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3994 |
also have "?X (Suc k) f n t * (xs!x - real_of_float c) * inverse (real k) + ?f 0 c = |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3995 |
(\<Sum> i = 0..<Suc n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) * ?f i c * (xs!x - c)^i) + |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3996 |
inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - c)^Suc n" (is "_ = ?T") |
64272 | 3997 |
unfolding funpow_Suc C_def[symmetric] sum_move0 prod_head_Suc |
35082 | 3998 |
by (auto simp add: algebra_simps) |
64267 | 3999 |
(simp only: mult.left_commute [of _ "inverse (real k)"] sum_distrib_left [symmetric]) |
49351 | 4000 |
finally have "?T \<in> {l .. u}" . |
4001 |
} |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4002 |
thus ?thesis using DERIV by blast |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4003 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4004 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4005 |
|
64272 | 4006 |
lemma prod_fact: "real (\<Prod> {1..<1 + k}) = fact (k :: nat)" |
4007 |
by (simp add: fact_prod atLeastLessThanSuc_atLeastAtMost) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4008 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4009 |
lemma approx_tse: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4010 |
assumes "bounded_by xs vs" |
60680 | 4011 |
and bnd_x: "vs ! x = Some (lx, ux)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4012 |
and bnd_c: "real_of_float c \<in> {lx .. ux}" |
49351 | 4013 |
and "x < length vs" and "x < length xs" |
4014 |
and ate: "Some (l, u) = approx_tse prec x s c 1 f vs" |
|
60680 | 4015 |
shows "interpret_floatarith f xs \<in> {l .. u}" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4016 |
proof - |
63040 | 4017 |
define F where [abs_def]: "F n z = |
4018 |
interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := z])" for n z |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4019 |
hence F0: "F 0 = (\<lambda> z. interpret_floatarith f (xs[x := z]))" by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4020 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4021 |
hence "bounded_by (xs[x := c]) vs" and "x < length vs" "x < length xs" |
60533 | 4022 |
using \<open>bounded_by xs vs\<close> bnd_x bnd_c \<open>x < length vs\<close> \<open>x < length xs\<close> |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4023 |
by (auto intro!: bounded_by_update_var) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4024 |
|
60533 | 4025 |
from approx_tse_generic[OF \<open>bounded_by xs vs\<close> this bnd_x ate] |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4026 |
obtain n |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4027 |
where DERIV: "\<forall> m z. m < n \<and> real_of_float lx \<le> z \<and> z \<le> real_of_float ux \<longrightarrow> DERIV (F m) z :> F (Suc m) z" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4028 |
and hyp: "\<And> (t::real). t \<in> {lx .. ux} \<Longrightarrow> |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
4029 |
(\<Sum> j = 0..<n. inverse(fact j) * F j c * (xs!x - c)^j) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
4030 |
inverse ((fact n)) * F n t * (xs!x - c)^n |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4031 |
\<in> {l .. u}" (is "\<And> t. _ \<Longrightarrow> ?taylor t \<in> _") |
64272 | 4032 |
unfolding F_def atLeastAtMost_iff[symmetric] prod_fact |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
4033 |
by blast |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4034 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4035 |
have bnd_xs: "xs ! x \<in> { lx .. ux }" |
60533 | 4036 |
using \<open>bounded_by xs vs\<close>[THEN bounded_byE, OF \<open>x < length vs\<close>] bnd_x by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4037 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4038 |
show ?thesis |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4039 |
proof (cases n) |
60680 | 4040 |
case 0 |
4041 |
thus ?thesis |
|
4042 |
using hyp[OF bnd_xs] unfolding F_def by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4043 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4044 |
case (Suc n') |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4045 |
show ?thesis |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4046 |
proof (cases "xs ! x = c") |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4047 |
case True |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4048 |
from True[symmetric] hyp[OF bnd_xs] Suc show ?thesis |
64267 | 4049 |
unfolding F_def Suc sum_head_upt_Suc[OF zero_less_Suc] sum_shift_bounds_Suc_ivl |
60680 | 4050 |
by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4051 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4052 |
case False |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4053 |
have "lx \<le> real_of_float c" "real_of_float c \<le> ux" "lx \<le> xs!x" "xs!x \<le> ux" |
60533 | 4054 |
using Suc bnd_c \<open>bounded_by xs vs\<close>[THEN bounded_byE, OF \<open>x < length vs\<close>] bnd_x by auto |
63570 | 4055 |
from taylor[OF zero_less_Suc, of F, OF F0 DERIV[unfolded Suc] this False] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4056 |
obtain t::real where t_bnd: "if xs ! x < c then xs ! x < t \<and> t < c else c < t \<and> t < xs ! x" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
4057 |
and fl_eq: "interpret_floatarith f (xs[x := xs ! x]) = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
4058 |
(\<Sum>m = 0..<Suc n'. F m c / (fact m) * (xs ! x - c) ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
4059 |
F (Suc n') t / (fact (Suc n')) * (xs ! x - c) ^ Suc n'" |
56195 | 4060 |
unfolding atLeast0LessThan by blast |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4061 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4062 |
from t_bnd bnd_xs bnd_c have *: "t \<in> {lx .. ux}" |
60680 | 4063 |
by (cases "xs ! x < c") auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4064 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4065 |
have "interpret_floatarith f (xs[x := xs ! x]) = ?taylor t" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
4066 |
unfolding fl_eq Suc by (auto simp add: algebra_simps divide_inverse) |
60680 | 4067 |
also have "\<dots> \<in> {l .. u}" |
4068 |
using * by (rule hyp) |
|
4069 |
finally show ?thesis |
|
4070 |
by simp |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4071 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4072 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4073 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4074 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4075 |
fun approx_tse_form' where |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4076 |
"approx_tse_form' prec t f 0 l u cmp = |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
4077 |
(case approx_tse prec 0 t ((l + u) * Float 1 (- 1)) 1 f [Some (l, u)] |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4078 |
of Some (l, u) \<Rightarrow> cmp l u | None \<Rightarrow> False)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4079 |
"approx_tse_form' prec t f (Suc s) l u cmp = |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
4080 |
(let m = (l + u) * Float 1 (- 1) |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4081 |
in (if approx_tse_form' prec t f s l m cmp then |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4082 |
approx_tse_form' prec t f s m u cmp else False))" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4083 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4084 |
lemma approx_tse_form': |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4085 |
fixes x :: real |
60680 | 4086 |
assumes "approx_tse_form' prec t f s l u cmp" |
4087 |
and "x \<in> {l .. u}" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4088 |
shows "\<exists>l' u' ly uy. x \<in> {l' .. u'} \<and> real_of_float l \<le> l' \<and> u' \<le> real_of_float u \<and> cmp ly uy \<and> |
60680 | 4089 |
approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" |
4090 |
using assms |
|
4091 |
proof (induct s arbitrary: l u) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4092 |
case 0 |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4093 |
then obtain ly uy |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
4094 |
where *: "approx_tse prec 0 t ((l + u) * Float 1 (- 1)) 1 f [Some (l, u)] = Some (ly, uy)" |
55413
a8e96847523c
adapted theories to '{case,rec}_{list,option}' names
blanchet
parents:
54782
diff
changeset
|
4095 |
and **: "cmp ly uy" by (auto elim!: case_optionE) |
46545 | 4096 |
with 0 show ?case by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4097 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4098 |
case (Suc s) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
4099 |
let ?m = "(l + u) * Float 1 (- 1)" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4100 |
from Suc.prems |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4101 |
have l: "approx_tse_form' prec t f s l ?m cmp" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4102 |
and u: "approx_tse_form' prec t f s ?m u cmp" |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4103 |
by (auto simp add: Let_def lazy_conj) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4104 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4105 |
have m_l: "real_of_float l \<le> ?m" and m_u: "?m \<le> real_of_float u" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
4106 |
unfolding less_eq_float_def using Suc.prems by auto |
60680 | 4107 |
with \<open>x \<in> { l .. u }\<close> consider "x \<in> { l .. ?m}" | "x \<in> {?m .. u}" |
4108 |
by atomize_elim auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4109 |
thus ?case |
60680 | 4110 |
proof cases |
4111 |
case 1 |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4112 |
from Suc.hyps[OF l this] |
60680 | 4113 |
obtain l' u' ly uy where |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4114 |
"x \<in> {l' .. u'} \<and> real_of_float l \<le> l' \<and> real_of_float u' \<le> ?m \<and> cmp ly uy \<and> |
60680 | 4115 |
approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" |
4116 |
by blast |
|
4117 |
with m_u show ?thesis |
|
4118 |
by (auto intro!: exI) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4119 |
next |
60680 | 4120 |
case 2 |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4121 |
from Suc.hyps[OF u this] |
60680 | 4122 |
obtain l' u' ly uy where |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4123 |
"x \<in> { l' .. u' } \<and> ?m \<le> real_of_float l' \<and> u' \<le> real_of_float u \<and> cmp ly uy \<and> |
60680 | 4124 |
approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" |
4125 |
by blast |
|
4126 |
with m_u show ?thesis |
|
4127 |
by (auto intro!: exI) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4128 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4129 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4130 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4131 |
lemma approx_tse_form'_less: |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4132 |
fixes x :: real |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4133 |
assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 < l)" |
60680 | 4134 |
and x: "x \<in> {l .. u}" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4135 |
shows "interpret_floatarith b [x] < interpret_floatarith a [x]" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4136 |
proof - |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4137 |
from approx_tse_form'[OF tse x] |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4138 |
obtain l' u' ly uy |
60680 | 4139 |
where x': "x \<in> {l' .. u'}" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4140 |
and "real_of_float l \<le> real_of_float l'" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4141 |
and "real_of_float u' \<le> real_of_float u" and "0 < ly" |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
4142 |
and tse: "approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4143 |
by blast |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4144 |
|
60680 | 4145 |
hence "bounded_by [x] [Some (l', u')]" |
4146 |
by (auto simp add: bounded_by_def) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4147 |
from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x' |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4148 |
have "ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
4149 |
by auto |
60680 | 4150 |
from order_less_le_trans[OF _ this, of 0] \<open>0 < ly\<close> show ?thesis |
4151 |
by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4152 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4153 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4154 |
lemma approx_tse_form'_le: |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4155 |
fixes x :: real |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4156 |
assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 \<le> l)" |
60680 | 4157 |
and x: "x \<in> {l .. u}" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4158 |
shows "interpret_floatarith b [x] \<le> interpret_floatarith a [x]" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4159 |
proof - |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4160 |
from approx_tse_form'[OF tse x] |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4161 |
obtain l' u' ly uy |
60680 | 4162 |
where x': "x \<in> {l' .. u'}" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4163 |
and "l \<le> real_of_float l'" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
4164 |
and "real_of_float u' \<le> u" and "0 \<le> ly" |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
4165 |
and tse: "approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4166 |
by blast |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4167 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4168 |
hence "bounded_by [x] [Some (l', u')]" by (auto simp add: bounded_by_def) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4169 |
from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x' |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4170 |
have "ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
4171 |
by auto |
60680 | 4172 |
from order_trans[OF _ this, of 0] \<open>0 \<le> ly\<close> show ?thesis |
4173 |
by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4174 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4175 |
|
58986 | 4176 |
fun approx_tse_concl where |
4177 |
"approx_tse_concl prec t (Less lf rt) s l u l' u' \<longleftrightarrow> |
|
4178 |
approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)" | |
|
4179 |
"approx_tse_concl prec t (LessEqual lf rt) s l u l' u' \<longleftrightarrow> |
|
4180 |
approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)" | |
|
4181 |
"approx_tse_concl prec t (AtLeastAtMost x lf rt) s l u l' u' \<longleftrightarrow> |
|
4182 |
(if approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l) then |
|
4183 |
approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l) else False)" | |
|
4184 |
"approx_tse_concl prec t (Conj f g) s l u l' u' \<longleftrightarrow> |
|
4185 |
approx_tse_concl prec t f s l u l' u' \<and> approx_tse_concl prec t g s l u l' u'" | |
|
4186 |
"approx_tse_concl prec t (Disj f g) s l u l' u' \<longleftrightarrow> |
|
4187 |
approx_tse_concl prec t f s l u l' u' \<or> approx_tse_concl prec t g s l u l' u'" | |
|
4188 |
"approx_tse_concl _ _ _ _ _ _ _ _ \<longleftrightarrow> False" |
|
4189 |
||
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4190 |
definition |
60680 | 4191 |
"approx_tse_form prec t s f = |
4192 |
(case f of |
|
4193 |
Bound x a b f \<Rightarrow> |
|
4194 |
x = Var 0 \<and> |
|
4195 |
(case (approx prec a [None], approx prec b [None]) of |
|
4196 |
(Some (l, u), Some (l', u')) \<Rightarrow> approx_tse_concl prec t f s l u l' u' |
|
4197 |
| _ \<Rightarrow> False) |
|
4198 |
| _ \<Rightarrow> False)" |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4199 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4200 |
lemma approx_tse_form: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4201 |
assumes "approx_tse_form prec t s f" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4202 |
shows "interpret_form f [x]" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4203 |
proof (cases f) |
60680 | 4204 |
case f_def: (Bound i a b f') |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4205 |
with assms obtain l u l' u' |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4206 |
where a: "approx prec a [None] = Some (l, u)" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4207 |
and b: "approx prec b [None] = Some (l', u')" |
55413
a8e96847523c
adapted theories to '{case,rec}_{list,option}' names
blanchet
parents:
54782
diff
changeset
|
4208 |
unfolding approx_tse_form_def by (auto elim!: case_optionE) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4209 |
|
60680 | 4210 |
from f_def assms have "i = Var 0" |
4211 |
unfolding approx_tse_form_def by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4212 |
hence i: "interpret_floatarith i [x] = x" by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4213 |
|
60680 | 4214 |
{ |
4215 |
let ?f = "\<lambda>z. interpret_floatarith z [x]" |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4216 |
assume "?f i \<in> { ?f a .. ?f b }" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4217 |
with approx[OF _ a[symmetric], of "[x]"] approx[OF _ b[symmetric], of "[x]"] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
4218 |
have bnd: "x \<in> { l .. u'}" unfolding bounded_by_def i by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4219 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4220 |
have "interpret_form f' [x]" |
60680 | 4221 |
using assms[unfolded f_def] |
58986 | 4222 |
proof (induct f') |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4223 |
case (Less lf rt) |
58986 | 4224 |
with a b |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4225 |
have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
4226 |
unfolding approx_tse_form_def by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4227 |
from approx_tse_form'_less[OF this bnd] |
58986 | 4228 |
show ?case using Less by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4229 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4230 |
case (LessEqual lf rt) |
60680 | 4231 |
with f_def a b assms |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4232 |
have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
4233 |
unfolding approx_tse_form_def by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4234 |
from approx_tse_form'_le[OF this bnd] |
58986 | 4235 |
show ?case using LessEqual by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4236 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4237 |
case (AtLeastAtMost x lf rt) |
60680 | 4238 |
with f_def a b assms |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4239 |
have "approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
4240 |
and "approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)" |
62390 | 4241 |
unfolding approx_tse_form_def lazy_conj by (auto split: if_split_asm) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4242 |
from approx_tse_form'_le[OF this(1) bnd] approx_tse_form'_le[OF this(2) bnd] |
58986 | 4243 |
show ?case using AtLeastAtMost by auto |
4244 |
qed (auto simp: f_def approx_tse_form_def elim!: case_optionE) |
|
60680 | 4245 |
} |
4246 |
thus ?thesis unfolding f_def by auto |
|
58986 | 4247 |
qed (insert assms, auto simp add: approx_tse_form_def) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
4248 |
|
60533 | 4249 |
text \<open>@{term approx_form_eval} is only used for the {\tt value}-command.\<close> |
32919
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4250 |
|
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4251 |
fun approx_form_eval :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option list" where |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4252 |
"approx_form_eval prec (Bound (Var n) a b f) bs = |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4253 |
(case (approx prec a bs, approx prec b bs) |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4254 |
of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)]) |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4255 |
| _ \<Rightarrow> bs)" | |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4256 |
"approx_form_eval prec (Assign (Var n) a f) bs = |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4257 |
(case (approx prec a bs) |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4258 |
of (Some (l, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)]) |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4259 |
| _ \<Rightarrow> bs)" | |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4260 |
"approx_form_eval prec (Less a b) bs = bs @ [approx prec a bs, approx prec b bs]" | |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4261 |
"approx_form_eval prec (LessEqual a b) bs = bs @ [approx prec a bs, approx prec b bs]" | |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4262 |
"approx_form_eval prec (AtLeastAtMost x a b) bs = |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4263 |
bs @ [approx prec x bs, approx prec a bs, approx prec b bs]" | |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4264 |
"approx_form_eval _ _ bs = bs" |
37adfa07b54b
approximation now fails earlier when using interval splitting; value [approximate] now supports bounded variables; renamed Var -> Atom for better readability
hoelzl
parents:
32650
diff
changeset
|
4265 |
|
60680 | 4266 |
|
60533 | 4267 |
subsection \<open>Implement proof method \texttt{approximation}\<close> |
29805 | 4268 |
|
60533 | 4269 |
oracle approximation_oracle = \<open>fn (thy, t) => |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4270 |
let |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4271 |
fun bad t = error ("Bad term: " ^ Syntax.string_of_term_global thy t); |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4272 |
|
38716
3c3b4ad683d5
approximation_oracle: actually match true/false in ML, not arbitrary values;
wenzelm
parents:
38558
diff
changeset
|
4273 |
fun term_of_bool true = @{term True} |
3c3b4ad683d5
approximation_oracle: actually match true/false in ML, not arbitrary values;
wenzelm
parents:
38558
diff
changeset
|
4274 |
| term_of_bool false = @{term False}; |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4275 |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
49962
diff
changeset
|
4276 |
val mk_int = HOLogic.mk_number @{typ int} o @{code integer_of_int}; |
58988 | 4277 |
fun dest_int (@{term int_of_integer} $ j) = @{code int_of_integer} (snd (HOLogic.dest_number j)) |
4278 |
| dest_int i = @{code int_of_integer} (snd (HOLogic.dest_number i)); |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
49962
diff
changeset
|
4279 |
|
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4280 |
fun term_of_float (@{code Float} (k, l)) = |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
49962
diff
changeset
|
4281 |
@{term Float} $ mk_int k $ mk_int l; |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4282 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4283 |
fun term_of_float_float_option NONE = @{term "None :: (float \<times> float) option"} |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4284 |
| term_of_float_float_option (SOME ff) = @{term "Some :: float \<times> float \<Rightarrow> _"} |
59058
a78612c67ec0
renamed "pairself" to "apply2", in accordance to @{apply 2};
wenzelm
parents:
58988
diff
changeset
|
4285 |
$ HOLogic.mk_prod (apply2 term_of_float ff); |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4286 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4287 |
val term_of_float_float_option_list = |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4288 |
HOLogic.mk_list @{typ "(float \<times> float) option"} o map term_of_float_float_option; |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4289 |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
49962
diff
changeset
|
4290 |
fun nat_of_term t = @{code nat_of_integer} |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
49962
diff
changeset
|
4291 |
(HOLogic.dest_nat t handle TERM _ => snd (HOLogic.dest_number t)); |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4292 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4293 |
fun float_of_term (@{term Float} $ k $ l) = |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
49962
diff
changeset
|
4294 |
@{code Float} (dest_int k, dest_int l) |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4295 |
| float_of_term t = bad t; |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4296 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4297 |
fun floatarith_of_term (@{term Add} $ a $ b) = @{code Add} (floatarith_of_term a, floatarith_of_term b) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4298 |
| floatarith_of_term (@{term Minus} $ a) = @{code Minus} (floatarith_of_term a) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4299 |
| floatarith_of_term (@{term Mult} $ a $ b) = @{code Mult} (floatarith_of_term a, floatarith_of_term b) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4300 |
| floatarith_of_term (@{term Inverse} $ a) = @{code Inverse} (floatarith_of_term a) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4301 |
| floatarith_of_term (@{term Cos} $ a) = @{code Cos} (floatarith_of_term a) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4302 |
| floatarith_of_term (@{term Arctan} $ a) = @{code Arctan} (floatarith_of_term a) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4303 |
| floatarith_of_term (@{term Abs} $ a) = @{code Abs} (floatarith_of_term a) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4304 |
| floatarith_of_term (@{term Max} $ a $ b) = @{code Max} (floatarith_of_term a, floatarith_of_term b) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4305 |
| floatarith_of_term (@{term Min} $ a $ b) = @{code Min} (floatarith_of_term a, floatarith_of_term b) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4306 |
| floatarith_of_term @{term Pi} = @{code Pi} |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4307 |
| floatarith_of_term (@{term Sqrt} $ a) = @{code Sqrt} (floatarith_of_term a) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4308 |
| floatarith_of_term (@{term Exp} $ a) = @{code Exp} (floatarith_of_term a) |
62200
67792e4a5486
Made Approximation work for powr again
Manuel Eberl <eberlm@in.tum.de>
parents:
61969
diff
changeset
|
4309 |
| floatarith_of_term (@{term Powr} $ a $ b) = @{code Powr} (floatarith_of_term a, floatarith_of_term b) |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4310 |
| floatarith_of_term (@{term Ln} $ a) = @{code Ln} (floatarith_of_term a) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4311 |
| floatarith_of_term (@{term Power} $ a $ n) = |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4312 |
@{code Power} (floatarith_of_term a, nat_of_term n) |
63263
c6c95d64607a
approximation, derivative, and continuity of floor and ceiling
immler
parents:
63248
diff
changeset
|
4313 |
| floatarith_of_term (@{term Floor} $ a) = @{code Floor} (floatarith_of_term a) |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4314 |
| floatarith_of_term (@{term Var} $ n) = @{code Var} (nat_of_term n) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4315 |
| floatarith_of_term (@{term Num} $ m) = @{code Num} (float_of_term m) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4316 |
| floatarith_of_term t = bad t; |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4317 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4318 |
fun form_of_term (@{term Bound} $ a $ b $ c $ p) = @{code Bound} |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4319 |
(floatarith_of_term a, floatarith_of_term b, floatarith_of_term c, form_of_term p) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4320 |
| form_of_term (@{term Assign} $ a $ b $ p) = @{code Assign} |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4321 |
(floatarith_of_term a, floatarith_of_term b, form_of_term p) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4322 |
| form_of_term (@{term Less} $ a $ b) = @{code Less} |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4323 |
(floatarith_of_term a, floatarith_of_term b) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4324 |
| form_of_term (@{term LessEqual} $ a $ b) = @{code LessEqual} |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4325 |
(floatarith_of_term a, floatarith_of_term b) |
58986 | 4326 |
| form_of_term (@{term Conj} $ a $ b) = @{code Conj} |
4327 |
(form_of_term a, form_of_term b) |
|
4328 |
| form_of_term (@{term Disj} $ a $ b) = @{code Disj} |
|
4329 |
(form_of_term a, form_of_term b) |
|
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4330 |
| form_of_term (@{term AtLeastAtMost} $ a $ b $ c) = @{code AtLeastAtMost} |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4331 |
(floatarith_of_term a, floatarith_of_term b, floatarith_of_term c) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4332 |
| form_of_term t = bad t; |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4333 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4334 |
fun float_float_option_of_term @{term "None :: (float \<times> float) option"} = NONE |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4335 |
| float_float_option_of_term (@{term "Some :: float \<times> float \<Rightarrow> _"} $ ff) = |
59058
a78612c67ec0
renamed "pairself" to "apply2", in accordance to @{apply 2};
wenzelm
parents:
58988
diff
changeset
|
4336 |
SOME (apply2 float_of_term (HOLogic.dest_prod ff)) |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4337 |
| float_float_option_of_term (@{term approx'} $ n $ a $ ffs) = @{code approx'} |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4338 |
(nat_of_term n) (floatarith_of_term a) (float_float_option_list_of_term ffs) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4339 |
| float_float_option_of_term t = bad t |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4340 |
and float_float_option_list_of_term |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4341 |
(@{term "replicate :: _ \<Rightarrow> (float \<times> float) option \<Rightarrow> _"} $ n $ @{term "None :: (float \<times> float) option"}) = |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4342 |
@{code replicate} (nat_of_term n) NONE |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4343 |
| float_float_option_list_of_term (@{term approx_form_eval} $ n $ p $ ffs) = |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4344 |
@{code approx_form_eval} (nat_of_term n) (form_of_term p) (float_float_option_list_of_term ffs) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4345 |
| float_float_option_list_of_term t = map float_float_option_of_term |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4346 |
(HOLogic.dest_list t); |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4347 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4348 |
val nat_list_of_term = map nat_of_term o HOLogic.dest_list ; |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4349 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4350 |
fun bool_of_term (@{term approx_form} $ n $ p $ ffs $ ms) = @{code approx_form} |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4351 |
(nat_of_term n) (form_of_term p) (float_float_option_list_of_term ffs) (nat_list_of_term ms) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4352 |
| bool_of_term (@{term approx_tse_form} $ m $ n $ q $ p) = |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4353 |
@{code approx_tse_form} (nat_of_term m) (nat_of_term n) (nat_of_term q) (form_of_term p) |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4354 |
| bool_of_term t = bad t; |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4355 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4356 |
fun eval t = case fastype_of t |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4357 |
of @{typ bool} => |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4358 |
(term_of_bool o bool_of_term) t |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4359 |
| @{typ "(float \<times> float) option"} => |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4360 |
(term_of_float_float_option o float_float_option_of_term) t |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4361 |
| @{typ "(float \<times> float) option list"} => |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4362 |
(term_of_float_float_option_list o float_float_option_list_of_term) t |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4363 |
| _ => bad t; |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4364 |
|
52131 | 4365 |
val normalize = eval o Envir.beta_norm o Envir.eta_long []; |
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4366 |
|
59621
291934bac95e
Thm.cterm_of and Thm.ctyp_of operate on local context;
wenzelm
parents:
59582
diff
changeset
|
4367 |
in Thm.global_cterm_of thy (Logic.mk_equals (t, normalize t)) end |
60533 | 4368 |
\<close> |
31099
03314c427b34
optimized Approximation by precompiling approx_inequality
hoelzl
parents:
31098
diff
changeset
|
4369 |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4370 |
lemma intervalE: "a \<le> x \<and> x \<le> b \<Longrightarrow> \<lbrakk> x \<in> { a .. b } \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4371 |
by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4372 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4373 |
lemma meta_eqE: "x \<equiv> a \<Longrightarrow> \<lbrakk> x = a \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4374 |
by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4375 |
|
63929
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4376 |
named_theorems approximation_preproc |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4377 |
|
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4378 |
lemma approximation_preproc_floatarith[approximation_preproc]: |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4379 |
"0 = real_of_float 0" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4380 |
"1 = real_of_float 1" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4381 |
"0 = Float 0 0" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4382 |
"1 = Float 1 0" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4383 |
"numeral a = Float (numeral a) 0" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4384 |
"numeral a = real_of_float (numeral a)" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4385 |
"x - y = x + - y" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4386 |
"x / y = x * inverse y" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4387 |
"ceiling x = - floor (- x)" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4388 |
"log x y = ln y * inverse (ln x)" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4389 |
"sin x = cos (pi / 2 - x)" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4390 |
"tan x = sin x / cos x" |
63931
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4391 |
by (simp_all add: inverse_eq_divide ceiling_def log_def sin_cos_eq tan_def real_of_float_eq) |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4392 |
|
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4393 |
lemma approximation_preproc_int[approximation_preproc]: |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4394 |
"real_of_int 0 = real_of_float 0" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4395 |
"real_of_int 1 = real_of_float 1" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4396 |
"real_of_int (i + j) = real_of_int i + real_of_int j" |
63929
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4397 |
"real_of_int (- i) = - real_of_int i" |
63931
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4398 |
"real_of_int (i - j) = real_of_int i - real_of_int j" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4399 |
"real_of_int (i * j) = real_of_int i * real_of_int j" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4400 |
"real_of_int (i div j) = real_of_int (floor (real_of_int i / real_of_int j))" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4401 |
"real_of_int (min i j) = min (real_of_int i) (real_of_int j)" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4402 |
"real_of_int (max i j) = max (real_of_int i) (real_of_int j)" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4403 |
"real_of_int (abs i) = abs (real_of_int i)" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4404 |
"real_of_int (i ^ n) = (real_of_int i) ^ n" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4405 |
"real_of_int (numeral a) = real_of_float (numeral a)" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4406 |
"i mod j = i - i div j * j" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4407 |
"i = j \<longleftrightarrow> real_of_int i = real_of_int j" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4408 |
"i \<le> j \<longleftrightarrow> real_of_int i \<le> real_of_int j" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4409 |
"i < j \<longleftrightarrow> real_of_int i < real_of_int j" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4410 |
"i \<in> {j .. k} \<longleftrightarrow> real_of_int i \<in> {real_of_int j .. real_of_int k}" |
64246 | 4411 |
by (simp_all add: floor_divide_of_int_eq minus_div_mult_eq_mod [symmetric]) |
63931
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4412 |
|
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4413 |
lemma approximation_preproc_nat[approximation_preproc]: |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4414 |
"real 0 = real_of_float 0" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4415 |
"real 1 = real_of_float 1" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4416 |
"real (i + j) = real i + real j" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4417 |
"real (i - j) = max (real i - real j) 0" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4418 |
"real (i * j) = real i * real j" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4419 |
"real (i div j) = real_of_int (floor (real i / real j))" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4420 |
"real (min i j) = min (real i) (real j)" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4421 |
"real (max i j) = max (real i) (real j)" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4422 |
"real (i ^ n) = (real i) ^ n" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4423 |
"real (numeral a) = real_of_float (numeral a)" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4424 |
"i mod j = i - i div j * j" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4425 |
"n = m \<longleftrightarrow> real n = real m" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4426 |
"n \<le> m \<longleftrightarrow> real n \<le> real m" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4427 |
"n < m \<longleftrightarrow> real n < real m" |
f17a1c60ac39
approximation: preprocessing for nat/int expressions
immler
parents:
63929
diff
changeset
|
4428 |
"n \<in> {m .. l} \<longleftrightarrow> real n \<in> {real m .. real l}" |
64243 | 4429 |
by (simp_all add: real_div_nat_eq_floor_of_divide minus_div_mult_eq_mod [symmetric]) |
63929
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4430 |
|
59850 | 4431 |
ML_file "approximation.ML" |
4432 |
||
60533 | 4433 |
method_setup approximation = \<open> |
60680 | 4434 |
let |
4435 |
val free = |
|
4436 |
Args.context -- Args.term >> (fn (_, Free (n, _)) => n | (ctxt, t) => |
|
4437 |
error ("Bad free variable: " ^ Syntax.string_of_term ctxt t)); |
|
59850 | 4438 |
in |
60680 | 4439 |
Scan.lift Parse.nat -- |
59850 | 4440 |
Scan.optional (Scan.lift (Args.$$$ "splitting" |-- Args.colon) |
60680 | 4441 |
|-- Parse.and_list' (free --| Scan.lift (Args.$$$ "=") -- Scan.lift Parse.nat)) [] -- |
4442 |
Scan.option (Scan.lift (Args.$$$ "taylor" |-- Args.colon) |-- |
|
4443 |
(free |-- Scan.lift (Args.$$$ "=") |-- Scan.lift Parse.nat)) >> |
|
59850 | 4444 |
(fn ((prec, splitting), taylor) => fn ctxt => |
4445 |
SIMPLE_METHOD' (Approximation.approximation_tac prec splitting taylor ctxt)) |
|
4446 |
end |
|
60533 | 4447 |
\<close> "real number approximation" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4448 |
|
58988 | 4449 |
|
4450 |
section "Quickcheck Generator" |
|
4451 |
||
63929
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4452 |
lemma approximation_preproc_push_neg[approximation_preproc]: |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4453 |
fixes a b::real |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4454 |
shows |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4455 |
"\<not> (a < b) \<longleftrightarrow> b \<le> a" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4456 |
"\<not> (a \<le> b) \<longleftrightarrow> b < a" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4457 |
"\<not> (a = b) \<longleftrightarrow> b < a \<or> a < b" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4458 |
"\<not> (p \<and> q) \<longleftrightarrow> \<not> p \<or> \<not> q" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4459 |
"\<not> (p \<or> q) \<longleftrightarrow> \<not> p \<and> \<not> q" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4460 |
"\<not> \<not> q \<longleftrightarrow> q" |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4461 |
by auto |
b673e7221b16
approximation: rewrite for reduction to base expressions
immler
parents:
63918
diff
changeset
|
4462 |
|
58988 | 4463 |
ML_file "approximation_generator.ML" |
4464 |
setup "Approximation_Generator.setup" |
|
4465 |
||
29805 | 4466 |
end |