author | paulson <lp15@cam.ac.uk> |
Fri, 13 Nov 2015 12:27:13 +0000 | |
changeset 61649 | 268d88ec9087 |
parent 61610 | 4f54d2759a0b |
child 61824 | dcbe9f756ae0 |
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 |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
34 |
unfolding setsum_right_distrib shift_pow uminus_add_conv_diff [symmetric] setsum_negf[symmetric] |
49351 | 35 |
setsum_head_upt_Suc[OF zero_less_Suc] |
57418 | 36 |
setsum.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
diff
changeset
|
172 |
else if odd n then |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
173 |
(- power_up_fl prec (abs l) n, |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
174 |
if u < 0 then - power_down_fl prec (abs u) n else power_up_fl prec u n) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
175 |
else if u < 0 then (power_down_fl prec (abs u) n, power_up_fl prec (abs l) n) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
176 |
else (0, power_up_fl prec (max (abs l) (abs u)) n))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
177 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
178 |
lemma le_minus_power_downI: "0 \<le> x \<Longrightarrow> x ^ n \<le> - a \<Longrightarrow> a \<le> - power_down prec x n" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
179 |
by (subst le_minus_iff) (auto intro: power_down_le power_mono_odd) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
180 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
181 |
lemma float_power_bnds: |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
182 |
"(l1, u1) = float_power_bnds prec n l u \<Longrightarrow> x \<in> {l .. u} \<Longrightarrow> (x::real) ^ n \<in> {l1..u1}" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
183 |
by (auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
184 |
simp: float_power_bnds_def max_def real_power_up_fl real_power_down_fl minus_le_iff |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
185 |
split: split_if_asm |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
186 |
intro!: power_up_le power_down_le le_minus_power_downI |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
187 |
intro: power_mono_odd power_mono power_mono_even zero_le_even_power) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
188 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
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> |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
191 |
l1 \<le> x ^ n \<and> x ^ n \<le> u1" |
29805 | 192 |
using float_power_bnds by auto |
193 |
||
60680 | 194 |
|
29805 | 195 |
section "Square root" |
196 |
||
60533 | 197 |
text \<open> |
29805 | 198 |
The square root computation is implemented as newton iteration. As first first step we use the |
199 |
nearest power of two greater than the square root. |
|
60533 | 200 |
\<close> |
29805 | 201 |
|
202 |
fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
203 |
"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
parents:
31148
diff
changeset
|
204 |
"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
205 |
in Float 1 (- 1) * float_plus_up prec y (float_divr prec x y))" |
29805 | 206 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
207 |
lemma compute_sqrt_iteration_base[code]: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
208 |
shows "sqrt_iteration prec n (Float m e) = |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
209 |
(if n = 0 then Float 1 ((if m = 0 then 0 else bitlen \<bar>m\<bar> + e) div 2 + 1) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
210 |
else (let y = sqrt_iteration prec (n - 1) (Float m e) in |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
211 |
Float 1 (- 1) * float_plus_up prec y (float_divr prec (Float m e) y)))" |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
212 |
using bitlen_Float by (cases n) simp_all |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
213 |
|
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
|
214 |
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
parents:
31148
diff
changeset
|
215 |
"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
|
216 |
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
parents:
31148
diff
changeset
|
217 |
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
|
218 |
"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
parents:
31148
diff
changeset
|
219 |
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
parents:
31148
diff
changeset
|
220 |
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
|
221 |
by pat_completeness auto |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55413
diff
changeset
|
222 |
termination by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto) |
29805 | 223 |
|
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
|
224 |
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
parents:
31148
diff
changeset
|
225 |
declare ub_sqrt.simps[simp del] |
29805 | 226 |
|
227 |
lemma sqrt_ub_pos_pos_1: |
|
228 |
assumes "sqrt x < b" and "0 < b" and "0 < x" |
|
229 |
shows "sqrt x < (b + x / b)/2" |
|
230 |
proof - |
|
53077 | 231 |
from assms have "0 < (b - sqrt x)\<^sup>2 " by simp |
232 |
also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + (sqrt x)\<^sup>2" by algebra |
|
233 |
also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + x" using assms by simp |
|
234 |
finally have "0 < b\<^sup>2 - 2 * b * sqrt x + x" . |
|
29805 | 235 |
hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms |
236 |
by (simp add: field_simps power2_eq_square) |
|
237 |
thus ?thesis by (simp add: field_simps) |
|
238 |
qed |
|
239 |
||
60680 | 240 |
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
|
241 |
assumes "0 < real_of_float x" |
54269 | 242 |
shows "sqrt x < sqrt_iteration prec n x" |
29805 | 243 |
proof (induct n) |
244 |
case 0 |
|
245 |
show ?case |
|
246 |
proof (cases x) |
|
247 |
case (Float m e) |
|
60680 | 248 |
hence "0 < m" |
249 |
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
|
250 |
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
|
251 |
by (meson not_less powr_ge_pzero) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
252 |
hence "0 < sqrt m" by auto |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
253 |
|
60680 | 254 |
have int_nat_bl: "(nat (bitlen m)) = bitlen m" |
255 |
using bitlen_nonneg by auto |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
256 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
257 |
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
|
258 |
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
|
259 |
also have "\<dots> < 1 * 2 powr (e + nat (bitlen m))" |
29805 | 260 |
proof (rule mult_strict_right_mono, auto) |
60680 | 261 |
show "m < 2^nat (bitlen m)" |
262 |
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
|
263 |
unfolding of_int_less_iff[of m, symmetric] by auto |
29805 | 264 |
qed |
60680 | 265 |
finally have "sqrt x < sqrt (2 powr (e + bitlen m))" |
266 |
unfolding int_nat_bl by auto |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
267 |
also have "\<dots> \<le> 2 powr ((e + bitlen m) div 2 + 1)" |
29805 | 268 |
proof - |
269 |
let ?E = "e + bitlen m" |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
270 |
have E_mod_pow: "2 powr (?E mod 2) < 4" |
29805 | 271 |
proof (cases "?E mod 2 = 1") |
60680 | 272 |
case True |
273 |
thus ?thesis by auto |
|
29805 | 274 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
275 |
case False |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
276 |
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
|
277 |
have "?E mod 2 < 2" by auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
278 |
from this[THEN zless_imp_add1_zle] |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
279 |
have "?E mod 2 \<le> 0" using False by auto |
60533 | 280 |
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
|
281 |
show ?thesis by auto |
29805 | 282 |
qed |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56813
diff
changeset
|
283 |
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
|
284 |
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
|
285 |
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
|
286 |
|
60680 | 287 |
have E_eq: "2 powr ?E = 2 powr (?E div 2 + ?E div 2 + ?E mod 2)" |
288 |
by auto |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
289 |
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
|
290 |
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
|
291 |
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
|
292 |
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
|
293 |
also have "\<dots> < 2 powr (?E div 2) * 2 powr 1" |
60680 | 294 |
by (rule mult_strict_left_mono) (auto intro: E_mod_pow) |
295 |
also have "\<dots> = 2 powr (?E div 2 + 1)" |
|
296 |
unfolding add.commute[of _ 1] powr_add[symmetric] by simp |
|
29805 | 297 |
finally show ?thesis by auto |
298 |
qed |
|
60533 | 299 |
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
|
300 |
unfolding Float |
47600 | 301 |
by (subst compute_sqrt_iteration_base) (simp add: ac_simps) |
29805 | 302 |
qed |
303 |
next |
|
304 |
case (Suc n) |
|
305 |
let ?b = "sqrt_iteration prec n x" |
|
60680 | 306 |
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
|
307 |
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
|
308 |
also have "\<dots> < real_of_float ?b" |
60680 | 309 |
using Suc . |
310 |
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
|
311 |
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
|
312 |
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
|
313 |
by (rule divide_right_mono, auto simp add: float_divr) |
60680 | 314 |
also have "\<dots> = (Float 1 (- 1)) * (?b + (float_divr prec x ?b))" |
315 |
by simp |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
316 |
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
|
317 |
by (auto simp add: algebra_simps float_plus_up_le) |
60680 | 318 |
finally show ?case |
319 |
unfolding sqrt_iteration.simps Let_def distrib_left . |
|
29805 | 320 |
qed |
321 |
||
60680 | 322 |
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
|
323 |
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
|
324 |
shows "0 < real_of_float (sqrt_iteration prec n x)" (is "0 < ?sqrt") |
29805 | 325 |
proof - |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
326 |
have "0 < sqrt x" using assms by auto |
29805 | 327 |
also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] . |
328 |
finally show ?thesis . |
|
329 |
qed |
|
330 |
||
60680 | 331 |
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
|
332 |
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
|
333 |
shows "0 \<le> real_of_float (lb_sqrt prec x)" |
29805 | 334 |
proof (cases "0 < x") |
60680 | 335 |
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
|
336 |
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
|
337 |
using \<open>0 \<le> real_of_float x\<close> by auto |
60680 | 338 |
hence "0 < sqrt_iteration prec prec x" |
339 |
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
|
340 |
hence "0 \<le> real_of_float (float_divl prec x (sqrt_iteration prec prec x))" |
60680 | 341 |
using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] unfolding less_eq_float_def by auto |
342 |
thus ?thesis |
|
343 |
unfolding lb_sqrt.simps using True by auto |
|
29805 | 344 |
next |
60680 | 345 |
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
|
346 |
with \<open>0 \<le> real_of_float x\<close> have "real_of_float x = 0" by auto |
60680 | 347 |
thus ?thesis |
348 |
unfolding lb_sqrt.simps by auto |
|
29805 | 349 |
qed |
350 |
||
49351 | 351 |
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
|
352 |
proof - |
60680 | 353 |
have lb: "lb_sqrt prec x \<le> sqrt x" if "0 < x" for x :: float |
354 |
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
|
355 |
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
|
356 |
hence sqrt_gt0: "0 < sqrt x" by auto |
60680 | 357 |
hence sqrt_ub: "sqrt x < sqrt_iteration prec prec x" |
358 |
using sqrt_iteration_bound by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
359 |
have "(float_divl prec x (sqrt_iteration prec prec x)) \<le> |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
360 |
x / (sqrt_iteration prec prec x)" by (rule float_divl) |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
361 |
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
|
362 |
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
|
363 |
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
|
364 |
also have "\<dots> = sqrt x" |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
365 |
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
|
366 |
sqrt_divide_self_eq[OF \<open>0 \<le> real_of_float x\<close>, symmetric] by auto |
60680 | 367 |
finally show ?thesis |
368 |
unfolding lb_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto |
|
369 |
qed |
|
370 |
have ub: "sqrt x \<le> ub_sqrt prec x" if "0 < x" for x :: float |
|
371 |
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
|
372 |
from that have "0 < real_of_float x" by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
373 |
hence "0 < sqrt x" by auto |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
374 |
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
|
375 |
using sqrt_iteration_bound by auto |
60680 | 376 |
then show ?thesis |
377 |
unfolding ub_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto |
|
378 |
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
|
379 |
show ?thesis |
54269 | 380 |
using lb[of "-x"] ub[of "-x"] lb[of x] ub[of x] |
381 |
by (auto simp add: lb_sqrt.simps ub_sqrt.simps real_sqrt_minus) |
|
29805 | 382 |
qed |
383 |
||
60680 | 384 |
lemma bnds_sqrt: "\<forall>(x::real) lx ux. |
385 |
(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
|
386 |
proof ((rule allI) +, rule impI, erule conjE, rule conjI) |
60680 | 387 |
fix x :: real |
388 |
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
|
389 |
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
|
390 |
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
|
391 |
hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto |
29805 | 392 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
393 |
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
|
394 |
from order_trans[OF _ this] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
395 |
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
|
396 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
397 |
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
|
398 |
from order_trans[OF this] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
399 |
show "sqrt x \<le> u" unfolding u using bnds_sqrt'[of ux prec] by auto |
29805 | 400 |
qed |
401 |
||
60680 | 402 |
|
29805 | 403 |
section "Arcus tangens and \<pi>" |
404 |
||
405 |
subsection "Compute arcus tangens series" |
|
406 |
||
60533 | 407 |
text \<open> |
29805 | 408 |
As first step we implement the computation of the arcus tangens series. This is only valid in the range |
409 |
@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens. |
|
60533 | 410 |
\<close> |
29805 | 411 |
|
412 |
fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
|
413 |
and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
414 |
"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
|
415 |
| "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
|
416 |
(rapprox_rat prec 1 k) (- float_round_down prec (x * (lb_arctan_horner prec n (k + 2) x)))" |
29805 | 417 |
| "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
|
418 |
| "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
|
419 |
(lapprox_rat prec 1 k) (- float_round_up prec (x * (ub_arctan_horner prec n (k + 2) x)))" |
29805 | 420 |
|
49351 | 421 |
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
|
422 |
assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1" |
60680 | 423 |
and "even n" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
424 |
shows "arctan (sqrt y) \<in> |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
425 |
{(sqrt y * lb_arctan_horner prec n 1 y) .. (sqrt y * ub_arctan_horner prec (Suc n) 1 y)}" |
29805 | 426 |
proof - |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
427 |
let ?c = "\<lambda>i. (- 1) ^ i * (1 / (i * 2 + (1::nat)) * sqrt y ^ (i * 2 + 1))" |
54269 | 428 |
let ?S = "\<lambda>n. \<Sum> i=0..<n. ?c i" |
29805 | 429 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
430 |
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
|
431 |
have "sqrt y \<le> 1" using assms by auto |
60533 | 432 |
from \<open>even n\<close> obtain m where "2 * m = n" by (blast elim: evenE) |
31809 | 433 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
434 |
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
|
435 |
proof (cases "sqrt y = 0") |
60680 | 436 |
case True |
437 |
then show ?thesis by simp |
|
438 |
next |
|
29805 | 439 |
case False |
60533 | 440 |
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
|
441 |
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
|
442 |
|
60533 | 443 |
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
|
444 |
from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this] |
60533 | 445 |
monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded \<open>2 * m = n\<close>] |
446 |
show ?thesis unfolding arctan_series[OF \<open>\<bar> sqrt y \<bar> \<le> 1\<close>] Suc_eq_plus1 atLeast0LessThan . |
|
60680 | 447 |
qed |
29805 | 448 |
note arctan_bounds = this[unfolded atLeastAtMost_iff] |
449 |
||
450 |
have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto |
|
451 |
||
31809 | 452 |
note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0 |
29805 | 453 |
and lb="\<lambda>n i k x. lb_arctan_horner prec n k x" |
31809 | 454 |
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
|
455 |
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
|
456 |
|
60680 | 457 |
have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> arctan (sqrt y)" |
458 |
proof - |
|
459 |
have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> ?S n" |
|
60533 | 460 |
using bounds(1) \<open>0 \<le> sqrt y\<close> |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
461 |
unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
462 |
unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult |
29805 | 463 |
by (auto intro!: mult_left_mono) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
464 |
also have "\<dots> \<le> arctan (sqrt y)" using arctan_bounds .. |
60680 | 465 |
finally show ?thesis . |
466 |
qed |
|
29805 | 467 |
moreover |
60680 | 468 |
have "arctan (sqrt y) \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)" |
469 |
proof - |
|
470 |
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
|
471 |
also have "\<dots> \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)" |
60533 | 472 |
using bounds(2)[of "Suc n"] \<open>0 \<le> sqrt y\<close> |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
473 |
unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric] |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
474 |
unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult |
29805 | 475 |
by (auto intro!: mult_left_mono) |
60680 | 476 |
finally show ?thesis . |
477 |
qed |
|
29805 | 478 |
ultimately show ?thesis by auto |
479 |
qed |
|
480 |
||
60680 | 481 |
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
|
482 |
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
|
483 |
shows "arctan (sqrt y) \<in> |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
484 |
{(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
|
485 |
(sqrt y * ub_arctan_horner prec (get_odd n) 1 y)}" |
54269 | 486 |
using |
487 |
arctan_0_1_bounds'[OF assms, of n prec] |
|
488 |
arctan_0_1_bounds'[OF assms, of "n + 1" prec] |
|
489 |
arctan_0_1_bounds'[OF assms, of "n - 1" prec] |
|
60680 | 490 |
by (auto simp: get_even_def get_odd_def odd_pos |
491 |
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
|
492 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
493 |
lemma arctan_lower_bound: |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
494 |
assumes "0 \<le> x" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
495 |
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
|
496 |
proof - |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
497 |
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
|
498 |
using assms |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
499 |
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
|
500 |
(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
|
501 |
thus ?thesis by simp |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
502 |
qed |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
503 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
504 |
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
|
505 |
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
|
506 |
(auto intro!: derivative_eq_intros divide_nonpos_nonneg |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
507 |
simp: inverse_eq_divide arctan_lower_bound) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
508 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
509 |
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
|
510 |
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
|
511 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
512 |
lemma arctan_mult_le: |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
513 |
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
|
514 |
shows "x * z \<le> arctan x" |
60680 | 515 |
proof (cases "x = 0") |
516 |
case True |
|
517 |
then show ?thesis by simp |
|
518 |
next |
|
519 |
case False |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
520 |
with assms have "z \<le> arctan y / y" by (simp add: field_simps) |
60533 | 521 |
also have "\<dots> \<le> arctan x / x" using assms \<open>x \<noteq> 0\<close> by (auto intro!: arctan_divide_mono) |
522 |
finally show ?thesis using assms \<open>x \<noteq> 0\<close> by (simp add: field_simps) |
|
60680 | 523 |
qed |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
524 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
525 |
lemma arctan_le_mult: |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
526 |
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
|
527 |
shows "arctan y \<le> y * z" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
528 |
proof - |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
529 |
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
|
530 |
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
|
531 |
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
|
532 |
qed |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
533 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
534 |
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
|
535 |
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
|
536 |
shows "arctan x \<in> |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
537 |
{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
|
538 |
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
|
539 |
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
|
540 |
"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
|
541 |
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
|
542 |
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
|
543 |
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
|
544 |
by simp |
60533 | 545 |
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
|
546 |
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
|
547 |
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
|
548 |
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
|
549 |
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
|
550 |
by simp |
60533 | 551 |
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
|
552 |
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
|
553 |
ultimately show ?thesis by simp |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
554 |
qed |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
555 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
556 |
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
|
557 |
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
|
558 |
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
|
559 |
{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
|
560 |
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
|
561 |
using assms |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
562 |
apply (cases "x > 0") |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
563 |
apply (intro arctan_0_1_bounds_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
564 |
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
|
565 |
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
|
566 |
mult_pos_pos) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
567 |
done |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
568 |
|
29805 | 569 |
|
570 |
subsection "Compute \<pi>" |
|
571 |
||
572 |
definition ub_pi :: "nat \<Rightarrow> float" where |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
573 |
"ub_pi prec = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
574 |
(let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
575 |
A = rapprox_rat prec 1 5 ; |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
576 |
B = lapprox_rat prec 1 239 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
577 |
in ((Float 1 2) * float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
578 |
((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
|
579 |
(float_round_down (Suc prec) (A * A))))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
580 |
(- 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
|
581 |
(float_round_up (Suc prec) (B * B)))))))" |
29805 | 582 |
|
583 |
definition lb_pi :: "nat \<Rightarrow> float" where |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
584 |
"lb_pi prec = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
585 |
(let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
586 |
A = lapprox_rat prec 1 5 ; |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
587 |
B = rapprox_rat prec 1 239 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
588 |
in ((Float 1 2) * float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
589 |
((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
|
590 |
(float_round_up (Suc prec) (A * A))))) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
591 |
(- 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
|
592 |
(float_round_down (Suc prec) (B * B)))))))" |
29805 | 593 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
594 |
lemma pi_boundaries: "pi \<in> {(lb_pi n) .. (ub_pi n)}" |
29805 | 595 |
proof - |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
596 |
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
|
597 |
unfolding machin[symmetric] by auto |
29805 | 598 |
|
60680 | 599 |
{ |
600 |
fix prec n :: nat |
|
601 |
fix k :: int |
|
602 |
assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto |
|
29805 | 603 |
let ?k = "rapprox_rat prec 1 k" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
604 |
let ?kl = "float_round_down (Suc prec) (?k * ?k)" |
60533 | 605 |
have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto |
606 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only 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 "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
|
608 |
have "real_of_float ?k \<le> 1" |
60533 | 609 |
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
|
610 |
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
|
611 |
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
|
612 |
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
|
613 |
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
|
614 |
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
|
615 |
by auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
616 |
finally have "arctan (1 / k) \<le> ?k * ub_arctan_horner prec (get_odd n) 1 ?kl" . |
29805 | 617 |
} note ub_arctan = this |
618 |
||
60680 | 619 |
{ |
620 |
fix prec n :: nat |
|
621 |
fix k :: int |
|
622 |
assume "1 < k" hence "0 \<le> k" and "0 < k" by auto |
|
29805 | 623 |
let ?k = "lapprox_rat prec 1 k" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
624 |
let ?ku = "float_round_up (Suc prec) (?k * ?k)" |
60533 | 625 |
have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto |
626 |
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
|
627 |
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 | 628 |
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
|
629 |
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
|
630 |
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
|
631 |
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
|
632 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
633 |
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
|
634 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
635 |
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
|
636 |
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
|
637 |
by auto |
60533 | 638 |
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
|
639 |
finally have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan (1 / k)" . |
29805 | 640 |
} note lb_arctan = this |
641 |
||
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
642 |
have "pi \<le> ub_pi n " |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
643 |
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
|
644 |
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
|
645 |
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
|
646 |
(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
|
647 |
moreover have "lb_pi n \<le> pi" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
648 |
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
|
649 |
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
|
650 |
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
|
651 |
(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
|
652 |
ultimately show ?thesis by auto |
29805 | 653 |
qed |
654 |
||
60680 | 655 |
|
29805 | 656 |
subsection "Compute arcus tangens in the entire domain" |
657 |
||
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
|
658 |
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
|
659 |
"lb_arctan prec x = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
660 |
(let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
661 |
ub_horner = \<lambda> x. float_round_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
662 |
(x * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
663 |
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
|
664 |
lb_horner = \<lambda> x. float_round_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
665 |
(x * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
666 |
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
|
667 |
in |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
668 |
if x < 0 then - ub_arctan prec (-x) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
669 |
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
|
670 |
else if x \<le> Float 1 1 then |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
671 |
Float 1 1 * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
672 |
lb_horner |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
673 |
(float_divl prec x |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
674 |
(float_plus_up prec 1 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
675 |
(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
|
676 |
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
|
677 |
if inv > 1 then 0 |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
678 |
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
|
679 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
680 |
| "ub_arctan prec x = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
681 |
(let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
682 |
lb_horner = \<lambda> x. float_round_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
683 |
(x * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
684 |
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
|
685 |
ub_horner = \<lambda> x. float_round_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
686 |
(x * |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
687 |
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
|
688 |
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
|
689 |
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
|
690 |
else if x \<le> Float 1 1 then |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
691 |
let y = float_divr prec x |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
692 |
(float_plus_down |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
693 |
(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
|
694 |
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
|
695 |
else float_plus_up prec (ub_pi prec * Float 1 (- 1)) ( - lb_horner (float_divl prec 1 x)))" |
29805 | 696 |
by pat_completeness auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
697 |
termination |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
698 |
by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto) |
29805 | 699 |
|
700 |
declare ub_arctan_horner.simps[simp del] |
|
701 |
declare lb_arctan_horner.simps[simp del] |
|
702 |
||
60680 | 703 |
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
|
704 |
assumes "0 \<le> real_of_float x" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
705 |
shows "lb_arctan prec x \<le> arctan x" |
29805 | 706 |
proof - |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
707 |
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
|
708 |
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
|
709 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
710 |
let "?ub_horner x" = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
711 |
"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
|
712 |
and "?lb_horner x" = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
713 |
"x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))" |
29805 | 714 |
|
715 |
show ?thesis |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
716 |
proof (cases "x \<le> Float 1 (- 1)") |
60680 | 717 |
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
|
718 |
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
|
719 |
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
|
720 |
show ?thesis |
60533 | 721 |
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
|
722 |
by (auto intro!: float_round_down_le) |
29805 | 723 |
next |
60680 | 724 |
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
|
725 |
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
|
726 |
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
|
727 |
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
|
728 |
let ?fR = "float_plus_up prec 1 (ub_sqrt prec ?sxx)" |
29805 | 729 |
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
|
730 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
731 |
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
|
732 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
733 |
have "sqrt (1 + x*x) \<le> sqrt ?sxx" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
734 |
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
|
735 |
also have "\<dots> \<le> ub_sqrt prec ?sxx" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
736 |
using bnds_sqrt'[of ?sxx prec] by auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
737 |
finally |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
738 |
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
|
739 |
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
|
740 |
hence "0 < ?fR" and "0 < real_of_float ?fR" using \<open>0 < ?R\<close> by auto |
29805 | 741 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
742 |
have monotone: "?DIV \<le> x / ?R" |
29805 | 743 |
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
|
744 |
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
|
745 |
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 | 746 |
finally show ?thesis . |
747 |
qed |
|
748 |
||
749 |
show ?thesis |
|
750 |
proof (cases "x \<le> Float 1 1") |
|
751 |
case True |
|
60680 | 752 |
have "x \<le> sqrt (1 + x * x)" |
753 |
using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto |
|
60533 | 754 |
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
|
755 |
finally have "real_of_float x \<le> ?fR" |
60680 | 756 |
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
|
757 |
moreover have "?DIV \<le> real_of_float x / ?fR" |
60680 | 758 |
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
|
759 |
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
|
760 |
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
|
761 |
|
77b453bd616f
Coercion "real" now has type nat => real only 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 |
have "0 \<le> real_of_float ?DIV" |
60680 | 763 |
using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] \<open>0 < ?fR\<close> |
764 |
unfolding less_eq_float_def by auto |
|
60533 | 765 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only 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 |
from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float (?DIV)\<close> \<open>real_of_float (?DIV) \<le> 1\<close>] |
60680 | 767 |
have "Float 1 1 * ?lb_horner ?DIV \<le> 2 * arctan ?DIV" |
768 |
by simp |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
769 |
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
|
770 |
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono arctan_monotone') |
60680 | 771 |
also have "2 * arctan (x / ?R) = arctan x" |
772 |
using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left . |
|
773 |
finally show ?thesis |
|
774 |
unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
775 |
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF True] |
|
776 |
by (auto simp: float_round_down.rep_eq |
|
777 |
intro!: order_trans[OF mult_left_mono[OF truncate_down]]) |
|
29805 | 778 |
next |
779 |
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
|
780 |
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
|
781 |
hence "1 \<le> real_of_float x" by auto |
29805 | 782 |
|
783 |
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
|
784 |
have "0 \<le> arctan x" using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>] |
60680 | 785 |
using arctan_tan[of 0, unfolded tan_zero] by auto |
29805 | 786 |
|
787 |
show ?thesis |
|
788 |
proof (cases "1 < ?invx") |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
789 |
case True |
60680 | 790 |
show ?thesis |
791 |
unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
792 |
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF False] if_P[OF True] |
|
60533 | 793 |
using \<open>0 \<le> arctan x\<close> by auto |
29805 | 794 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
795 |
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
|
796 |
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
|
797 |
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
|
798 |
by (rule order_trans[OF _ float_divr]) (auto simp add: \<open>0 \<le> real_of_float x\<close>) |
60680 | 799 |
|
800 |
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
|
801 |
using \<open>0 < real_of_float x\<close> by auto |
60680 | 802 |
|
803 |
have "arctan (1 / x) \<le> arctan ?invx" |
|
804 |
unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divr) |
|
805 |
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
|
806 |
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
|
807 |
by (auto intro!: float_round_up_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
808 |
also note float_round_up |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
809 |
finally have "pi / 2 - float_round_up prec (?ub_horner ?invx) \<le> arctan x" |
60533 | 810 |
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
|
811 |
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
|
812 |
moreover |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
813 |
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
|
814 |
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
|
815 |
ultimately |
60680 | 816 |
show ?thesis |
817 |
unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
818 |
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
|
819 |
by (auto intro!: float_plus_down_le) |
29805 | 820 |
qed |
821 |
qed |
|
822 |
qed |
|
823 |
qed |
|
824 |
||
60680 | 825 |
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
|
826 |
assumes "0 \<le> real_of_float x" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
827 |
shows "arctan x \<le> ub_arctan prec x" |
29805 | 828 |
proof - |
60680 | 829 |
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
|
830 |
using \<open>0 \<le> real_of_float x\<close> by auto |
60680 | 831 |
|
832 |
let "?ub_horner x" = |
|
833 |
"float_round_up prec (x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))" |
|
834 |
let "?lb_horner x" = |
|
835 |
"float_round_down prec (x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))" |
|
29805 | 836 |
|
837 |
show ?thesis |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
838 |
proof (cases "x \<le> Float 1 (- 1)") |
60680 | 839 |
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
|
840 |
hence "real_of_float x \<le> 1" by auto |
60680 | 841 |
show ?thesis |
842 |
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
|
843 |
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
|
844 |
by (auto intro!: float_round_up_le) |
29805 | 845 |
next |
60680 | 846 |
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
|
847 |
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
|
848 |
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
|
849 |
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
|
850 |
let ?fR = "float_plus_down (Suc prec) 1 (lb_sqrt prec ?sxx)" |
29805 | 851 |
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
|
852 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
853 |
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
|
854 |
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
|
855 |
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
|
856 |
|
29805 | 857 |
hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg) |
858 |
||
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
859 |
have "lb_sqrt prec ?sxx \<le> sqrt ?sxx" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
860 |
using bnds_sqrt'[of ?sxx] by auto |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
861 |
also have "\<dots> \<le> sqrt (1 + x*x)" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
862 |
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
|
863 |
finally have "lb_sqrt prec ?sxx \<le> sqrt (1 + x*x)" . |
60680 | 864 |
hence "?fR \<le> ?R" |
865 |
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
|
866 |
have "0 < real_of_float ?fR" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
867 |
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
|
868 |
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
|
869 |
truncate_down_nonneg add_nonneg_nonneg) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
870 |
have monotone: "x / ?R \<le> (float_divr prec x ?fR)" |
29805 | 871 |
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
|
872 |
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
|
873 |
have "x / ?R \<le> x / ?fR" . |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
874 |
also have "\<dots> \<le> ?DIV" by (rule float_divr) |
29805 | 875 |
finally show ?thesis . |
876 |
qed |
|
877 |
||
878 |
show ?thesis |
|
879 |
proof (cases "x \<le> Float 1 1") |
|
880 |
case True |
|
881 |
show ?thesis |
|
882 |
proof (cases "?DIV > 1") |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
883 |
case True |
60680 | 884 |
have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)" |
885 |
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
|
886 |
from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le] |
60680 | 887 |
show ?thesis |
888 |
unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
889 |
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 | 890 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
891 |
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
|
892 |
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
|
893 |
|
60680 | 894 |
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
|
895 |
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
|
896 |
hence "0 \<le> real_of_float ?DIV" |
60680 | 897 |
using monotone by (rule order_trans) |
898 |
||
899 |
have "arctan x = 2 * arctan (x / ?R)" |
|
900 |
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
|
901 |
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
|
902 |
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
|
903 |
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
|
904 |
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
|
905 |
by (auto intro!: float_round_up_le) |
60680 | 906 |
finally show ?thesis |
907 |
unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
908 |
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 | 909 |
qed |
910 |
next |
|
911 |
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
|
912 |
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
|
913 |
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
|
914 |
hence "0 < real_of_float x" by auto |
47600 | 915 |
hence "0 < x" by auto |
29805 | 916 |
|
917 |
let "?invx" = "float_divl prec 1 x" |
|
60680 | 918 |
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
|
919 |
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
|
920 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
921 |
have "real_of_float ?invx \<le> 1" |
60680 | 922 |
unfolding less_float_def |
923 |
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
|
924 |
(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
|
925 |
have "0 \<le> real_of_float ?invx" |
60680 | 926 |
using \<open>0 < x\<close> by (intro float_divl_lower_bound) auto |
927 |
||
928 |
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
|
929 |
using \<open>0 < real_of_float x\<close> by auto |
60680 | 930 |
|
931 |
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
|
932 |
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
|
933 |
by (auto intro!: float_round_down_le) |
60680 | 934 |
also have "\<dots> \<le> arctan (1 / x)" |
935 |
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
|
936 |
finally have "arctan x \<le> pi / 2 - (?lb_horner ?invx)" |
60533 | 937 |
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
|
938 |
unfolding sgn_pos[OF \<open>0 < 1 / x\<close>] le_diff_eq by auto |
29805 | 939 |
moreover |
60680 | 940 |
have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)" |
941 |
unfolding Float_num times_divide_eq_right mult_1_right |
|
942 |
using pi_boundaries by auto |
|
29805 | 943 |
ultimately |
60680 | 944 |
show ?thesis |
945 |
unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] |
|
946 |
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
|
947 |
by (auto intro!: float_round_up_le float_plus_up_le) |
29805 | 948 |
qed |
949 |
qed |
|
950 |
qed |
|
951 |
||
60680 | 952 |
lemma arctan_boundaries: "arctan x \<in> {(lb_arctan prec x) .. (ub_arctan prec x)}" |
29805 | 953 |
proof (cases "0 \<le> x") |
60680 | 954 |
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
|
955 |
hence "0 \<le> real_of_float x" by auto |
60680 | 956 |
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
|
957 |
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 | 958 |
unfolding atLeastAtMost_iff by auto |
29805 | 959 |
next |
60680 | 960 |
case False |
29805 | 961 |
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
|
962 |
from False have "x < 0" and "0 \<le> real_of_float ?mx" |
60680 | 963 |
by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
964 |
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
|
965 |
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 | 966 |
show ?thesis |
967 |
unfolding minus_float.rep_eq arctan_minus lb_arctan.simps[where x=x] |
|
968 |
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
|
969 |
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
|
970 |
by (simp add: arctan_minus) |
29805 | 971 |
qed |
972 |
||
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
973 |
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 | 974 |
proof (rule allI, rule allI, rule allI, rule impI) |
60680 | 975 |
fix x :: real |
976 |
fix lx ux |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
977 |
assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux}" |
60680 | 978 |
hence l: "lb_arctan prec lx = l " |
979 |
and u: "ub_arctan prec ux = u" |
|
980 |
and x: "x \<in> {lx .. ux}" |
|
981 |
by auto |
|
982 |
show "l \<le> arctan x \<and> arctan x \<le> u" |
|
983 |
proof |
|
984 |
show "l \<le> arctan x" |
|
985 |
proof - |
|
986 |
from arctan_boundaries[of lx prec, unfolded l] |
|
987 |
have "l \<le> arctan lx" by (auto simp del: lb_arctan.simps) |
|
988 |
also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone') |
|
989 |
finally show ?thesis . |
|
990 |
qed |
|
991 |
show "arctan x \<le> u" |
|
992 |
proof - |
|
993 |
have "arctan x \<le> arctan ux" using x by (auto intro: arctan_monotone') |
|
994 |
also have "\<dots> \<le> u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps) |
|
995 |
finally show ?thesis . |
|
996 |
qed |
|
997 |
qed |
|
29805 | 998 |
qed |
999 |
||
60680 | 1000 |
|
29805 | 1001 |
section "Sinus and Cosinus" |
1002 |
||
1003 |
subsection "Compute the cosinus and sinus series" |
|
1004 |
||
1005 |
fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
|
1006 |
and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
1007 |
"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
|
1008 |
| "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
|
1009 |
(rapprox_rat prec 1 k) (- |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1010 |
float_round_down prec (x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))" |
29805 | 1011 |
| "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
|
1012 |
| "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
|
1013 |
(lapprox_rat prec 1 k) (- |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1014 |
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
|
1015 |
|
29805 | 1016 |
lemma cos_aux: |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1017 |
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
|
1018 |
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 | 1019 |
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
|
1020 |
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
|
1021 |
let "?f n" = "fact (2 * n) :: nat" |
60680 | 1022 |
have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)" for n |
1023 |
proof - |
|
1024 |
have "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto |
|
1025 |
then show ?thesis by auto |
|
1026 |
qed |
|
31809 | 1027 |
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
|
1028 |
OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps] |
60680 | 1029 |
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
|
1030 |
by (auto simp add: power_mult power2_eq_square[of "real_of_float x"]) |
29805 | 1031 |
qed |
1032 |
||
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1033 |
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
|
1034 |
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
|
1035 |
(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
|
1036 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1037 |
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
|
1038 |
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
|
1039 |
|
60680 | 1040 |
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
|
1041 |
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
|
1042 |
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
|
1043 |
proof (cases "real_of_float x = 0") |
60680 | 1044 |
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
|
1045 |
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
|
1046 |
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
|
1047 |
using \<open>0 \<le> real_of_float x\<close> by auto |
60680 | 1048 |
have "0 < x * x" |
1049 |
using \<open>0 < x\<close> by simp |
|
1050 |
||
1051 |
have morph_to_if_power: "(\<Sum> i=0..<n. (-1::real) ^ i * (1/(fact (2 * i))) * x ^ (2 * i)) = |
|
1052 |
(\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * x ^ i)" |
|
1053 |
(is "?sum = ?ifsum") for x n |
|
29805 | 1054 |
proof - |
1055 |
have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto |
|
31809 | 1056 |
also have "\<dots> = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1057 |
(\<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
|
1058 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then (- 1) ^ (i div 2) / ((fact i)) * x ^ i else 0)" |
56195 | 1059 |
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
|
1060 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then (- 1) ^ (i div 2) / ((fact i)) else 0) * x ^ i)" |
57418 | 1061 |
by (rule setsum.cong) auto |
60680 | 1062 |
finally show ?thesis . |
1063 |
qed |
|
29805 | 1064 |
|
1065 |
{ fix n :: nat assume "0 < n" |
|
1066 |
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
|
1067 |
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
|
1068 |
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
|
1069 |
+ (cos (t + 1/2 * (2 * n) * pi) / (fact (2*n))) * (real_of_float x)^(2*n)" |
29805 | 1070 |
(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
|
1071 |
using Maclaurin_cos_expansion2[OF \<open>0 < real_of_float x\<close> \<open>0 < 2 * n\<close>] |
56195 | 1072 |
unfolding cos_coeff_def atLeast0LessThan by auto |
29805 | 1073 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1074 |
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
|
1075 |
also have "\<dots> = cos (t + n * pi)" by (simp add: cos_add) |
29805 | 1076 |
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
|
1077 |
finally have "cos t * (- 1) ^ n = ?rest" . |
29805 | 1078 |
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
|
1079 |
have "t \<le> pi / 2" using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto |
60533 | 1080 |
hence "0 \<le> cos t" using \<open>0 < t\<close> and cos_ge_zero by auto |
29805 | 1081 |
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto |
1082 |
||
1083 |
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
|
1084 |
have "0 < ?pow" using \<open>0 < real_of_float x\<close> by auto |
29805 | 1085 |
|
1086 |
{ |
|
1087 |
assume "even n" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1088 |
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
|
1089 |
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
|
1090 |
also have "\<dots> \<le> cos x" |
29805 | 1091 |
proof - |
60533 | 1092 |
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
|
1093 |
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
|
1094 |
thus ?thesis unfolding cos_eq by auto |
29805 | 1095 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1096 |
finally have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos x" . |
29805 | 1097 |
} note lb = this |
1098 |
||
1099 |
{ |
|
1100 |
assume "odd n" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1101 |
have "cos x \<le> ?SUM" |
29805 | 1102 |
proof - |
60533 | 1103 |
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
|
1104 |
have "0 \<le> (- ?rest) / ?fact * ?pow" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1105 |
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
|
1106 |
thus ?thesis unfolding cos_eq by auto |
29805 | 1107 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1108 |
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
|
1109 |
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
|
1110 |
finally have "cos x \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" . |
29805 | 1111 |
} note ub = this and lb |
1112 |
} note ub = this(1) and lb = this(2) |
|
1113 |
||
60680 | 1114 |
have "cos x \<le> (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" |
1115 |
using ub[OF odd_pos[OF get_odd] get_odd] . |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1116 |
moreover have "(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos x" |
29805 | 1117 |
proof (cases "0 < get_even n") |
60680 | 1118 |
case True |
1119 |
show ?thesis using lb[OF True get_even] . |
|
29805 | 1120 |
next |
1121 |
case False |
|
1122 |
hence "get_even n = 0" by auto |
|
60680 | 1123 |
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
|
1124 |
by (rule order_trans[OF _ \<open>0 < real_of_float x\<close>[THEN less_imp_le]]) auto |
60680 | 1125 |
with \<open>x \<le> pi / 2\<close> show ?thesis |
1126 |
unfolding \<open>get_even n = 0\<close> lb_sin_cos_aux.simps minus_float.rep_eq zero_float.rep_eq |
|
1127 |
using cos_ge_zero by auto |
|
29805 | 1128 |
qed |
1129 |
ultimately show ?thesis by auto |
|
1130 |
next |
|
1131 |
case True |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1132 |
hence "x = 0" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1133 |
by transfer |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1134 |
thus ?thesis |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1135 |
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
|
1136 |
by simp |
29805 | 1137 |
qed |
1138 |
||
60680 | 1139 |
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
|
1140 |
assumes "0 \<le> real_of_float x" |
60680 | 1141 |
shows "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> |
1142 |
(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb") |
|
1143 |
and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1)) \<le> |
|
1144 |
(x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub") |
|
29805 | 1145 |
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
|
1146 |
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
|
1147 |
let "?f n" = "fact (2 * n + 1) :: nat" |
60680 | 1148 |
have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)" for n |
1149 |
proof - |
|
45129
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
44821
diff
changeset
|
1150 |
have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto |
60680 | 1151 |
show ?thesis |
1152 |
unfolding F by auto |
|
1153 |
qed |
|
29805 | 1154 |
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
|
1155 |
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
|
1156 |
show "?lb" and "?ub" using \<open>0 \<le> real_of_float x\<close> |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
1157 |
unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_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
|
1158 |
unfolding mult.commute[where 'a=real] of_nat_fact |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1159 |
by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real_of_float x"]) |
29805 | 1160 |
qed |
1161 |
||
60680 | 1162 |
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
|
1163 |
assumes "0 \<le> real_of_float x" |
60680 | 1164 |
and "x \<le> pi / 2" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1165 |
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
|
1166 |
proof (cases "real_of_float x = 0") |
60680 | 1167 |
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
|
1168 |
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
|
1169 |
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
|
1170 |
using \<open>0 \<le> real_of_float x\<close> by auto |
60680 | 1171 |
have "0 < x * x" |
1172 |
using \<open>0 < x\<close> by simp |
|
1173 |
||
1174 |
have setsum_morph: "(\<Sum>j = 0 ..< n. (- 1) ^ (((2 * j + 1) - Suc 0) div 2) / ((fact (2 * j + 1))) * x ^(2 * j + 1)) = |
|
1175 |
(\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * x ^ i)" |
|
1176 |
(is "?SUM = _") for x :: real and n |
|
1177 |
proof - |
|
1178 |
have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" |
|
1179 |
by auto |
|
1180 |
have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" |
|
1181 |
by auto |
|
1182 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i)) * x ^ i)" |
|
1183 |
unfolding sum_split_even_odd atLeast0LessThan .. |
|
1184 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i))) * x ^ i)" |
|
1185 |
by (rule setsum.cong) auto |
|
1186 |
finally show ?thesis . |
|
1187 |
qed |
|
29805 | 1188 |
|
1189 |
{ fix n :: nat assume "0 < n" |
|
1190 |
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
|
1191 |
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
|
1192 |
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
|
1193 |
+ (sin (t + 1/2 * (2 * n + 1) * pi) / (fact (2*n + 1))) * (real_of_float x)^(2*n + 1)" |
29805 | 1194 |
(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
|
1195 |
using Maclaurin_sin_expansion3[OF \<open>0 < 2 * n + 1\<close> \<open>0 < real_of_float x\<close>] |
56195 | 1196 |
unfolding sin_coeff_def atLeast0LessThan by auto |
29805 | 1197 |
|
60680 | 1198 |
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
|
1199 |
unfolding sin_add cos_add of_nat_add distrib_right distrib_left by auto |
29805 | 1200 |
moreover |
60680 | 1201 |
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
|
1202 |
using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto |
60680 | 1203 |
hence "0 \<le> cos t" |
1204 |
using \<open>0 < t\<close> and cos_ge_zero by auto |
|
1205 |
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest" |
|
1206 |
by auto |
|
1207 |
||
1208 |
have "0 < ?fact" |
|
1209 |
by (simp del: fact_Suc) |
|
1210 |
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
|
1211 |
using \<open>0 < real_of_float x\<close> by (rule zero_less_power) |
29805 | 1212 |
|
1213 |
{ |
|
1214 |
assume "even n" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1215 |
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
|
1216 |
(\<Sum> i = 0 ..< 2 * n. (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
|
1217 |
using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding setsum_morph[symmetric] by auto |
29805 | 1218 |
also have "\<dots> \<le> ?SUM" by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1219 |
also have "\<dots> \<le> sin x" |
29805 | 1220 |
proof - |
60533 | 1221 |
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
|
1222 |
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
|
1223 |
thus ?thesis unfolding sin_eq by auto |
29805 | 1224 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1225 |
finally have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin x" . |
29805 | 1226 |
} note lb = this |
1227 |
||
1228 |
{ |
|
1229 |
assume "odd n" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1230 |
have "sin x \<le> ?SUM" |
29805 | 1231 |
proof - |
60533 | 1232 |
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
|
1233 |
have "0 \<le> (- ?rest) / ?fact * ?pow" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1234 |
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
|
1235 |
thus ?thesis unfolding sin_eq by auto |
29805 | 1236 |
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
|
1237 |
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
|
1238 |
by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1239 |
also have "\<dots> \<le> (x * ub_sin_cos_aux prec 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
|
1240 |
using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding setsum_morph[symmetric] by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1241 |
finally have "sin x \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" . |
29805 | 1242 |
} note ub = this and lb |
1243 |
} note ub = this(1) and lb = this(2) |
|
1244 |
||
60680 | 1245 |
have "sin x \<le> (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" |
1246 |
using ub[OF odd_pos[OF get_odd] get_odd] . |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1247 |
moreover have "(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin x" |
29805 | 1248 |
proof (cases "0 < get_even n") |
60680 | 1249 |
case True |
1250 |
show ?thesis |
|
1251 |
using lb[OF True get_even] . |
|
29805 | 1252 |
next |
1253 |
case False |
|
1254 |
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
|
1255 |
with \<open>x \<le> pi / 2\<close> \<open>0 \<le> real_of_float x\<close> |
60680 | 1256 |
show ?thesis |
1257 |
unfolding \<open>get_even n = 0\<close> ub_sin_cos_aux.simps minus_float.rep_eq |
|
1258 |
using sin_ge_zero by auto |
|
29805 | 1259 |
qed |
1260 |
ultimately show ?thesis by auto |
|
1261 |
next |
|
1262 |
case True |
|
1263 |
show ?thesis |
|
1264 |
proof (cases "n = 0") |
|
31809 | 1265 |
case True |
60680 | 1266 |
thus ?thesis |
1267 |
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
|
1268 |
using \<open>real_of_float x = 0\<close> lapprox_rat[where x="-1" and y=1] by auto |
29805 | 1269 |
next |
60680 | 1270 |
case False |
1271 |
with not0_implies_Suc obtain m where "n = Suc m" by blast |
|
1272 |
thus ?thesis |
|
1273 |
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
|
1274 |
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 | 1275 |
by (cases "even (Suc m)") auto |
29805 | 1276 |
qed |
1277 |
qed |
|
1278 |
||
60680 | 1279 |
|
29805 | 1280 |
subsection "Compute the cosinus in the entire domain" |
1281 |
||
1282 |
definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
1283 |
"lb_cos prec x = (let |
|
1284 |
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
|
1285 |
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
|
1286 |
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
|
1287 |
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
|
1288 |
else half (half (horner (x * Float 1 (- 2)))))" |
29805 | 1289 |
|
1290 |
definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
1291 |
"ub_cos prec x = (let |
|
1292 |
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
|
1293 |
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
|
1294 |
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
|
1295 |
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
|
1296 |
else half (half (horner (x * Float 1 (- 2)))))" |
29805 | 1297 |
|
60680 | 1298 |
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
|
1299 |
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
|
1300 |
shows "cos x \<in> {(lb_cos prec x) .. (ub_cos prec x)}" (is "?cos x \<in> {(?lb x) .. (?ub x) }") |
29805 | 1301 |
proof - |
60680 | 1302 |
have x_half[symmetric]: "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" for x :: real |
1303 |
proof - |
|
1304 |
have "cos x = cos (x / 2 + x / 2)" |
|
1305 |
by auto |
|
29805 | 1306 |
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" |
1307 |
unfolding cos_add by auto |
|
60680 | 1308 |
also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" |
1309 |
by algebra |
|
1310 |
finally show ?thesis . |
|
1311 |
qed |
|
29805 | 1312 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1313 |
have "\<not> x < 0" using \<open>0 \<le> real_of_float x\<close> by auto |
29805 | 1314 |
let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)" |
1315 |
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
|
1316 |
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
|
1317 |
let "?lb_half x" = "if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)" |
29805 | 1318 |
|
1319 |
show ?thesis |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1320 |
proof (cases "x < Float 1 (- 1)") |
60680 | 1321 |
case True |
1322 |
hence "x \<le> pi / 2" |
|
1323 |
using pi_ge_two by auto |
|
1324 |
show ?thesis |
|
1325 |
unfolding lb_cos_def[where x=x] ub_cos_def[where x=x] |
|
1326 |
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
|
1327 |
using cos_boundaries[OF \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi / 2\<close>] . |
29805 | 1328 |
next |
1329 |
case False |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1330 |
{ fix y x :: float let ?x2 = "(x * Float 1 (- 1))" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1331 |
assume "y \<le> cos ?x2" and "-pi \<le> x" and "x \<le> pi" |
60680 | 1332 |
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" |
1333 |
using pi_ge_two unfolding Float_num by auto |
|
1334 |
hence "0 \<le> cos ?x2" |
|
1335 |
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
|
1336 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1337 |
have "(?lb_half y) \<le> cos x" |
29805 | 1338 |
proof (cases "y < 0") |
60680 | 1339 |
case True |
1340 |
show ?thesis |
|
1341 |
using cos_ge_minus_one unfolding if_P[OF True] by auto |
|
29805 | 1342 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1343 |
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
|
1344 |
hence "0 \<le> real_of_float y" by auto |
60533 | 1345 |
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
|
1346 |
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
|
1347 |
hence "2 * real_of_float y * real_of_float y \<le> 2 * cos ?x2 * cos ?x2" |
60680 | 1348 |
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
|
1349 |
hence "2 * real_of_float y * real_of_float y - 1 \<le> 2 * cos (x / 2) * cos (x / 2) - 1" |
60680 | 1350 |
unfolding Float_num by auto |
1351 |
thus ?thesis |
|
1352 |
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
|
1353 |
by (auto intro!: float_plus_down_le) |
29805 | 1354 |
qed |
1355 |
} 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
|
1356 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1357 |
{ fix y x :: float let ?x2 = "(x * Float 1 (- 1))" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1358 |
assume ub: "cos ?x2 \<le> y" and "- pi \<le> x" and "x \<le> pi" |
60680 | 1359 |
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" |
1360 |
using pi_ge_two unfolding Float_num by auto |
|
29805 | 1361 |
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
|
1362 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1363 |
have "cos x \<le> (?ub_half y)" |
29805 | 1364 |
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
|
1365 |
have "0 \<le> real_of_float y" |
60680 | 1366 |
using \<open>0 \<le> cos ?x2\<close> ub by (rule order_trans) |
60533 | 1367 |
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
|
1368 |
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
|
1369 |
hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real_of_float y * real_of_float y" |
60680 | 1370 |
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
|
1371 |
hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real_of_float y * real_of_float y - 1" |
60680 | 1372 |
unfolding Float_num by auto |
1373 |
thus ?thesis |
|
1374 |
unfolding x_half Float_num |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1375 |
by (auto intro!: float_plus_up_le) |
29805 | 1376 |
qed |
1377 |
} 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
|
1378 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
1379 |
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
|
1380 |
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
|
1381 |
|
60680 | 1382 |
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
|
1383 |
using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] \<open>0 \<le> real_of_float x\<close> |
60680 | 1384 |
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
|
1385 |
|
29805 | 1386 |
show ?thesis |
1387 |
proof (cases "x < 1") |
|
60680 | 1388 |
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
|
1389 |
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
|
1390 |
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
|
1391 |
using pi_ge_two \<open>0 \<le> real_of_float x\<close> using assms by auto |
29805 | 1392 |
from cos_boundaries[OF this] |
60680 | 1393 |
have lb: "(?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> (?ub_horner ?x2)" |
1394 |
by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1395 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1396 |
have "(?lb x) \<le> ?cos x" |
29805 | 1397 |
proof - |
60533 | 1398 |
from lb_half[OF lb \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>] |
60680 | 1399 |
show ?thesis |
1400 |
unfolding lb_cos_def[where x=x] Let_def |
|
1401 |
using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto |
|
29805 | 1402 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1403 |
moreover have "?cos x \<le> (?ub x)" |
29805 | 1404 |
proof - |
60533 | 1405 |
from ub_half[OF ub \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>] |
60680 | 1406 |
show ?thesis |
1407 |
unfolding ub_cos_def[where x=x] Let_def |
|
1408 |
using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto |
|
29805 | 1409 |
qed |
1410 |
ultimately show ?thesis by auto |
|
1411 |
next |
|
1412 |
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
|
1413 |
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
|
1414 |
using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> unfolding Float_num by auto |
29805 | 1415 |
from cos_boundaries[OF this] |
60680 | 1416 |
have lb: "(?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> (?ub_horner ?x4)" |
1417 |
by auto |
|
1418 |
||
1419 |
have eq_4: "?x2 * Float 1 (- 1) = x * Float 1 (- 2)" |
|
1420 |
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
|
1421 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1422 |
have "(?lb x) \<le> ?cos x" |
29805 | 1423 |
proof - |
60680 | 1424 |
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
|
1425 |
using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> by auto |
60533 | 1426 |
from lb_half[OF lb_half[OF lb this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4] |
60680 | 1427 |
show ?thesis |
1428 |
unfolding lb_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>] |
|
1429 |
if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def . |
|
29805 | 1430 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1431 |
moreover have "?cos x \<le> (?ub x)" |
29805 | 1432 |
proof - |
60680 | 1433 |
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
|
1434 |
using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open> x \<le> pi\<close> by auto |
60533 | 1435 |
from ub_half[OF ub_half[OF ub this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4] |
60680 | 1436 |
show ?thesis |
1437 |
unfolding ub_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>] |
|
1438 |
if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def . |
|
29805 | 1439 |
qed |
1440 |
ultimately show ?thesis by auto |
|
1441 |
qed |
|
1442 |
qed |
|
1443 |
qed |
|
1444 |
||
60680 | 1445 |
lemma lb_cos_minus: |
1446 |
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
|
1447 |
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
|
1448 |
shows "cos (real_of_float(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}" |
29805 | 1449 |
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
|
1450 |
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
|
1451 |
using \<open>-pi \<le> x\<close> \<open>real_of_float x \<le> 0\<close> by auto |
29805 | 1452 |
from lb_cos[OF this] show ?thesis . |
1453 |
qed |
|
1454 |
||
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
|
1455 |
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
|
1456 |
"bnds_cos prec lx ux = (let |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1457 |
lpi = float_round_down prec (lb_pi prec) ; |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1458 |
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
|
1459 |
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
|
1460 |
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
|
1461 |
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
|
1462 |
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
|
1463 |
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
|
1464 |
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
|
1465 |
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
|
1466 |
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
|
1467 |
else (Float (- 1) 0, Float 1 0))" |
29805 | 1468 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1469 |
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
|
1470 |
by (simp add: floor_fl_def) |
29805 | 1471 |
|
60680 | 1472 |
lemma cos_periodic_nat[simp]: |
1473 |
fixes n :: nat |
|
1474 |
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
|
1475 |
proof (induct n arbitrary: x) |
60680 | 1476 |
case 0 |
1477 |
then show ?case by simp |
|
1478 |
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
|
1479 |
case (Suc n) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1480 |
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
|
1481 |
unfolding Suc_eq_plus1 of_nat_add of_int_1 distrib_right by auto |
60680 | 1482 |
show ?case |
1483 |
unfolding split_pi_off using Suc by auto |
|
1484 |
qed |
|
1485 |
||
1486 |
lemma cos_periodic_int[simp]: |
|
1487 |
fixes i :: int |
|
1488 |
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
|
1489 |
proof (cases "0 \<le> i") |
60680 | 1490 |
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
|
1491 |
hence i_nat: "real_of_int i = nat i" by auto |
60680 | 1492 |
show ?thesis |
1493 |
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
|
1494 |
next |
60680 | 1495 |
case False |
1496 |
hence i_nat: "i = - real (nat (-i))" by auto |
|
1497 |
have "cos x = cos (x + i * (2 * pi) - i * (2 * pi))" |
|
1498 |
by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1499 |
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
|
1500 |
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
|
1501 |
finally show ?thesis by auto |
29805 | 1502 |
qed |
1503 |
||
60680 | 1504 |
lemma bnds_cos: "\<forall>(x::real) lx ux. (l, u) = |
1505 |
bnds_cos prec lx ux \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> cos x \<and> cos x \<le> u" |
|
1506 |
proof (rule allI | rule impI | erule conjE)+ |
|
1507 |
fix x :: real |
|
1508 |
fix lx ux |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1509 |
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
|
1510 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1511 |
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
|
1512 |
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
|
1513 |
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
|
1514 |
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
|
1515 |
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
|
1516 |
let ?lx = "float_plus_down prec lx ?lx2" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1517 |
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
|
1518 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1519 |
obtain k :: int where k: "k = real_of_float ?k" |
60680 | 1520 |
by (rule floor_int) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1521 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1522 |
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
|
1523 |
using float_round_up[of "ub_pi prec" prec] pi_boundaries[of prec] |
60680 | 1524 |
float_round_down[of prec "lb_pi prec"] |
1525 |
by auto |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1526 |
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
|
1527 |
using x |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1528 |
by (cases "k = 0") |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1529 |
(auto intro!: add_mono |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1530 |
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
|
1531 |
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
|
1532 |
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
|
1533 |
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
|
1534 |
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
|
1535 |
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
|
1536 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1537 |
{ 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
|
1538 |
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
|
1539 |
have pi_lx: "- pi \<le> ?lx" and lx_0: "real_of_float ?lx \<le> 0" |
47600 | 1540 |
by simp_all |
29805 | 1541 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1542 |
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
|
1543 |
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
|
1544 |
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
|
1545 |
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
|
1546 |
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
|
1547 |
cos_minus mult_minus_left) simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1548 |
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
|
1549 |
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
|
1550 |
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
|
1551 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1552 |
{ 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
|
1553 |
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
|
1554 |
have pi_lx: "?lx \<le> pi" and lx_0: "0 \<le> real_of_float ?lx" |
47600 | 1555 |
by auto |
29805 | 1556 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1557 |
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
|
1558 |
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
|
1559 |
by (simp only: of_int_minus |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
1560 |
cos_minus mult_minus_left) simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1561 |
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
|
1562 |
using lb_cos[OF lx_0 pi_lx] by simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1563 |
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
|
1564 |
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
|
1565 |
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
|
1566 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1567 |
{ 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
|
1568 |
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
|
1569 |
have pi_ux: "- pi \<le> ?ux" and ux_0: "real_of_float ?ux \<le> 0" |
47600 | 1570 |
by simp_all |
29805 | 1571 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1572 |
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
|
1573 |
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
|
1574 |
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
|
1575 |
cos_minus mult_minus_left) simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1576 |
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
|
1577 |
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
|
1578 |
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
|
1579 |
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
|
1580 |
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
|
1581 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1582 |
{ 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
|
1583 |
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
|
1584 |
have pi_ux: "?ux \<le> pi" and ux_0: "0 \<le> real_of_float ?ux" |
47600 | 1585 |
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
|
1586 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1587 |
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
|
1588 |
using lb_cos[OF ux_0 pi_ux] by simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1589 |
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
|
1590 |
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
|
1591 |
by (simp only: of_int_minus |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
1592 |
cos_minus mult_minus_left) simp |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1593 |
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
|
1594 |
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
|
1595 |
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
|
1596 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1597 |
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
|
1598 |
proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0") |
60680 | 1599 |
case True |
1600 |
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
|
1601 |
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
|
1602 |
from True lpi[THEN le_imp_neg_le] lx ux |
60680 | 1603 |
have "- pi \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> 0" |
47600 | 1604 |
by auto |
60680 | 1605 |
with True negative_ux negative_lx show ?thesis |
1606 |
unfolding l u by simp |
|
1607 |
next |
|
1608 |
case 1: False |
|
1609 |
show ?thesis |
|
1610 |
proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi") |
|
1611 |
case True with bnds 1 |
|
1612 |
have l: "l = lb_cos prec ?ux" |
|
1613 |
and u: "u = ub_cos prec ?lx" |
|
1614 |
by (auto simp add: bnds_cos_def Let_def) |
|
1615 |
from True lpi lx ux |
|
1616 |
have "0 \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> pi" |
|
1617 |
by auto |
|
1618 |
with True positive_ux positive_lx show ?thesis |
|
1619 |
unfolding l u by simp |
|
29805 | 1620 |
next |
60680 | 1621 |
case 2: False |
1622 |
show ?thesis |
|
1623 |
proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi") |
|
1624 |
case Cond: True |
|
1625 |
with bnds 1 2 have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)" |
|
1626 |
and u: "u = Float 1 0" |
|
1627 |
by (auto simp add: bnds_cos_def Let_def) |
|
1628 |
show ?thesis |
|
1629 |
unfolding u l using negative_lx positive_ux Cond |
|
1630 |
by (cases "x - k * (2 * pi) < 0") (auto simp add: real_of_float_min) |
|
1631 |
next |
|
1632 |
case 3: False |
|
1633 |
show ?thesis |
|
1634 |
proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi") |
|
1635 |
case Cond: True |
|
1636 |
with bnds 1 2 3 |
|
1637 |
have l: "l = Float (- 1) 0" |
|
1638 |
and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))" |
|
1639 |
by (auto simp add: bnds_cos_def Let_def) |
|
1640 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1641 |
have "cos x \<le> real_of_float u" |
60680 | 1642 |
proof (cases "x - k * (2 * pi) < pi") |
1643 |
case True |
|
1644 |
hence "x - k * (2 * pi) \<le> pi" by simp |
|
1645 |
from positive_lx[OF Cond[THEN conjunct1] this] show ?thesis |
|
1646 |
unfolding u by (simp add: real_of_float_max) |
|
1647 |
next |
|
1648 |
case False |
|
1649 |
hence "pi \<le> x - k * (2 * pi)" by simp |
|
1650 |
hence pi_x: "- pi \<le> x - k * (2 * pi) - 2 * pi" by simp |
|
1651 |
||
1652 |
have "?ux \<le> 2 * pi" |
|
1653 |
using Cond lpi by auto |
|
1654 |
hence "x - k * (2 * pi) - 2 * pi \<le> 0" |
|
1655 |
using ux by simp |
|
1656 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1657 |
have ux_0: "real_of_float (?ux - 2 * ?lpi) \<le> 0" |
60680 | 1658 |
using Cond by auto |
1659 |
||
1660 |
from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto |
|
1661 |
hence "- ?lpi \<le> ?ux - 2 * ?lpi" by auto |
|
1662 |
hence pi_ux: "- pi \<le> (?ux - 2 * ?lpi)" |
|
1663 |
using lpi[THEN le_imp_neg_le] by auto |
|
1664 |
||
1665 |
have x_le_ux: "x - k * (2 * pi) - 2 * pi \<le> (?ux - 2 * ?lpi)" |
|
1666 |
using ux lpi by auto |
|
1667 |
have "cos x = cos (x + (-k) * (2 * pi) + (-1::int) * (2 * pi))" |
|
1668 |
unfolding cos_periodic_int .. |
|
1669 |
also have "\<dots> \<le> cos ((?ux - 2 * ?lpi))" |
|
1670 |
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
|
1671 |
by (simp only: minus_float.rep_eq of_int_minus of_int_1 |
60680 | 1672 |
mult_minus_left mult_1_left) simp |
1673 |
also have "\<dots> = cos ((- (?ux - 2 * ?lpi)))" |
|
1674 |
unfolding uminus_float.rep_eq cos_minus .. |
|
1675 |
also have "\<dots> \<le> (ub_cos prec (- (?ux - 2 * ?lpi)))" |
|
1676 |
using lb_cos_minus[OF pi_ux ux_0] by simp |
|
1677 |
finally show ?thesis unfolding u by (simp add: real_of_float_max) |
|
1678 |
qed |
|
1679 |
thus ?thesis unfolding l by auto |
|
1680 |
next |
|
1681 |
case 4: False |
|
1682 |
show ?thesis |
|
1683 |
proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0") |
|
1684 |
case Cond: True |
|
1685 |
with bnds 1 2 3 4 have l: "l = Float (- 1) 0" |
|
1686 |
and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))" |
|
1687 |
by (auto simp add: bnds_cos_def Let_def) |
|
1688 |
||
1689 |
have "cos x \<le> u" |
|
1690 |
proof (cases "-pi < x - k * (2 * pi)") |
|
1691 |
case True |
|
1692 |
hence "-pi \<le> x - k * (2 * pi)" by simp |
|
1693 |
from negative_ux[OF this Cond[THEN conjunct2]] show ?thesis |
|
1694 |
unfolding u by (simp add: real_of_float_max) |
|
1695 |
next |
|
1696 |
case False |
|
1697 |
hence "x - k * (2 * pi) \<le> -pi" by simp |
|
1698 |
hence pi_x: "x - k * (2 * pi) + 2 * pi \<le> pi" by simp |
|
1699 |
||
1700 |
have "-2 * pi \<le> ?lx" using Cond lpi by auto |
|
1701 |
||
1702 |
hence "0 \<le> x - k * (2 * pi) + 2 * pi" using lx by simp |
|
1703 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1704 |
have lx_0: "0 \<le> real_of_float (?lx + 2 * ?lpi)" |
60680 | 1705 |
using Cond lpi by auto |
1706 |
||
1707 |
from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto |
|
1708 |
hence "?lx + 2 * ?lpi \<le> ?lpi" by auto |
|
1709 |
hence pi_lx: "(?lx + 2 * ?lpi) \<le> pi" |
|
1710 |
using lpi[THEN le_imp_neg_le] by auto |
|
1711 |
||
1712 |
have lx_le_x: "(?lx + 2 * ?lpi) \<le> x - k * (2 * pi) + 2 * pi" |
|
1713 |
using lx lpi by auto |
|
1714 |
||
1715 |
have "cos x = cos (x + (-k) * (2 * pi) + (1 :: int) * (2 * pi))" |
|
1716 |
unfolding cos_periodic_int .. |
|
1717 |
also have "\<dots> \<le> cos ((?lx + 2 * ?lpi))" |
|
1718 |
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
|
1719 |
by (simp only: minus_float.rep_eq of_int_minus of_int_1 |
60680 | 1720 |
mult_minus_left mult_1_left) simp |
1721 |
also have "\<dots> \<le> (ub_cos prec (?lx + 2 * ?lpi))" |
|
1722 |
using lb_cos[OF lx_0 pi_lx] by simp |
|
1723 |
finally show ?thesis unfolding u by (simp add: real_of_float_max) |
|
1724 |
qed |
|
1725 |
thus ?thesis unfolding l by auto |
|
1726 |
next |
|
1727 |
case False |
|
1728 |
with bnds 1 2 3 4 show ?thesis |
|
1729 |
by (auto simp add: bnds_cos_def Let_def) |
|
1730 |
qed |
|
1731 |
qed |
|
1732 |
qed |
|
29805 | 1733 |
qed |
60680 | 1734 |
qed |
29805 | 1735 |
qed |
1736 |
||
60680 | 1737 |
|
29805 | 1738 |
section "Exponential function" |
1739 |
||
1740 |
subsection "Compute the series of the exponential function" |
|
1741 |
||
60680 | 1742 |
fun ub_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
1743 |
and lb_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
|
1744 |
where |
|
29805 | 1745 |
"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
|
1746 |
"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
|
1747 |
(rapprox_rat prec 1 (int k)) (float_round_up prec (x * lb_exp_horner prec n (i + 1) (k * i) x))" | |
29805 | 1748 |
"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
|
1749 |
"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
|
1750 |
(lapprox_rat prec 1 (int k)) (float_round_down prec (x * ub_exp_horner prec n (i + 1) (k * i) x))" |
29805 | 1751 |
|
60680 | 1752 |
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
|
1753 |
assumes "real_of_float x \<le> 0" |
60680 | 1754 |
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 | 1755 |
proof - |
60680 | 1756 |
have f_eq: "fact (Suc n) = fact n * ((\<lambda>i::nat. i + 1) ^^ n) 1" for n |
1757 |
proof - |
|
1758 |
have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m" |
|
1759 |
by (induct n) auto |
|
1760 |
show ?thesis |
|
1761 |
unfolding F by auto |
|
1762 |
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
|
1763 |
|
29805 | 1764 |
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, |
1765 |
OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps] |
|
1766 |
||
60680 | 1767 |
have "lb_exp_horner prec (get_even n) 1 1 x \<le> exp x" |
1768 |
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
|
1769 |
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 | 1770 |
using bounds(1) by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1771 |
also have "\<dots> \<le> exp x" |
29805 | 1772 |
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
|
1773 |
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 | 1774 |
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
|
1775 |
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
|
1776 |
by (auto simp: zero_le_even_power) |
56536 | 1777 |
ultimately show ?thesis using get_odd exp_gt_zero by auto |
29805 | 1778 |
qed |
60680 | 1779 |
finally show ?thesis . |
1780 |
qed |
|
1781 |
moreover |
|
1782 |
have "exp x \<le> ub_exp_horner prec (get_odd n) 1 1 x" |
|
1783 |
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
|
1784 |
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
|
1785 |
proof (cases "real_of_float x = 0") |
29805 | 1786 |
case True |
1787 |
have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto |
|
1788 |
thus ?thesis unfolding True power_0_left by auto |
|
1789 |
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
|
1790 |
case False hence "real_of_float x < 0" using \<open>real_of_float x \<le> 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
|
1791 |
show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq \<open>real_of_float x < 0\<close>) |
29805 | 1792 |
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
|
1793 |
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
|
1794 |
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 | 1795 |
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
|
1796 |
moreover have "exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n) \<le> 0" |
46545 | 1797 |
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
|
1798 |
ultimately have "exp x \<le> (\<Sum>j = 0..<get_odd n. 1 / (fact j) * real_of_float x ^ j)" |
56536 | 1799 |
using get_odd exp_gt_zero by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1800 |
also have "\<dots> \<le> ub_exp_horner prec (get_odd n) 1 1 x" |
29805 | 1801 |
using bounds(2) by auto |
60680 | 1802 |
finally show ?thesis . |
1803 |
qed |
|
1804 |
ultimately show ?thesis by auto |
|
29805 | 1805 |
qed |
1806 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1807 |
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
|
1808 |
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
|
1809 |
using bnds_exp_horner[of x prec n] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1810 |
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
|
1811 |
|
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1812 |
|
29805 | 1813 |
subsection "Compute the exponential function on the entire domain" |
1814 |
||
1815 |
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
|
1816 |
"lb_exp prec x = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1817 |
(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
|
1818 |
else |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1819 |
let |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1820 |
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
|
1821 |
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
|
1822 |
in |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1823 |
if x < - 1 then |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1824 |
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
|
1825 |
else horner x)" | |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1826 |
"ub_exp prec x = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1827 |
(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
|
1828 |
else if x < - 1 then |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1829 |
power_up_fl prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1830 |
(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
|
1831 |
(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
|
1832 |
else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)" |
60680 | 1833 |
by pat_completeness auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1834 |
termination |
60680 | 1835 |
by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if 0 < x then 1 else 0))") auto |
29805 | 1836 |
|
1837 |
lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)" |
|
1838 |
proof - |
|
1839 |
have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto |
|
60680 | 1840 |
have "1 / 4 = (Float 1 (- 2))" |
1841 |
unfolding Float_num by auto |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1842 |
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
|
1843 |
by (subst less_eq_float.rep_eq [symmetric]) code_simp |
60680 | 1844 |
also have "\<dots> \<le> exp (- 1 :: float)" |
1845 |
using bnds_exp_horner[where x="- 1"] by auto |
|
1846 |
finally show ?thesis |
|
1847 |
by simp |
|
29805 | 1848 |
qed |
1849 |
||
60680 | 1850 |
lemma lb_exp_pos: |
1851 |
assumes "\<not> 0 < x" |
|
1852 |
shows "0 < lb_exp prec x" |
|
29805 | 1853 |
proof - |
1854 |
let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x" |
|
60680 | 1855 |
let "?horner x" = "let y = ?lb_horner x in if y \<le> 0 then Float 1 (- 2) else y" |
1856 |
have pos_horner: "0 < ?horner x" for x |
|
1857 |
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
|
1858 |
moreover have "0 < real_of_float ((?horner x) ^ num)" for x :: float and num :: nat |
60680 | 1859 |
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
|
1860 |
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
|
1861 |
also have "\<dots> = (?horner x) ^ num" by auto |
60680 | 1862 |
finally show ?thesis . |
1863 |
qed |
|
29805 | 1864 |
ultimately show ?thesis |
60533 | 1865 |
unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] Let_def |
60680 | 1866 |
by (cases "floor_fl x", cases "x < - 1") |
1867 |
(auto simp: real_power_up_fl real_power_down_fl intro!: power_up_less power_down_pos) |
|
29805 | 1868 |
qed |
1869 |
||
60680 | 1870 |
lemma exp_boundaries': |
1871 |
assumes "x \<le> 0" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1872 |
shows "exp x \<in> { (lb_exp prec x) .. (ub_exp prec x)}" |
29805 | 1873 |
proof - |
1874 |
let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x" |
|
1875 |
let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x" |
|
1876 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1877 |
have "real_of_float x \<le> 0" and "\<not> x > 0" |
60680 | 1878 |
using \<open>x \<le> 0\<close> by auto |
29805 | 1879 |
show ?thesis |
1880 |
proof (cases "x < - 1") |
|
60680 | 1881 |
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
|
1882 |
hence "- 1 \<le> real_of_float x" by auto |
29805 | 1883 |
show ?thesis |
1884 |
proof (cases "?lb_exp_horner x \<le> 0") |
|
60680 | 1885 |
case True |
1886 |
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
|
1887 |
have "- 1 \<le> real_of_float x" by auto |
60680 | 1888 |
hence "exp (- 1) \<le> exp x" |
1889 |
unfolding exp_le_cancel_iff . |
|
1890 |
from order_trans[OF exp_m1_ge_quarter this] have "Float 1 (- 2) \<le> exp x" |
|
1891 |
unfolding Float_num . |
|
1892 |
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
|
1893 |
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 | 1894 |
next |
60680 | 1895 |
case False |
1896 |
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
|
1897 |
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 | 1898 |
qed |
1899 |
next |
|
1900 |
case True |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1901 |
let ?num = "nat (- int_floor_fl x)" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1902 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1903 |
have "real_of_int (int_floor_fl x) < - 1" |
60680 | 1904 |
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
|
1905 |
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
|
1906 |
hence "int_floor_fl x < 0" by auto |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1907 |
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
|
1908 |
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
|
1909 |
hence "0 < ?num" by auto |
29805 | 1910 |
hence "real ?num \<noteq> 0" by auto |
60680 | 1911 |
have num_eq: "real ?num = - int_floor_fl x" |
1912 |
using \<open>0 < nat (- int_floor_fl x)\<close> by auto |
|
1913 |
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
|
1914 |
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
|
1915 |
hence "real_of_int (int_floor_fl x) < 0" |
60680 | 1916 |
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
|
1917 |
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
|
1918 |
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
|
1919 |
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
|
1920 |
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
|
1921 |
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
|
1922 |
by (simp add: floor_fl_def int_floor_fl_def) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1923 |
have "exp x \<le> ub_exp prec x" |
29805 | 1924 |
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
|
1925 |
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
|
1926 |
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
|
1927 |
unfolding less_eq_float_def zero_float.rep_eq . |
31809 | 1928 |
|
60680 | 1929 |
have "exp x = exp (?num * (x / ?num))" |
1930 |
using \<open>real ?num \<noteq> 0\<close> by auto |
|
1931 |
also have "\<dots> = exp (x / ?num) ^ ?num" |
|
1932 |
unfolding exp_real_of_nat_mult .. |
|
1933 |
also have "\<dots> \<le> exp (float_divr prec x (- floor_fl x)) ^ ?num" |
|
1934 |
unfolding num_eq fl_eq |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1935 |
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
|
1936 |
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
|
1937 |
unfolding real_of_float_power |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1938 |
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
|
1939 |
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
|
1940 |
by (auto simp add: real_power_up_fl intro!: power_up ub_exp_horner_nonneg div_less_zero) |
60680 | 1941 |
finally show ?thesis |
1942 |
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 | 1943 |
qed |
31809 | 1944 |
moreover |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1945 |
have "lb_exp prec x \<le> exp x" |
29805 | 1946 |
proof - |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
1947 |
let ?divl = "float_divl prec x (- floor_fl x)" |
29805 | 1948 |
let ?horner = "?lb_exp_horner ?divl" |
31809 | 1949 |
|
29805 | 1950 |
show ?thesis |
1951 |
proof (cases "?horner \<le> 0") |
|
60680 | 1952 |
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
|
1953 |
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
|
1954 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
1955 |
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
|
1956 |
using \<open>real_of_float (floor_fl x) < 0\<close> \<open>real_of_float x \<le> 0\<close> |
60680 | 1957 |
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
|
1958 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
1959 |
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
|
1960 |
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
|
1961 |
using \<open>0 \<le> real_of_float ?horner\<close>[unfolded floor_fl_def[symmetric]] |
60680 | 1962 |
bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] |
1963 |
by (auto intro!: power_mono) |
|
1964 |
also have "\<dots> \<le> exp (x / ?num) ^ ?num" |
|
1965 |
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
|
1966 |
using float_divl by (auto intro!: power_mono simp del: uminus_float.rep_eq) |
60680 | 1967 |
also have "\<dots> = exp (?num * (x / ?num))" |
1968 |
unfolding exp_real_of_nat_mult .. |
|
1969 |
also have "\<dots> = exp x" |
|
1970 |
using \<open>real ?num \<noteq> 0\<close> by auto |
|
1971 |
finally show ?thesis |
|
1972 |
using False |
|
1973 |
unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>] |
|
1974 |
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
|
1975 |
by (auto simp: real_power_down_fl intro!: power_down_le) |
29805 | 1976 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
1977 |
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
|
1978 |
have "power_down_fl prec (Float 1 (- 2)) ?num \<le> (Float 1 (- 2)) ^ ?num" |
60680 | 1979 |
by (metis Float_le_zero_iff less_imp_le linorder_not_less |
1980 |
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
|
1981 |
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
|
1982 |
by simp |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
1983 |
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
|
1984 |
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
|
1985 |
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
|
1986 |
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 | 1987 |
have "- 1 \<le> x / (- floor_fl x)" |
1988 |
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
|
1989 |
from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]] |
60680 | 1990 |
have "Float 1 (- 2) \<le> exp (x / (- floor_fl x))" |
1991 |
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
|
1992 |
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
|
1993 |
by (metis Float_num(5) power_mono zero_le_divide_1_iff zero_le_numeral) |
60680 | 1994 |
also have "\<dots> = exp x" |
1995 |
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
|
1996 |
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
|
1997 |
finally show ?thesis |
60680 | 1998 |
unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>] |
1999 |
int_floor_fl_def Let_def if_P[OF True] real_of_float_power . |
|
29805 | 2000 |
qed |
2001 |
qed |
|
2002 |
ultimately show ?thesis by auto |
|
2003 |
qed |
|
2004 |
qed |
|
2005 |
||
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2006 |
lemma exp_boundaries: "exp x \<in> { lb_exp prec x .. ub_exp prec x }" |
29805 | 2007 |
proof - |
2008 |
show ?thesis |
|
2009 |
proof (cases "0 < x") |
|
60680 | 2010 |
case False |
2011 |
hence "x \<le> 0" by auto |
|
29805 | 2012 |
from exp_boundaries'[OF this] show ?thesis . |
2013 |
next |
|
60680 | 2014 |
case True |
2015 |
hence "-x \<le> 0" by auto |
|
31809 | 2016 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2017 |
have "lb_exp prec x \<le> exp x" |
29805 | 2018 |
proof - |
60533 | 2019 |
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
|
2020 |
have ub_exp: "exp (- real_of_float x) \<le> ub_exp prec (-x)" |
60680 | 2021 |
unfolding atLeastAtMost_iff minus_float.rep_eq by auto |
2022 |
||
2023 |
have "float_divl prec 1 (ub_exp prec (-x)) \<le> 1 / ub_exp prec (-x)" |
|
2024 |
using float_divl[where x=1] by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2025 |
also have "\<dots> \<le> exp x" |
60680 | 2026 |
using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] |
2027 |
exp_gt_zero, symmetric]] |
|
2028 |
unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide |
|
2029 |
by auto |
|
2030 |
finally show ?thesis |
|
2031 |
unfolding lb_exp.simps if_P[OF True] . |
|
29805 | 2032 |
qed |
2033 |
moreover |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2034 |
have "exp x \<le> ub_exp prec x" |
29805 | 2035 |
proof - |
60533 | 2036 |
have "\<not> 0 < -x" using \<open>0 < x\<close> by auto |
2037 |
||
2038 |
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
|
2039 |
have lb_exp: "lb_exp prec (-x) \<le> exp (- real_of_float x)" |
60680 | 2040 |
unfolding atLeastAtMost_iff minus_float.rep_eq by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2041 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2042 |
have "exp x \<le> (1 :: float) / lb_exp prec (-x)" |
60533 | 2043 |
using lb_exp lb_exp_pos[OF \<open>\<not> 0 < -x\<close>, of prec] |
47600 | 2044 |
by (simp del: lb_exp.simps add: exp_minus inverse_eq_divide field_simps) |
60680 | 2045 |
also have "\<dots> \<le> float_divr prec 1 (lb_exp prec (-x))" |
2046 |
using float_divr . |
|
2047 |
finally show ?thesis |
|
2048 |
unfolding ub_exp.simps if_P[OF True] . |
|
29805 | 2049 |
qed |
60680 | 2050 |
ultimately show ?thesis |
2051 |
by auto |
|
29805 | 2052 |
qed |
2053 |
qed |
|
2054 |
||
60680 | 2055 |
lemma bnds_exp: "\<forall>(x::real) lx ux. (l, u) = |
2056 |
(lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> exp x \<and> exp x \<le> u" |
|
29805 | 2057 |
proof (rule allI, rule allI, rule allI, rule impI) |
60680 | 2058 |
fix x :: real and lx ux |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2059 |
assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux}" |
60680 | 2060 |
hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {lx .. ux}" |
2061 |
by auto |
|
2062 |
show "l \<le> exp x \<and> exp x \<le> u" |
|
2063 |
proof |
|
2064 |
show "l \<le> exp x" |
|
2065 |
proof - |
|
2066 |
from exp_boundaries[of lx prec, unfolded l] |
|
2067 |
have "l \<le> exp lx" by (auto simp del: lb_exp.simps) |
|
2068 |
also have "\<dots> \<le> exp x" using x by auto |
|
2069 |
finally show ?thesis . |
|
2070 |
qed |
|
2071 |
show "exp x \<le> u" |
|
2072 |
proof - |
|
2073 |
have "exp x \<le> exp ux" using x by auto |
|
2074 |
also have "\<dots> \<le> u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps) |
|
2075 |
finally show ?thesis . |
|
2076 |
qed |
|
2077 |
qed |
|
29805 | 2078 |
qed |
2079 |
||
60680 | 2080 |
|
29805 | 2081 |
section "Logarithm" |
2082 |
||
2083 |
subsection "Compute the logarithm series" |
|
2084 |
||
31809 | 2085 |
fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
29805 | 2086 |
and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
2087 |
"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
|
2088 |
"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
|
2089 |
(rapprox_rat prec 1 (int i)) (- float_round_down prec (x * lb_ln_horner prec n (Suc i) x))" | |
29805 | 2090 |
"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
|
2091 |
"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
|
2092 |
(lapprox_rat prec 1 (int i)) (- float_round_up prec (x * ub_ln_horner prec n (Suc i) x))" |
29805 | 2093 |
|
2094 |
lemma ln_bounds: |
|
60680 | 2095 |
assumes "0 \<le> x" |
2096 |
and "x < 1" |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2097 |
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
|
2098 |
and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub") |
29805 | 2099 |
proof - |
30952
7ab2716dd93b
power operation on functions with syntax o^; power operation on relations with syntax ^^
haftmann
parents:
30886
diff
changeset
|
2100 |
let "?a n" = "(1/real (n +1)) * x ^ (Suc n)" |
29805 | 2101 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2102 |
have ln_eq: "(\<Sum> i. (- 1) ^ i * ?a i) = ln (x + 1)" |
60533 | 2103 |
using ln_series[of "x + 1"] \<open>0 \<le> x\<close> \<open>x < 1\<close> by auto |
29805 | 2104 |
|
2105 |
have "norm x < 1" using assms by auto |
|
31809 | 2106 |
have "?a ----> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric] |
60533 | 2107 |
using tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF \<open>norm x < 1\<close>]]] by auto |
60680 | 2108 |
have "0 \<le> ?a n" for n |
2109 |
by (rule mult_nonneg_nonneg) (auto simp: \<open>0 \<le> x\<close>) |
|
2110 |
have "?a (Suc n) \<le> ?a n" for n |
|
2111 |
unfolding inverse_eq_divide[symmetric] |
|
2112 |
proof (rule mult_mono) |
|
2113 |
show "0 \<le> x ^ Suc (Suc n)" |
|
2114 |
by (auto simp add: \<open>0 \<le> x\<close>) |
|
2115 |
have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" |
|
2116 |
unfolding power_Suc2 mult.assoc[symmetric] |
|
2117 |
by (rule mult_left_mono, fact less_imp_le[OF \<open>x < 1\<close>]) (auto simp: \<open>0 \<le> x\<close>) |
|
2118 |
thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto |
|
2119 |
qed auto |
|
60533 | 2120 |
from summable_Leibniz'(2,4)[OF \<open>?a ----> 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 | 2121 |
show ?lb and ?ub |
2122 |
unfolding atLeast0LessThan by auto |
|
29805 | 2123 |
qed |
2124 |
||
31809 | 2125 |
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
|
2126 |
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
|
2127 |
and "real_of_float x < 1" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2128 |
shows "x * lb_ln_horner prec (get_even n) 1 x \<le> ln (x + 1)" (is "?lb \<le> ?ln") |
60680 | 2129 |
and "ln (x + 1) \<le> x * ub_ln_horner prec (get_odd n) 1 x" (is "?ln \<le> ?ub") |
29805 | 2130 |
proof - |
2131 |
obtain ev where ev: "get_even n = 2 * ev" using get_even_double .. |
|
2132 |
obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double .. |
|
2133 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2134 |
let "?s n" = "(- 1) ^ n * (1 / real (1 + n)) * (real_of_float x)^(Suc n)" |
29805 | 2135 |
|
60680 | 2136 |
have "?lb \<le> setsum ?s {0 ..< 2 * ev}" |
2137 |
unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_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
|
2138 |
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
|
2139 |
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
|
2140 |
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
|
2141 |
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
|
2142 |
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
|
2143 |
unfolding real_of_float_power |
29805 | 2144 |
by (rule mult_right_mono) |
60680 | 2145 |
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
|
2146 |
using ln_bounds(1)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto |
31809 | 2147 |
finally show "?lb \<le> ?ln" . |
29805 | 2148 |
|
60680 | 2149 |
have "?ln \<le> setsum ?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
|
2150 |
using ln_bounds(2)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto |
60680 | 2151 |
also have "\<dots> \<le> ?ub" |
2152 |
unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_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
|
2153 |
unfolding mult.commute[of "real_of_float x"] od |
29805 | 2154 |
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
|
2155 |
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
|
2156 |
unfolding real_of_float_power |
29805 | 2157 |
by (rule mult_right_mono) |
31809 | 2158 |
finally show "?ln \<le> ?ub" . |
29805 | 2159 |
qed |
2160 |
||
60680 | 2161 |
lemma ln_add: |
2162 |
fixes x :: real |
|
2163 |
assumes "0 < x" and "0 < y" |
|
2164 |
shows "ln (x + y) = ln x + ln (1 + y / x)" |
|
29805 | 2165 |
proof - |
2166 |
have "x \<noteq> 0" using assms by auto |
|
60680 | 2167 |
have "x + y = x * (1 + y / x)" |
2168 |
unfolding distrib_left times_divide_eq_right nonzero_mult_divide_cancel_left[OF \<open>x \<noteq> 0\<close>] |
|
2169 |
by auto |
|
31809 | 2170 |
moreover |
56541 | 2171 |
have "0 < y / x" using assms by auto |
29805 | 2172 |
hence "0 < 1 + y / x" by auto |
60680 | 2173 |
ultimately show ?thesis |
2174 |
using ln_mult assms by auto |
|
29805 | 2175 |
qed |
2176 |
||
60680 | 2177 |
|
29805 | 2178 |
subsection "Compute the logarithm of 2" |
2179 |
||
31809 | 2180 |
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
|
2181 |
in float_plus_up prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2182 |
((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
|
2183 |
(float_round_up prec (third * ub_ln_horner prec (get_odd prec) 1 third)))" |
31809 | 2184 |
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
|
2185 |
in float_plus_down prec |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2186 |
((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
|
2187 |
(float_round_down prec (third * lb_ln_horner prec (get_even prec) 1 third)))" |
29805 | 2188 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2189 |
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
|
2190 |
and lb_ln2: "lb_ln2 prec \<le> ln 2" (is "?lb_ln2") |
29805 | 2191 |
proof - |
2192 |
let ?uthird = "rapprox_rat (max prec 1) 1 3" |
|
2193 |
let ?lthird = "lapprox_rat prec 1 3" |
|
2194 |
||
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
|
2195 |
have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1::real)" |
29805 | 2196 |
using ln_add[of "3 / 2" "1 / 2"] by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2197 |
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
|
2198 |
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
|
2199 |
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
|
2200 |
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
|
2201 |
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
|
2202 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2203 |
have lb2: "0 \<le> real_of_float (Float 1 (- 1))" and ub2: "real_of_float (Float 1 (- 1)) < 1" |
60680 | 2204 |
unfolding Float_num by auto |
29805 | 2205 |
|
2206 |
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
|
2207 |
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
|
2208 |
by (simp add: Float.compute_rapprox_rat Float.compute_lapprox_rat rapprox_posrat_less1) |
29805 | 2209 |
|
2210 |
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
|
2211 |
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
|
2212 |
have lthird_gt0: "0 < real_of_float ?lthird + 1" using lb3_lb by auto |
29805 | 2213 |
|
60680 | 2214 |
show ?ub_ln2 |
2215 |
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
|
2216 |
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
|
2217 |
have "ln (1 / 3 + 1) \<le> ln (real_of_float ?uthird + 1)" |
60680 | 2218 |
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
|
2219 |
also have "\<dots> \<le> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird" |
29805 | 2220 |
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
|
2221 |
also note float_round_up |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2222 |
finally show "ln (1 / 3 + 1) \<le> float_round_up prec (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" . |
29805 | 2223 |
qed |
60680 | 2224 |
show ?lb_ln2 |
2225 |
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
|
2226 |
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
|
2227 |
have "?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird \<le> ln (real_of_float ?lthird + 1)" |
29805 | 2228 |
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
|
2229 |
note float_round_down_le[OF this] |
60680 | 2230 |
also have "\<dots> \<le> ln (1 / 3 + 1)" |
2231 |
unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] |
|
2232 |
using lb3 by auto |
|
2233 |
finally show "float_round_down prec (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> |
|
2234 |
ln (1 / 3 + 1)" . |
|
29805 | 2235 |
qed |
2236 |
qed |
|
2237 |
||
60680 | 2238 |
|
29805 | 2239 |
subsection "Compute the logarithm in the entire domain" |
2240 |
||
2241 |
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
|
2242 |
"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
|
2243 |
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
|
2244 |
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
|
2245 |
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
|
2246 |
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
|
2247 |
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
|
2248 |
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
|
2249 |
"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
|
2250 |
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
|
2251 |
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
|
2252 |
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
|
2253 |
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
|
2254 |
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
|
2255 |
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
|
2256 |
Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))" |
60680 | 2257 |
by pat_completeness auto |
2258 |
||
2259 |
termination |
|
2260 |
proof (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 1 then 1 else 0))", auto) |
|
2261 |
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
|
2262 |
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
|
2263 |
hence "0 < real_of_float x" "1 \<le> max prec (Suc 0)" "real_of_float x < 1" |
60680 | 2264 |
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
|
2265 |
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 | 2266 |
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
|
2267 |
using \<open>real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1\<close> by auto |
29805 | 2268 |
next |
60680 | 2269 |
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
|
2270 |
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 | 2271 |
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
|
2272 |
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
|
2273 |
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
|
2274 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2275 |
|
60680 | 2276 |
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
|
2277 |
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
|
2278 |
apply (auto simp add: zero_less_mult_iff zero_float_def dest: less_zeroE) |
60680 | 2279 |
apply (metis not_le powr_ge_pzero) |
2280 |
done |
|
2281 |
||
2282 |
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
|
2283 |
using powr_gt_zero[of 2 "e"] |
54269 | 2284 |
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
|
2285 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2286 |
lemma Float_representation_aux: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2287 |
fixes m e |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2288 |
defines "x \<equiv> Float m e" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2289 |
assumes "x > 0" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2290 |
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
|
2291 |
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
|
2292 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2293 |
from assms have mantissa_pos: "m > 0" "mantissa x > 0" |
47600 | 2294 |
using Float_pos_eq_mantissa_pos[of m e] float_pos_eq_mantissa_pos[of x] by simp_all |
60680 | 2295 |
thus ?th1 |
2296 |
using bitlen_Float[of m e] assms |
|
2297 |
by (auto simp add: zero_less_mult_iff intro!: arg_cong2[where f=Float]) |
|
47600 | 2298 |
have "x \<noteq> float_of 0" |
60533 | 2299 |
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
|
2300 |
from denormalize_shift[OF assms(1) this] guess i . note i = this |
47600 | 2301 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2302 |
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
|
2303 |
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
|
2304 |
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
|
2305 |
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
|
2306 |
(real_of_int (mantissa x) * 2 ^ i) * 2 powr (1 - real_of_int (bitlen (mantissa x * 2 ^ i)))" |
60533 | 2307 |
using \<open>mantissa x > 0\<close> by (simp add: powr_realpow) |
47600 | 2308 |
then show ?th2 |
2309 |
unfolding i by transfer auto |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2310 |
qed |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2311 |
|
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2312 |
lemma compute_ln[code]: |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2313 |
fixes m e |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2314 |
defines "x \<equiv> Float m e" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2315 |
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
|
2316 |
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
|
2317 |
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
|
2318 |
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
|
2319 |
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
|
2320 |
else let l = bitlen m - 1 in |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2321 |
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
|
2322 |
(is ?th1) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2323 |
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
|
2324 |
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
|
2325 |
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
|
2326 |
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
|
2327 |
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
|
2328 |
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
|
2329 |
else let l = bitlen m - 1 in |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2330 |
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
|
2331 |
(is ?th2) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2332 |
proof - |
60680 | 2333 |
from assms Float_pos_eq_mantissa_pos have "x > 0 \<Longrightarrow> m > 0" |
2334 |
by simp |
|
2335 |
thus ?th1 ?th2 |
|
2336 |
using Float_representation_aux[of m e] |
|
2337 |
unfolding x_def[symmetric] |
|
47600 | 2338 |
by (auto dest: not_leE) |
29805 | 2339 |
qed |
2340 |
||
60680 | 2341 |
lemma ln_shifted_float: |
2342 |
assumes "0 < m" |
|
2343 |
shows "ln (Float m e) = ln 2 * (e + (bitlen m - 1)) + ln (Float m (- (bitlen m - 1)))" |
|
29805 | 2344 |
proof - |
2345 |
let ?B = "2^nat (bitlen m - 1)" |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2346 |
def bl \<equiv> "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
|
2347 |
have "0 < real_of_int m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" |
60680 | 2348 |
using assms by auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2349 |
hence "0 \<le> bl" by (simp add: bitlen_def bl_def) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2350 |
show ?thesis |
29805 | 2351 |
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
|
2352 |
case True |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2353 |
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
|
2354 |
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
|
2355 |
apply (simp add: ln_mult) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2356 |
apply (cases "e=0") |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2357 |
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
|
2358 |
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
|
2359 |
done |
29805 | 2360 |
next |
60680 | 2361 |
case False |
2362 |
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
|
2363 |
have lne: "ln (2 powr real_of_int e) = ln (inverse (2 powr - e))" |
60680 | 2364 |
by (simp add: powr_minus) |
2365 |
hence pow_gt0: "(0::real) < 2^nat (-e)" |
|
2366 |
by auto |
|
2367 |
hence inv_gt0: "(0::real) < inverse (2^nat (-e))" |
|
2368 |
by auto |
|
2369 |
show ?thesis |
|
2370 |
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
|
2371 |
using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close> |
56483 | 2372 |
by (auto simp add: lne ln_mult ln_powr ln_div field_simps) |
29805 | 2373 |
qed |
2374 |
qed |
|
2375 |
||
60680 | 2376 |
lemma ub_ln_lb_ln_bounds': |
2377 |
assumes "1 \<le> x" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2378 |
shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)" |
60680 | 2379 |
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub") |
29805 | 2380 |
proof (cases "x < Float 1 1") |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2381 |
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
|
2382 |
hence "real_of_float (x - 1) < 1" and "real_of_float x < 2" by auto |
60533 | 2383 |
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
|
2384 |
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
|
2385 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2386 |
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
|
2387 |
|
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2388 |
show ?thesis |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2389 |
proof (cases "x \<le> Float 3 (- 1)") |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2390 |
case True |
60680 | 2391 |
show ?thesis |
2392 |
unfolding lb_ln.simps |
|
2393 |
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
|
2394 |
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 | 2395 |
\<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
|
2396 |
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
|
2397 |
next |
60680 | 2398 |
case False |
2399 |
hence *: "3 / 2 < x" by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2400 |
|
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2401 |
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
|
2402 |
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
|
2403 |
by (auto simp add: algebra_simps diff_divide_distrib) |
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2404 |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2405 |
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
|
2406 |
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
|
2407 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2408 |
{ 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
|
2409 |
by (rule rapprox_rat_le1) simp_all |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2410 |
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
|
2411 |
by (rule order_trans[OF _ rapprox_rat]) simp |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2412 |
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
|
2413 |
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
|
2414 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2415 |
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
|
2416 |
\<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
|
2417 |
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
|
2418 |
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
|
2419 |
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
|
2420 |
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
|
2421 |
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
|
2422 |
qed |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2423 |
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
|
2424 |
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
|
2425 |
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
|
2426 |
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
|
2427 |
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
|
2428 |
qed |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2429 |
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
|
2430 |
+ ?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
|
2431 |
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
|
2432 |
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
|
2433 |
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
|
2434 |
} |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2435 |
moreover |
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2436 |
{ 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
|
2437 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2438 |
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
|
2439 |
by (rule order_trans[OF lapprox_rat], simp) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2440 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only 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 |
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
|
2442 |
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
|
2443 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2444 |
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
|
2445 |
\<le> ln (real_of_float ?max + 1)" |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2446 |
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
|
2447 |
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
|
2448 |
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
|
2449 |
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
|
2450 |
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
|
2451 |
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
|
2452 |
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
|
2453 |
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
|
2454 |
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
|
2455 |
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
|
2456 |
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
|
2457 |
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
|
2458 |
auto simp add: max_def) |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2459 |
qed |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2460 |
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
|
2461 |
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
|
2462 |
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
|
2463 |
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
|
2464 |
} |
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2465 |
ultimately |
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
2466 |
show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def |
60533 | 2467 |
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
|
2468 |
qed |
29805 | 2469 |
next |
2470 |
case False |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2471 |
hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 (- 1)" |
60533 | 2472 |
using \<open>1 \<le> x\<close> by auto |
29805 | 2473 |
show ?thesis |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2474 |
proof - |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2475 |
def m \<equiv> "mantissa x" |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2476 |
def e \<equiv> "exponent x" |
60680 | 2477 |
from Float_mantissa_exponent[of x] have Float: "x = Float m e" |
2478 |
by (simp add: m_def e_def) |
|
29805 | 2479 |
let ?s = "Float (e + (bitlen m - 1)) 0" |
2480 |
let ?x = "Float m (- (bitlen m - 1))" |
|
2481 |
||
60680 | 2482 |
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
|
2483 |
apply (auto simp add: zero_less_mult_iff) |
60680 | 2484 |
using not_le powr_ge_pzero apply blast |
2485 |
done |
|
2486 |
def bl \<equiv> "bitlen m - 1" |
|
2487 |
hence "bl \<ge> 0" |
|
2488 |
using \<open>m > 0\<close> by (simp add: bitlen_def) |
|
2489 |
have "1 \<le> Float m e" |
|
2490 |
using \<open>1 \<le> x\<close> Float unfolding less_eq_float_def by auto |
|
60533 | 2491 |
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
|
2492 |
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
|
2493 |
unfolding bl_def[symmetric] |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2494 |
by (auto simp: powr_realpow[symmetric] field_simps inverse_eq_divide) |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2495 |
(auto simp : powr_minus field_simps inverse_eq_divide) |
29805 | 2496 |
|
2497 |
{ |
|
60680 | 2498 |
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
|
2499 |
(is "real_of_float ?lb2 \<le> _") |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2500 |
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
|
2501 |
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
|
2502 |
using lb_ln2[of prec] |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2503 |
proof (rule mult_mono) |
60533 | 2504 |
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
|
2505 |
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
|
2506 |
qed auto |
29805 | 2507 |
moreover |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2508 |
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
|
2509 |
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
|
2510 |
by (auto intro!: float_round_down_le) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2511 |
ultimately have "float_plus_down prec ?lb2 ?lb_horner \<le> ln x" |
60533 | 2512 |
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
|
2513 |
} |
29805 | 2514 |
moreover |
2515 |
{ |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2516 |
from ln_float_bounds(2)[OF x_bnds] |
60680 | 2517 |
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
|
2518 |
(is "_ \<le> real_of_float ?ub_horner") |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2519 |
by (auto intro!: float_round_up_le) |
29805 | 2520 |
moreover |
60680 | 2521 |
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
|
2522 |
(is "_ \<le> real_of_float ?ub2") |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2523 |
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
|
2524 |
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
|
2525 |
using ub_ln2[of prec] |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2526 |
proof (rule mult_mono) |
60533 | 2527 |
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
|
2528 |
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
|
2529 |
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
|
2530 |
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
|
2531 |
qed auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2532 |
ultimately have "ln x \<le> float_plus_up prec ?ub2 ?ub_horner" |
60533 | 2533 |
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
|
2534 |
by (auto intro!: float_plus_up_le) |
29805 | 2535 |
} |
60680 | 2536 |
ultimately show ?thesis |
2537 |
unfolding lb_ln.simps |
|
2538 |
unfolding ub_ln.simps |
|
2539 |
unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] |
|
2540 |
if_not_P[OF False] if_not_P[OF \<open>\<not> x \<le> Float 3 (- 1)\<close>] Let_def |
|
2541 |
unfolding plus_float.rep_eq e_def[symmetric] m_def[symmetric] |
|
2542 |
by simp |
|
29805 | 2543 |
qed |
2544 |
qed |
|
2545 |
||
49351 | 2546 |
lemma ub_ln_lb_ln_bounds: |
2547 |
assumes "0 < x" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2548 |
shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)" |
60680 | 2549 |
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub") |
29805 | 2550 |
proof (cases "x < 1") |
60680 | 2551 |
case False |
2552 |
hence "1 \<le> x" |
|
2553 |
unfolding less_float_def less_eq_float_def by auto |
|
2554 |
show ?thesis |
|
2555 |
using ub_ln_lb_ln_bounds'[OF \<open>1 \<le> x\<close>] . |
|
29805 | 2556 |
next |
60680 | 2557 |
case True |
2558 |
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
|
2559 |
from True have "real_of_float x \<le> 1" "x \<le> 1" |
60680 | 2560 |
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
|
2561 |
have "0 < real_of_float x" and "real_of_float x \<noteq> 0" |
60680 | 2562 |
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
|
2563 |
hence A: "0 < 1 / real_of_float x" by auto |
29805 | 2564 |
|
2565 |
{ |
|
2566 |
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
|
2567 |
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
|
2568 |
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
|
2569 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2570 |
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
|
2571 |
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
|
2572 |
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
|
2573 |
have "?ln \<le> - the (lb_ln prec ?divl)" unfolding uminus_float.rep_eq by (rule order_trans) |
29805 | 2574 |
} moreover |
2575 |
{ |
|
2576 |
let ?divr = "float_divr prec 1 x" |
|
60533 | 2577 |
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
|
2578 |
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
|
2579 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2580 |
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
|
2581 |
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 | 2582 |
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
|
2583 |
have "- the (ub_ln prec ?divr) \<le> ?ln" unfolding uminus_float.rep_eq by (rule order_trans) |
29805 | 2584 |
} |
2585 |
ultimately show ?thesis unfolding lb_ln.simps[where x=x] ub_ln.simps[where x=x] |
|
60533 | 2586 |
unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_P[OF True] by auto |
29805 | 2587 |
qed |
2588 |
||
49351 | 2589 |
lemma lb_ln: |
2590 |
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
|
2591 |
shows "y \<le> ln x" and "0 < real_of_float x" |
29805 | 2592 |
proof - |
2593 |
have "0 < x" |
|
2594 |
proof (rule ccontr) |
|
60680 | 2595 |
assume "\<not> 0 < x" |
2596 |
hence "x \<le> 0" |
|
2597 |
unfolding less_eq_float_def less_float_def by auto |
|
2598 |
thus False |
|
2599 |
using assms by auto |
|
29805 | 2600 |
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
|
2601 |
thus "0 < real_of_float x" by auto |
60680 | 2602 |
have "the (lb_ln prec x) \<le> ln x" |
2603 |
using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] .. |
|
2604 |
thus "y \<le> ln x" |
|
2605 |
unfolding assms[symmetric] by auto |
|
29805 | 2606 |
qed |
2607 |
||
49351 | 2608 |
lemma ub_ln: |
2609 |
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
|
2610 |
shows "ln x \<le> y" and "0 < real_of_float x" |
29805 | 2611 |
proof - |
2612 |
have "0 < x" |
|
2613 |
proof (rule ccontr) |
|
60680 | 2614 |
assume "\<not> 0 < x" |
2615 |
hence "x \<le> 0" by auto |
|
2616 |
thus False |
|
2617 |
using assms by auto |
|
29805 | 2618 |
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
|
2619 |
thus "0 < real_of_float x" by auto |
60680 | 2620 |
have "ln x \<le> the (ub_ln prec x)" |
2621 |
using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] .. |
|
2622 |
thus "ln x \<le> y" |
|
2623 |
unfolding assms[symmetric] by auto |
|
29805 | 2624 |
qed |
2625 |
||
60680 | 2626 |
lemma bnds_ln: "\<forall>(x::real) lx ux. (Some l, Some u) = |
2627 |
(lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> ln x \<and> ln x \<le> u" |
|
29805 | 2628 |
proof (rule allI, rule allI, rule allI, rule impI) |
60680 | 2629 |
fix x :: real |
2630 |
fix lx ux |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2631 |
assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux}" |
60680 | 2632 |
hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {lx .. ux}" |
2633 |
by auto |
|
2634 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2635 |
have "ln ux \<le> u" and "0 < real_of_float ux" |
60680 | 2636 |
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
|
2637 |
have "l \<le> ln lx" and "0 < real_of_float lx" and "0 < x" |
60680 | 2638 |
using lb_ln[OF l] x by auto |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2639 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only 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 |
from ln_le_cancel_iff[OF \<open>0 < real_of_float lx\<close> \<open>0 < x\<close>] \<open>l \<le> ln lx\<close> |
60680 | 2641 |
have "l \<le> ln x" |
2642 |
using x unfolding atLeastAtMost_iff by auto |
|
29805 | 2643 |
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
|
2644 |
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 | 2645 |
have "ln x \<le> u" |
2646 |
using x unfolding atLeastAtMost_iff by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2647 |
ultimately show "l \<le> ln x \<and> ln x \<le> u" .. |
29805 | 2648 |
qed |
2649 |
||
60680 | 2650 |
|
29805 | 2651 |
section "Implement floatarith" |
2652 |
||
2653 |
subsection "Define syntax and semantics" |
|
2654 |
||
58310 | 2655 |
datatype floatarith |
29805 | 2656 |
= Add floatarith floatarith |
2657 |
| Minus floatarith |
|
2658 |
| Mult floatarith floatarith |
|
2659 |
| Inverse floatarith |
|
2660 |
| Cos floatarith |
|
2661 |
| Arctan floatarith |
|
2662 |
| Abs floatarith |
|
2663 |
| Max floatarith floatarith |
|
2664 |
| Min floatarith floatarith |
|
2665 |
| Pi |
|
2666 |
| Sqrt floatarith |
|
2667 |
| Exp floatarith |
|
2668 |
| Ln floatarith |
|
2669 |
| Power floatarith nat |
|
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
|
2670 |
| Var nat |
29805 | 2671 |
| Num float |
2672 |
||
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
2673 |
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
|
2674 |
"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
|
2675 |
"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
|
2676 |
"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
|
2677 |
"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
|
2678 |
"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
|
2679 |
"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
|
2680 |
"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
|
2681 |
"interpret_floatarith (Max a b) vs = max (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
|
2682 |
"interpret_floatarith (Abs a) vs = abs (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
|
2683 |
"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
|
2684 |
"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
|
2685 |
"interpret_floatarith (Exp a) vs = exp (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
|
2686 |
"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
|
2687 |
"interpret_floatarith (Power a n) vs = (interpret_floatarith a vs)^n" | |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2688 |
"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
|
2689 |
"interpret_floatarith (Var n) vs = vs ! n" |
29805 | 2690 |
|
60680 | 2691 |
lemma interpret_floatarith_divide: |
2692 |
"interpret_floatarith (Mult a (Inverse b)) vs = |
|
2693 |
(interpret_floatarith a vs) / (interpret_floatarith b vs)" |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2694 |
unfolding divide_inverse interpret_floatarith.simps .. |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2695 |
|
60680 | 2696 |
lemma interpret_floatarith_diff: |
2697 |
"interpret_floatarith (Add a (Minus b)) vs = |
|
2698 |
(interpret_floatarith a vs) - (interpret_floatarith b vs)" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53077
diff
changeset
|
2699 |
unfolding interpret_floatarith.simps by simp |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2700 |
|
60680 | 2701 |
lemma interpret_floatarith_sin: |
2702 |
"interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) vs = |
|
2703 |
sin (interpret_floatarith a vs)" |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2704 |
unfolding sin_cos_eq interpret_floatarith.simps |
60680 | 2705 |
interpret_floatarith_divide interpret_floatarith_diff |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2706 |
by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2707 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2708 |
lemma interpret_floatarith_tan: |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2709 |
"interpret_floatarith (Mult (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) (Inverse (Cos a))) vs = |
60680 | 2710 |
tan (interpret_floatarith a vs)" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2711 |
unfolding interpret_floatarith.simps(3,4) interpret_floatarith_sin tan_def divide_inverse |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2712 |
by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2713 |
|
60680 | 2714 |
lemma interpret_floatarith_log: |
2715 |
"interpret_floatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = |
|
2716 |
log (interpret_floatarith b vs) (interpret_floatarith x vs)" |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2717 |
unfolding log_def interpret_floatarith.simps divide_inverse .. |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2718 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2719 |
lemma interpret_floatarith_num: |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2720 |
shows "interpret_floatarith (Num (Float 0 0)) vs = 0" |
60680 | 2721 |
and "interpret_floatarith (Num (Float 1 0)) vs = 1" |
2722 |
and "interpret_floatarith (Num (Float (- 1) 0)) vs = - 1" |
|
2723 |
and "interpret_floatarith (Num (Float (numeral a) 0)) vs = numeral a" |
|
2724 |
and "interpret_floatarith (Num (Float (- numeral a) 0)) vs = - numeral a" |
|
2725 |
by auto |
|
2726 |
||
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2727 |
|
29805 | 2728 |
subsection "Implement approximation function" |
2729 |
||
2730 |
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 |
|
2731 |
"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" | |
|
2732 |
"lift_bin' a b f = None" |
|
2733 |
||
2734 |
fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where |
|
2735 |
"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u) |
|
2736 |
| t \<Rightarrow> None)" | |
|
2737 |
"lift_un b f = None" |
|
2738 |
||
2739 |
fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where |
|
2740 |
"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" | |
|
2741 |
"lift_un' b f = None" |
|
2742 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
2743 |
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
|
2744 |
"bounded_by xs vs \<longleftrightarrow> |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2745 |
(\<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
|
2746 |
| 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
|
2747 |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2748 |
lemma bounded_byE: |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2749 |
assumes "bounded_by xs vs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2750 |
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
|
2751 |
| 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
|
2752 |
using assms bounded_by_def by blast |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2753 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2754 |
lemma bounded_by_update: |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2755 |
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
|
2756 |
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
|
2757 |
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
|
2758 |
proof - |
60680 | 2759 |
{ |
2760 |
fix j |
|
2761 |
let ?vs = "vs[i := Some (l,u)]" |
|
2762 |
assume "j < length ?vs" |
|
2763 |
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
|
2764 |
have "case ?vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> xs ! j \<in> { real_of_float l .. real_of_float u }" |
60680 | 2765 |
proof (cases "?vs ! j") |
2766 |
case (Some b) |
|
2767 |
thus ?thesis |
|
2768 |
proof (cases "i = j") |
|
2769 |
case True |
|
2770 |
thus ?thesis using \<open>?vs ! j = Some b\<close> and bnd by auto |
|
2771 |
next |
|
2772 |
case False |
|
2773 |
thus ?thesis using \<open>bounded_by xs vs\<close> unfolding bounded_by_def by auto |
|
2774 |
qed |
|
2775 |
qed auto |
|
2776 |
} |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2777 |
thus ?thesis unfolding bounded_by_def by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2778 |
qed |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2779 |
|
60680 | 2780 |
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
|
2781 |
unfolding bounded_by_def by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2782 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
2783 |
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
|
2784 |
"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
|
2785 |
"approx prec (Add a b) bs = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2786 |
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
|
2787 |
(\<lambda> l1 u1 l2 u2. (float_plus_down prec l1 l2, float_plus_up prec u1 u2))" | |
29805 | 2788 |
"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
|
2789 |
"approx prec (Mult a b) bs = |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2790 |
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
|
2791 |
(\<lambda> a1 a2 b1 b2. |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2792 |
(float_plus_down prec (nprt a1 * pprt b2) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2793 |
(float_plus_down prec (nprt a2 * nprt b2) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2794 |
(float_plus_down prec (pprt a1 * pprt b1) (pprt a2 * nprt b1))), |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2795 |
float_plus_up prec (pprt a2 * pprt b2) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2796 |
(float_plus_up prec (pprt a1 * nprt b2) |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
2797 |
(float_plus_up prec (nprt a2 * pprt b1) (nprt a1 * nprt b1)))))" | |
29805 | 2798 |
"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))" | |
2799 |
"approx prec (Cos a) bs = lift_un' (approx' prec a bs) (bnds_cos prec)" | |
|
2800 |
"approx prec Pi bs = Some (lb_pi prec, ub_pi prec)" | |
|
2801 |
"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))" | |
|
2802 |
"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))" | |
|
2803 |
"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>))" | |
|
2804 |
"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
|
2805 |
"approx prec (Sqrt a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" | |
29805 | 2806 |
"approx prec (Exp a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" | |
2807 |
"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
|
2808 |
"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds prec n)" | |
29805 | 2809 |
"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
|
2810 |
"approx prec (Var i) bs = (if i < length bs then bs ! i else None)" |
29805 | 2811 |
|
2812 |
lemma lift_bin'_ex: |
|
2813 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f" |
|
2814 |
shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b" |
|
2815 |
proof (cases a) |
|
60680 | 2816 |
case None |
2817 |
hence "None = lift_bin' a b f" |
|
2818 |
unfolding None lift_bin'.simps .. |
|
2819 |
thus ?thesis |
|
2820 |
using lift_bin'_Some by auto |
|
29805 | 2821 |
next |
2822 |
case (Some a') |
|
2823 |
show ?thesis |
|
2824 |
proof (cases b) |
|
60680 | 2825 |
case None |
2826 |
hence "None = lift_bin' a b f" |
|
2827 |
unfolding None lift_bin'.simps .. |
|
29805 | 2828 |
thus ?thesis using lift_bin'_Some by auto |
2829 |
next |
|
2830 |
case (Some b') |
|
60680 | 2831 |
obtain la ua where a': "a' = (la, ua)" |
2832 |
by (cases a') auto |
|
2833 |
obtain lb ub where b': "b' = (lb, ub)" |
|
2834 |
by (cases b') auto |
|
2835 |
thus ?thesis |
|
2836 |
unfolding \<open>a = Some a'\<close> \<open>b = Some b'\<close> a' b' by auto |
|
29805 | 2837 |
qed |
2838 |
qed |
|
2839 |
||
2840 |
lemma lift_bin'_f: |
|
2841 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f" |
|
60680 | 2842 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" |
2843 |
and Pb: "\<And>l u. Some (l, u) = g b \<Longrightarrow> P l u b" |
|
29805 | 2844 |
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)" |
2845 |
proof - |
|
2846 |
obtain l1 u1 l2 u2 |
|
60680 | 2847 |
where Sa: "Some (l1, u1) = g a" |
2848 |
and Sb: "Some (l2, u2) = g b" |
|
2849 |
using lift_bin'_ex[OF assms(1)] by auto |
|
2850 |
have lu: "(l, u) = f l1 u1 l2 u2" |
|
2851 |
using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto |
|
2852 |
have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" |
|
2853 |
unfolding lu[symmetric] by auto |
|
2854 |
thus ?thesis |
|
2855 |
using Pa[OF Sa] Pb[OF Sb] by auto |
|
29805 | 2856 |
qed |
2857 |
||
2858 |
lemma approx_approx': |
|
60680 | 2859 |
assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> |
2860 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
|
2861 |
and approx': "Some (l, u) = approx' prec a vs" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2862 |
shows "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
29805 | 2863 |
proof - |
2864 |
obtain l' u' where S: "Some (l', u') = approx prec a vs" |
|
60680 | 2865 |
using approx' unfolding approx'.simps by (cases "approx prec a vs") auto |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2866 |
have l': "l = float_round_down prec l'" and u': "u = float_round_up prec u'" |
29805 | 2867 |
using approx' unfolding approx'.simps S[symmetric] by auto |
31809 | 2868 |
show ?thesis unfolding l' u' |
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2869 |
using order_trans[OF Pa[OF S, THEN conjunct2] float_round_up[of u']] |
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
2870 |
using order_trans[OF float_round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto |
29805 | 2871 |
qed |
2872 |
||
2873 |
lemma lift_bin': |
|
2874 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f" |
|
60680 | 2875 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
2876 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
|
2877 |
and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> |
|
2878 |
l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u" |
|
2879 |
shows "\<exists>l1 u1 l2 u2. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and> |
|
2880 |
(l2 \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u2) \<and> |
|
2881 |
l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)" |
|
29805 | 2882 |
proof - |
2883 |
{ fix l u assume "Some (l, u) = approx' prec a bs" |
|
2884 |
with approx_approx'[of prec a bs, OF _ this] Pa |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2885 |
have "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" by auto } note Pa = this |
29805 | 2886 |
{ fix l u assume "Some (l, u) = approx' prec b bs" |
2887 |
with approx_approx'[of prec b bs, OF _ this] Pb |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2888 |
have "l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u" by auto } note Pb = this |
29805 | 2889 |
|
2890 |
from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb] |
|
2891 |
show ?thesis by auto |
|
2892 |
qed |
|
2893 |
||
2894 |
lemma lift_un'_ex: |
|
2895 |
assumes lift_un'_Some: "Some (l, u) = lift_un' a f" |
|
2896 |
shows "\<exists> l u. Some (l, u) = a" |
|
2897 |
proof (cases a) |
|
60680 | 2898 |
case None |
2899 |
hence "None = lift_un' a f" |
|
2900 |
unfolding None lift_un'.simps .. |
|
2901 |
thus ?thesis |
|
2902 |
using lift_un'_Some by auto |
|
29805 | 2903 |
next |
2904 |
case (Some a') |
|
60680 | 2905 |
obtain la ua where a': "a' = (la, ua)" |
2906 |
by (cases a') auto |
|
2907 |
thus ?thesis |
|
2908 |
unfolding \<open>a = Some a'\<close> a' by auto |
|
29805 | 2909 |
qed |
2910 |
||
2911 |
lemma lift_un'_f: |
|
2912 |
assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f" |
|
60680 | 2913 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" |
29805 | 2914 |
shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)" |
2915 |
proof - |
|
60680 | 2916 |
obtain l1 u1 where Sa: "Some (l1, u1) = g a" |
2917 |
using lift_un'_ex[OF assms(1)] by auto |
|
2918 |
have lu: "(l, u) = f l1 u1" |
|
2919 |
using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto |
|
2920 |
have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" |
|
2921 |
unfolding lu[symmetric] by auto |
|
2922 |
thus ?thesis |
|
2923 |
using Pa[OF Sa] by auto |
|
29805 | 2924 |
qed |
2925 |
||
2926 |
lemma lift_un': |
|
2927 |
assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f" |
|
60680 | 2928 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
2929 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
|
2930 |
(is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
|
2931 |
shows "\<exists>l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and> |
|
2932 |
l = fst (f l1 u1) \<and> u = snd (f l1 u1)" |
|
29805 | 2933 |
proof - |
60680 | 2934 |
have Pa: "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
2935 |
if "Some (l, u) = approx' prec a bs" for l u |
|
2936 |
using approx_approx'[of prec a bs, OF _ that] Pa |
|
2937 |
by auto |
|
29805 | 2938 |
from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa] |
2939 |
show ?thesis by auto |
|
2940 |
qed |
|
2941 |
||
2942 |
lemma lift_un'_bnds: |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
2943 |
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 | 2944 |
and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f" |
2945 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
|
2946 |
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
|
2947 |
shows "real_of_float l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real_of_float u" |
29805 | 2948 |
proof - |
2949 |
from lift_un'[OF lift_un'_Some Pa] |
|
60680 | 2950 |
obtain l1 u1 where "l1 \<le> interpret_floatarith a xs" |
2951 |
and "interpret_floatarith a xs \<le> u1" |
|
2952 |
and "l = fst (f l1 u1)" |
|
2953 |
and "u = snd (f l1 u1)" |
|
2954 |
by blast |
|
2955 |
hence "(l, u) = f l1 u1" and "interpret_floatarith a xs \<in> {l1 .. u1}" |
|
2956 |
by auto |
|
2957 |
thus ?thesis |
|
2958 |
using bnds by auto |
|
29805 | 2959 |
qed |
2960 |
||
2961 |
lemma lift_un_ex: |
|
2962 |
assumes lift_un_Some: "Some (l, u) = lift_un a f" |
|
60680 | 2963 |
shows "\<exists>l u. Some (l, u) = a" |
29805 | 2964 |
proof (cases a) |
60680 | 2965 |
case None |
2966 |
hence "None = lift_un a f" |
|
2967 |
unfolding None lift_un.simps .. |
|
2968 |
thus ?thesis |
|
2969 |
using lift_un_Some by auto |
|
29805 | 2970 |
next |
2971 |
case (Some a') |
|
60680 | 2972 |
obtain la ua where a': "a' = (la, ua)" |
2973 |
by (cases a') auto |
|
2974 |
thus ?thesis |
|
2975 |
unfolding \<open>a = Some a'\<close> a' by auto |
|
29805 | 2976 |
qed |
2977 |
||
2978 |
lemma lift_un_f: |
|
2979 |
assumes lift_un_Some: "Some (l, u) = lift_un (g a) f" |
|
60680 | 2980 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" |
29805 | 2981 |
shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)" |
2982 |
proof - |
|
60680 | 2983 |
obtain l1 u1 where Sa: "Some (l1, u1) = g a" |
2984 |
using lift_un_ex[OF assms(1)] by auto |
|
29805 | 2985 |
have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None" |
2986 |
proof (rule ccontr) |
|
2987 |
assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)" |
|
2988 |
hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto |
|
31809 | 2989 |
hence "lift_un (g a) f = None" |
29805 | 2990 |
proof (cases "fst (f l1 u1) = None") |
2991 |
case True |
|
60680 | 2992 |
then obtain b where b: "f l1 u1 = (None, b)" |
2993 |
by (cases "f l1 u1") auto |
|
2994 |
thus ?thesis |
|
2995 |
unfolding Sa[symmetric] lift_un.simps b by auto |
|
29805 | 2996 |
next |
60680 | 2997 |
case False |
2998 |
hence "snd (f l1 u1) = None" |
|
2999 |
using or by auto |
|
3000 |
with False obtain b where b: "f l1 u1 = (Some b, None)" |
|
3001 |
by (cases "f l1 u1") auto |
|
3002 |
thus ?thesis |
|
3003 |
unfolding Sa[symmetric] lift_un.simps b by auto |
|
29805 | 3004 |
qed |
60680 | 3005 |
thus False |
3006 |
using lift_un_Some by auto |
|
29805 | 3007 |
qed |
60680 | 3008 |
then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" |
3009 |
by (cases "f l1 u1") auto |
|
29805 | 3010 |
from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f] |
60680 | 3011 |
have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" |
3012 |
unfolding f by auto |
|
3013 |
thus ?thesis |
|
3014 |
unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto |
|
29805 | 3015 |
qed |
3016 |
||
3017 |
lemma lift_un: |
|
3018 |
assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f" |
|
60680 | 3019 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
3020 |
l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
|
3021 |
(is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
|
3022 |
shows "\<exists>l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and> |
|
29805 | 3023 |
Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)" |
3024 |
proof - |
|
60680 | 3025 |
have Pa: "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" |
3026 |
if "Some (l, u) = approx' prec a bs" for l u |
|
3027 |
using approx_approx'[of prec a bs, OF _ that] Pa by auto |
|
29805 | 3028 |
from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa] |
3029 |
show ?thesis by auto |
|
3030 |
qed |
|
3031 |
||
3032 |
lemma lift_un_bnds: |
|
60680 | 3033 |
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" |
3034 |
and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f" |
|
3035 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> |
|
3036 |
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
|
3037 |
shows "real_of_float l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real_of_float u" |
29805 | 3038 |
proof - |
3039 |
from lift_un[OF lift_un_Some Pa] |
|
60680 | 3040 |
obtain l1 u1 where "l1 \<le> interpret_floatarith a xs" |
3041 |
and "interpret_floatarith a xs \<le> u1" |
|
3042 |
and "Some l = fst (f l1 u1)" |
|
3043 |
and "Some u = snd (f l1 u1)" |
|
3044 |
by blast |
|
3045 |
hence "(Some l, Some u) = f l1 u1" and "interpret_floatarith a xs \<in> {l1 .. u1}" |
|
3046 |
by auto |
|
3047 |
thus ?thesis |
|
3048 |
using bnds by auto |
|
29805 | 3049 |
qed |
3050 |
||
3051 |
lemma approx: |
|
3052 |
assumes "bounded_by xs vs" |
|
60680 | 3053 |
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
|
3054 |
shows "l \<le> interpret_floatarith arith xs \<and> interpret_floatarith arith xs \<le> u" (is "?P l u arith") |
60533 | 3055 |
using \<open>Some (l, u) = approx prec arith vs\<close> |
45129
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
44821
diff
changeset
|
3056 |
proof (induct arith arbitrary: l u) |
29805 | 3057 |
case (Add a b) |
3058 |
from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps |
|
60680 | 3059 |
obtain l1 u1 l2 u2 where "l = float_plus_down prec l1 l2" |
3060 |
and "u = float_plus_up prec u1 u2" "l1 \<le> interpret_floatarith a xs" |
|
3061 |
and "interpret_floatarith a xs \<le> u1" "l2 \<le> interpret_floatarith b xs" |
|
3062 |
and "interpret_floatarith b xs \<le> u2" |
|
3063 |
unfolding fst_conv snd_conv by blast |
|
3064 |
thus ?case |
|
3065 |
unfolding interpret_floatarith.simps by (auto intro!: float_plus_up_le float_plus_down_le) |
|
29805 | 3066 |
next |
3067 |
case (Minus a) |
|
3068 |
from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps |
|
60680 | 3069 |
obtain l1 u1 where "l = -u1" "u = -l1" |
3070 |
and "l1 \<le> interpret_floatarith a xs" "interpret_floatarith a xs \<le> u1" |
|
3071 |
unfolding fst_conv snd_conv by blast |
|
3072 |
thus ?case |
|
3073 |
unfolding interpret_floatarith.simps using minus_float.rep_eq by auto |
|
29805 | 3074 |
next |
3075 |
case (Mult a b) |
|
3076 |
from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps |
|
31809 | 3077 |
obtain l1 u1 l2 u2 |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3078 |
where l: "l = float_plus_down prec (nprt l1 * pprt u2) (float_plus_down prec (nprt u1 * nprt u2) (float_plus_down prec (pprt l1 * pprt l2) (pprt u1 * nprt l2)))" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3079 |
and u: "u = float_plus_up prec (pprt u1 * pprt u2) (float_plus_up prec (pprt l1 * nprt u2) (float_plus_up prec (nprt u1 * pprt l2) (nprt l1 * nprt l2)))" |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3080 |
and "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1" |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3081 |
and "l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> u2" unfolding fst_conv snd_conv by blast |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3082 |
hence bnds: |
60680 | 3083 |
"nprt l1 * pprt u2 + nprt u1 * nprt u2 + pprt l1 * pprt l2 + pprt u1 * nprt l2 \<le> |
3084 |
interpret_floatarith (Mult a b) xs" (is "?l \<le> _") |
|
3085 |
"interpret_floatarith (Mult a b) xs \<le> |
|
3086 |
pprt u1 * pprt u2 + pprt l1 * nprt u2 + nprt u1 * pprt l2 + nprt l1 * nprt l2" (is "_ \<le> ?u") |
|
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3087 |
unfolding interpret_floatarith.simps l u |
29805 | 3088 |
using mult_le_prts mult_ge_prts by auto |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3089 |
from l u have "l \<le> ?l" "?u \<le> u" |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3090 |
by (auto intro!: float_plus_up_le float_plus_down_le) |
60680 | 3091 |
thus ?case |
3092 |
using bnds by simp |
|
29805 | 3093 |
next |
3094 |
case (Inverse a) |
|
3095 |
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 | 3096 |
obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)" |
29805 | 3097 |
and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)" |
60680 | 3098 |
and l1: "l1 \<le> interpret_floatarith a xs" |
3099 |
and u1: "interpret_floatarith a xs \<le> u1" |
|
3100 |
by blast |
|
3101 |
have either: "0 < l1 \<or> u1 < 0" |
|
3102 |
proof (rule ccontr) |
|
3103 |
assume P: "\<not> (0 < l1 \<or> u1 < 0)" |
|
3104 |
show False |
|
3105 |
using l' unfolding if_not_P[OF P] by auto |
|
3106 |
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
|
3107 |
moreover have l1_le_u1: "real_of_float l1 \<le> real_of_float u1" |
60680 | 3108 |
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
|
3109 |
ultimately have "real_of_float l1 \<noteq> 0" and "real_of_float u1 \<noteq> 0" |
60680 | 3110 |
by auto |
29805 | 3111 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3112 |
have inv: "inverse u1 \<le> inverse (interpret_floatarith a xs) |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3113 |
\<and> inverse (interpret_floatarith a xs) \<le> inverse l1" |
29805 | 3114 |
proof (cases "0 < l1") |
60680 | 3115 |
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
|
3116 |
hence "0 < real_of_float u1" and "0 < real_of_float l1" "0 < interpret_floatarith a xs" |
47600 | 3117 |
using l1_le_u1 l1 by auto |
29805 | 3118 |
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
|
3119 |
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
|
3120 |
inverse_le_iff_le[OF \<open>0 < interpret_floatarith a xs\<close> \<open>0 < real_of_float l1\<close>] |
29805 | 3121 |
using l1 u1 by auto |
3122 |
next |
|
60680 | 3123 |
case False |
3124 |
hence "u1 < 0" |
|
3125 |
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
|
3126 |
hence "real_of_float u1 < 0" and "real_of_float l1 < 0" "interpret_floatarith a xs < 0" |
47600 | 3127 |
using l1_le_u1 u1 by auto |
29805 | 3128 |
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
|
3129 |
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
|
3130 |
inverse_le_iff_le_neg[OF \<open>interpret_floatarith a xs < 0\<close> \<open>real_of_float l1 < 0\<close>] |
29805 | 3131 |
using l1 u1 by auto |
3132 |
qed |
|
31468
b8267feaf342
Approximation: Corrected precision of ln on all real values
hoelzl
parents:
31467
diff
changeset
|
3133 |
|
60680 | 3134 |
from l' have "l = float_divl prec 1 u1" |
3135 |
by (cases "0 < l1 \<or> u1 < 0") auto |
|
3136 |
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
|
3137 |
unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float u1 \<noteq> 0\<close>] |
60680 | 3138 |
using float_divl[of prec 1 u1] by auto |
3139 |
also have "\<dots> \<le> inverse (interpret_floatarith a xs)" |
|
3140 |
using inv by auto |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3141 |
finally have "l \<le> inverse (interpret_floatarith a xs)" . |
29805 | 3142 |
moreover |
60680 | 3143 |
from u' have "u = float_divr prec 1 l1" |
3144 |
by (cases "0 < l1 \<or> u1 < 0") auto |
|
3145 |
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
|
3146 |
unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float l1 \<noteq> 0\<close>] |
60680 | 3147 |
using float_divr[of 1 l1 prec] by auto |
3148 |
hence "inverse (interpret_floatarith a xs) \<le> u" |
|
3149 |
by (rule order_trans[OF inv[THEN conjunct2]]) |
|
3150 |
ultimately show ?case |
|
3151 |
unfolding interpret_floatarith.simps using l1 u1 by auto |
|
29805 | 3152 |
next |
3153 |
case (Abs x) |
|
3154 |
from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps |
|
60680 | 3155 |
obtain l1 u1 where l': "l = (if l1 < 0 \<and> 0 < u1 then 0 else min \<bar>l1\<bar> \<bar>u1\<bar>)" |
3156 |
and u': "u = max \<bar>l1\<bar> \<bar>u1\<bar>" |
|
3157 |
and l1: "l1 \<le> interpret_floatarith x xs" |
|
3158 |
and u1: "interpret_floatarith x xs \<le> u1" |
|
3159 |
by blast |
|
3160 |
thus ?case |
|
3161 |
unfolding l' u' |
|
3162 |
by (cases "l1 < 0 \<and> 0 < u1") (auto simp add: real_of_float_min real_of_float_max) |
|
29805 | 3163 |
next |
3164 |
case (Min a b) |
|
3165 |
from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps |
|
3166 |
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
|
3167 |
and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1" |
60680 | 3168 |
and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> u2" |
3169 |
by blast |
|
3170 |
thus ?case |
|
3171 |
unfolding l' u' by (auto simp add: real_of_float_min) |
|
29805 | 3172 |
next |
3173 |
case (Max a b) |
|
3174 |
from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps |
|
3175 |
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
|
3176 |
and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1" |
60680 | 3177 |
and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> u2" |
3178 |
by blast |
|
3179 |
thus ?case |
|
3180 |
unfolding l' u' by (auto simp add: real_of_float_max) |
|
3181 |
next |
|
3182 |
case (Cos a) |
|
3183 |
with lift_un'_bnds[OF bnds_cos] show ?case by auto |
|
3184 |
next |
|
3185 |
case (Arctan a) |
|
3186 |
with lift_un'_bnds[OF bnds_arctan] show ?case by auto |
|
3187 |
next |
|
3188 |
case Pi |
|
3189 |
with pi_boundaries show ?case by auto |
|
3190 |
next |
|
3191 |
case (Sqrt a) |
|
3192 |
with lift_un'_bnds[OF bnds_sqrt] show ?case by auto |
|
3193 |
next |
|
3194 |
case (Exp a) |
|
3195 |
with lift_un'_bnds[OF bnds_exp] show ?case by auto |
|
3196 |
next |
|
3197 |
case (Ln a) |
|
3198 |
with lift_un_bnds[OF bnds_ln] show ?case by auto |
|
3199 |
next |
|
3200 |
case (Power a n) |
|
3201 |
with lift_un'_bnds[OF bnds_power] show ?case by auto |
|
3202 |
next |
|
3203 |
case (Num f) |
|
3204 |
thus ?case by auto |
|
29805 | 3205 |
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
|
3206 |
case (Var n) |
60533 | 3207 |
from this[symmetric] \<open>bounded_by xs vs\<close>[THEN bounded_byE, of n] |
60680 | 3208 |
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
|
3209 |
qed |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3210 |
|
58310 | 3211 |
datatype form = Bound floatarith floatarith floatarith form |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3212 |
| Assign floatarith floatarith form |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3213 |
| Less floatarith floatarith |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3214 |
| LessEqual floatarith floatarith |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3215 |
| AtLeastAtMost floatarith floatarith floatarith |
58986 | 3216 |
| Conj form form |
3217 |
| Disj form form |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3218 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3219 |
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
|
3220 |
"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
|
3221 |
"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
|
3222 |
"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
|
3223 |
"interpret_form (LessEqual a b) vs = (interpret_floatarith a vs \<le> interpret_floatarith b vs)" | |
58986 | 3224 |
"interpret_form (AtLeastAtMost x a b) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs })" | |
3225 |
"interpret_form (Conj f g) vs \<longleftrightarrow> interpret_form f vs \<and> interpret_form g vs" | |
|
3226 |
"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
|
3227 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3228 |
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
|
3229 |
"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
|
3230 |
"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
|
3231 |
(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
|
3232 |
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
|
3233 |
"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
|
3234 |
(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
|
3235 |
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
|
3236 |
| _ \<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
|
3237 |
"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
|
3238 |
(case (approx prec a bs) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3239 |
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
|
3240 |
| _ \<Rightarrow> False)" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3241 |
"approx_form prec (Less a b) bs ss = |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3242 |
(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
|
3243 |
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
|
3244 |
| _ \<Rightarrow> False)" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3245 |
"approx_form prec (LessEqual a b) bs ss = |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3246 |
(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
|
3247 |
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
|
3248 |
| _ \<Rightarrow> False)" | |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3249 |
"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
|
3250 |
(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
|
3251 |
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
|
3252 |
| _ \<Rightarrow> False)" | |
58986 | 3253 |
"approx_form prec (Conj a b) bs ss \<longleftrightarrow> approx_form prec a bs ss \<and> approx_form prec b bs ss" | |
3254 |
"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
|
3255 |
"approx_form _ _ _ _ = False" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3256 |
|
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
|
3257 |
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
|
3258 |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3259 |
lemma approx_form_approx_form': |
60680 | 3260 |
assumes "approx_form' prec f s n l u bs ss" |
3261 |
and "(x::real) \<in> { l .. u }" |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3262 |
obtains l' u' where "x \<in> { l' .. u' }" |
49351 | 3263 |
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
|
3264 |
using assms proof (induct s arbitrary: l u) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3265 |
case 0 |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3266 |
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
|
3267 |
show thesis by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3268 |
next |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3269 |
case (Suc s) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3270 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
3271 |
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
|
3272 |
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
|
3273 |
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
|
3274 |
|
60533 | 3275 |
with \<open>x \<in> { l .. u }\<close> |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3276 |
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
|
3277 |
thus thesis |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3278 |
proof (rule disjE) |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3279 |
assume *: "x \<in> { l .. ?m }" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3280 |
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
|
3281 |
show thesis by (simp add: Let_def lazy_conj) |
29805 | 3282 |
next |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3283 |
assume *: "x \<in> { ?m .. u }" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3284 |
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
|
3285 |
show thesis by (simp add: Let_def lazy_conj) |
29805 | 3286 |
qed |
3287 |
qed |
|
3288 |
||
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3289 |
lemma approx_form_aux: |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3290 |
assumes "approx_form prec f vs ss" |
49351 | 3291 |
and "bounded_by xs vs" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3292 |
shows "interpret_form f xs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3293 |
using assms proof (induct f arbitrary: vs) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3294 |
case (Bound x a b f) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3295 |
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
|
3296 |
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
|
3297 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3298 |
with Bound.prems obtain l u' l' u |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3299 |
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
|
3300 |
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
|
3301 |
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
|
3302 |
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
|
3303 |
|
60680 | 3304 |
have "interpret_form f xs" |
3305 |
if "xs ! n \<in> { interpret_floatarith a xs .. interpret_floatarith b xs }" |
|
3306 |
proof - |
|
3307 |
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
|
3308 |
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
|
3309 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3310 |
from approx_form_approx_form'[OF approx_form' this] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3311 |
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
|
3312 |
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
|
3313 |
|
60680 | 3314 |
from \<open>bounded_by xs vs\<close> bnds have "bounded_by xs (vs[n := Some (lx, ux)])" |
3315 |
by (rule bounded_by_update) |
|
3316 |
with Bound.hyps[OF approx_form] show ?thesis |
|
3317 |
by blast |
|
3318 |
qed |
|
3319 |
thus ?case |
|
3320 |
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
|
3321 |
next |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3322 |
case (Assign x a f) |
60680 | 3323 |
then obtain n where x_eq: "x = Var n" |
3324 |
by (cases x) auto |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3325 |
|
47599
400b158f1589
replace the float datatype by a type with unique representation
hoelzl
parents:
47108
diff
changeset
|
3326 |
with Assign.prems obtain l u |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3327 |
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
|
3328 |
and x_eq: "x = Var n" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3329 |
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
|
3330 |
by (cases "approx prec a vs") auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3331 |
|
60680 | 3332 |
have "interpret_form f xs" |
3333 |
if bnds: "xs ! n = interpret_floatarith a xs" |
|
3334 |
proof - |
|
3335 |
from approx[OF Assign.prems(2) bnd_eq] bnds |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3336 |
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
|
3337 |
from approx_form_approx_form'[OF approx_form' this] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3338 |
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
|
3339 |
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
|
3340 |
|
60680 | 3341 |
from \<open>bounded_by xs vs\<close> bnds have "bounded_by xs (vs[n := Some (lx, ux)])" |
3342 |
by (rule bounded_by_update) |
|
3343 |
with Assign.hyps[OF approx_form] show ?thesis |
|
3344 |
by blast |
|
3345 |
qed |
|
3346 |
thus ?case |
|
3347 |
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
|
3348 |
next |
29805 | 3349 |
case (Less a b) |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3350 |
then obtain l u l' u' |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3351 |
where l_eq: "Some (l, u) = approx prec a vs" |
49351 | 3352 |
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
|
3353 |
and inequality: "real_of_float (float_plus_up prec u (-l')) < 0" |
60680 | 3354 |
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
|
3355 |
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
|
3356 |
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
|
3357 |
show ?case by auto |
29805 | 3358 |
next |
3359 |
case (LessEqual a b) |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3360 |
then obtain l u l' u' |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3361 |
where l_eq: "Some (l, u) = approx prec a vs" |
49351 | 3362 |
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
|
3363 |
and inequality: "real_of_float (float_plus_up prec u (-l')) \<le> 0" |
60680 | 3364 |
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
|
3365 |
from order_trans[OF float_plus_up inequality] |
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3366 |
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
|
3367 |
show ?case by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3368 |
next |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3369 |
case (AtLeastAtMost x a b) |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3370 |
then obtain lx ux l u l' u' |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3371 |
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
|
3372 |
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
|
3373 |
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
|
3374 |
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
|
3375 |
by (cases "approx prec x vs", auto, |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3376 |
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
|
3377 |
cases "approx prec b vs", auto) |
58985
bf498e0af9e3
truncate intermediate results in horner to improve performance of approximate;
immler
parents:
58982
diff
changeset
|
3378 |
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
|
3379 |
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
|
3380 |
show ?case by auto |
58986 | 3381 |
qed auto |
29805 | 3382 |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3383 |
lemma approx_form: |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3384 |
assumes "n = length xs" |
60680 | 3385 |
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
|
3386 |
shows "interpret_form f xs" |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
3387 |
using approx_form_aux[OF _ bounded_by_None] assms by auto |
29805 | 3388 |
|
60680 | 3389 |
|
60533 | 3390 |
subsection \<open>Implementing Taylor series expansion\<close> |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3391 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3392 |
fun isDERIV :: "nat \<Rightarrow> floatarith \<Rightarrow> real list \<Rightarrow> bool" where |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3393 |
"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
|
3394 |
"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
|
3395 |
"isDERIV x (Minus a) vs = isDERIV x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3396 |
"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
|
3397 |
"isDERIV x (Cos a) vs = isDERIV x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3398 |
"isDERIV x (Arctan a) vs = isDERIV x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3399 |
"isDERIV x (Min a b) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3400 |
"isDERIV x (Max a b) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3401 |
"isDERIV x (Abs a) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3402 |
"isDERIV x Pi vs = True" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3403 |
"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
|
3404 |
"isDERIV x (Exp a) vs = isDERIV x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3405 |
"isDERIV x (Ln a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3406 |
"isDERIV x (Power a 0) vs = True" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3407 |
"isDERIV x (Power a (Suc n)) vs = isDERIV x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3408 |
"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
|
3409 |
"isDERIV x (Var n) vs = True" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3410 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3411 |
fun DERIV_floatarith :: "nat \<Rightarrow> floatarith \<Rightarrow> floatarith" where |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3412 |
"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
|
3413 |
"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
|
3414 |
"DERIV_floatarith x (Minus a) = Minus (DERIV_floatarith x a)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3415 |
"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
|
3416 |
"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
|
3417 |
"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
|
3418 |
"DERIV_floatarith x (Min a b) = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3419 |
"DERIV_floatarith x (Max a b) = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3420 |
"DERIV_floatarith x (Abs a) = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3421 |
"DERIV_floatarith x Pi = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3422 |
"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
|
3423 |
"DERIV_floatarith x (Exp a) = Mult (Exp a) (DERIV_floatarith x a)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3424 |
"DERIV_floatarith x (Ln a) = Mult (Inverse a) (DERIV_floatarith x a)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3425 |
"DERIV_floatarith x (Power a 0) = Num 0" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3426 |
"DERIV_floatarith x (Power a (Suc n)) = Mult (Num (Float (int (Suc n)) 0)) (Mult (Power a n) (DERIV_floatarith x a))" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3427 |
"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
|
3428 |
"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
|
3429 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3430 |
lemma DERIV_floatarith: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3431 |
assumes "n < length vs" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3432 |
assumes isDERIV: "isDERIV n f (vs[n := x])" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3433 |
shows "DERIV (\<lambda> x'. interpret_floatarith f (vs[n := x'])) x :> |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3434 |
interpret_floatarith (DERIV_floatarith n f) (vs[n := x])" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3435 |
(is "DERIV (?i f) x :> _") |
49351 | 3436 |
using isDERIV |
3437 |
proof (induct f arbitrary: x) |
|
3438 |
case (Inverse a) |
|
3439 |
thus ?case |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56195
diff
changeset
|
3440 |
by (auto intro!: derivative_eq_intros simp add: algebra_simps power2_eq_square) |
49351 | 3441 |
next |
3442 |
case (Cos a) |
|
3443 |
thus ?case |
|
56382 | 3444 |
by (auto intro!: derivative_eq_intros |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3445 |
simp del: interpret_floatarith.simps(5) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3446 |
simp add: interpret_floatarith_sin interpret_floatarith.simps(5)[of a]) |
49351 | 3447 |
next |
3448 |
case (Power a n) |
|
3449 |
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
|
3450 |
by (cases n) (auto intro!: derivative_eq_intros simp del: power_Suc) |
49351 | 3451 |
next |
3452 |
case (Ln a) |
|
56382 | 3453 |
thus ?case by (auto intro!: derivative_eq_intros simp add: divide_inverse) |
49351 | 3454 |
next |
3455 |
case (Var i) |
|
60533 | 3456 |
thus ?case using \<open>n < length vs\<close> by auto |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56195
diff
changeset
|
3457 |
qed (auto intro!: derivative_eq_intros) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3458 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3459 |
declare approx.simps[simp del] |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3460 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3461 |
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
|
3462 |
"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
|
3463 |
"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
|
3464 |
"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
|
3465 |
"isDERIV_approx prec x (Inverse a) vs = |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3466 |
(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
|
3467 |
"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
|
3468 |
"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
|
3469 |
"isDERIV_approx prec x (Min a b) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3470 |
"isDERIV_approx prec x (Max a b) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3471 |
"isDERIV_approx prec x (Abs a) vs = False" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3472 |
"isDERIV_approx prec x Pi vs = True" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3473 |
"isDERIV_approx prec x (Sqrt a) vs = |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3474 |
(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
|
3475 |
"isDERIV_approx prec x (Exp a) vs = isDERIV_approx prec x a vs" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3476 |
"isDERIV_approx prec x (Ln a) vs = |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3477 |
(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
|
3478 |
"isDERIV_approx prec x (Power a 0) vs = True" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3479 |
"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
|
3480 |
"isDERIV_approx prec 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
|
3481 |
"isDERIV_approx prec x (Var n) vs = True" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3482 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3483 |
lemma isDERIV_approx: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3484 |
assumes "bounded_by xs vs" |
49351 | 3485 |
and isDERIV_approx: "isDERIV_approx prec x f vs" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3486 |
shows "isDERIV x f xs" |
49351 | 3487 |
using isDERIV_approx |
3488 |
proof (induct f) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3489 |
case (Inverse a) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3490 |
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
|
3491 |
and *: "0 < l \<or> u < 0" |
49351 | 3492 |
by (cases "approx prec a vs") auto |
60533 | 3493 |
with approx[OF \<open>bounded_by xs vs\<close> approx_Some] |
47600 | 3494 |
have "interpret_floatarith a xs \<noteq> 0" by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3495 |
thus ?case using Inverse by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3496 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3497 |
case (Ln a) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3498 |
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
|
3499 |
and *: "0 < l" |
49351 | 3500 |
by (cases "approx prec a vs") auto |
60533 | 3501 |
with approx[OF \<open>bounded_by xs vs\<close> approx_Some] |
47600 | 3502 |
have "0 < interpret_floatarith a xs" by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3503 |
thus ?case using Ln by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3504 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3505 |
case (Sqrt a) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3506 |
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
|
3507 |
and *: "0 < l" |
49351 | 3508 |
by (cases "approx prec a vs") auto |
60533 | 3509 |
with approx[OF \<open>bounded_by xs vs\<close> approx_Some] |
47600 | 3510 |
have "0 < interpret_floatarith a xs" by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3511 |
thus ?case using Sqrt by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3512 |
next |
60680 | 3513 |
case (Power a n) |
3514 |
thus ?case by (cases n) auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3515 |
qed auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3516 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3517 |
lemma bounded_by_update_var: |
60680 | 3518 |
assumes "bounded_by xs vs" |
3519 |
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
|
3520 |
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
|
3521 |
shows "bounded_by (xs[i := x]) vs" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3522 |
proof (cases "i < length xs") |
49351 | 3523 |
case False |
60680 | 3524 |
thus ?thesis |
3525 |
using \<open>bounded_by xs vs\<close> by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3526 |
next |
60680 | 3527 |
case True |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3528 |
let ?xs = "xs[i := x]" |
60680 | 3529 |
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
|
3530 |
have "case vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> ?xs ! j \<in> {real_of_float l .. real_of_float u}" |
60680 | 3531 |
if "j < length vs" for j |
3532 |
proof (cases "vs ! j") |
|
3533 |
case None |
|
3534 |
then show ?thesis by simp |
|
3535 |
next |
|
3536 |
case (Some b) |
|
3537 |
thus ?thesis |
|
3538 |
proof (cases "i = j") |
|
3539 |
case True |
|
3540 |
thus ?thesis using \<open>vs ! i = Some (l, u)\<close> Some and bnd \<open>i < length ?xs\<close> |
|
3541 |
by auto |
|
3542 |
next |
|
3543 |
case False |
|
49351 | 3544 |
thus ?thesis |
60680 | 3545 |
using \<open>bounded_by xs vs\<close>[THEN bounded_byE, OF \<open>j < length vs\<close>] Some by auto |
3546 |
qed |
|
3547 |
qed |
|
3548 |
thus ?thesis |
|
3549 |
unfolding bounded_by_def by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3550 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3551 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3552 |
lemma isDERIV_approx': |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3553 |
assumes "bounded_by xs vs" |
60680 | 3554 |
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
|
3555 |
and X_in: "X \<in> {real_of_float l .. real_of_float u}" |
49351 | 3556 |
and approx: "isDERIV_approx prec x f vs" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3557 |
shows "isDERIV x f (xs[x := X])" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3558 |
proof - |
60680 | 3559 |
from bounded_by_update_var[OF \<open>bounded_by xs vs\<close> vs_x X_in] approx |
3560 |
show ?thesis by (rule isDERIV_approx) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3561 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3562 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3563 |
lemma DERIV_approx: |
60680 | 3564 |
assumes "n < length xs" |
3565 |
and bnd: "bounded_by xs vs" |
|
49351 | 3566 |
and isD: "isDERIV_approx prec n f vs" |
3567 |
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
|
3568 |
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
|
3569 |
DERIV (\<lambda> x. interpret_floatarith f (xs[n := x])) (xs!n) :> x" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3570 |
(is "\<exists> x. _ \<and> _ \<and> DERIV (?i f) _ :> _") |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3571 |
proof (rule exI[of _ "?i ?D (xs!n)"], rule conjI[OF _ conjI]) |
60680 | 3572 |
let "?i f" = "\<lambda>x. interpret_floatarith f (xs[n := x])" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3573 |
from approx[OF bnd app] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3574 |
show "l \<le> ?i ?D (xs!n)" and "?i ?D (xs!n) \<le> u" |
60533 | 3575 |
using \<open>n < length xs\<close> by auto |
3576 |
from DERIV_floatarith[OF \<open>n < length xs\<close>, of f "xs!n"] isDERIV_approx[OF bnd isD] |
|
60680 | 3577 |
show "DERIV (?i f) (xs!n) :> (?i ?D (xs!n))" |
3578 |
by simp |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3579 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3580 |
|
49351 | 3581 |
fun lift_bin :: "(float * float) option \<Rightarrow> |
3582 |
(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float) option) \<Rightarrow> |
|
3583 |
(float * float) option" where |
|
3584 |
"lift_bin (Some (l1, u1)) (Some (l2, u2)) f = f l1 u1 l2 u2" |
|
3585 |
| "lift_bin a b f = None" |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3586 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3587 |
lemma lift_bin: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3588 |
assumes lift_bin_Some: "Some (l, u) = lift_bin a b f" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3589 |
obtains l1 u1 l2 u2 |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3590 |
where "a = Some (l1, u1)" |
49351 | 3591 |
and "b = Some (l2, u2)" |
3592 |
and "f l1 u1 l2 u2 = Some (l, u)" |
|
3593 |
using assms by (cases a, simp, cases b, simp, auto) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3594 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3595 |
fun approx_tse where |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3596 |
"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
|
3597 |
"approx_tse prec n (Suc s) c k f bs = |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3598 |
(if isDERIV_approx prec n f bs then |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3599 |
lift_bin (approx prec f (bs[n := Some (c,c)])) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3600 |
(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
|
3601 |
(\<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
|
3602 |
(Add (Var 0) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3603 |
(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
|
3604 |
(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
|
3605 |
(Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), bs!n]) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3606 |
else approx prec f bs)" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3607 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3608 |
lemma bounded_by_Cons: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3609 |
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
|
3610 |
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
|
3611 |
shows "bounded_by (x#xs) ((Some (l, u))#vs)" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3612 |
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
|
3613 |
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 | 3614 |
if *: "i < length ((Some (l, u))#vs)" for i |
3615 |
proof (cases i) |
|
3616 |
case 0 |
|
3617 |
with x show ?thesis by auto |
|
3618 |
next |
|
3619 |
case (Suc i) |
|
3620 |
with * have "i < length vs" by auto |
|
3621 |
from bnd[THEN bounded_byE, OF this] |
|
3622 |
show ?thesis unfolding Suc nth_Cons_Suc . |
|
3623 |
qed |
|
3624 |
thus ?thesis |
|
3625 |
by (auto simp add: bounded_by_def) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3626 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3627 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3628 |
lemma approx_tse_generic: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3629 |
assumes "bounded_by xs vs" |
60680 | 3630 |
and bnd_c: "bounded_by (xs[x := c]) vs" |
3631 |
and "x < length vs" and "x < length xs" |
|
49351 | 3632 |
and bnd_x: "vs ! x = Some (lx, ux)" |
3633 |
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
|
3634 |
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
|
3635 |
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
|
3636 |
(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
|
3637 |
\<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
|
3638 |
interpret_floatarith ((DERIV_floatarith x ^^ i) f) (xs[x := c]) * |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3639 |
(xs!x - c)^i) + |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3640 |
inverse (real (\<Prod> j \<in> {k..<k+n}. j)) * |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3641 |
interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := t]) * |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3642 |
(xs!x - c)^n \<in> {l .. u})" (is "\<exists> n. ?taylor f k l u n") |
60680 | 3643 |
using ate |
3644 |
proof (induct s arbitrary: k f l u) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3645 |
case 0 |
49351 | 3646 |
{ |
3647 |
fix t::real assume "t \<in> {lx .. ux}" |
|
60533 | 3648 |
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
|
3649 |
from approx[OF this 0[unfolded approx_tse.simps]] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3650 |
have "(interpret_floatarith f (xs[x := t])) \<in> {l .. u}" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3651 |
by (auto simp add: algebra_simps) |
49351 | 3652 |
} |
3653 |
thus ?case by (auto intro!: exI[of _ 0]) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3654 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3655 |
case (Suc s) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3656 |
show ?case |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3657 |
proof (cases "isDERIV_approx prec x f vs") |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3658 |
case False |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3659 |
note ap = Suc.prems[unfolded approx_tse.simps if_not_P[OF False]] |
49351 | 3660 |
{ |
3661 |
fix t::real assume "t \<in> {lx .. ux}" |
|
60533 | 3662 |
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
|
3663 |
from approx[OF this ap] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3664 |
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
|
3665 |
by (auto simp add: algebra_simps) |
49351 | 3666 |
} |
3667 |
thus ?thesis by (auto intro!: exI[of _ 0]) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3668 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3669 |
case True |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3670 |
with Suc.prems |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3671 |
obtain l1 u1 l2 u2 |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3672 |
where a: "Some (l1, u1) = approx prec f (vs[x := Some (c,c)])" |
49351 | 3673 |
and ate: "Some (l2, u2) = approx_tse prec x s c (Suc k) (DERIV_floatarith x f) vs" |
3674 |
and final: "Some (l, u) = approx prec |
|
3675 |
(Add (Var 0) |
|
3676 |
(Mult (Inverse (Num (Float (int k) 0))) |
|
3677 |
(Mult (Add (Var (Suc (Suc 0))) (Minus (Num c))) |
|
3678 |
(Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), vs!x]" |
|
56073
29e308b56d23
enhanced simplifier solver for preconditions of rewrite rule, can now deal with conjunctions
nipkow
parents:
55506
diff
changeset
|
3679 |
by (auto elim!: lift_bin) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3680 |
|
60533 | 3681 |
from bnd_c \<open>x < length xs\<close> |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3682 |
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
|
3683 |
by (auto intro!: bounded_by_update) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3684 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3685 |
from approx[OF this a] |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3686 |
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
|
3687 |
(is "?f 0 (real_of_float c) \<in> _") |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3688 |
by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3689 |
|
60680 | 3690 |
have funpow_Suc[symmetric]: "(f ^^ Suc n) x = (f ^^ n) (f x)" |
3691 |
for f :: "'a \<Rightarrow> 'a" and n :: nat and x :: 'a |
|
3692 |
by (induct n) auto |
|
3693 |
from Suc.hyps[OF ate, unfolded this] obtain n |
|
3694 |
where DERIV_hyp: "\<And>m z. \<lbrakk> m < n ; (z::real) \<in> { lx .. ux } \<rbrakk> \<Longrightarrow> |
|
3695 |
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
|
3696 |
and hyp: "\<forall>t \<in> {real_of_float lx .. real_of_float ux}. |
60680 | 3697 |
(\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {Suc k..<Suc k + i}. j)) * ?f (Suc i) c * (xs!x - c)^i) + |
3698 |
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
|
3699 |
(is "\<forall> t \<in> _. ?X (Suc k) f n t \<in> _") |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3700 |
by blast |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3701 |
|
60680 | 3702 |
have DERIV: "DERIV (?f m) z :> ?f (Suc m) z" |
3703 |
if "m < Suc n" and bnd_z: "z \<in> { lx .. ux }" for m and z::real |
|
3704 |
proof (cases m) |
|
3705 |
case 0 |
|
3706 |
with DERIV_floatarith[OF \<open>x < length xs\<close> |
|
3707 |
isDERIV_approx'[OF \<open>bounded_by xs vs\<close> bnd_x bnd_z True]] |
|
3708 |
show ?thesis by simp |
|
3709 |
next |
|
3710 |
case (Suc m') |
|
3711 |
hence "m' < n" |
|
3712 |
using \<open>m < Suc n\<close> by auto |
|
3713 |
from DERIV_hyp[OF this bnd_z] show ?thesis |
|
3714 |
using Suc by simp |
|
3715 |
qed |
|
3716 |
||
3717 |
have "\<And>k i. k < i \<Longrightarrow> {k ..< i} = insert k {Suc k ..< i}" by auto |
|
3718 |
hence setprod_head_Suc: "\<And>k i. \<Prod>{k ..< k + Suc i} = k * \<Prod>{Suc k ..< Suc k + i}" |
|
3719 |
by auto |
|
3720 |
have setsum_move0: "\<And>k F. setsum F {0..<Suc k} = F 0 + setsum (\<lambda> k. F (Suc k)) {0..<k}" |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3721 |
unfolding setsum_shift_bounds_Suc_ivl[symmetric] |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3722 |
unfolding setsum_head_upt_Suc[OF zero_less_Suc] .. |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3723 |
def C \<equiv> "xs!x - c" |
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3724 |
|
49351 | 3725 |
{ |
3726 |
fix t::real assume t: "t \<in> {lx .. ux}" |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3727 |
hence "bounded_by [xs!x] [vs!x]" |
60533 | 3728 |
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
|
3729 |
by (cases "vs!x", auto simp add: bounded_by_def) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3730 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3731 |
with hyp[THEN bspec, OF t] f_c |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3732 |
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
|
3733 |
by (auto intro!: bounded_by_Cons) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3734 |
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
|
3735 |
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
|
3736 |
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
|
3737 |
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
|
3738 |
(\<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
|
3739 |
inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - c)^Suc n" (is "_ = ?T") |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32920
diff
changeset
|
3740 |
unfolding funpow_Suc C_def[symmetric] setsum_move0 setprod_head_Suc |
35082 | 3741 |
by (auto simp add: algebra_simps) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
3742 |
(simp only: mult.left_commute [of _ "inverse (real k)"] setsum_right_distrib [symmetric]) |
49351 | 3743 |
finally have "?T \<in> {l .. u}" . |
3744 |
} |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3745 |
thus ?thesis using DERIV by blast |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3746 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3747 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3748 |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3749 |
lemma setprod_fact: "real (\<Prod> {1..<1 + k}) = fact (k :: 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
|
3750 |
by (metis Suc_eq_plus1_left atLeastLessThanSuc_atLeastAtMost fact_altdef_nat of_nat_fact) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3751 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3752 |
lemma approx_tse: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3753 |
assumes "bounded_by xs vs" |
60680 | 3754 |
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
|
3755 |
and bnd_c: "real_of_float c \<in> {lx .. ux}" |
49351 | 3756 |
and "x < length vs" and "x < length xs" |
3757 |
and ate: "Some (l, u) = approx_tse prec x s c 1 f vs" |
|
60680 | 3758 |
shows "interpret_floatarith f xs \<in> {l .. u}" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3759 |
proof - |
60680 | 3760 |
def F \<equiv> "\<lambda>n z. interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := z])" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3761 |
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
|
3762 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3763 |
hence "bounded_by (xs[x := c]) vs" and "x < length vs" "x < length xs" |
60533 | 3764 |
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
|
3765 |
by (auto intro!: bounded_by_update_var) |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3766 |
|
60533 | 3767 |
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
|
3768 |
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
|
3769 |
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
|
3770 |
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
|
3771 |
(\<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
|
3772 |
inverse ((fact n)) * F n t * (xs!x - c)^n |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3773 |
\<in> {l .. u}" (is "\<And> t. _ \<Longrightarrow> ?taylor t \<in> _") |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3774 |
unfolding F_def atLeastAtMost_iff[symmetric] setprod_fact |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3775 |
by blast |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3776 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3777 |
have bnd_xs: "xs ! x \<in> { lx .. ux }" |
60533 | 3778 |
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
|
3779 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3780 |
show ?thesis |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3781 |
proof (cases n) |
60680 | 3782 |
case 0 |
3783 |
thus ?thesis |
|
3784 |
using hyp[OF bnd_xs] unfolding F_def by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3785 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3786 |
case (Suc n') |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3787 |
show ?thesis |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3788 |
proof (cases "xs ! x = c") |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3789 |
case True |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3790 |
from True[symmetric] hyp[OF bnd_xs] Suc show ?thesis |
60680 | 3791 |
unfolding F_def Suc setsum_head_upt_Suc[OF zero_less_Suc] setsum_shift_bounds_Suc_ivl |
3792 |
by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3793 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3794 |
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
|
3795 |
have "lx \<le> real_of_float c" "real_of_float c \<le> ux" "lx \<le> xs!x" "xs!x \<le> ux" |
60533 | 3796 |
using Suc bnd_c \<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
|
3797 |
from Taylor.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
|
3798 |
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
|
3799 |
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
|
3800 |
(\<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
|
3801 |
F (Suc n') t / (fact (Suc n')) * (xs ! x - c) ^ Suc n'" |
56195 | 3802 |
unfolding atLeast0LessThan by blast |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3803 |
|
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3804 |
from t_bnd bnd_xs bnd_c have *: "t \<in> {lx .. ux}" |
60680 | 3805 |
by (cases "xs ! x < c") auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3806 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3807 |
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
|
3808 |
unfolding fl_eq Suc by (auto simp add: algebra_simps divide_inverse) |
60680 | 3809 |
also have "\<dots> \<in> {l .. u}" |
3810 |
using * by (rule hyp) |
|
3811 |
finally show ?thesis |
|
3812 |
by simp |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3813 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3814 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3815 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3816 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3817 |
fun approx_tse_form' where |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3818 |
"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
|
3819 |
(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
|
3820 |
of Some (l, u) \<Rightarrow> cmp l u | None \<Rightarrow> False)" | |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3821 |
"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
|
3822 |
(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
|
3823 |
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
|
3824 |
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
|
3825 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3826 |
lemma approx_tse_form': |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3827 |
fixes x :: real |
60680 | 3828 |
assumes "approx_tse_form' prec t f s l u cmp" |
3829 |
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
|
3830 |
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 | 3831 |
approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" |
3832 |
using assms |
|
3833 |
proof (induct s arbitrary: l u) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3834 |
case 0 |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3835 |
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
|
3836 |
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
|
3837 |
and **: "cmp ly uy" by (auto elim!: case_optionE) |
46545 | 3838 |
with 0 show ?case by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3839 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3840 |
case (Suc s) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
3841 |
let ?m = "(l + u) * Float 1 (- 1)" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3842 |
from Suc.prems |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3843 |
have l: "approx_tse_form' prec t f s l ?m cmp" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3844 |
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
|
3845 |
by (auto simp add: Let_def lazy_conj) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3846 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
60680
diff
changeset
|
3847 |
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
|
3848 |
unfolding less_eq_float_def using Suc.prems by auto |
60680 | 3849 |
with \<open>x \<in> { l .. u }\<close> consider "x \<in> { l .. ?m}" | "x \<in> {?m .. u}" |
3850 |
by atomize_elim auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3851 |
thus ?case |
60680 | 3852 |
proof cases |
3853 |
case 1 |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3854 |
from Suc.hyps[OF l this] |
60680 | 3855 |
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
|
3856 |
"x \<in> {l' .. u'} \<and> real_of_float l \<le> l' \<and> real_of_float u' \<le> ?m \<and> cmp ly uy \<and> |
60680 | 3857 |
approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" |
3858 |
by blast |
|
3859 |
with m_u show ?thesis |
|
3860 |
by (auto intro!: exI) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3861 |
next |
60680 | 3862 |
case 2 |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3863 |
from Suc.hyps[OF u this] |
60680 | 3864 |
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
|
3865 |
"x \<in> { l' .. u' } \<and> ?m \<le> real_of_float l' \<and> u' \<le> real_of_float u \<and> cmp ly uy \<and> |
60680 | 3866 |
approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" |
3867 |
by blast |
|
3868 |
with m_u show ?thesis |
|
3869 |
by (auto intro!: exI) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3870 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3871 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3872 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3873 |
lemma approx_tse_form'_less: |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3874 |
fixes x :: real |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3875 |
assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 < l)" |
60680 | 3876 |
and x: "x \<in> {l .. u}" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3877 |
shows "interpret_floatarith b [x] < interpret_floatarith a [x]" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3878 |
proof - |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3879 |
from approx_tse_form'[OF tse x] |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3880 |
obtain l' u' ly uy |
60680 | 3881 |
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
|
3882 |
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
|
3883 |
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
|
3884 |
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
|
3885 |
by blast |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3886 |
|
60680 | 3887 |
hence "bounded_by [x] [Some (l', u')]" |
3888 |
by (auto simp add: bounded_by_def) |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3889 |
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
|
3890 |
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
|
3891 |
by auto |
60680 | 3892 |
from order_less_le_trans[OF _ this, of 0] \<open>0 < ly\<close> show ?thesis |
3893 |
by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3894 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3895 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3896 |
lemma approx_tse_form'_le: |
40881
e84f82418e09
Use coercions in Approximation (by Dmitriy Traytel).
hoelzl
parents:
39556
diff
changeset
|
3897 |
fixes x :: real |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3898 |
assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 \<le> l)" |
60680 | 3899 |
and x: "x \<in> {l .. u}" |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3900 |
shows "interpret_floatarith b [x] \<le> interpret_floatarith a [x]" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3901 |
proof - |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3902 |
from approx_tse_form'[OF tse x] |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3903 |
obtain l' u' ly uy |
60680 | 3904 |
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
|
3905 |
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
|
3906 |
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
|
3907 |
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
|
3908 |
by blast |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3909 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3910 |
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
|
3911 |
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
|
3912 |
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
|
3913 |
by auto |
60680 | 3914 |
from order_trans[OF _ this, of 0] \<open>0 \<le> ly\<close> show ?thesis |
3915 |
by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3916 |
qed |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3917 |
|
58986 | 3918 |
fun approx_tse_concl where |
3919 |
"approx_tse_concl prec t (Less lf rt) s l u l' u' \<longleftrightarrow> |
|
3920 |
approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)" | |
|
3921 |
"approx_tse_concl prec t (LessEqual lf rt) s l u l' u' \<longleftrightarrow> |
|
3922 |
approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)" | |
|
3923 |
"approx_tse_concl prec t (AtLeastAtMost x lf rt) s l u l' u' \<longleftrightarrow> |
|
3924 |
(if approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l) then |
|
3925 |
approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l) else False)" | |
|
3926 |
"approx_tse_concl prec t (Conj f g) s l u l' u' \<longleftrightarrow> |
|
3927 |
approx_tse_concl prec t f s l u l' u' \<and> approx_tse_concl prec t g s l u l' u'" | |
|
3928 |
"approx_tse_concl prec t (Disj f g) s l u l' u' \<longleftrightarrow> |
|
3929 |
approx_tse_concl prec t f s l u l' u' \<or> approx_tse_concl prec t g s l u l' u'" | |
|
3930 |
"approx_tse_concl _ _ _ _ _ _ _ _ \<longleftrightarrow> False" |
|
3931 |
||
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3932 |
definition |
60680 | 3933 |
"approx_tse_form prec t s f = |
3934 |
(case f of |
|
3935 |
Bound x a b f \<Rightarrow> |
|
3936 |
x = Var 0 \<and> |
|
3937 |
(case (approx prec a [None], approx prec b [None]) of |
|
3938 |
(Some (l, u), Some (l', u')) \<Rightarrow> approx_tse_concl prec t f s l u l' u' |
|
3939 |
| _ \<Rightarrow> False) |
|
3940 |
| _ \<Rightarrow> False)" |
|
31863
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 |
lemma approx_tse_form: |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3943 |
assumes "approx_tse_form prec t s f" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3944 |
shows "interpret_form f [x]" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3945 |
proof (cases f) |
60680 | 3946 |
case f_def: (Bound i a b f') |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3947 |
with assms obtain l u l' u' |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3948 |
where a: "approx prec a [None] = Some (l, u)" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3949 |
and b: "approx prec b [None] = Some (l', u')" |
55413
a8e96847523c
adapted theories to '{case,rec}_{list,option}' names
blanchet
parents:
54782
diff
changeset
|
3950 |
unfolding approx_tse_form_def by (auto elim!: case_optionE) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3951 |
|
60680 | 3952 |
from f_def assms have "i = Var 0" |
3953 |
unfolding approx_tse_form_def by auto |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3954 |
hence i: "interpret_floatarith i [x] = x" by auto |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3955 |
|
60680 | 3956 |
{ |
3957 |
let ?f = "\<lambda>z. interpret_floatarith z [x]" |
|
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3958 |
assume "?f i \<in> { ?f a .. ?f b }" |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3959 |
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
|
3960 |
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
|
3961 |
|
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3962 |
have "interpret_form f' [x]" |
60680 | 3963 |
using assms[unfolded f_def] |
58986 | 3964 |
proof (induct f') |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3965 |
case (Less lf rt) |
58986 | 3966 |
with a b |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3967 |
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
|
3968 |
unfolding approx_tse_form_def by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3969 |
from approx_tse_form'_less[OF this bnd] |
58986 | 3970 |
show ?case using Less by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3971 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3972 |
case (LessEqual lf rt) |
60680 | 3973 |
with f_def a b assms |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3974 |
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
|
3975 |
unfolding approx_tse_form_def by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3976 |
from approx_tse_form'_le[OF this bnd] |
58986 | 3977 |
show ?case using LessEqual by auto |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3978 |
next |
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3979 |
case (AtLeastAtMost x lf rt) |
60680 | 3980 |
with f_def a b assms |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3981 |
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
|
3982 |
and "approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)" |
58986 | 3983 |
unfolding approx_tse_form_def lazy_conj by (auto split: split_if_asm) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3984 |
from approx_tse_form'_le[OF this(1) bnd] approx_tse_form'_le[OF this(2) bnd] |
58986 | 3985 |
show ?case using AtLeastAtMost by auto |
3986 |
qed (auto simp: f_def approx_tse_form_def elim!: case_optionE) |
|
60680 | 3987 |
} |
3988 |
thus ?thesis unfolding f_def by auto |
|
58986 | 3989 |
qed (insert assms, auto simp add: approx_tse_form_def) |
31863
e391eee8bf14
Implemented taylor series expansion for approximation
hoelzl
parents:
31811
diff
changeset
|
3990 |
|
60533 | 3991 |
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
|
3992 |
|
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
|
3993 |
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
|
3994 |
"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
|
3995 |
(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
|
3996 |
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
|
3997 |
| _ \<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
|
3998 |
"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
|
3999 |
(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
|
4000 |
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
|
4001 |
| _ \<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
|
4002 |
"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
|
4003 |
"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
|
4004 |
"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
|
4005 |
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
|
4006 |
"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
|
4007 |
|
60680 | 4008 |
|
60533 | 4009 |
subsection \<open>Implement proof method \texttt{approximation}\<close> |
29805 | 4010 |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4011 |
lemmas interpret_form_equations = interpret_form.simps interpret_floatarith.simps interpret_floatarith_num |
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
|
4012 |
interpret_floatarith_divide interpret_floatarith_diff interpret_floatarith_tan interpret_floatarith_log |
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
|
4013 |
interpret_floatarith_sin |
29805 | 4014 |
|
60533 | 4015 |
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
|
4016 |
let |
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4017 |
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
|
4018 |
|
38716
3c3b4ad683d5
approximation_oracle: actually match true/false in ML, not arbitrary values;
wenzelm
parents:
38558
diff
changeset
|
4019 |
fun term_of_bool true = @{term True} |
3c3b4ad683d5
approximation_oracle: actually match true/false in ML, not arbitrary values;
wenzelm
parents:
38558
diff
changeset
|
4020 |
| 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
|
4021 |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
49962
diff
changeset
|
4022 |
val mk_int = HOLogic.mk_number @{typ int} o @{code integer_of_int}; |
58988 | 4023 |
fun dest_int (@{term int_of_integer} $ j) = @{code int_of_integer} (snd (HOLogic.dest_number j)) |
4024 |
| 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
|
4025 |
|
36985
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4026 |
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
|
4027 |
@{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
|
4028 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4029 |
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
|
4030 |
| 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
|
4031 |
$ 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
|
4032 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4033 |
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
|
4034 |
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
|
4035 |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
49962
diff
changeset
|
4036 |
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
|
4037 |
(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
|
4038 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4039 |
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
|
4040 |
@{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
|
4041 |
| 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
|
4042 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4043 |
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
|
4044 |
| 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
|
4045 |
| 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
|
4046 |
| 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
|
4047 |
| 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
|
4048 |
| 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
|
4049 |
| 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
|
4050 |
| 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
|
4051 |
| 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
|
4052 |
| 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
|
4053 |
| 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
|
4054 |
| floatarith_of_term (@{term Exp} $ a) = @{code Exp} (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
|
4055 |
| 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
|
4056 |
| 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
|
4057 |
@{code Power} (floatarith_of_term a, 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
|
4058 |
| 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
|
4059 |
| 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
|
4060 |
| 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
|
4061 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4062 |
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
|
4063 |
(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
|
4064 |
| 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
|
4065 |
(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
|
4066 |
| 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
|
4067 |
(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
|
4068 |
| 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
|
4069 |
(floatarith_of_term a, floatarith_of_term b) |
58986 | 4070 |
| form_of_term (@{term Conj} $ a $ b) = @{code Conj} |
4071 |
(form_of_term a, form_of_term b) |
|
4072 |
| form_of_term (@{term Disj} $ a $ b) = @{code Disj} |
|
4073 |
(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
|
4074 |
| 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
|
4075 |
(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
|
4076 |
| 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
|
4077 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4078 |
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
|
4079 |
| 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
|
4080 |
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
|
4081 |
| 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
|
4082 |
(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
|
4083 |
| 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
|
4084 |
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
|
4085 |
(@{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
|
4086 |
@{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
|
4087 |
| 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
|
4088 |
@{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
|
4089 |
| 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
|
4090 |
(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
|
4091 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4092 |
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
|
4093 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4094 |
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
|
4095 |
(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
|
4096 |
| 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
|
4097 |
@{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
|
4098 |
| 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
|
4099 |
|
41c5d4002f60
spelt out normalizer explicitly -- avoid dynamic reference to code generator configuration; avoid using old Codegen.eval_term
haftmann
parents:
36960
diff
changeset
|
4100 |
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
|
4101 |
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
|
4102 |
(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
|
4103 |
| @{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
|
4104 |
(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
|
4105 |
| @{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
|
4106 |
(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
|
4107 |
| _ => 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
|
4108 |
|
52131 | 4109 |
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
|
4110 |
|
59621
291934bac95e
Thm.cterm_of and Thm.ctyp_of operate on local context;
wenzelm
parents:
59582
diff
changeset
|
4111 |
in Thm.global_cterm_of thy (Logic.mk_equals (t, normalize t)) end |
60533 | 4112 |
\<close> |
31099
03314c427b34
optimized Approximation by precompiling approx_inequality
hoelzl
parents:
31098
diff
changeset
|
4113 |
|
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4114 |
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
|
4115 |
by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4116 |
|
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4117 |
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
|
4118 |
by auto |
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4119 |
|
59850 | 4120 |
ML_file "approximation.ML" |
4121 |
||
60533 | 4122 |
method_setup approximation = \<open> |
60680 | 4123 |
let |
4124 |
val free = |
|
4125 |
Args.context -- Args.term >> (fn (_, Free (n, _)) => n | (ctxt, t) => |
|
4126 |
error ("Bad free variable: " ^ Syntax.string_of_term ctxt t)); |
|
59850 | 4127 |
in |
60680 | 4128 |
Scan.lift Parse.nat -- |
59850 | 4129 |
Scan.optional (Scan.lift (Args.$$$ "splitting" |-- Args.colon) |
60680 | 4130 |
|-- Parse.and_list' (free --| Scan.lift (Args.$$$ "=") -- Scan.lift Parse.nat)) [] -- |
4131 |
Scan.option (Scan.lift (Args.$$$ "taylor" |-- Args.colon) |-- |
|
4132 |
(free |-- Scan.lift (Args.$$$ "=") |-- Scan.lift Parse.nat)) >> |
|
59850 | 4133 |
(fn ((prec, splitting), taylor) => fn ctxt => |
4134 |
SIMPLE_METHOD' (Approximation.approximation_tac prec splitting taylor ctxt)) |
|
4135 |
end |
|
60533 | 4136 |
\<close> "real number approximation" |
31811
64dea9a15031
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl
parents:
31810
diff
changeset
|
4137 |
|
58988 | 4138 |
|
4139 |
section "Quickcheck Generator" |
|
4140 |
||
4141 |
ML_file "approximation_generator.ML" |
|
4142 |
setup "Approximation_Generator.setup" |
|
4143 |
||
29805 | 4144 |
end |