Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 2,643 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 |
(*
Author: René Thiemann
Akihisa Yamada
License: BSD
*)
subsection \<open>Compare Instance for Complex Numbers\<close>
text \<open>We define some code equations for complex numbers, provide a comparator for complex
numbers, and register complex numbers for the container framework.\<close>
theory Compare_Complex
imports
HOL.Complex
Polynomial_Interpolation.Missing_Unsorted
Deriving.Compare_Real
Containers.Set_Impl
begin
declare [[code drop: Gcd_fin]]
declare [[code drop: Lcm_fin]]
definition gcds :: "'a::semiring_gcd list \<Rightarrow> 'a"
where [simp, code_abbrev]: "gcds xs = gcd_list xs"
lemma [code]:
"gcds xs = fold gcd xs 0"
by (simp add: Gcd_fin.set_eq_fold)
definition lcms :: "'a::semiring_gcd list \<Rightarrow> 'a"
where [simp, code_abbrev]: "lcms xs = lcm_list xs"
lemma [code]:
"lcms xs = fold lcm xs 1"
by (simp add: Lcm_fin.set_eq_fold)
lemma in_reals_code [code_unfold]:
"x \<in> \<real> \<longleftrightarrow> Im x = 0"
by (fact complex_is_Real_iff)
definition is_norm_1 :: "complex \<Rightarrow> bool" where
"is_norm_1 z = ((Re z)\<^sup>2 + (Im z)\<^sup>2 = 1)"
lemma is_norm_1[simp]: "is_norm_1 x = (norm x = 1)"
unfolding is_norm_1_def norm_complex_def by simp
definition is_norm_le_1 :: "complex \<Rightarrow> bool" where
"is_norm_le_1 z = ((Re z)\<^sup>2 + (Im z)\<^sup>2 \<le> 1)"
lemma is_norm_le_1[simp]: "is_norm_le_1 x = (norm x \<le> 1)"
unfolding is_norm_le_1_def norm_complex_def by simp
instantiation complex :: finite_UNIV
begin
definition "finite_UNIV = Phantom(complex) False"
instance
by (intro_classes, unfold finite_UNIV_complex_def, simp add: infinite_UNIV_char_0)
end
instantiation complex :: compare
begin
definition compare_complex :: "complex \<Rightarrow> complex \<Rightarrow> order" where
"compare_complex x y = compare (Re x, Im x) (Re y, Im y)"
instance
proof (intro_classes, unfold_locales; unfold compare_complex_def)
fix x y z :: complex
let ?c = "compare :: (real \<times> real) comparator"
interpret comparator ?c by (rule comparator_compare)
show "invert_order (?c (Re x, Im x) (Re y, Im y)) = ?c (Re y, Im y) (Re x, Im x)"
by (rule sym)
{
assume "?c (Re x, Im x) (Re y, Im y) = Lt"
"?c (Re y, Im y) (Re z, Im z) = Lt"
thus "?c (Re x, Im x) (Re z, Im z) = Lt"
by (rule comp_trans)
}
{
assume "?c (Re x, Im x) (Re y, Im y) = Eq"
from weak_eq[OF this] show "x = y" unfolding complex_eq_iff by auto
}
qed
end
derive (eq) ceq complex real
derive (compare) ccompare complex
derive (compare) ccompare real
derive (dlist) set_impl complex real
end
|