equal
deleted
inserted
replaced
23 lemma complex_eqI [intro?]: "Re x = Re y \<Longrightarrow> Im x = Im y \<Longrightarrow> x = y" |
23 lemma complex_eqI [intro?]: "Re x = Re y \<Longrightarrow> Im x = Im y \<Longrightarrow> x = y" |
24 by (rule complex.expand) simp |
24 by (rule complex.expand) simp |
25 |
25 |
26 lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" |
26 lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" |
27 by (auto intro: complex.expand) |
27 by (auto intro: complex.expand) |
|
28 |
28 |
29 |
29 |
30 |
30 subsection \<open>Addition and Subtraction\<close> |
31 subsection \<open>Addition and Subtraction\<close> |
31 |
32 |
32 instantiation complex :: ab_group_add |
33 instantiation complex :: ab_group_add |
652 shows "((\<lambda>z. cnj (f z)) has_vector_derivative cnj f') (at z within A)" |
653 shows "((\<lambda>z. cnj (f z)) has_vector_derivative cnj f') (at z within A)" |
653 using assms by (auto simp: has_vector_derivative_complex_iff intro: derivative_intros) |
654 using assms by (auto simp: has_vector_derivative_complex_iff intro: derivative_intros) |
654 |
655 |
655 |
656 |
656 subsection \<open>Basic Lemmas\<close> |
657 subsection \<open>Basic Lemmas\<close> |
|
658 |
|
659 lemma complex_of_real_code[code_unfold]: "of_real = (\<lambda>x. Complex x 0)" |
|
660 by (intro ext, auto simp: complex_eq_iff) |
657 |
661 |
658 lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0" |
662 lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0" |
659 by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff) |
663 by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff) |
660 |
664 |
661 lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0" |
665 lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0" |