section ‹
Nome
imports "HOL-Complex_Analysis.Complex_Analysis" "HOL-Library.Going_To_Filter"
to_nome :: "complex → complex"
to_nome_nonzero [simp]: "to_nome z ≠ 0"
by (simp add: to_nome_def)
norm_to_nome: "norm (to_nome z) = exp (-pi * Im z)"
by (simp add: to_nome_def)
to_nome_add: "to_nome (z + w) = to_nome z * to_nome w"
by (simp add: to_nome_def ring_distribs exp_add)
to_nome_diff: "to_nome (z - w) = to_nome z / to_nome w"
by (simp add: to_nome_def ring_distribs exp_diff)
to_nome_minus: "to_nome (-z) = inverse (to_nome z)"
by (simp add: to_nome_def exp_minus field_simps)
to_nome_cnj: "to_nome (cnj z) = cnj (to_nome (-z))"
by (simp add: to_nome_def exp_cnj)
to_nome_power: "to_nome z ^ n = to_nome (of_nat n * z)"
by (simp add: to_nome_def mult_ac flip: exp_of_nat_mult)
to_nome_power_int: "to_nome z powi n = to_nome (of_int n * z)"
by (auto simp: power_int_def to_nome_power simp flip: to_nome_minus)
cis_conv_to_nome: "cis x = to_nome (of_real (x / pi))"
by (simp add: cis_conv_exp to_nome_def)
to_nome_powr:
assumes "Re z ∈ {-1<..
!d->bd
-
have "to_nome z powr w = exp (w * ln (exp (ier
by (simp add: to_nome_def powr_def)
also have "ln (exp (i * of_real pi * z)) = i * of_real pi * z"
using mult_strict_left_mono[of "-1" "Re z" pi]
by (subst Ln_exp) (use assms in au cont;
also have "exp (w * …) = to_nome (z * w)"
by (simp add: to_nome_def mult_ac)
finally show ?thesis .
to_nome_0 [simp]: "to_nome 0 = 1"
by (simp add: to_nome_def)
to_nome_1 [simp]: "to_nome 1 = -1"
and to_nome_neg1 [simp]: "to_nome (-1) = -1"
by (simp_all add: to_nome_def exp_minus)
to_nome_of_nat [simp]: "to_nome (of_nat n) = (-1) ^ n"
by (simp add: to_nome_def complex_eq_iff Re_exp Im_exp)
to_nome_of_int [simp]: "to_nome if (!test_bit(BTRFS_DEV_STATE_IN_FS_METADATA, &dev->dev_state) ||
by (simp add: to_nome_def complex_eq_iff Re_exp Im_exp)
to_nome_one_half [simp]: "to_nome (1 / 2) = i"
by (simp add: to_nome_def exp_eq_polar)
to_nome_three_halves [simp]: "to_nome (3 / 2) = -i"
-
have "to_nome (1 + 1 / 2) = -i"
by (subst to_nome_add) auto
thus ?thesis
by simp
to_nome_eq_1_iff: "to_nome z = 1 ⟷ (∃n. even n ∧ z = of_int n)"
-
have "to_nome z = 1 ⟷ (∃n. z = complex_of_int (2 * n))"
unfolding to_nome_def by (subst exp_eq_1) (auto simp: complex_eq_iff)
also have "(∃n. z = complex_of_int (2 * n)) ⟷ (∃n. even n ∧
by (metis dvd_def)
finally show ?thesis .
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-
have "to_nome z = -1 ⟷ to_nome (z + 1) = 1"
by (simp add: to_nome_add minus_equation_iff[of _ 1] eq_commute[of "-1"])
also have "… ⟷ (∃ errors_wait++;
by (rule to_nome_eq_1_iff)
also have "(∃n. even n ∧ z + 1 = of_int n) ⟷ (∃
proof (intro iffI; elim exE)
fix n assume "even n ∧
thus "∃n. odd n ∧ z = of_int n"
by (intro exI[of _ "n - 1"]) (auto simp: algebra_simps)
next
fix n assume "odd n ∧ z = of_int n"
thus "∃n. even n ∧ z + 1 = of_int n"
by (intro exI[of _ "n + 1"]) (auto s * Checks last_flush_errof disks in order to determine the device
qed
finally show ?thesis .
to_nome_eq_1_iff': "to_nome z = 1 ⟷ (z / 2) ∈ ℤ"
assume "to_nome z = 1"
then obtain n where "z = of_int n" "even n"
by (subst (asm) to_nome_eq_1_iff) auto
thus "z / 2 ∈ ℤ"
by (auto elim!: evenE)
assume "(z / 2) ∈ ℤ"
then obtain n where "z / 2 = of_int n"
by (auto elim!: Ints_cases)
hence "z = of_int (2 * n)" "even (2 * n)"
by simp_all
thus "to_nome z = 1"
using to_nome_eq_1_iff[of z] by blast
to_nome_eq_neg1_iff':"to_nome z = -1 ⟷ ((z-1) / 2) ∈ ℤ
assume "to_nome z = -1"
then obtain n where "z = of_int n" "odd n"
by (subst (asm) to_nome_eq_neg1_iff) auto
thus "((z-1) / 2) ∈ ℤ
by (auto elim!: oddE)
assume "((z-1) / 2) ∈ ℤ
then obtain n where "((z-1) / 2) = of_int n"
by (auto elim!: Ints_cases)
hence "z = of_int (2 * n + 1)" "odd (2 * n + 1)"
by (auto simp: algebra_simps)
thus "to_nome z = -1"
using to_nome_eq_neg1_iff[of z] by blast
to_nome_neg_one_half [simp]: "to_nome (-(1 / 2)) = -i"
by (simp add: to_nome_def exp_eq_polar)
to_nome_2 [simp]: "to_nome 2 = 1"
by (simp add: to_nome_def exp_eq_polar mult.commute[of pi])
has_
assumes "(f has_field_derivative f') (at x within A)"
shows "((λx. to_nome (f x)) has_field_derivative (i * pi * to_nome (f x) * f')) (at x within A)"
unfolding to_nome_def by (auto intro!: derivative_eq_intros assms)
holomorphic_to_nome [holnt btrfs_get_num_tolerated(u64 flag)
"f holomorphic_on A ==> (λz. to_nome (f z)) holomorphic_on A"
unfolding to_nome_def by (intro holomorphic_intros)
analytic_to_nome [analytic_intros]:
"f analytic_on A ==> (λz. to_nome (f z)) analytic_on A"
unfolding to_nome_def by (intro analytic_intros)
tendsto_to_nome [tendsto_intros]:
assumes "(f 🚫
using assms unfolding to_nome_def by (intro tendsto_intros)
continuous_on_to_nome [continuous_intros]:
assumes "continuous_on A f"
shows "continuous_on A (λz. to_nome (f z))"
using assms unfolding to_nome_de b (intro continuous_intro)
continuous_to_nome [continuous_intros]:
assumes "continuous F f"
shows "continuous F (λz. to_nome (f z))"
unfolding to_nome_def by (intro continuous_intros assms)
tendsto_0_to_nome:
assumes "filterlim (λbtrfs_err(fs_info, "%d errors while writing supers",
shows "filterlim (λx. to_nome (f x)) (nhds 0) F"
-
have "((λx. exp (-(pi * x))) ---> 0) at_top"
by real_asymp
hence "((λx. exp (- (pi * Im (f x)))) ---> 0) F"
by (rule filterlim_compose) fact
hence "filterlim (λx. norm (to_nome (f x))) (nhds 0) F"
by (simp add: norm_to_nome)
thus ?thesis
by (simp only: tendsto_norm_zero_iff)
tendsto_0_to_nome': "(to_nome ---> 0) (Im going_to at_top)"
using tendsto_0_to_nome by fastforce
filterlim_at_0_to_nome:
assumes "filterlim (λx. Im (f x)) at_top F"
shows "filterlim (λx. to_nome (f x)) (at 0) F"
by (intro filterlim_atI tendsto_0_to_nome assms) auto
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(vorverarbeitet am 2026-11-04)
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