# Theory HOL-Library.Periodic_Fun

```(*  Title:    HOL/Library/Periodic_Fun.thy
Author:   Manuel Eberl, TU München
*)

section ‹Periodic Functions›

theory Periodic_Fun
imports Complex_Main
begin

text ‹
A locale for periodic functions. The idea is that one proves \$f(x + p) = f(x)\$
for some period \$p\$ and gets derived results like \$f(x - p) = f(x)\$ and \$f(x + 2p) = f(x)\$

\<^term>‹g› and \<^term>‹gm› are ``plus/minus k periods'' functions.
\<^term>‹g1› and \<^term>‹gn1› are ``plus/minus one period'' functions.
This is useful e.g. if the period is one; the lemmas one gets are then
\<^term>‹f (x + 1) = f x› instead of \<^term>‹f (x + 1 * 1) = f x› etc.
›
locale periodic_fun =
fixes f :: "('a :: {ring_1}) ⇒ 'b" and g gm :: "'a ⇒ 'a ⇒ 'a" and g1 gn1 :: "'a ⇒ 'a"
assumes plus_1: "f (g1 x) = f x"
assumes periodic_arg_plus_0: "g x 0 = x"
assumes periodic_arg_plus_distrib: "g x (of_int (m + n)) = g (g x (of_int n)) (of_int m)"
assumes plus_1_eq: "g x 1 = g1 x" and minus_1_eq: "g x (-1) = gn1 x"
and minus_eq: "g x (-y) = gm x y"
begin

lemma plus_of_nat: "f (g x (of_nat n)) = f x"
by (induction n) (insert periodic_arg_plus_distrib[of _ 1 "int n" for n],

lemma minus_of_nat: "f (gm x (of_nat n)) = f x"
proof -
have "f (g x (- of_nat n)) = f (g (g x (- of_nat n)) (of_nat n))"
by (rule plus_of_nat[symmetric])
also have "… = f (g (g x (of_int (- of_nat n))) (of_int (of_nat n)))" by simp
also have "… = f x"
by (subst periodic_arg_plus_distrib [symmetric]) (simp add: periodic_arg_plus_0)
finally show ?thesis by (simp add: minus_eq)
qed

lemma plus_of_int: "f (g x (of_int n)) = f x"
by (induction n) (simp_all add: plus_of_nat minus_of_nat minus_eq del: of_nat_Suc)

lemma minus_of_int: "f (gm x (of_int n)) = f x"
using plus_of_int[of x "of_int (-n)"] by (simp add: minus_eq)

lemma plus_numeral: "f (g x (numeral n)) = f x"
by (subst of_nat_numeral[symmetric], subst plus_of_nat) (rule refl)

lemma minus_numeral: "f (gm x (numeral n)) = f x"
by (subst of_nat_numeral[symmetric], subst minus_of_nat) (rule refl)

lemma minus_1: "f (gn1 x) = f x"
using minus_of_nat[of x 1] by (simp flip: minus_1_eq minus_eq)

lemmas periodic_simps = plus_of_nat minus_of_nat plus_of_int minus_of_int
plus_numeral minus_numeral plus_1 minus_1

end

text ‹
Specialised case of the \<^term>‹periodic_fun› locale for periods that are not 1.
Gives lemmas \<^term>‹f (x - period) = f x› etc.
›
locale periodic_fun_simple =
fixes f :: "('a :: {ring_1}) ⇒ 'b" and period :: 'a
assumes plus_period: "f (x + period) = f x"
begin
sublocale periodic_fun f "λz x. z + x * period" "λz x. z - x * period"
"λz. z + period" "λz. z - period"
by standard (simp_all add: ring_distribs plus_period)
end

text ‹
Specialised case of the \<^term>‹periodic_fun› locale for period 1.
Gives lemmas \<^term>‹f (x - 1) = f x› etc.
›
locale periodic_fun_simple' =
fixes f :: "('a :: {ring_1}) ⇒ 'b"
assumes plus_period: "f (x + 1) = f x"
begin
sublocale periodic_fun f "λz x. z + x" "λz x. z - x" "λz. z + 1" "λz. z - 1"
by standard (simp_all add: ring_distribs plus_period)

lemma of_nat: "f (of_nat n) = f 0" using plus_of_nat[of 0 n] by simp
lemma uminus_of_nat: "f (-of_nat n) = f 0" using minus_of_nat[of 0 n] by simp
lemma of_int: "f (of_int n) = f 0" using plus_of_int[of 0 n] by simp
lemma uminus_of_int: "f (-of_int n) = f 0" using minus_of_int[of 0 n] by simp
lemma of_numeral: "f (numeral n) = f 0" using plus_numeral[of 0 n] by simp
lemma of_neg_numeral: "f (-numeral n) = f 0" using minus_numeral[of 0 n] by simp
lemma of_1: "f 1 = f 0" using plus_of_nat[of 0 1] by simp
lemma of_neg_1: "f (-1) = f 0" using minus_of_nat[of 0 1] by simp

lemmas periodic_simps' =
of_nat uminus_of_nat of_int uminus_of_int of_numeral of_neg_numeral of_1 of_neg_1

end

lemma sin_plus_pi: "sin ((z :: 'a :: {real_normed_field,banach}) + of_real pi) = - sin z"

lemma cos_plus_pi: "cos ((z :: 'a :: {real_normed_field,banach}) + of_real pi) = - cos z"

interpretation sin: periodic_fun_simple sin "2 * of_real pi :: 'a :: {real_normed_field,banach}"
proof
fix z :: 'a
have "sin (z + 2 * of_real pi) = sin (z + of_real pi + of_real pi)" by (simp add: ac_simps)
also have "… = sin z" by (simp only: sin_plus_pi) simp
finally show "sin (z + 2 * of_real pi) = sin z" .
qed

interpretation cos: periodic_fun_simple cos "2 * of_real pi :: 'a :: {real_normed_field,banach}"
proof
fix z :: 'a
have "cos (z + 2 * of_real pi) = cos (z + of_real pi + of_real pi)" by (simp add: ac_simps)
also have "… = cos z" by (simp only: cos_plus_pi) simp
finally show "cos (z + 2 * of_real pi) = cos z" .
qed

interpretation tan: periodic_fun_simple tan "2 * of_real pi :: 'a :: {real_normed_field,banach}"
by standard (simp only: tan_def [abs_def] sin.plus_1 cos.plus_1)

interpretation cot: periodic_fun_simple cot "2 * of_real pi :: 'a :: {real_normed_field,banach}"
by standard (simp only: cot_def [abs_def] sin.plus_1 cos.plus_1)

lemma cos_eq_neg_periodic_intro:
assumes "x - y = 2*(of_int k)*pi + pi ∨ x + y = 2*(of_int k)*pi + pi"
shows "cos x = - cos y" using assms
proof
assume "x - y = 2 * (of_int k) * pi + pi"
then show ?thesis
using cos.periodic_simps[of "y+pi"]
next
assume "x + y = 2 * real_of_int k * pi + pi "
then show ?thesis
using cos.periodic_simps[of "-y+pi"]
by (clarsimp simp add: algebra_simps) (smt (verit))
qed

lemma cos_eq_periodic_intro:
assumes "x - y = 2*(of_int k)*pi ∨ x + y = 2*(of_int k)*pi"
shows "cos x = cos y"
by (smt (verit, best) assms cos_eq_neg_periodic_intro cos_minus_pi cos_periodic_pi)

lemma cos_eq_arccos_Ex:
"cos x = y ⟷ -1≤y ∧ y≤1 ∧ (∃k::int. x = arccos y + 2*k*pi ∨ x = - arccos y + 2*k*pi)" (is "?L=?R")
proof
assume ?R then show "cos x = y"
by (metis cos.plus_of_int cos_arccos cos_minus id_apply mult.assoc mult.left_commute of_real_eq_id)
next
assume L: ?L
let ?goal = "(∃k::int. x = arccos y + 2*k*pi ∨ x = - arccos y + 2*k*pi)"
obtain k::int where k: "-pi < x - k*(2*pi)" "x - k*(2*pi) ≤ pi"
using ceiling_divide_lower [of "2*pi" "x-pi"] ceiling_divide_upper [of "2*pi" "x-pi"]
by (simp add: divide_simps algebra_simps) (metis mult.commute)
have *: "cos (x - k * 2*pi) = y"
using cos.periodic_simps(3)[of x "-k"] L by (auto simp add:field_simps)
then have **: ?goal when "x-k*2*pi ≥ 0"
using arccos_cos k that by force
then show "-1≤y ∧ y≤1 ∧ ?goal"
using "*" arccos_cos2 k(1) by force
qed

end
```