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 $f(x + 2p) = f(x)$
for free.
@{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],
simp_all add: plus_1 periodic_arg_plus_0 plus_1_eq)
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 add: minus_1_eq minus_eq[symmetric])
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"
by (simp add: sin_add)
lemma cos_plus_pi: "cos ((z :: 'a :: {real_normed_field,banach}) + of_real pi) = - cos z"
by (simp add: cos_add)
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)
end