# Theory Formal_Power_Series

theory Formal_Power_Series
imports Complex_Main Euclidean_Algorithm
```(*  Title:      HOL/Computational_Algebra/Formal_Power_Series.thy
Author:     Amine Chaieb, University of Cambridge
*)

section ‹A formalization of formal power series›

theory Formal_Power_Series
imports
Complex_Main
Euclidean_Algorithm
begin

subsection ‹The type of formal power series›

typedef 'a fps = "{f :: nat ⇒ 'a. True}"
morphisms fps_nth Abs_fps
by simp

notation fps_nth (infixl "\$" 75)

lemma expand_fps_eq: "p = q ⟷ (∀n. p \$ n = q \$ n)"
by (simp add: fps_nth_inject [symmetric] fun_eq_iff)

lemma fps_ext: "(⋀n. p \$ n = q \$ n) ⟹ p = q"

lemma fps_nth_Abs_fps [simp]: "Abs_fps f \$ n = f n"

text ‹Definition of the basic elements 0 and 1 and the basic operations of addition,
negation and multiplication.›

instantiation fps :: (zero) zero
begin
definition fps_zero_def: "0 = Abs_fps (λn. 0)"
instance ..
end

lemma fps_zero_nth [simp]: "0 \$ n = 0"
unfolding fps_zero_def by simp

instantiation fps :: ("{one, zero}") one
begin
definition fps_one_def: "1 = Abs_fps (λn. if n = 0 then 1 else 0)"
instance ..
end

lemma fps_one_nth [simp]: "1 \$ n = (if n = 0 then 1 else 0)"
unfolding fps_one_def by simp

instantiation fps :: (plus) plus
begin
definition fps_plus_def: "(+) = (λf g. Abs_fps (λn. f \$ n + g \$ n))"
instance ..
end

lemma fps_add_nth [simp]: "(f + g) \$ n = f \$ n + g \$ n"
unfolding fps_plus_def by simp

instantiation fps :: (minus) minus
begin
definition fps_minus_def: "(-) = (λf g. Abs_fps (λn. f \$ n - g \$ n))"
instance ..
end

lemma fps_sub_nth [simp]: "(f - g) \$ n = f \$ n - g \$ n"
unfolding fps_minus_def by simp

instantiation fps :: (uminus) uminus
begin
definition fps_uminus_def: "uminus = (λf. Abs_fps (λn. - (f \$ n)))"
instance ..
end

lemma fps_neg_nth [simp]: "(- f) \$ n = - (f \$ n)"
unfolding fps_uminus_def by simp

instantiation fps :: ("{comm_monoid_add, times}") times
begin
definition fps_times_def: "( * ) = (λf g. Abs_fps (λn. ∑i=0..n. f \$ i * g \$ (n - i)))"
instance ..
end

lemma fps_mult_nth: "(f * g) \$ n = (∑i=0..n. f\$i * g\$(n - i))"
unfolding fps_times_def by simp

lemma fps_mult_nth_0 [simp]: "(f * g) \$ 0 = f \$ 0 * g \$ 0"
unfolding fps_times_def by simp

declare atLeastAtMost_iff [presburger]
declare Bex_def [presburger]
declare Ball_def [presburger]

lemma mult_delta_left:
fixes x y :: "'a::mult_zero"
shows "(if b then x else 0) * y = (if b then x * y else 0)"
by simp

lemma mult_delta_right:
fixes x y :: "'a::mult_zero"
shows "x * (if b then y else 0) = (if b then x * y else 0)"
by simp

lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
by auto

subsection ‹Formal power series form a commutative ring with unity, if the range of sequences
they represent is a commutative ring with unity›

proof
fix a b c :: "'a fps"
show "a + b + c = a + (b + c)"
qed

proof
fix a b :: "'a fps"
show "a + b = b + a"
qed

lemma fps_mult_assoc_lemma:
fixes k :: nat
and f :: "nat ⇒ nat ⇒ nat ⇒ 'a::comm_monoid_add"
shows "(∑j=0..k. ∑i=0..j. f i (j - i) (n - j)) =
(∑j=0..k. ∑i=0..k - j. f j i (n - j - i))"

instance fps :: (semiring_0) semigroup_mult
proof
fix a b c :: "'a fps"
show "(a * b) * c = a * (b * c)"
proof (rule fps_ext)
fix n :: nat
have "(∑j=0..n. ∑i=0..j. a\$i * b\$(j - i) * c\$(n - j)) =
(∑j=0..n. ∑i=0..n - j. a\$j * b\$i * c\$(n - j - i))"
by (rule fps_mult_assoc_lemma)
then show "((a * b) * c) \$ n = (a * (b * c)) \$ n"
by (simp add: fps_mult_nth sum_distrib_left sum_distrib_right mult.assoc)
qed
qed

lemma fps_mult_commute_lemma:
fixes n :: nat
and f :: "nat ⇒ nat ⇒ 'a::comm_monoid_add"
shows "(∑i=0..n. f i (n - i)) = (∑i=0..n. f (n - i) i)"
by (rule sum.reindex_bij_witness[where i="(-) n" and j="(-) n"]) auto

instance fps :: (comm_semiring_0) ab_semigroup_mult
proof
fix a b :: "'a fps"
show "a * b = b * a"
proof (rule fps_ext)
fix n :: nat
have "(∑i=0..n. a\$i * b\$(n - i)) = (∑i=0..n. a\$(n - i) * b\$i)"
by (rule fps_mult_commute_lemma)
then show "(a * b) \$ n = (b * a) \$ n"
qed
qed

proof
fix a :: "'a fps"
show "0 + a = a" by (simp add: fps_ext)
show "a + 0 = a" by (simp add: fps_ext)
qed

proof
fix a :: "'a fps"
show "0 + a = a" by (simp add: fps_ext)
qed

instance fps :: (semiring_1) monoid_mult
proof
fix a :: "'a fps"
show "1 * a = a"
by (simp add: fps_ext fps_mult_nth mult_delta_left sum.delta)
show "a * 1 = a"
by (simp add: fps_ext fps_mult_nth mult_delta_right sum.delta')
qed

proof
fix a b c :: "'a fps"
show "b = c" if "a + b = a + c"
using that by (simp add: expand_fps_eq)
show "b = c" if "b + a = c + a"
using that by (simp add: expand_fps_eq)
qed

proof
fix a b c :: "'a fps"
show "a + b - a = b"
show "a - b - c = a - (b + c)"
qed

proof
fix a b :: "'a fps"
show "- a + a = 0" by (simp add: fps_ext)
show "a + - b = a - b" by (simp add: fps_ext)
qed

proof
fix a b :: "'a fps"
show "- a + a = 0" by (simp add: fps_ext)
show "a - b = a + - b" by (simp add: fps_ext)
qed

instance fps :: (zero_neq_one) zero_neq_one

instance fps :: (semiring_0) semiring
proof
fix a b c :: "'a fps"
show "(a + b) * c = a * c + b * c"
by (simp add: expand_fps_eq fps_mult_nth distrib_right sum.distrib)
show "a * (b + c) = a * b + a * c"
by (simp add: expand_fps_eq fps_mult_nth distrib_left sum.distrib)
qed

instance fps :: (semiring_0) semiring_0
proof
fix a :: "'a fps"
show "0 * a = 0"
show "a * 0 = 0"
qed

instance fps :: (semiring_0_cancel) semiring_0_cancel ..

instance fps :: (semiring_1) semiring_1 ..

subsection ‹Selection of the nth power of the implicit variable in the infinite sum›

lemma fps_square_nth: "(f^2) \$ n = (∑k≤n. f \$ k * f \$ (n - k))"
by (simp add: power2_eq_square fps_mult_nth atLeast0AtMost)

lemma fps_nonzero_nth: "f ≠ 0 ⟷ (∃ n. f \$n ≠ 0)"

lemma fps_nonzero_nth_minimal: "f ≠ 0 ⟷ (∃n. f \$ n ≠ 0 ∧ (∀m < n. f \$ m = 0))"
(is "?lhs ⟷ ?rhs")
proof
let ?n = "LEAST n. f \$ n ≠ 0"
show ?rhs if ?lhs
proof -
from that have "∃n. f \$ n ≠ 0"
then have "f \$ ?n ≠ 0"
by (rule LeastI_ex)
moreover have "∀m<?n. f \$ m = 0"
by (auto dest: not_less_Least)
ultimately have "f \$ ?n ≠ 0 ∧ (∀m<?n. f \$ m = 0)" ..
then show ?thesis ..
qed
show ?lhs if ?rhs
using that by (auto simp add: expand_fps_eq)
qed

lemma fps_eq_iff: "f = g ⟷ (∀n. f \$ n = g \$n)"
by (rule expand_fps_eq)

lemma fps_sum_nth: "sum f S \$ n = sum (λk. (f k) \$ n) S"
proof (cases "finite S")
case True
then show ?thesis by (induct set: finite) auto
next
case False
then show ?thesis by simp
qed

subsection ‹Injection of the basic ring elements and multiplication by scalars›

definition "fps_const c = Abs_fps (λn. if n = 0 then c else 0)"

lemma fps_nth_fps_const [simp]: "fps_const c \$ n = (if n = 0 then c else 0)"
unfolding fps_const_def by simp

lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0"

lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1"

lemma fps_const_neg [simp]: "- (fps_const (c::'a::ring)) = fps_const (- c)"

lemma fps_const_add [simp]: "fps_const (c::'a::monoid_add) + fps_const d = fps_const (c + d)"

lemma fps_const_sub [simp]: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"

lemma fps_const_mult[simp]: "fps_const (c::'a::ring) * fps_const d = fps_const (c * d)"
by (simp add: fps_eq_iff fps_mult_nth sum.neutral)

Abs_fps (λn. if n = 0 then c + f\$0 else f\$n)"

Abs_fps (λn. if n = 0 then f\$0 + c else f\$n)"

lemma fps_const_mult_left: "fps_const (c::'a::semiring_0) * f = Abs_fps (λn. c * f\$n)"
unfolding fps_eq_iff fps_mult_nth
by (simp add: fps_const_def mult_delta_left sum.delta)

lemma fps_const_mult_right: "f * fps_const (c::'a::semiring_0) = Abs_fps (λn. f\$n * c)"
unfolding fps_eq_iff fps_mult_nth
by (simp add: fps_const_def mult_delta_right sum.delta')

lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)\$n = c* f\$n"
by (simp add: fps_mult_nth mult_delta_left sum.delta)

lemma fps_mult_right_const_nth [simp]: "(f * fps_const (c::'a::semiring_1))\$n = f\$n * c"
by (simp add: fps_mult_nth mult_delta_right sum.delta')

subsection ‹Formal power series form an integral domain›

instance fps :: (ring) ring ..

instance fps :: (ring_1) ring_1
by (intro_classes, auto simp add: distrib_right)

instance fps :: (comm_ring_1) comm_ring_1
by (intro_classes, auto simp add: distrib_right)

instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors
proof
fix a b :: "'a fps"
assume "a ≠ 0" and "b ≠ 0"
then obtain i j where i: "a \$ i ≠ 0" "∀k<i. a \$ k = 0" and j: "b \$ j ≠ 0" "∀k<j. b \$ k =0"
unfolding fps_nonzero_nth_minimal
by blast+
have "(a * b) \$ (i + j) = (∑k=0..i+j. a \$ k * b \$ (i + j - k))"
by (rule fps_mult_nth)
also have "… = (a \$ i * b \$ (i + j - i)) + (∑k∈{0..i+j} - {i}. a \$ k * b \$ (i + j - k))"
by (rule sum.remove) simp_all
also have "(∑k∈{0..i+j}-{i}. a \$ k * b \$ (i + j - k)) = 0"
proof (rule sum.neutral [rule_format])
fix k assume "k ∈ {0..i+j} - {i}"
then have "k < i ∨ i+j-k < j"
by auto
then show "a \$ k * b \$ (i + j - k) = 0"
using i j by auto
qed
also have "a \$ i * b \$ (i + j - i) + 0 = a \$ i * b \$ j"
by simp
also have "a \$ i * b \$ j ≠ 0"
using i j by simp
finally have "(a*b) \$ (i+j) ≠ 0" .
then show "a * b ≠ 0"
unfolding fps_nonzero_nth by blast
qed

instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors ..

instance fps :: (idom) idom ..

lemma numeral_fps_const: "numeral k = fps_const (numeral k)"
by (induct k) (simp_all only: numeral.simps fps_const_1_eq_1

lemma neg_numeral_fps_const:
"(- numeral k :: 'a :: ring_1 fps) = fps_const (- numeral k)"

lemma fps_numeral_nth: "numeral n \$ i = (if i = 0 then numeral n else 0)"

lemma fps_numeral_nth_0 [simp]: "numeral n \$ 0 = numeral n"

lemma fps_of_nat: "fps_const (of_nat c) = of_nat c"

lemma numeral_neq_fps_zero [simp]: "(numeral f :: 'a :: field_char_0 fps) ≠ 0"
proof
assume "numeral f = (0 :: 'a fps)"
from arg_cong[of _ _ "λF. F \$ 0", OF this] show False by simp
qed

subsection ‹The efps_Xtractor series fps_X›

lemma minus_one_power_iff: "(- (1::'a::comm_ring_1)) ^ n = (if even n then 1 else - 1)"
by (induct n) auto

definition "fps_X = Abs_fps (λn. if n = 1 then 1 else 0)"

lemma fps_X_mult_nth [simp]:
"(fps_X * (f :: 'a::semiring_1 fps)) \$n = (if n = 0 then 0 else f \$ (n - 1))"
proof (cases "n = 0")
case False
have "(fps_X * f) \$n = (∑i = 0..n. fps_X \$ i * f \$ (n - i))"
also have "… = f \$ (n - 1)"
using False by (simp add: fps_X_def mult_delta_left sum.delta)
finally show ?thesis
using False by simp
next
case True
then show ?thesis
qed

lemma fps_X_mult_right_nth[simp]:
"((a::'a::semiring_1 fps) * fps_X) \$ n = (if n = 0 then 0 else a \$ (n - 1))"
proof -
have "(a * fps_X) \$ n = (∑i = 0..n. a \$ i * (if n - i = Suc 0 then 1 else 0))"
also have "… = (∑i = 0..n. if i = n - 1 then if n = 0 then 0 else a \$ i else 0)"
by (intro sum.cong) auto
also have "… = (if n = 0 then 0 else a \$ (n - 1))" by (simp add: sum.delta)
finally show ?thesis .
qed

lemma fps_mult_fps_X_commute: "fps_X * (a :: 'a :: semiring_1 fps) = a * fps_X"

lemma fps_X_power_iff: "fps_X ^ n = Abs_fps (λm. if m = n then 1 else 0)"
by (induction n) (auto simp: fps_eq_iff)

lemma fps_X_nth[simp]: "fps_X\$n = (if n = 1 then 1 else 0)"

lemma fps_X_power_nth[simp]: "(fps_X^k) \$n = (if n = k then 1 else 0::'a::comm_ring_1)"

lemma fps_X_power_mult_nth: "(fps_X^k * (f :: 'a::comm_ring_1 fps)) \$n = (if n < k then 0 else f \$ (n - k))"
apply (induct k arbitrary: n)
apply simp
unfolding power_Suc mult.assoc
apply (case_tac n)
apply auto
done

lemma fps_X_power_mult_right_nth:
"((f :: 'a::comm_ring_1 fps) * fps_X^k) \$n = (if n < k then 0 else f \$ (n - k))"
by (metis fps_X_power_mult_nth mult.commute)

lemma fps_X_neq_fps_const [simp]: "(fps_X :: 'a :: zero_neq_one fps) ≠ fps_const c"
proof
assume "(fps_X::'a fps) = fps_const (c::'a)"
hence "fps_X\$1 = (fps_const (c::'a))\$1" by (simp only:)
thus False by auto
qed

lemma fps_X_neq_zero [simp]: "(fps_X :: 'a :: zero_neq_one fps) ≠ 0"
by (simp only: fps_const_0_eq_0[symmetric] fps_X_neq_fps_const) simp

lemma fps_X_neq_one [simp]: "(fps_X :: 'a :: zero_neq_one fps) ≠ 1"
by (simp only: fps_const_1_eq_1[symmetric] fps_X_neq_fps_const) simp

lemma fps_X_neq_numeral [simp]: "(fps_X :: 'a :: {semiring_1,zero_neq_one} fps) ≠ numeral c"
by (simp only: numeral_fps_const fps_X_neq_fps_const) simp

lemma fps_X_pow_eq_fps_X_pow_iff [simp]:
"(fps_X :: ('a :: {comm_ring_1}) fps) ^ m = fps_X ^ n ⟷ m = n"
proof
assume "(fps_X :: 'a fps) ^ m = fps_X ^ n"
hence "(fps_X :: 'a fps) ^ m \$ m = fps_X ^ n \$ m" by (simp only:)
thus "m = n" by (simp split: if_split_asm)
qed simp_all

subsection ‹Subdegrees›

definition subdegree :: "('a::zero) fps ⇒ nat" where
"subdegree f = (if f = 0 then 0 else LEAST n. f\$n ≠ 0)"

lemma subdegreeI:
assumes "f \$ d ≠ 0" and "⋀i. i < d ⟹ f \$ i = 0"
shows   "subdegree f = d"
proof-
from assms(1) have "f ≠ 0" by auto
moreover from assms(1) have "(LEAST i. f \$ i ≠ 0) = d"
proof (rule Least_equality)
fix e assume "f \$ e ≠ 0"
with assms(2) have "¬(e < d)" by blast
thus "e ≥ d" by simp
qed
ultimately show ?thesis unfolding subdegree_def by simp
qed

lemma nth_subdegree_nonzero [simp,intro]: "f ≠ 0 ⟹ f \$ subdegree f ≠ 0"
proof-
assume "f ≠ 0"
hence "subdegree f = (LEAST n. f \$ n ≠ 0)" by (simp add: subdegree_def)
also from ‹f ≠ 0› have "∃n. f\$n ≠ 0" using fps_nonzero_nth by blast
from LeastI_ex[OF this] have "f \$ (LEAST n. f \$ n ≠ 0) ≠ 0" .
finally show ?thesis .
qed

lemma nth_less_subdegree_zero [dest]: "n < subdegree f ⟹ f \$ n = 0"
proof (cases "f = 0")
assume "f ≠ 0" and less: "n < subdegree f"
note less
also from ‹f ≠ 0› have "subdegree f = (LEAST n. f \$ n ≠ 0)" by (simp add: subdegree_def)
finally show "f \$ n = 0" using not_less_Least by blast
qed simp_all

lemma subdegree_geI:
assumes "f ≠ 0" "⋀i. i < n ⟹ f\$i = 0"
shows   "subdegree f ≥ n"
proof (rule ccontr)
assume "¬(subdegree f ≥ n)"
with assms(2) have "f \$ subdegree f = 0" by simp
moreover from assms(1) have "f \$ subdegree f ≠ 0" by simp
qed

lemma subdegree_greaterI:
assumes "f ≠ 0" "⋀i. i ≤ n ⟹ f\$i = 0"
shows   "subdegree f > n"
proof (rule ccontr)
assume "¬(subdegree f > n)"
with assms(2) have "f \$ subdegree f = 0" by simp
moreover from assms(1) have "f \$ subdegree f ≠ 0" by simp
qed

lemma subdegree_leI:
"f \$ n ≠ 0 ⟹ subdegree f ≤ n"
by (rule leI) auto

lemma subdegree_0 [simp]: "subdegree 0 = 0"

lemma subdegree_1 [simp]: "subdegree (1 :: ('a :: zero_neq_one) fps) = 0"
by (auto intro!: subdegreeI)

lemma subdegree_fps_X [simp]: "subdegree (fps_X :: ('a :: zero_neq_one) fps) = 1"
by (auto intro!: subdegreeI simp: fps_X_def)

lemma subdegree_fps_const [simp]: "subdegree (fps_const c) = 0"
by (cases "c = 0") (auto intro!: subdegreeI)

lemma subdegree_numeral [simp]: "subdegree (numeral n) = 0"

lemma subdegree_eq_0_iff: "subdegree f = 0 ⟷ f = 0 ∨ f \$ 0 ≠ 0"
proof (cases "f = 0")
assume "f ≠ 0"
thus ?thesis
using nth_subdegree_nonzero[OF ‹f ≠ 0›] by (fastforce intro!: subdegreeI)
qed simp_all

lemma subdegree_eq_0 [simp]: "f \$ 0 ≠ 0 ⟹ subdegree f = 0"

lemma nth_subdegree_mult [simp]:
fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps"
shows "(f * g) \$ (subdegree f + subdegree g) = f \$ subdegree f * g \$ subdegree g"
proof-
let ?n = "subdegree f + subdegree g"
have "(f * g) \$ ?n = (∑i=0..?n. f\$i * g\$(?n-i))"
also have "... = (∑i=0..?n. if i = subdegree f then f\$i * g\$(?n-i) else 0)"
proof (intro sum.cong)
fix x assume x: "x ∈ {0..?n}"
hence "x = subdegree f ∨ x < subdegree f ∨ ?n - x < subdegree g" by auto
thus "f \$ x * g \$ (?n - x) = (if x = subdegree f then f \$ x * g \$ (?n - x) else 0)"
by (elim disjE conjE) auto
qed auto
also have "... = f \$ subdegree f * g \$ subdegree g" by (simp add: sum.delta)
finally show ?thesis .
qed

lemma subdegree_mult [simp]:
assumes "f ≠ 0" "g ≠ 0"
shows "subdegree ((f :: ('a :: {ring_no_zero_divisors}) fps) * g) = subdegree f + subdegree g"
proof (rule subdegreeI)
let ?n = "subdegree f + subdegree g"
have "(f * g) \$ ?n = (∑i=0..?n. f\$i * g\$(?n-i))" by (simp add: fps_mult_nth)
also have "... = (∑i=0..?n. if i = subdegree f then f\$i * g\$(?n-i) else 0)"
proof (intro sum.cong)
fix x assume x: "x ∈ {0..?n}"
hence "x = subdegree f ∨ x < subdegree f ∨ ?n - x < subdegree g" by auto
thus "f \$ x * g \$ (?n - x) = (if x = subdegree f then f \$ x * g \$ (?n - x) else 0)"
by (elim disjE conjE) auto
qed auto
also have "... = f \$ subdegree f * g \$ subdegree g" by (simp add: sum.delta)
also from assms have "... ≠ 0" by auto
finally show "(f * g) \$ (subdegree f + subdegree g) ≠ 0" .
next
fix m assume m: "m < subdegree f + subdegree g"
have "(f * g) \$ m = (∑i=0..m. f\$i * g\$(m-i))" by (simp add: fps_mult_nth)
also have "... = (∑i=0..m. 0)"
proof (rule sum.cong)
fix i assume "i ∈ {0..m}"
with m have "i < subdegree f ∨ m - i < subdegree g" by auto
thus "f\$i * g\$(m-i) = 0" by (elim disjE) auto
qed auto
finally show "(f * g) \$ m = 0" by simp
qed

lemma subdegree_power [simp]:
"subdegree ((f :: ('a :: ring_1_no_zero_divisors) fps) ^ n) = n * subdegree f"
by (cases "f = 0"; induction n) simp_all

lemma subdegree_uminus [simp]:
"subdegree (-(f::('a::group_add) fps)) = subdegree f"

lemma subdegree_minus_commute [simp]:
"subdegree (f-(g::('a::group_add) fps)) = subdegree (g - f)"
proof -
have "f - g = -(g - f)" by simp
also have "subdegree ... = subdegree (g - f)" by (simp only: subdegree_uminus)
finally show ?thesis .
qed

assumes "f ≠ -(g :: ('a :: {group_add}) fps)"
shows   "subdegree (f + g) ≥ min (subdegree f) (subdegree g)"
proof (rule subdegree_geI)
from assms show "f + g ≠ 0" by (subst (asm) eq_neg_iff_add_eq_0)
next
fix i assume "i < min (subdegree f) (subdegree g)"
hence "f \$ i = 0" and "g \$ i = 0" by auto
thus "(f + g) \$ i = 0" by force
qed

assumes "f ≠ 0"
assumes "subdegree f < subdegree (g :: ('a :: {group_add}) fps)"
shows   "subdegree (f + g) = subdegree f"
proof (rule antisym[OF subdegree_leI])
from assms show "subdegree (f + g) ≥ subdegree f"
by (intro order.trans[OF min.boundedI subdegree_add_ge]) auto
from assms have "f \$ subdegree f ≠ 0" "g \$ subdegree f = 0" by auto
thus "(f + g) \$ subdegree f ≠ 0" by simp
qed

assumes "g ≠ 0"
assumes "subdegree g < subdegree (f :: ('a :: {ab_group_add}) fps)"
shows   "subdegree (f + g) = subdegree g"

lemma subdegree_diff_eq1:
assumes "f ≠ 0"
assumes "subdegree f < subdegree (g :: ('a :: {ab_group_add}) fps)"
shows   "subdegree (f - g) = subdegree f"

lemma subdegree_diff_eq2:
assumes "g ≠ 0"
assumes "subdegree g < subdegree (f :: ('a :: {ab_group_add}) fps)"
shows   "subdegree (f - g) = subdegree g"

lemma subdegree_diff_ge [simp]:
assumes "f ≠ (g :: ('a :: {group_add}) fps)"
shows   "subdegree (f - g) ≥ min (subdegree f) (subdegree g)"
using assms subdegree_add_ge[of f "-g"] by simp

subsection ‹Shifting and slicing›

definition fps_shift :: "nat ⇒ 'a fps ⇒ 'a fps" where
"fps_shift n f = Abs_fps (λi. f \$ (i + n))"

lemma fps_shift_nth [simp]: "fps_shift n f \$ i = f \$ (i + n)"

lemma fps_shift_0 [simp]: "fps_shift 0 f = f"
by (intro fps_ext) (simp add: fps_shift_def)

lemma fps_shift_zero [simp]: "fps_shift n 0 = 0"
by (intro fps_ext) (simp add: fps_shift_def)

lemma fps_shift_one: "fps_shift n 1 = (if n = 0 then 1 else 0)"
by (intro fps_ext) (simp add: fps_shift_def)

lemma fps_shift_fps_const: "fps_shift n (fps_const c) = (if n = 0 then fps_const c else 0)"
by (intro fps_ext) (simp add: fps_shift_def)

lemma fps_shift_numeral: "fps_shift n (numeral c) = (if n = 0 then numeral c else 0)"

lemma fps_shift_fps_X_power [simp]:
"n ≤ m ⟹ fps_shift n (fps_X ^ m) = (fps_X ^ (m - n) ::'a::comm_ring_1 fps)"
by (intro fps_ext) (auto simp: fps_shift_def )

lemma fps_shift_times_fps_X_power:
"n ≤ subdegree f ⟹ fps_shift n f * fps_X ^ n = (f :: 'a :: comm_ring_1 fps)"
by (intro fps_ext) (auto simp: fps_X_power_mult_right_nth nth_less_subdegree_zero)

lemma fps_shift_times_fps_X_power' [simp]:
"fps_shift n (f * fps_X^n) = (f :: 'a :: comm_ring_1 fps)"
by (intro fps_ext) (auto simp: fps_X_power_mult_right_nth nth_less_subdegree_zero)

lemma fps_shift_times_fps_X_power'':
"m ≤ n ⟹ fps_shift n (f * fps_X^m) = fps_shift (n - m) (f :: 'a :: comm_ring_1 fps)"
by (intro fps_ext) (auto simp: fps_X_power_mult_right_nth nth_less_subdegree_zero)

lemma fps_shift_subdegree [simp]:
"n ≤ subdegree f ⟹ subdegree (fps_shift n f) = subdegree (f :: 'a :: comm_ring_1 fps) - n"
by (cases "f = 0") (force intro: nth_less_subdegree_zero subdegreeI)+

lemma subdegree_decompose:
"f = fps_shift (subdegree f) f * fps_X ^ subdegree (f :: ('a :: comm_ring_1) fps)"
by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth)

lemma subdegree_decompose':
"n ≤ subdegree (f :: ('a :: comm_ring_1) fps) ⟹ f = fps_shift n f * fps_X^n"
by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth intro!: nth_less_subdegree_zero)

lemma fps_shift_fps_shift:
"fps_shift (m + n) f = fps_shift m (fps_shift n f)"

"fps_shift n (f + g) = fps_shift n f + fps_shift n g"

lemma fps_shift_mult:
assumes "n ≤ subdegree (g :: 'b :: {comm_ring_1} fps)"
shows   "fps_shift n (h*g) = h * fps_shift n g"
proof -
from assms have "g = fps_shift n g * fps_X^n" by (rule subdegree_decompose')
also have "h * ... = (h * fps_shift n g) * fps_X^n" by simp
also have "fps_shift n ... = h * fps_shift n g" by simp
finally show ?thesis .
qed

lemma fps_shift_mult_right:
assumes "n ≤ subdegree (g :: 'b :: {comm_ring_1} fps)"
shows   "fps_shift n (g*h) = h * fps_shift n g"
by (subst mult.commute, subst fps_shift_mult) (simp_all add: assms)

lemma nth_subdegree_zero_iff [simp]: "f \$ subdegree f = 0 ⟷ f = 0"
by (cases "f = 0") auto

lemma fps_shift_subdegree_zero_iff [simp]:
"fps_shift (subdegree f) f = 0 ⟷ f = 0"
by (subst (1) nth_subdegree_zero_iff[symmetric], cases "f = 0")
(simp_all del: nth_subdegree_zero_iff)

definition "fps_cutoff n f = Abs_fps (λi. if i < n then f\$i else 0)"

lemma fps_cutoff_nth [simp]: "fps_cutoff n f \$ i = (if i < n then f\$i else 0)"
unfolding fps_cutoff_def by simp

lemma fps_cutoff_zero_iff: "fps_cutoff n f = 0 ⟷ (f = 0 ∨ n ≤ subdegree f)"
proof
assume A: "fps_cutoff n f = 0"
thus "f = 0 ∨ n ≤ subdegree f"
proof (cases "f = 0")
assume "f ≠ 0"
with A have "n ≤ subdegree f"
by (intro subdegree_geI) (auto simp: fps_eq_iff split: if_split_asm)
thus ?thesis ..
qed simp
qed (auto simp: fps_eq_iff intro: nth_less_subdegree_zero)

lemma fps_cutoff_0 [simp]: "fps_cutoff 0 f = 0"

lemma fps_cutoff_zero [simp]: "fps_cutoff n 0 = 0"

lemma fps_cutoff_one: "fps_cutoff n 1 = (if n = 0 then 0 else 1)"

lemma fps_cutoff_fps_const: "fps_cutoff n (fps_const c) = (if n = 0 then 0 else fps_const c)"

lemma fps_cutoff_numeral: "fps_cutoff n (numeral c) = (if n = 0 then 0 else numeral c)"

lemma fps_shift_cutoff:
"fps_shift n (f :: ('a :: comm_ring_1) fps) * fps_X^n + fps_cutoff n f = f"

subsection ‹Formal Power series form a metric space›

instantiation fps :: (comm_ring_1) dist
begin

definition
dist_fps_def: "dist (a :: 'a fps) b = (if a = b then 0 else inverse (2 ^ subdegree (a - b)))"

lemma dist_fps_ge0: "dist (a :: 'a fps) b ≥ 0"

lemma dist_fps_sym: "dist (a :: 'a fps) b = dist b a"

instance ..

end

instantiation fps :: (comm_ring_1) metric_space
begin

definition uniformity_fps_def [code del]:
"(uniformity :: ('a fps × 'a fps) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"

definition open_fps_def' [code del]:
"open (U :: 'a fps set) ⟷ (∀x∈U. eventually (λ(x', y). x' = x ⟶ y ∈ U) uniformity)"

instance
proof
show th: "dist a b = 0 ⟷ a = b" for a b :: "'a fps"
by (simp add: dist_fps_def split: if_split_asm)
then have th'[simp]: "dist a a = 0" for a :: "'a fps" by simp

fix a b c :: "'a fps"
consider "a = b" | "c = a ∨ c = b" | "a ≠ b" "a ≠ c" "b ≠ c" by blast
then show "dist a b ≤ dist a c + dist b c"
proof cases
case 1
then show ?thesis by (simp add: dist_fps_def)
next
case 2
then show ?thesis
by (cases "c = a") (simp_all add: th dist_fps_sym)
next
case neq: 3
have False if "dist a b > dist a c + dist b c"
proof -
let ?n = "subdegree (a - b)"
from neq have "dist a b > 0" "dist b c > 0" and "dist a c > 0" by (simp_all add: dist_fps_def)
with that have "dist a b > dist a c" and "dist a b > dist b c" by simp_all
with neq have "?n < subdegree (a - c)" and "?n < subdegree (b - c)"
hence "(a - c) \$ ?n = 0" and "(b - c) \$ ?n = 0"
by (simp_all only: nth_less_subdegree_zero)
hence "(a - b) \$ ?n = 0" by simp
moreover from neq have "(a - b) \$ ?n ≠ 0" by (intro nth_subdegree_nonzero) simp_all
qed
thus ?thesis by (auto simp add: not_le[symmetric])
qed
qed (rule open_fps_def' uniformity_fps_def)+

end

declare uniformity_Abort[where 'a="'a :: comm_ring_1 fps", code]

lemma open_fps_def: "open (S :: 'a::comm_ring_1 fps set) = (∀a ∈ S. ∃r. r >0 ∧ {y. dist y a < r} ⊆ S)"
unfolding open_dist subset_eq by simp

text ‹The infinite sums and justification of the notation in textbooks.›

lemma reals_power_lt_ex:
fixes x y :: real
assumes xp: "x > 0"
and y1: "y > 1"
shows "∃k>0. (1/y)^k < x"
proof -
have yp: "y > 0"
using y1 by simp
from reals_Archimedean2[of "max 0 (- log y x) + 1"]
obtain k :: nat where k: "real k > max 0 (- log y x) + 1"
by blast
from k have kp: "k > 0"
by simp
from k have "real k > - log y x"
by simp
then have "ln y * real k > - ln x"
unfolding log_def
using ln_gt_zero_iff[OF yp] y1
by (simp add: minus_divide_left field_simps del: minus_divide_left[symmetric])
then have "ln y * real k + ln x > 0"
by simp
then have "exp (real k * ln y + ln x) > exp 0"
then have "y ^ k * x > 1"
unfolding exp_zero exp_add exp_of_nat_mult exp_ln [OF xp] exp_ln [OF yp]
by simp
then have "x > (1 / y)^k" using yp
then show ?thesis
using kp by blast
qed

lemma fps_sum_rep_nth: "(sum (λi. fps_const(a\$i)*fps_X^i) {0..m})\$n =
(if n ≤ m then a\$n else 0::'a::comm_ring_1)"
by (auto simp add: fps_sum_nth cond_value_iff cong del: if_weak_cong)

lemma fps_notation: "(λn. sum (λi. fps_const(a\$i) * fps_X^i) {0..n}) ⇢ a"
(is "?s ⇢ a")
proof -
have "∃n0. ∀n ≥ n0. dist (?s n) a < r" if "r > 0" for r
proof -
obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0"
using reals_power_lt_ex[OF ‹r > 0›, of 2] by auto
show ?thesis
proof -
have "dist (?s n) a < r" if nn0: "n ≥ n0" for n
proof -
from that have thnn0: "(1/2)^n ≤ (1/2 :: real)^n0"
show ?thesis
proof (cases "?s n = a")
case True
then show ?thesis
unfolding dist_eq_0_iff[of "?s n" a, symmetric]
using ‹r > 0› by (simp del: dist_eq_0_iff)
next
case False
from False have dth: "dist (?s n) a = (1/2)^subdegree (?s n - a)"
from False have kn: "subdegree (?s n - a) > n"
by (intro subdegree_greaterI) (simp_all add: fps_sum_rep_nth)
then have "dist (?s n) a < (1/2)^n"
also have "… ≤ (1/2)^n0"
using nn0 by (simp add: divide_simps)
also have "… < r"
using n0 by simp
finally show ?thesis .
qed
qed
then show ?thesis by blast
qed
qed
then show ?thesis
unfolding lim_sequentially by blast
qed

subsection ‹Inverses of formal power series›

declare sum.cong[fundef_cong]

begin

fun natfun_inverse:: "'a fps ⇒ nat ⇒ 'a"
where
"natfun_inverse f 0 = inverse (f\$0)"
| "natfun_inverse f n = - inverse (f\$0) * sum (λi. f\$i * natfun_inverse f (n - i)) {1..n}"

definition fps_inverse_def: "inverse f = (if f \$ 0 = 0 then 0 else Abs_fps (natfun_inverse f))"

definition fps_divide_def:
"f div g = (if g = 0 then 0 else
let n = subdegree g; h = fps_shift n g
in  fps_shift n (f * inverse h))"

instance ..

end

lemma fps_inverse_zero [simp]:
"inverse (0 :: 'a::{comm_monoid_add,inverse,times,uminus} fps) = 0"

lemma fps_inverse_one [simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"
apply (auto simp add: expand_fps_eq fps_inverse_def)
apply (case_tac n)
apply auto
done

lemma inverse_mult_eq_1 [intro]:
assumes f0: "f\$0 ≠ (0::'a::field)"
shows "inverse f * f = 1"
proof -
have c: "inverse f * f = f * inverse f"
from f0 have ifn: "⋀n. inverse f \$ n = natfun_inverse f n"
from f0 have th0: "(inverse f * f) \$ 0 = 1"
have "(inverse f * f)\$n = 0" if np: "n > 0" for n
proof -
from np have eq: "{0..n} = {0} ∪ {1 .. n}"
by auto
have d: "{0} ∩ {1 .. n} = {}"
by auto
from f0 np have th0: "- (inverse f \$ n) =
(sum (λi. f\$i * natfun_inverse f (n - i)) {1..n}) / (f\$0)"
by (cases n) (simp_all add: divide_inverse fps_inverse_def)
from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]
have th1: "sum (λi. f\$i * natfun_inverse f (n - i)) {1..n} = - (f\$0) * (inverse f)\$n"
have "(f * inverse f) \$ n = (∑i = 0..n. f \$i * natfun_inverse f (n - i))"
unfolding fps_mult_nth ifn ..
also have "… = f\$0 * natfun_inverse f n + (∑i = 1..n. f\$i * natfun_inverse f (n-i))"
also have "… = 0"
unfolding th1 ifn by simp
finally show ?thesis unfolding c .
qed
with th0 show ?thesis
qed

lemma fps_inverse_0_iff[simp]: "(inverse f) \$ 0 = (0::'a::division_ring) ⟷ f \$ 0 = 0"

lemma fps_inverse_nth_0 [simp]: "inverse f \$ 0 = inverse (f \$ 0 :: 'a :: division_ring)"

lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::division_ring) fps) ⟷ f \$ 0 = 0"
proof
assume A: "inverse f = 0"
have "0 = inverse f \$ 0" by (subst A) simp
thus "f \$ 0 = 0" by simp

lemma fps_inverse_idempotent[intro, simp]:
assumes f0: "f\$0 ≠ (0::'a::field)"
shows "inverse (inverse f) = f"
proof -
from f0 have if0: "inverse f \$ 0 ≠ 0" by simp
from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0]
have "inverse f * f = inverse f * inverse (inverse f)"
then show ?thesis
using f0 unfolding mult_cancel_left by simp
qed

lemma fps_inverse_unique:
assumes fg: "(f :: 'a :: field fps) * g = 1"
shows   "inverse f = g"
proof -
have f0: "f \$ 0 ≠ 0"
proof
assume "f \$ 0 = 0"
hence "0 = (f * g) \$ 0" by simp
also from fg have "(f * g) \$ 0 = 1" by simp
finally show False by simp
qed
from inverse_mult_eq_1[OF this] fg
have th0: "inverse f * f = g * f"
then show ?thesis
using f0
unfolding mult_cancel_right
qed

lemma fps_inverse_eq_0: "f\$0 = 0 ⟹ inverse (f :: 'a :: division_ring fps) = 0"
by simp

lemma sum_zero_lemma:
fixes n::nat
assumes "0 < n"
shows "(∑i = 0..n. if n = i then 1 else if n - i = 1 then - 1 else 0) = (0::'a::field)"
proof -
let ?f = "λi. if n = i then 1 else if n - i = 1 then - 1 else 0"
let ?g = "λi. if i = n then 1 else if i = n - 1 then - 1 else 0"
let ?h = "λi. if i=n - 1 then - 1 else 0"
have th1: "sum ?f {0..n} = sum ?g {0..n}"
by (rule sum.cong) auto
have th2: "sum ?g {0..n - 1} = sum ?h {0..n - 1}"
apply (rule sum.cong)
using assms
apply auto
done
have eq: "{0 .. n} = {0.. n - 1} ∪ {n}"
by auto
from assms have d: "{0.. n - 1} ∩ {n} = {}"
by auto
have f: "finite {0.. n - 1}" "finite {n}"
by auto
show ?thesis
unfolding th1
apply (simp add: sum.union_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
unfolding th2
done
qed

lemma fps_inverse_mult: "inverse (f * g :: 'a::field fps) = inverse f * inverse g"
proof (cases "f\$0 = 0 ∨ g\$0 = 0")
assume "¬(f\$0 = 0 ∨ g\$0 = 0)"
hence [simp]: "f\$0 ≠ 0" "g\$0 ≠ 0" by simp_all
show ?thesis
proof (rule fps_inverse_unique)
have "f * g * (inverse f * inverse g) = (inverse f * f) * (inverse g * g)" by simp
also have "... = 1" by (subst (1 2) inverse_mult_eq_1) simp_all
finally show "f * g * (inverse f * inverse g) = 1" .
qed
next
assume A: "f\$0 = 0 ∨ g\$0 = 0"
hence "inverse (f * g) = 0" by simp
also from A have "... = inverse f * inverse g" by auto
finally show "inverse (f * g) = inverse f * inverse g" .
qed

lemma fps_inverse_gp: "inverse (Abs_fps(λn. (1::'a::field))) =
Abs_fps (λn. if n= 0 then 1 else if n=1 then - 1 else 0)"
apply (rule fps_inverse_unique)
apply (simp_all add: fps_eq_iff fps_mult_nth sum_zero_lemma)
done

lemma subdegree_inverse [simp]: "subdegree (inverse (f::'a::field fps)) = 0"
proof (cases "f\$0 = 0")
assume nz: "f\$0 ≠ 0"
hence "subdegree (inverse f) + subdegree f = subdegree (inverse f * f)"
by (subst subdegree_mult) auto
also from nz have "subdegree f = 0" by (simp add: subdegree_eq_0_iff)
also from nz have "inverse f * f = 1" by (rule inverse_mult_eq_1)
finally show "subdegree (inverse f) = 0" by simp

lemma fps_is_unit_iff [simp]: "(f :: 'a :: field fps) dvd 1 ⟷ f \$ 0 ≠ 0"
proof
assume "f dvd 1"
then obtain g where "1 = f * g" by (elim dvdE)
from this[symmetric] have "(f*g) \$ 0 = 1" by simp
thus "f \$ 0 ≠ 0" by auto
next
assume A: "f \$ 0 ≠ 0"
thus "f dvd 1" by (simp add: inverse_mult_eq_1[OF A, symmetric])
qed

lemma subdegree_eq_0' [simp]: "(f :: 'a :: field fps) dvd 1 ⟹ subdegree f = 0"
by simp

lemma fps_unit_dvd [simp]: "(f \$ 0 :: 'a :: field) ≠ 0 ⟹ f dvd g"
by (rule dvd_trans, subst fps_is_unit_iff) simp_all

instantiation fps :: (field) normalization_semidom
begin

definition fps_unit_factor_def [simp]:
"unit_factor f = fps_shift (subdegree f) f"

definition fps_normalize_def [simp]:
"normalize f = (if f = 0 then 0 else fps_X ^ subdegree f)"

instance proof
fix f :: "'a fps"
show "unit_factor f * normalize f = f"
next
fix f g :: "'a fps"
show "unit_factor (f * g) = unit_factor f * unit_factor g"
proof (cases "f = 0 ∨ g = 0")
assume "¬(f = 0 ∨ g = 0)"
thus "unit_factor (f * g) = unit_factor f * unit_factor g"
unfolding fps_unit_factor_def
by (auto simp: fps_shift_fps_shift fps_shift_mult fps_shift_mult_right)
qed auto
next
fix f g :: "'a fps"
assume "g ≠ 0"
then have "f * (fps_shift (subdegree g) g * inverse (fps_shift (subdegree g) g)) = f"
by (metis add_cancel_right_left fps_shift_nth inverse_mult_eq_1 mult.commute mult_cancel_left2 nth_subdegree_nonzero)
then have "fps_shift (subdegree g) (g * (f * inverse (fps_shift (subdegree g) g))) = f"
with ‹g ≠ 0› show "f * g / g = f"
by (simp add: fps_divide_def Let_def ac_simps)
qed (auto simp add: fps_divide_def Let_def)

end

instantiation fps :: (field) idom_modulo
begin

definition fps_mod_def:
"f mod g = (if g = 0 then f else
let n = subdegree g; h = fps_shift n g
in  fps_cutoff n (f * inverse h) * h)"

lemma fps_mod_eq_zero:
assumes "g ≠ 0" and "subdegree f ≥ subdegree g"
shows   "f mod g = 0"
using assms by (cases "f = 0") (auto simp: fps_cutoff_zero_iff fps_mod_def Let_def)

lemma fps_times_divide_eq:
assumes "g ≠ 0" and "subdegree f ≥ subdegree (g :: 'a fps)"
shows   "f div g * g = f"
proof (cases "f = 0")
assume nz: "f ≠ 0"
define n where "n = subdegree g"
define h where "h = fps_shift n g"
from assms have [simp]: "h \$ 0 ≠ 0" unfolding h_def by (simp add: n_def)

from assms nz have "f div g * g = fps_shift n (f * inverse h) * g"
by (simp add: fps_divide_def Let_def h_def n_def)
also have "... = fps_shift n (f * inverse h) * fps_X^n * h" unfolding h_def n_def
by (subst subdegree_decompose[of g]) simp
also have "fps_shift n (f * inverse h) * fps_X^n = f * inverse h"
by (rule fps_shift_times_fps_X_power) (simp_all add: nz assms n_def)
also have "... * h = f * (inverse h * h)" by simp
also have "inverse h * h = 1" by (rule inverse_mult_eq_1) simp
finally show ?thesis by simp

lemma
assumes "g\$0 ≠ 0"
shows   fps_divide_unit: "f div g = f * inverse g" and fps_mod_unit [simp]: "f mod g = 0"
proof -
from assms have [simp]: "subdegree g = 0" by (simp add: subdegree_eq_0_iff)
from assms show "f div g = f * inverse g"
by (auto simp: fps_divide_def Let_def subdegree_eq_0_iff)
from assms show "f mod g = 0" by (intro fps_mod_eq_zero) auto
qed

instance proof
fix f g :: "'a fps"
define n where "n = subdegree g"
define h where "h = fps_shift n g"

show "f div g * g + f mod g = f"
proof (cases "g = 0 ∨ f = 0")
assume "¬(g = 0 ∨ f = 0)"
hence nz [simp]: "f ≠ 0" "g ≠ 0" by simp_all
show ?thesis
proof (rule disjE[OF le_less_linear])
assume "subdegree f ≥ subdegree g"
with nz show ?thesis by (simp add: fps_mod_eq_zero fps_times_divide_eq)
next
assume "subdegree f < subdegree g"
have g_decomp: "g = h * fps_X^n" unfolding h_def n_def by (rule subdegree_decompose)
have "f div g * g + f mod g =
fps_shift n (f * inverse h) * g + fps_cutoff n (f * inverse h) * h"
by (simp add: fps_mod_def fps_divide_def Let_def n_def h_def)
also have "... = h * (fps_shift n (f * inverse h) * fps_X^n + fps_cutoff n (f * inverse h))"
by (subst g_decomp) (simp add: algebra_simps)
also have "... = f * (inverse h * h)"
by (subst fps_shift_cutoff) simp
also have "inverse h * h = 1" by (rule inverse_mult_eq_1) (simp add: h_def n_def)
finally show ?thesis by simp
qed
qed (auto simp: fps_mod_def fps_divide_def Let_def)
qed

end

lemma subdegree_mod:
assumes "f ≠ 0" "subdegree f < subdegree g"
shows   "subdegree (f mod g) = subdegree f"
proof (cases "f div g * g = 0")
assume "f div g * g ≠ 0"
hence [simp]: "f div g ≠ 0" "g ≠ 0" by auto
from div_mult_mod_eq[of f g] have "f mod g = f - f div g * g" by (simp add: algebra_simps)
also from assms have "subdegree ... = subdegree f"
by (intro subdegree_diff_eq1) simp_all
finally show ?thesis .
next
assume zero: "f div g * g = 0"
from div_mult_mod_eq[of f g] have "f mod g = f - f div g * g" by (simp add: algebra_simps)
also note zero
finally show ?thesis by simp
qed

lemma fps_divide_nth_0 [simp]: "g \$ 0 ≠ 0 ⟹ (f div g) \$ 0 = f \$ 0 / (g \$ 0 :: _ :: field)"

lemma dvd_imp_subdegree_le:
"(f :: 'a :: idom fps) dvd g ⟹ g ≠ 0 ⟹ subdegree f ≤ subdegree g"
by (auto elim: dvdE)

lemma fps_dvd_iff:
assumes "(f :: 'a :: field fps) ≠ 0" "g ≠ 0"
shows   "f dvd g ⟷ subdegree f ≤ subdegree g"
proof
assume "subdegree f ≤ subdegree g"
with assms have "g mod f = 0"
by (simp add: fps_mod_def Let_def fps_cutoff_zero_iff)
thus "f dvd g" by (simp add: dvd_eq_mod_eq_0)

lemma fps_shift_altdef:
"fps_shift n f = (f :: 'a :: field fps) div fps_X^n"

lemma fps_div_fps_X_power_nth: "((f :: 'a :: field fps) div fps_X^n) \$ k = f \$ (k + n)"

lemma fps_div_fps_X_nth: "((f :: 'a :: field fps) div fps_X) \$ k = f \$ Suc k"
using fps_div_fps_X_power_nth[of f 1] by simp

lemma fps_const_inverse: "inverse (fps_const (a::'a::field)) = fps_const (inverse a)"
by (cases "a ≠ 0", rule fps_inverse_unique) (auto simp: fps_eq_iff)

lemma fps_const_divide: "fps_const (x :: _ :: field) / fps_const y = fps_const (x / y)"
by (cases "y = 0") (simp_all add: fps_divide_unit fps_const_inverse divide_inverse)

lemma inverse_fps_numeral:
"inverse (numeral n :: ('a :: field_char_0) fps) = fps_const (inverse (numeral n))"
by (intro fps_inverse_unique fps_ext) (simp_all add: fps_numeral_nth)

lemma fps_numeral_divide_divide:
"x / numeral b / numeral c = (x / numeral (b * c) :: 'a :: field fps)"
by (cases "numeral b = (0::'a)"; cases "numeral c = (0::'a)")
(simp_all add: fps_divide_unit fps_inverse_mult [symmetric] numeral_fps_const numeral_mult
del: numeral_mult [symmetric])

lemma fps_numeral_mult_divide:
"numeral b * x / numeral c = (numeral b / numeral c * x :: 'a :: field fps)"
by (cases "numeral c = (0::'a)") (simp_all add: fps_divide_unit numeral_fps_const)

lemmas fps_numeral_simps =
fps_numeral_divide_divide fps_numeral_mult_divide inverse_fps_numeral neg_numeral_fps_const

lemma subdegree_div:
assumes "q dvd p"
shows   "subdegree ((p :: 'a :: field fps) div q) = subdegree p - subdegree q"
proof (cases "p = 0")
case False
from assms have "p = p div q * q" by simp
also from assms False have "subdegree … = subdegree (p div q) + subdegree q"
by (intro subdegree_mult) (auto simp: dvd_div_eq_0_iff)
finally show ?thesis by simp
qed simp_all

lemma subdegree_div_unit:
assumes "q \$ 0 ≠ 0"
shows   "subdegree ((p :: 'a :: field fps) div q) = subdegree p"
using assms by (subst subdegree_div) simp_all

subsection ‹Formal power series form a Euclidean ring›

instantiation fps :: (field) euclidean_ring_cancel
begin

definition fps_euclidean_size_def:
"euclidean_size f = (if f = 0 then 0 else 2 ^ subdegree f)"

context
begin

private lemma fps_divide_cancel_aux1:
assumes "h\$0 ≠ (0 :: 'a :: field)"
shows   "(h * f) div (h * g) = f div g"
proof (cases "g = 0")
assume "g ≠ 0"
from assms have "h ≠ 0" by auto
note nz [simp] = ‹g ≠ 0› ‹h ≠ 0›
from assms have [simp]: "subdegree h = 0" by (simp add: subdegree_eq_0_iff)

have "(h * f) div (h * g) =
fps_shift (subdegree g) (h * f * inverse (fps_shift (subdegree g) (h*g)))"
also have "h * f * inverse (fps_shift (subdegree g) (h*g)) =
(inverse h * h) * f * inverse (fps_shift (subdegree g) g)"
by (subst fps_shift_mult) (simp_all add: algebra_simps fps_inverse_mult)
also from assms have "inverse h * h = 1" by (rule inverse_mult_eq_1)
finally show "(h * f) div (h * g) = f div g" by (simp_all add: fps_divide_def Let_def)

private lemma fps_divide_cancel_aux2:
"(f * fps_X^m) div (g * fps_X^m) = f div (g :: 'a :: field fps)"
proof (cases "g = 0")
assume [simp]: "g ≠ 0"
have "(f * fps_X^m) div (g * fps_X^m) =
fps_shift (subdegree g + m) (f*inverse (fps_shift (subdegree g + m) (g*fps_X^m))*fps_X^m)"
by (simp add: fps_divide_def Let_def algebra_simps)
also have "... = f div g"
by (simp add: fps_shift_times_fps_X_power'' fps_divide_def Let_def)
finally show ?thesis .

instance proof
fix f g :: "'a fps" assume [simp]: "g ≠ 0"
show "euclidean_size f ≤ euclidean_size (f * g)"
by (cases "f = 0") (auto simp: fps_euclidean_size_def)
show "euclidean_size (f mod g) < euclidean_size g"
apply (cases "f = 0", simp add: fps_euclidean_size_def)
apply (rule disjE[OF le_less_linear[of "subdegree g" "subdegree f"]])
apply (simp_all add: fps_mod_eq_zero fps_euclidean_size_def subdegree_mod)
done
show "(h * f) div (h * g) = f div g" if "h ≠ 0"
for f g h :: "'a fps"
proof -
define m where "m = subdegree h"
define h' where "h' = fps_shift m h"
have h_decomp: "h = h' * fps_X ^ m" unfolding h'_def m_def by (rule subdegree_decompose)
from ‹h ≠ 0› have [simp]: "h'\$0 ≠ 0" by (simp add: h'_def m_def)
have "(h * f) div (h * g) = (h' * f * fps_X^m) div (h' * g * fps_X^m)"
also have "... = f div g"
finally show ?thesis .
qed
show "(f + g * h) div h = g + f div h"
if "h ≠ 0" for f g h :: "'a fps"
proof -
define n h' where dfs: "n = subdegree h" "h' = fps_shift n h"
have "(f + g * h) div h = fps_shift n (f * inverse h') + fps_shift n (g * (h * inverse h'))"
also have "h * inverse h' = (inverse h' * h') * fps_X^n"
by (subst subdegree_decompose) (simp_all add: dfs)
also have "... = fps_X^n"
by (subst inverse_mult_eq_1) (simp_all add: dfs that)
also have "fps_shift n (g * fps_X^n) = g" by simp
also have "fps_shift n (f * inverse h') = f div h"
by (simp add: fps_divide_def Let_def dfs)
finally show ?thesis by simp
qed

end

end

instance fps :: (field) normalization_euclidean_semiring ..

instantiation fps :: (field) euclidean_ring_gcd
begin
definition fps_gcd_def: "(gcd :: 'a fps ⇒ _) = Euclidean_Algorithm.gcd"
definition fps_lcm_def: "(lcm :: 'a fps ⇒ _) = Euclidean_Algorithm.lcm"
definition fps_Gcd_def: "(Gcd :: 'a fps set ⇒ _) = Euclidean_Algorithm.Gcd"
definition fps_Lcm_def: "(Lcm :: 'a fps set ⇒ _) = Euclidean_Algorithm.Lcm"
instance by standard (simp_all add: fps_gcd_def fps_lcm_def fps_Gcd_def fps_Lcm_def)
end

lemma fps_gcd:
assumes [simp]: "f ≠ 0" "g ≠ 0"
shows   "gcd f g = fps_X ^ min (subdegree f) (subdegree g)"
proof -
let ?m = "min (subdegree f) (subdegree g)"
show "gcd f g = fps_X ^ ?m"
proof (rule sym, rule gcdI)
fix d assume "d dvd f" "d dvd g"
thus "d dvd fps_X ^ ?m" by (cases "d = 0") (auto simp: fps_dvd_iff)
qed

lemma fps_gcd_altdef: "gcd (f :: 'a :: field fps) g =
(if f = 0 ∧ g = 0 then 0 else
if f = 0 then fps_X ^ subdegree g else
if g = 0 then fps_X ^ subdegree f else
fps_X ^ min (subdegree f) (subdegree g))"

lemma fps_lcm:
assumes [simp]: "f ≠ 0" "g ≠ 0"
shows   "lcm f g = fps_X ^ max (subdegree f) (subdegree g)"
proof -
let ?m = "max (subdegree f) (subdegree g)"
show "lcm f g = fps_X ^ ?m"
proof (rule sym, rule lcmI)
fix d assume "f dvd d" "g dvd d"
thus "fps_X ^ ?m dvd d" by (cases "d = 0") (auto simp: fps_dvd_iff)
qed

lemma fps_lcm_altdef: "lcm (f :: 'a :: field fps) g =
(if f = 0 ∨ g = 0 then 0 else fps_X ^ max (subdegree f) (subdegree g))"

lemma fps_Gcd:
assumes "A - {0} ≠ {}"
shows   "Gcd A = fps_X ^ (INF f:A-{0}. subdegree f)"
proof (rule sym, rule GcdI)
fix f assume "f ∈ A"
thus "fps_X ^ (INF f:A - {0}. subdegree f) dvd f"
by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cINF_lower)
next
fix d assume d: "⋀f. f ∈ A ⟹ d dvd f"
from assms obtain f where "f ∈ A - {0}" by auto
with d[of f] have [simp]: "d ≠ 0" by auto
from d assms have "subdegree d ≤ (INF f:A-{0}. subdegree f)"
by (intro cINF_greatest) (auto simp: fps_dvd_iff[symmetric])
with d assms show "d dvd fps_X ^ (INF f:A-{0}. subdegree f)" by (simp add: fps_dvd_iff)
qed simp_all

lemma fps_Gcd_altdef: "Gcd (A :: 'a :: field fps set) =
(if A ⊆ {0} then 0 else fps_X ^ (INF f:A-{0}. subdegree f))"
using fps_Gcd by auto

lemma fps_Lcm:
assumes "A ≠ {}" "0 ∉ A" "bdd_above (subdegree`A)"
shows   "Lcm A = fps_X ^ (SUP f:A. subdegree f)"
proof (rule sym, rule LcmI)
fix f assume "f ∈ A"
moreover from assms(3) have "bdd_above (subdegree ` A)" by auto
ultimately show "f dvd fps_X ^ (SUP f:A. subdegree f)" using assms(2)
by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cSUP_upper)
next
fix d assume d: "⋀f. f ∈ A ⟹ f dvd d"
from assms obtain f where f: "f ∈ A" "f ≠ 0" by auto
show "fps_X ^ (SUP f:A. subdegree f) dvd d"
proof (cases "d = 0")
assume "d ≠ 0"
moreover from d have "⋀f. f ∈ A ⟹ f ≠ 0 ⟹ f dvd d" by blast
ultimately have "subdegree d ≥ (SUP f:A. subdegree f)" using assms
by (intro cSUP_least) (auto simp: fps_dvd_iff)
with ‹d ≠ 0› show ?thesis by (simp add: fps_dvd_iff)
qed simp_all
qed simp_all

lemma fps_Lcm_altdef:
"Lcm (A :: 'a :: field fps set) =
(if 0 ∈ A ∨ ¬bdd_above (subdegree`A) then 0 else
if A = {} then 1 else fps_X ^ (SUP f:A. subdegree f))"
proof (cases "bdd_above (subdegree`A)")
assume unbounded: "¬bdd_above (subdegree`A)"
have "Lcm A = 0"
proof (rule ccontr)
assume "Lcm A ≠ 0"
from unbounded obtain f where f: "f ∈ A" "subdegree (Lcm A) < subdegree f"
unfolding bdd_above_def by (auto simp: not_le)
moreover from f and ‹Lcm A ≠ 0› have "subdegree f ≤ subdegree (Lcm A)"
by (intro dvd_imp_subdegree_le dvd_Lcm) simp_all
ultimately show False by simp
qed
with unbounded show ?thesis by simp

subsection ‹Formal Derivatives, and the MacLaurin theorem around 0›

definition "fps_deriv f = Abs_fps (λn. of_nat (n + 1) * f \$ (n + 1))"

lemma fps_deriv_nth[simp]: "fps_deriv f \$ n = of_nat (n +1) * f \$ (n + 1)"

lemma fps_0th_higher_deriv:
"(fps_deriv ^^ n) f \$ 0 = (fact n * f \$ n :: 'a :: {comm_ring_1, semiring_char_0})"
by (induction n arbitrary: f) (simp_all del: funpow.simps add: funpow_Suc_right algebra_simps)

lemma fps_deriv_linear[simp]:
"fps_deriv (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) =
fps_const a * fps_deriv f + fps_const b * fps_deriv g"

lemma fps_deriv_mult[simp]:
fixes f :: "'a::comm_ring_1 fps"
shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
proof -
let ?D = "fps_deriv"
have "(f * ?D g + ?D f * g) \$ n = ?D (f*g) \$ n" for n
proof -
let ?Zn = "{0 ..n}"
let ?Zn1 = "{0 .. n + 1}"
let ?g = "λi. of_nat (i+1) * g \$ (i+1) * f \$ (n - i) +
of_nat (i+1)* f \$ (i+1) * g \$ (n - i)"
let ?h = "λi. of_nat i * g \$ i * f \$ ((n+1) - i) +
of_nat i* f \$ i * g \$ ((n + 1) - i)"
have s0: "sum (λi. of_nat i * f \$ i * g \$ (n + 1 - i)) ?Zn1 =
sum (λi. of_nat (n + 1 - i) * f \$ (n + 1 - i) * g \$ i) ?Zn1"
by (rule sum.reindex_bij_witness[where i="(-) (n + 1)" and j="(-) (n + 1)"]) auto
have s1: "sum (λi. f \$ i * g \$ (n + 1 - i)) ?Zn1 =
sum (λi. f \$ (n + 1 - i) * g \$ i) ?Zn1"
by (rule sum.reindex_bij_witness[where i="(-) (n + 1)" and j="(-) (n + 1)"]) auto
have "(f * ?D g + ?D f * g)\$n = (?D g * f + ?D f * g)\$n"
by (simp only: mult.commute)
also have "… = (∑i = 0..n. ?g i)"
also have "… = sum ?h {0..n+1}"
by (rule sum.reindex_bij_witness_not_neutral
[where S'="{}" and T'="{0}" and j="Suc" and i="λi. i - 1"]) auto
also have "… = (fps_deriv (f * g)) \$ n"
apply (simp only: fps_deriv_nth fps_mult_nth sum.distrib)
unfolding s0 s1
unfolding sum.distrib[symmetric] sum_distrib_left
apply (rule sum.cong)
apply (auto simp add: of_nat_diff field_simps)
done
finally show ?thesis .
qed
then show ?thesis
unfolding fps_eq_iff by auto
qed

lemma fps_deriv_fps_X[simp]: "fps_deriv fps_X = 1"
by (simp add: fps_deriv_def fps_X_def fps_eq_iff)

lemma fps_deriv_neg[simp]:
"fps_deriv (- (f:: 'a::comm_ring_1 fps)) = - (fps_deriv f)"

"fps_deriv ((f:: 'a::comm_ring_1 fps) + g) = fps_deriv f + fps_deriv g"
using fps_deriv_linear[of 1 f 1 g] by simp

lemma fps_deriv_sub[simp]:
"fps_deriv ((f:: 'a::comm_ring_1 fps) - g) = fps_deriv f - fps_deriv g"
using fps_deriv_add [of f "- g"] by simp

lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"
by (simp add: fps_ext fps_deriv_def fps_const_def)

lemma fps_deriv_of_nat [simp]: "fps_deriv (of_nat n) = 0"

lemma fps_deriv_numeral [simp]: "fps_deriv (numeral n) = 0"

lemma fps_deriv_mult_const_left[simp]:
"fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f"
by simp

lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"

lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
by (simp add: fps_deriv_def fps_eq_iff )

lemma fps_deriv_mult_const_right[simp]:
"fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c"
by simp

lemma fps_deriv_sum:
"fps_deriv (sum f S) = sum (λi. fps_deriv (f i :: 'a::comm_ring_1 fps)) S"
proof (cases "finite S")
case False
then show ?thesis by simp
next
case True
show ?thesis by (induct rule: finite_induct [OF True]) simp_all
qed

lemma fps_deriv_eq_0_iff [simp]:
"fps_deriv f = 0 ⟷ f = fps_const (f\$0 :: 'a::{idom,semiring_char_0})"
(is "?lhs ⟷ ?rhs")
proof
show ?lhs if ?rhs
proof -
from that have "fps_deriv f = fps_deriv (fps_const (f\$0))"
by simp
then show ?thesis
by simp
qed
show ?rhs if ?lhs
proof -
from that have "∀n. (fps_deriv f)\$n = 0"
by simp
then have "∀n. f\$(n+1) = 0"
by (simp del: of_nat_Suc of_nat_add One_nat_def)
then show ?thesis
apply (clarsimp simp add: fps_eq_iff fps_const_def)
apply (erule_tac x="n - 1" in allE)
apply simp
done
qed
qed

lemma fps_deriv_eq_iff:
fixes f :: "'a::{idom,semiring_char_0} fps"
shows "fps_deriv f = fps_deriv g ⟷ (f = fps_const(f\$0 - g\$0) + g)"
proof -
have "fps_deriv f = fps_deriv g ⟷ fps_deriv (f - g) = 0"
by simp
also have "… ⟷ f - g = fps_const ((f - g) \$ 0)"
unfolding fps_deriv_eq_0_iff ..
finally show ?thesis
qed

lemma fps_deriv_eq_iff_ex:
"(fps_deriv f = fps_deriv g) ⟷ (∃c::'a::{idom,semiring_char_0}. f = fps_const c + g)"
by (auto simp: fps_deriv_eq_iff)

fun fps_nth_deriv :: "nat ⇒ 'a::semiring_1 fps ⇒ 'a fps"
where
"fps_nth_deriv 0 f = f"
| "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"

lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"
by (induct n arbitrary: f) auto

lemma fps_nth_deriv_linear[simp]:
"fps_nth_deriv n (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) =
fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g"
by (induct n arbitrary: f g) (auto simp add: fps_nth_deriv_commute)

lemma fps_nth_deriv_neg[simp]:
"fps_nth_deriv n (- (f :: 'a::comm_ring_1 fps)) = - (fps_nth_deriv n f)"
by (induct n arbitrary: f) simp_all

"fps_nth_deriv n ((f :: 'a::comm_ring_1 fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g"
using fps_nth_deriv_linear[of n 1 f 1 g] by simp

lemma fps_nth_deriv_sub[simp]:
"fps_nth_deriv n ((f :: 'a::comm_ring_1 fps) - g) = fps_nth_deriv n f - fps_nth_deriv n g"
using fps_nth_deriv_add [of n f "- g"] by simp

lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"
by (induct n) simp_all

lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"
by (induct n) simp_all

lemma fps_nth_deriv_const[simp]:
"fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"
by (cases n) simp_all

lemma fps_nth_deriv_mult_const_left[simp]:
"fps_nth_deriv n (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_nth_deriv n f"
using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp

lemma fps_nth_deriv_mult_const_right[simp]:
"fps_nth_deriv n (f * fps_const (c::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c"
using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult.commute)

lemma fps_nth_deriv_sum:
"fps_nth_deriv n (sum f S) = sum (λi. fps_nth_deriv n (f i :: 'a::comm_ring_1 fps)) S"
proof (cases "finite S")
case True
show ?thesis by (induct rule: finite_induct [OF True]) simp_all
next
case False
then show ?thesis by simp
qed

lemma fps_deriv_maclauren_0:
"(fps_nth_deriv k (f :: 'a::comm_semiring_1 fps)) \$ 0 = of_nat (fact k) * f \$ k"
by (induct k arbitrary: f) (auto simp add: field_simps)

subsection ‹Powers›

lemma fps_power_zeroth_eq_one: "a\$0 =1 ⟹ a^n \$ 0 = (1::'a::semiring_1)"
by (induct n) (auto simp add: expand_fps_eq fps_mult_nth)

lemma fps_power_first_eq: "(a :: 'a::comm_ring_1 fps) \$ 0 =1 ⟹ a^n \$ 1 = of_nat n * a\$1"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
show ?case unfolding power_Suc fps_mult_nth
using Suc.hyps[OF ‹a\$0 = 1›] ‹a\$0 = 1› fps_power_zeroth_eq_one[OF ‹a\$0=1›]
qed

lemma startsby_one_power:"a \$ 0 = (1::'a::comm_ring_1) ⟹ a^n \$ 0 = 1"
by (induct n) (auto simp add: fps_mult_nth)

lemma startsby_zero_power:"a \$0 = (0::'a::comm_ring_1) ⟹ n > 0 ⟹ a^n \$0 = 0"
by (induct n) (auto simp add: fps_mult_nth)

lemma startsby_power:"a \$0 = (v::'a::comm_ring_1) ⟹ a^n \$0 = v^n"
by (induct n) (auto simp add: fps_mult_nth)

lemma startsby_zero_power_iff[simp]: "a^n \$0 = (0::'a::idom) ⟷ n ≠ 0 ∧ a\$0 = 0"
apply (rule iffI)
apply (induct n)
apply (rule startsby_zero_power, simp_all)
done

lemma startsby_zero_power_prefix:
assumes a0: "a \$ 0 = (0::'a::idom)"
shows "∀n < k. a ^ k \$ n = 0"
using a0
proof (induct k rule: nat_less_induct)
fix k
assume H: "∀m<k. a \$0 =  0 ⟶ (∀n<m. a ^ m \$ n = 0)" and a0: "a \$ 0 = 0"
show "∀m<k. a ^ k \$ m = 0"
proof (cases k)
case 0
then show ?thesis by simp
next
case (Suc l)
have "a^k \$ m = 0" if mk: "m < k" for m
proof (cases "m = 0")
case True
then show ?thesis
using startsby_zero_power[of a k] Suc a0 by simp
next
case False
have "a ^k \$ m = (a^l * a) \$m"
also have "… = (∑i = 0..m. a ^ l \$ i * a \$ (m - i))"
also have "… = 0"
apply (rule sum.neutral)
apply auto
apply (case_tac "x = m")
using a0 apply simp
apply (rule H[rule_format])
using a0 Suc mk apply auto
done
finally show ?thesis .
qed
then show ?thesis by blast
qed
qed

lemma startsby_zero_sum_depends:
assumes a0: "a \$0 = (0::'a::idom)"
and kn: "n ≥ k"
shows "sum (λi. (a ^ i)\$k) {0 .. n} = sum (λi. (a ^ i)\$k) {0 .. k}"
apply (rule sum.mono_neutral_right)
using kn
apply auto
apply (rule startsby_zero_power_prefix[rule_format, OF a0])
apply arith
done

lemma startsby_zero_power_nth_same:
assumes a0: "a\$0 = (0::'a::idom)"
shows "a^n \$ n = (a\$1) ^ n"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "a ^ Suc n \$ (Suc n) = (a^n * a)\$(Suc n)"
also have "… = sum (λi. a^n\$i * a \$ (Suc n - i)) {0.. Suc n}"
also have "… = sum (λi. a^n\$i * a \$ (Suc n - i)) {n .. Suc n}"
apply (rule sum.mono_neutral_right)
apply simp
apply clarsimp
apply clarsimp
apply (rule startsby_zero_power_prefix[rule_format, OF a0])
apply arith
done
also have "… = a^n \$ n * a\$1"
using a0 by simp
finally show ?case
using Suc.hyps by simp
qed

lemma fps_inverse_power:
fixes a :: "'a::field fps"
shows "inverse (a^n) = inverse a ^ n"
by (induction n) (simp_all add: fps_inverse_mult)

lemma fps_deriv_power:
"fps_deriv (a ^ n) = fps_const (of_nat n :: 'a::comm_ring_1) * fps_deriv a * a ^ (n - 1)"
apply (induct n)
apply (case_tac n)
done

lemma fps_inverse_deriv:
fixes a :: "'a::field fps"
assumes a0: "a\$0 ≠ 0"
shows "fps_deriv (inverse a) = - fps_deriv a * (inverse a)⇧2"
proof -
from inverse_mult_eq_1[OF a0]
have "fps_deriv (inverse a * a) = 0" by simp
then have "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0"
by simp
then have "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0"
by simp
with inverse_mult_eq_1[OF a0]
have "(inverse a)⇧2 * fps_deriv a + fps_deriv (inverse a) = 0"
unfolding power2_eq_square
done
then have "(inverse a)⇧2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * (inverse a)⇧2 =
0 - fps_deriv a * (inverse a)⇧2"
by simp
then show "fps_deriv (inverse a) = - fps_deriv a * (inverse a)⇧2"
qed

lemma fps_inverse_deriv':
fixes a :: "'a::field fps"
assumes a0: "a \$ 0 ≠ 0"
shows "fps_deriv (inverse a) = - fps_deriv a / a⇧2"
using fps_inverse_deriv[OF a0] a0
by (simp add: fps_divide_unit power2_eq_square fps_inverse_mult)

lemma inverse_mult_eq_1':
assumes f0: "f\$0 ≠ (0::'a::field)"
shows "f * inverse f = 1"
by (metis mult.commute inverse_mult_eq_1 f0)

lemma fps_inverse_minus [simp]: "inverse (-f) = -inverse (f :: 'a :: field fps)"
by (cases "f\$0 = 0") (auto intro: fps_inverse_unique simp: inverse_mult_eq_1' fps_inverse_eq_0)

lemma divide_fps_const [simp]: "f / fps_const (c :: 'a :: field) = fps_const (inverse c) * f"
by (cases "c = 0") (simp_all add: fps_divide_unit fps_const_inverse)

(* FIfps_XME: The last part of this proof should go through by simp once we have a proper
theorem collection for simplifying division on rings *)
lemma fps_divide_deriv:
assumes "b dvd (a :: 'a :: field fps)"
shows   "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b^2"
proof -
have eq_divide_imp: "c ≠ 0 ⟹ a * c = b ⟹ a = b div c" for a b c :: "'a :: field fps"
by (drule sym) (simp add: mult.assoc)
from assms have "a = a / b * b" by simp
also have "fps_deriv (a / b * b) = fps_deriv (a / b) * b + a / b * fps_deriv b" by simp
finally have "fps_deriv (a / b) * b^2 = fps_deriv a * b - a * fps_deriv b" using assms
thus ?thesis by (cases "b = 0") (auto simp: eq_divide_imp)
qed

lemma fps_inverse_gp': "inverse (Abs_fps (λn. 1::'a::field)) = 1 - fps_X"
by (simp add: fps_inverse_gp fps_eq_iff fps_X_def)

lemma fps_one_over_one_minus_fps_X_squared:
"inverse ((1 - fps_X)^2 :: 'a :: field fps) = Abs_fps (λn. of_nat (n+1))"
proof -
have "inverse ((1 - fps_X)^2 :: 'a fps) = fps_deriv (inverse (1 - fps_X))"
by (subst fps_inverse_deriv) (simp_all add: fps_inverse_power)
also have "inverse (1 - fps_X :: 'a fps) = Abs_fps (λ_. 1)"
by (subst fps_inverse_gp' [symmetric]) simp
also have "fps_deriv … = Abs_fps (λn. of_nat (n + 1))"
finally show ?thesis .
qed

lemma fps_nth_deriv_fps_X[simp]: "fps_nth_deriv n fps_X = (if n = 0 then fps_X else if n=1 then 1 else 0)"
by (cases n) simp_all

lemma fps_inverse_fps_X_plus1: "inverse (1 + fps_X) = Abs_fps (λn. (- (1::'a::field)) ^ n)"
(is "_ = ?r")
proof -
have eq: "(1 + fps_X) * ?r = 1"
unfolding minus_one_power_iff
by (auto simp add: field_simps fps_eq_iff)
show ?thesis
by (auto simp add: eq intro: fps_inverse_unique)
qed

subsection ‹Integration›

definition fps_integral :: "'a::field_char_0 fps ⇒ 'a ⇒ 'a fps"
where "fps_integral a a0 = Abs_fps (λn. if n = 0 then a0 else (a\$(n - 1) / of_nat n))"

lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a"
unfolding fps_integral_def fps_deriv_def
by (simp add: fps_eq_iff del: of_nat_Suc)

lemma fps_integral_linear:
"fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) =
fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0"
(is "?l = ?r")
proof -
have "fps_deriv ?l = fps_deriv ?r"
moreover have "?l\$0 = ?r\$0"
ultimately show ?thesis
unfolding fps_deriv_eq_iff by auto
qed

subsection ‹Composition of FPSs›

definition fps_compose :: "'a::semiring_1 fps ⇒ 'a fps ⇒ 'a fps"  (infixl "oo" 55)
where "a oo b = Abs_fps (λn. sum (λi. a\$i * (b^i\$n)) {0..n})"

lemma fps_compose_nth: "(a oo b)\$n = sum (λi. a\$i * (b^i\$n)) {0..n}"

lemma fps_compose_nth_0 [simp]: "(f oo g) \$ 0 = f \$ 0"

lemma fps_compose_fps_X[simp]: "a oo fps_X = (a :: 'a::comm_ring_1 fps)"
by (simp add: fps_ext fps_compose_def mult_delta_right sum.delta')

lemma fps_const_compose[simp]: "fps_const (a::'a::comm_ring_1) oo b = fps_const a"
by (simp add: fps_eq_iff fps_compose_nth mult_delta_left sum.delta)

lemma numeral_compose[simp]: "(numeral k :: 'a::comm_ring_1 fps) oo b = numeral k"
unfolding numeral_fps_const by simp

lemma neg_numeral_compose[simp]: "(- numeral k :: 'a::comm_ring_1 fps) oo b = - numeral k"
unfolding neg_numeral_fps_const by simp

lemma fps_X_fps_compose_startby0[simp]: "a\$0 = 0 ⟹ fps_X oo a = (a :: 'a::comm_ring_1 fps)"
by (simp add: fps_eq_iff fps_compose_def mult_delta_left sum.delta not_le)

subsection ‹Rules from Herbert Wilf's Generatingfunctionology›

subsubsection ‹Rule 1›
(* {a_{n+k}}_0^infty Corresponds to (f - sum (λi. a_i * x^i))/x^h, for h>0*)

lemma fps_power_mult_eq_shift:
"fps_X^Suc k * Abs_fps (λn. a (n + Suc k)) =
Abs_fps a - sum (λi. fps_const (a i :: 'a::comm_ring_1) * fps_X^i) {0 .. k}"
(is "?lhs = ?rhs")
proof -
have "?lhs \$ n = ?rhs \$ n" for n :: nat
proof -
have "?lhs \$ n = (if n < Suc k then 0 else a n)"
unfolding fps_X_power_mult_nth by auto
also have "… = ?rhs \$ n"
proof (induct k)
case 0
then show ?case
next
case (Suc k)
have "(Abs_fps a - sum (λi. fps_const (a i :: 'a) * fps_X^i) {0 .. Suc k})\$n =
(Abs_fps a - sum (λi. fps_const (a i :: 'a) * fps_X^i) {0 .. k} -
fps_const (a (Suc k)) * fps_X^ Suc k) \$ n"
also have "… = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * fps_X^ Suc k)\$n"
using Suc.hyps[symmetric] unfolding fps_sub_nth by simp
also have "… = (if n < Suc (Suc k) then 0 else a n)"
unfolding fps_X_power_mult_right_nth
apply (auto simp add: not_less fps_const_def)
apply (rule cong[of a a, OF refl])
apply arith
done
finally show ?case
by simp
qed
finally show ?thesis .
qed
then show ?thesis
qed

subsubsection ‹Rule 2›

(* We can not reach the form of Wilf, but still near to it using rewrite rules*)
(* If f reprents {a_n} and P is a polynomial, then
P(xD) f represents {P(n) a_n}*)

definition "fps_XD = ( * ) fps_X ∘ fps_deriv"

lemma fps_XD_add[simp]:"fps_XD (a + b) = fps_XD a + fps_XD (b :: 'a::comm_ring_1 fps)"

lemma fps_XD_mult_const[simp]:"fps_XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * fps_XD a"

lemma fps_XD_linear[simp]: "fps_XD (fps_const c * a + fps_const d * b) =
fps_const c * fps_XD a + fps_const d * fps_XD (b :: 'a::comm_ring_1 fps)"
by simp

lemma fps_XDN_linear:
"(fps_XD ^^ n) (fps_const c * a + fps_const d * b) =
fps_const c * (fps_XD ^^ n) a + fps_const d * (fps_XD ^^ n) (b :: 'a::comm_ring_1 fps)"
by (induct n) simp_all

lemma fps_mult_fps_X_deriv_shift: "fps_X* fps_deriv a = Abs_fps (λn. of_nat n* a\$n)"

lemma fps_mult_fps_XD_shift:
"(fps_XD ^^ k) (a :: 'a::comm_ring_1 fps) = Abs_fps (λn. (of_nat n ^ k) * a\$n)"
by (induct k arbitrary: a) (simp_all add: fps_XD_def fps_eq_iff field_simps del: One_nat_def)

subsubsection ‹Rule 3›

text ‹Rule 3 is trivial and is given by ‹fps_times_def›.›

subsubsection ‹Rule 5 --- summation and "division" by (1 - fps_X)›

lemma fps_divide_fps_X_minus1_sum_lemma:
"a = ((1::'a::comm_ring_1 fps) - fps_X) * Abs_fps (λn. sum (λi. a \$ i) {0..n})"
proof -
let ?sa = "Abs_fps (λn. sum (λi. a \$ i) {0..n})"
have th0: "⋀i. (1 - (fps_X::'a fps)) \$ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)"
by simp
have "a\$n = ((1 - fps_X) * ?sa) \$ n" for n
proof (cases "n = 0")
case True
then show ?thesis
next
case False
then have u: "{0} ∪ ({1} ∪ {2..n}) = {0..n}" "{1} ∪ {2..n} = {1..n}"
"{0..n - 1} ∪ {n} = {0..n}"
by (auto simp: set_eq_iff)
have d: "{0} ∩ ({1} ∪ {2..n}) = {}" "{1} ∩ {2..n} = {}" "{0..n - 1} ∩ {n} = {}"
using False by simp_all
have f: "finite {0}" "finite {1}" "finite {2 .. n}"
"finite {0 .. n - 1}" "finite {n}" by simp_all
have "((1 - fps_X) * ?sa) \$ n = sum (λi. (1 - fps_X)\$ i * ?sa \$ (n - i)) {0 .. n}"
also have "… = a\$n"
unfolding th0
unfolding sum.union_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
unfolding sum.union_disjoint[OF f(2) f(3) d(2)]
apply (simp)
unfolding sum.union_disjoint[OF f(4,5) d(3), unfolded u(3)]
apply simp
done
finally show ?thesis
by simp
qed
then show ?thesis
unfolding fps_eq_iff by blast
qed

lemma fps_divide_fps_X_minus1_sum:
"a /((1::'a::field fps) - fps_X) = Abs_fps (λn. sum (λi. a \$ i) {0..n})"
proof -
let ?fps_X = "1 - (fps_X::'a fps)"
have th0: "?fps_X \$ 0 ≠ 0"
by simp
have "a /?fps_X = ?fps_X *  Abs_fps (λn::nat. sum ((\$) a) {0..n}) * inverse ?fps_X"
using fps_divide_fps_X_minus1_sum_lemma[of a, symmetric] th0
also have "… = (inverse ?fps_X * ?fps_X) * Abs_fps (λn::nat. sum ((\$) a) {0..n}) "
finally show ?thesis
qed

subsubsection ‹Rule 4 in its more general form: generalizes Rule 3 for an arbitrary
finite product of FPS, also the relvant instance of powers of a FPS›

definition "natpermute n k = {l :: nat list. length l = k ∧ sum_list l = n}"

lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
apply (case_tac x)
apply auto
done

lemma append_natpermute_less_eq:
assumes "xs @ ys ∈ natpermute n k"
shows "sum_list xs ≤ n"
and "sum_list ys ≤ n"
proof -
from assms have "sum_list (xs @ ys) = n"
then have "sum_list xs + sum_list ys = n"
by simp
then show "sum_list xs ≤ n" and "sum_list ys ≤ n"
by simp_all
qed

lemma natpermute_split:
assumes "h ≤ k"
shows "natpermute n k =
(⋃m ∈{0..n}. {l1 @ l2 |l1 l2. l1 ∈ natpermute m h ∧ l2 ∈ natpermute (n - m) (k - h)})"
(is "?L = ?R" is "_ = (⋃m ∈{0..n}. ?S m)")
proof
show "?R ⊆ ?L"
proof
fix l
assume l: "l ∈ ?R"
from l obtain m xs ys where h: "m ∈ {0..n}"
and xs: "xs ∈ natpermute m h"
and ys: "ys ∈ natpermute (n - m) (k - h)"
and leq: "l = xs@ys" by blast
from xs have xs': "sum_list xs = m"
from ys have ys': "sum_list ys = n - m"
show "l ∈ ?L" using leq xs ys h
unfolding xs' ys'
using assms xs ys
unfolding natpermute_def
apply simp
done
qed
show "?L ⊆ ?R"
proof
fix l
assume l: "l ∈ natpermute n k"
let ?xs = "take h l"
let ?ys = "drop h l"
let ?m = "sum_list ?xs"
from l have ls: "sum_list (?xs @ ?ys) = n"
have xs: "?xs ∈ natpermute ?m h" using l assms
have l_take_drop: "sum_list l = sum_list (take h l @ drop h l)"
by simp
then have ys: "?ys ∈ natpermute (n - ?m) (k - h)"
using l assms ls by (auto simp add: natpermute_def simp del: append_take_drop_id)
from ls have m: "?m ∈ {0..n}"
by (simp add: l_take_drop del: append_take_drop_id)
from xs ys ls show "l ∈ ?R"
apply auto
apply (rule bexI [where x = "?m"])
apply (rule exI [where x = "?xs"])
apply (rule exI [where x = "?ys"])
using ls l
apply (auto simp add: natpermute_def l_take_drop simp del: append_take_drop_id)
apply simp
done
qed
qed

lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"

lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"
apply (auto simp add: set_replicate_conv_if natpermute_def)
apply (rule nth_equalityI)
apply simp_all
done

lemma natpermute_finite: "finite (natpermute n k)"
proof (induct k arbitrary: n)
case 0
then show ?case
apply (subst natpermute_split[of 0 0, simplified])
done
next
case (Suc k)
then show ?case unfolding natpermute_split [of k "Suc k", simplified]
apply -
apply (rule finite_UN_I)
apply simp
unfolding One_nat_def[symmetric] natlist_trivial_1
apply simp
done
qed

lemma natpermute_contain_maximal:
"{xs ∈ natpermute n (k + 1). n ∈ set xs} = (⋃i∈{0 .. k}. {(replicate (k + 1) 0) [i:=n]})"
(is "?A = ?B")
proof
show "?A ⊆ ?B"
proof
fix xs
assume "xs ∈ ?A"
then have H: "xs ∈ natpermute n (k + 1)" and n: "n ∈ set xs"
by blast+
then obtain i where i: "i ∈ {0.. k}" "xs!i = n"
unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def)
have eqs: "({0..k} - {i}) ∪ {i} = {0..k}"
using i by auto
have f: "finite({0..k} - {i})" "finite {i}"
by auto
have d: "({0..k} - {i}) ∩ {i} = {}"
using i by auto
from H have "n = sum (nth xs) {0..k}"
apply (auto simp add: atLeastLessThanSuc_atLeastAtMost sum_list_sum_nth)
done
also have "… = n + sum (nth xs) ({0..k} - {i})"
unfolding sum.union_disjoint[OF f d, unfolded eqs] using i by simp
finally have zxs: "∀ j∈ {0..k} - {i}. xs!j = 0"
by auto
from H have xsl: "length xs = k+1"
from i have i': "i < length (replicate (k+1) 0)"   "i < k+1"
unfolding length_replicate by presburger+
have "xs = replicate (k+1) 0 [i := n]"
apply (rule nth_equalityI)
unfolding xsl length_list_update length_replicate
apply simp
apply clarify
unfolding nth_list_update[OF i'(1)]
using i zxs
apply (case_tac "ia = i")
apply (auto simp del: replicate.simps)
done
then show "xs ∈ ?B" using i by blast
qed
show "?B ⊆ ?A"
proof
fix xs
assume "xs ∈ ?B"
then obtain i where i: "i ∈ {0..k}" and xs: "xs = replicate (k + 1) 0 [i:=n]"
by auto
have nxs: "n ∈ set xs"
unfolding xs
apply (rule set_update_memI)
using i apply simp
done
have xsl: "length xs = k + 1"
by (simp only: xs length_replicate length_list_update)
have "sum_list xs = sum (nth xs) {0..<k+1}"
unfolding sum_list_sum_nth xsl ..
also have "… = sum (λj. if j = i then n else 0) {0..< k+1}"
by (rule sum.cong) (simp_all add: xs del: replicate.simps)
also have "… = n" using i by (simp add: sum.delta)
finally have "xs ∈ natpermute n (k + 1)"
using xsl unfolding natpermute_def mem_Collect_eq by blast
then show "xs ∈ ?A"
using nxs by blast
qed
qed

text ‹The general form.›
lemma fps_prod_nth:
fixes m :: nat
and a :: "nat ⇒ 'a::comm_ring_1 fps"
shows "(prod a {0 .. m}) \$ n =
sum (λv. prod (λj. (a j) \$ (v!j)) {0..m}) (natpermute n (m+1))"
(is "?P m n")
proof (induct m arbitrary: n rule: nat_less_induct)
fix m n assume H: "∀m' < m. ∀n. ?P m' n"
show "?P m n"
proof (cases m)
case 0
then show ?thesis
apply simp
unfolding natlist_trivial_1[where n = n, unfolded One_nat_def]
apply simp
done
next
case (Suc k)
then have km: "k < m" by arith
have u0: "{0 .. k} ∪ {m} = {0..m}"
using Suc by (simp add: set_eq_iff) presburger
have f0: "finite {0 .. k}" "finite {m}" by auto
have d0: "{0 .. k} ∩ {m} = {}" using Suc by auto
have "(prod a {0 .. m}) \$ n = (prod a {0 .. k} * a m) \$ n"
unfolding prod.union_disjoint[OF f0 d0, unfolded u0] by simp
also have "… = (∑i = 0..n. (∑v∈natpermute i (k + 1). ∏j∈{0..k}. a j \$ v ! j) * a m \$ (n - i))"
unfolding fps_mult_nth H[rule_format, OF km] ..
also have "… = (∑v∈natpermute n (m + 1). ∏j∈{0..m}. a j \$ v ! j)"
unfolding natpermute_split[of m "m + 1", simplified, of n,
unfolded natlist_trivial_1[unfolded One_nat_def] Suc]
apply (subst sum.UNION_disjoint)
apply simp
apply simp
unfolding image_Collect[symmetric]
apply clarsimp
apply (rule finite_imageI)
apply (rule natpermute_finite)
apply auto
apply (rule sum.cong)
apply (rule refl)
unfolding sum_distrib_right
apply (rule sym)
apply (rule_tac l = "λxs. xs @ [n - x]" in sum.reindex_cong)
apply auto
unfolding prod.union_disjoint[OF f0 d0, unfolded u0, unfolded Suc]
apply (clarsimp simp add: natpermute_def nth_append)
done
finally show ?thesis .
qed
qed

text ‹The special form for powers.›
lemma fps_power_nth_Suc:
fixes m :: nat
and a :: "'a::comm_ring_1 fps"
shows "(a ^ Suc m)\$n = sum (λv. prod (λj. a \$ (v!j)) {0..m}) (natpermute n (m+1))"
proof -
have th0: "a^Suc m = prod (λi. a) {0..m}"
show ?thesis unfolding th0 fps_prod_nth ..
qed

lemma fps_power_nth:
fixes m :: nat
and a :: "'a::comm_ring_1 fps"
shows "(a ^m)\$n =
(if m=0 then 1\$n else sum (λv. prod (λj. a \$ (v!j)) {0..m - 1}) (natpermute n m))"
by (cases m) (simp_all add: fps_power_nth_Suc del: power_Suc)

lemma fps_nth_power_0:
fixes m :: nat
and a :: "'a::comm_ring_1 fps"
shows "(a ^m)\$0 = (a\$0) ^ m"
proof (cases m)
case 0
then show ?thesis by simp
next
case (Suc n)
then have c: "m = card {0..n}" by simp
have "(a ^m)\$0 = prod (λi. a\$0) {0..n}"
by (simp add: Suc fps_power_nth del: replicate.simps power_Suc)
also have "… = (a\$0) ^ m"
unfolding c by (rule prod_constant)
finally show ?thesis .
qed

lemma natpermute_max_card:
assumes n0: "n ≠ 0"
shows "card {xs ∈ natpermute n (k + 1). n ∈ set xs} = k + 1"
unfolding natpermute_contain_maximal
proof -
let ?A = "λi. {replicate (k + 1) 0[i := n]}"
let ?K = "{0 ..k}"
have fK: "finite ?K"
by simp
have fAK: "∀i∈?K. finite (?A i)"
by auto
have d: "∀i∈ ?K. ∀j∈ ?K. i ≠ j ⟶
{replicate (k + 1) 0[i := n]} ∩ {replicate (k + 1) 0[j := n]} = {}"
proof clarify
fix i j
assume i: "i ∈ ?K" and j: "j ∈ ?K" and ij: "i ≠ j"
have False if eq: "replicate (k+1) 0 [i:=n] = replicate (k+1) 0 [j:= n]"
proof -
have "(replicate (k+1) 0 [i:=n] ! i) = n"
using i by (simp del: replicate.simps)
moreover
have "(replicate (k+1) 0 [j:=n] ! i) = 0"
using i ij by (simp del: replicate.simps)
ultimately show ?thesis
using eq n0 by (simp del: replicate.simps)
qed
then show "{replicate (k + 1) 0[i := n]} ∩ {replicate (k + 1) 0[j := n]} = {}"
by auto
qed
from card_UN_disjoint[OF fK fAK d]
show "card (⋃i∈{0..k}. {replicate (k + 1) 0[i := n]}) = k + 1"
by simp
qed

lemma fps_power_Suc_nth:
fixes f :: "'a :: comm_ring_1 fps"
assumes k: "k > 0"
shows "(f ^ Suc m) \$ k =
of_nat (Suc m) * (f \$ k * (f \$ 0) ^ m) +
(∑v∈{v∈natpermute k (m+1). k ∉ set v}. ∏j = 0..m. f \$ v ! j)"
proof -
define A B
where "A = {v∈natpermute k (m+1). k ∈ set v}"
and  "B = {v∈natpermute k (m+1). k ∉ set v}"
have [simp]: "finite A" "finite B" "A ∩ B = {}" by (auto simp: A_def B_def natpermute_finite)

from natpermute_max_card[of k m] k have card_A: "card A = m + 1" by (simp add: A_def)
{
fix v assume v: "v ∈ A"
from v have [simp]: "length v = Suc m" by (simp add: A_def natpermute_def)
from v have "∃j. j ≤ m ∧ v ! j = k"
by (auto simp: set_conv_nth A_def natpermute_def less_Suc_eq_le)
then guess j by (elim exE conjE) note j = this

from v have "k = sum_list v" by (simp add: A_def natpermute_def)
also have "… = (∑i=0..m. v ! i)"
by (simp add: sum_list_sum_nth atLeastLessThanSuc_atLeastAtMost del: sum_op_ivl_Suc)
also from j have "{0..m} = insert j ({0..m}-{j})" by auto
also from j have "(∑i∈…. v ! i) = k + (∑i∈{0..m}-{j}. v ! i)"
by (subst sum.insert) simp_all
finally have "(∑i∈{0..m}-{j}. v ! i) = 0" by simp
hence zero: "v ! i = 0" if "i ∈ {0..m}-{j}" for i using that
by (subst (asm) sum_eq_0_iff) auto

from j have "{0..m} = insert j ({0..m} - {j})" by auto
also from j have "(∏i∈…. f \$ (v ! i)) = f \$ k * (∏i∈{0..m} - {j}. f \$ (v ! i))"
by (subst prod.insert) auto
also have "(∏i∈{0..m} - {j}. f \$ (v ! i)) = (∏i∈{0..m} - {j}. f \$ 0)"
by (intro prod.cong) (simp_all add: zero)
also from j have "… = (f \$ 0) ^ m" by (subst prod_constant) simp_all
finally have "(∏j = 0..m. f \$ (v ! j)) = f \$ k * (f \$ 0) ^ m" .
} note A = this

have "(f ^ Suc m) \$ k = (∑v∈natpermute k (m + 1). ∏j = 0..m. f \$ v ! j)"
by (rule fps_power_nth_Suc)
also have "natpermute k (m+1) = A ∪ B" unfolding A_def B_def by blast
also have "(∑v∈…. ∏j = 0..m. f \$ (v ! j)) =
(∑v∈A. ∏j = 0..m. f \$ (v ! j)) + (∑v∈B. ∏j = 0..m. f \$ (v ! j))"
by (intro sum.union_disjoint) simp_all
also have "(∑v∈A. ∏j = 0..m. f \$ (v ! j)) = of_nat (Suc m) * (f \$ k * (f \$ 0) ^ m)"
finally show ?thesis by (simp add: B_def)
qed

lemma fps_power_Suc_eqD:
fixes f g :: "'a :: {idom,semiring_char_0} fps"
assumes "f ^ Suc m = g ^ Suc m" "f \$ 0 = g \$ 0" "f \$ 0 ≠ 0"
shows   "f = g"
proof (rule fps_ext)
fix k :: nat
show "f \$ k = g \$ k"
proof (induction k rule: less_induct)
case (less k)
show ?case
proof (cases "k = 0")
case False
let ?h = "λf. (∑v | v ∈ natpermute k (m + 1) ∧ k ∉ set v. ∏j = 0..m. f \$ v ! j)"
from False fps_power_Suc_nth[of k f m] fps_power_Suc_nth[of k g m]
have "f \$ k * (of_nat (Suc m) * (f \$ 0) ^ m) + ?h f =
g \$ k * (of_nat (Suc m) * (f \$ 0) ^ m) + ?h g" using assms
by (simp add: mult_ac del: power_Suc of_nat_Suc)
also have "v ! i < k" if "v ∈ {v∈natpermute k (m+1). k ∉ set v}" "i ≤ m" for v i
using that elem_le_sum_list[of i v] unfolding natpermute_def
by (auto simp: set_conv_nth dest!: spec[of _ i])
hence "?h f = ?h g"
by (intro sum.cong refl prod.cong less lessI) (auto simp: natpermute_def)
finally have "f \$ k * (of_nat (Suc m) * (f \$ 0) ^ m) = g \$ k * (of_nat (Suc m) * (f \$ 0) ^ m)"
by simp
with assms show "f \$ k = g \$ k"
by (subst (asm) mult_right_cancel) (auto simp del: of_nat_Suc)
qed
qed

lemma fps_power_Suc_eqD':
fixes f g :: "'a :: {idom,semiring_char_0} fps"
assumes "f ^ Suc m = g ^ Suc m" "f \$ subdegree f = g \$ subdegree g"
shows   "f = g"
proof (cases "f = 0")
case False
have "Suc m * subdegree f = subdegree (f ^ Suc m)"
by (rule subdegree_power [symmetric])
also have "f ^ Suc m = g ^ Suc m" by fact
also have "subdegree … = Suc m * subdegree g" by (rule subdegree_power)
finally have [simp]: "subdegree f = subdegree g"
by (subst (asm) Suc_mult_cancel1)
have "fps_shift (subdegree f) f * fps_X ^ subdegree f = f"
by (rule subdegree_decompose [symmetric])
also have "… ^ Suc m = g ^ Suc m" by fact
also have "g = fps_shift (subdegree g) g * fps_X ^ subdegree g"
by (rule subdegree_decompose)
also have "subdegree f = subdegree g" by fact
finally have "fps_shift (subdegree g) f ^ Suc m = fps_shift (subdegree g) g ^ Suc m"
by (simp add: algebra_simps power_mult_distrib del: power_Suc)
hence "fps_shift (subdegree g) f = fps_shift (subdegree g) g"
by (rule fps_power_Suc_eqD) (insert assms False, auto)
with subdegree_decompose[of f] subdegree_decompose[of g] show ?thesis by simp
qed (insert assms, simp_all)

lemma fps_power_eqD':
fixes f g :: "'a :: {idom,semiring_char_0} fps"
assumes "f ^ m = g ^ m" "f \$ subdegree f = g \$ subdegree g" "m > 0"
shows   "f = g"
using fps_power_Suc_eqD'[of f "m-1" g] assms by simp

lemma fps_power_eqD:
fixes f g :: "'a :: {idom,semiring_char_0} fps"
assumes "f ^ m = g ^ m" "f \$ 0 = g \$ 0" "f \$ 0 ≠ 0" "m > 0"
shows   "f = g"
by (rule fps_power_eqD'[of f m g]) (insert assms, simp_all)

lemma fps_compose_inj_right:
assumes a0: "a\$0 = (0::'a::idom)"
and a1: "a\$1 ≠ 0"
shows "(b oo a = c oo a) ⟷ b = c"
(is "?lhs ⟷?rhs")
proof
show ?lhs if ?rhs using that by simp
show ?rhs if ?lhs
proof -
have "b\$n = c\$n" for n
proof (induct n rule: nat_less_induct)
fix n
assume H: "∀m<n. b\$m = c\$m"
show "b\$n = c\$n"
proof (cases n)
case 0
from ‹?lhs› have "(b oo a)\$n = (c oo a)\$n"
by simp
then show ?thesis
using 0 by (simp add: fps_compose_nth)
next
case (Suc n1)
have f: "finite {0 .. n1}" "finite {n}" by simp_all
have eq: "{0 .. n1} ∪ {n} = {0 .. n}" using Suc by auto
have d: "{0 .. n1} ∩ {n} = {}" using Suc by auto
have seq: "(∑i = 0..n1. b \$ i * a ^ i \$ n) = (∑i = 0..n1. c \$ i * a ^ i \$ n)"
apply (rule sum.cong)
using H Suc
apply auto
done
have th0: "(b oo a) \$n = (∑i = 0..n1. c \$ i * a ^ i \$ n) + b\$n * (a\$1)^n"
unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq] seq
using startsby_zero_power_nth_same[OF a0]
by simp
have th1: "(c oo a) \$n = (∑i = 0..n1. c \$ i * a ^ i \$ n) + c\$n * (a\$1)^n"
unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq]
using startsby_zero_power_nth_same[OF a0]
by simp
from ‹?lhs›[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
show ?thesis by auto
qed
qed
then show ?rhs by (simp add: fps_eq_iff)
qed
qed

declare prod.cong [fundef_cong]

function radical :: "(nat ⇒ 'a ⇒ 'a) ⇒ nat ⇒ 'a::field fps ⇒ nat ⇒ 'a"
where
"radical r 0 a 0 = 1"
| "radical r 0 a (Suc n) = 0"
| "radical r (Suc k) a 0 = r (Suc k) (a\$0)"
| "radical r (Suc k) a (Suc n) =
(a\$ Suc n - sum (λxs. prod (λj. radical r (Suc k) a (xs ! j)) {0..k})
{xs. xs ∈ natpermute (Suc n) (Suc k) ∧ Suc n ∉ set xs}) /
(of_nat (Suc k) * (radical r (Suc k) a 0)^k)"
by pat_completeness auto

proof
let ?R = "measure (λ(r, k, a, n). n)"
{
show "wf ?R" by auto
next
fix r k a n xs i
assume xs: "xs ∈ {xs ∈ natpermute (Suc n) (Suc k). Suc n ∉ set xs}" and i: "i ∈ {0..k}"
have False if c: "Suc n ≤ xs ! i"
proof -
from xs i have "xs !i ≠ Suc n"
by (auto simp add: in_set_conv_nth natpermute_def)
with c have c': "Suc n < xs!i" by arith
have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}"
by simp_all
have d: "{0 ..< i} ∩ ({i} ∪ {i+1 ..< Suc k}) = {}" "{i} ∩ {i+1..< Suc k} = {}"
by auto
have eqs: "{0..<Suc k} = {0 ..< i} ∪ ({i} ∪ {i+1 ..< Suc k})"
using i by auto
from xs have "Suc n = sum_list xs"
also have "… = sum (nth xs) {0..<Suc k}" using xs
also have "… = xs!i + sum (nth xs) {0..<i} + sum (nth xs) {i+1..<Suc k}"
unfolding eqs  sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)]
by simp
finally show ?thesis using c' by simp
qed
then show "((r, Suc k, a, xs!i), r, Suc k, a, Suc n) ∈ ?R"
apply auto
apply (metis not_less)
done
next
fix r k a n
show "((r, Suc k, a, 0), r, Suc k, a, Suc n) ∈ ?R" by simp
}
qed

apply (case_tac n)
apply auto
done

lemma fps_radical_nth_0[simp]: "fps_radical r n a \$ 0 = (if n = 0 then 1 else r n (a\$0))"

assumes r: "(r k (a\$0)) ^ k = a\$0"
shows "fps_radical r k a ^ k \$ 0 = (if k = 0 then 1 else a\$0)"
proof (cases k)
case 0
then show ?thesis by simp
next
case (Suc h)
have eq1: "fps_radical r k a ^ k \$ 0 = (∏j∈{0..h}. fps_radical r k a \$ (replicate k 0) ! j)"
unfolding fps_power_nth Suc by simp
also have "… = (∏j∈{0..h}. r k (a\$0))"
apply (rule prod.cong)
apply simp
using Suc
apply (subgoal_tac "replicate k 0 ! x = 0")
apply (auto intro: nth_replicate simp del: replicate.simps)
done
also have "… = a\$0"
using r Suc by (simp add: prod_constant)
finally show ?thesis
using Suc by simp
qed

fixes a:: "'a::field_char_0 fps"
assumes a0: "a\$0 ≠ 0"
shows "(r (Suc k) (a\$0)) ^ Suc k = a\$0 ⟷ (fps_radical r (Suc k) a) ^ (Suc k) = a"
(is "?lhs ⟷ ?rhs")
proof
let ?r = "fps_radical r (Suc k) a"
show ?rhs if r0: ?lhs
proof -
from a0 r0 have r00: "r (Suc k) (a\$0) ≠ 0" by auto
have "?r ^ Suc k \$ z = a\$z" for z
proof (induct z rule: nat_less_induct)
fix n
assume H: "∀m<n. ?r ^ Suc k \$ m = a\$m"
show "?r ^ Suc k \$ n = a \$n"
proof (cases n)
case 0
then show ?thesis
using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp
next
case (Suc n1)
then have "n ≠ 0" by simp
let ?Pnk = "natpermute n (k + 1)"
let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}"
let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}"
have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast
have d: "?Pnkn ∩ ?Pnknn = {}" by blast
have f: "finite ?Pnkn" "finite ?Pnknn"
using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
by (metis natpermute_finite)+
let ?f = "λv. ∏j∈{0..k}. ?r \$ v ! j"
have "sum ?f ?Pnkn = sum (λv. ?r \$ n * r (Suc k) (a \$ 0) ^ k) ?Pnkn"
proof (rule sum.cong)
fix v assume v: "v ∈ {xs ∈ natpermute n (k + 1). n ∈ set xs}"
let ?ths = "(∏j∈{0..k}. fps_radical r (Suc k) a \$ v ! j) =
fps_radical r (Suc k) a \$ n * r (Suc k) (a \$ 0) ^ k"
from v obtain i where i: "i ∈ {0..k}" "v = replicate (k+1) 0 [i:= n]"
unfolding natpermute_contain_maximal by auto
have "(∏j∈{0..k}. fps_radical r (Suc k) a \$ v ! j) =
(∏j∈{0..k}. if j = i then fps_radical r (Suc k) a \$ n else r (Suc k) (a\$0))"
apply (rule prod.cong, simp)
using i r0
apply (simp del: replicate.simps)
done
also have "… = (fps_radical r (Suc k) a \$ n) * r (Suc k) (a\$0) ^ k"
using i r0 by (simp add: prod_gen_delta)
finally show ?ths .
qed rule
then have "sum ?f ?Pnkn = of_nat (k+1) * ?r \$ n * r (Suc k) (a \$ 0) ^ k"
by (simp add: natpermute_max_card[OF ‹n ≠ 0›, simplified])
also have "… = a\$n - sum ?f ?Pnknn"
unfolding Suc using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc)
finally have fn: "sum ?f ?Pnkn = a\$n - sum ?f ?Pnknn" .
have "(?r ^ Suc k)\$n = sum ?f ?Pnkn + sum ?f ?Pnknn"
unfolding fps_power_nth_Suc sum.union_disjoint[OF f d, unfolded eq] ..
also have "… = a\$n" unfolding fn by simp
finally show ?thesis .
qed
qed
then show ?thesis using r0 by (simp add: fps_eq_iff)
qed
show ?lhs if ?rhs
proof -
from that have "((fps_radical r (Suc k) a) ^ (Suc k))\$0 = a\$0"
by simp
then show ?thesis
unfolding fps_power_nth_Suc
by (simp add: prod_constant del: replicate.simps)
qed
qed

(*
fixes a:: "'a::field_char_0 fps"
assumes r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0" and a0: "a\$0 ≠ 0"
shows "(fps_radical r (Suc k) a) ^ (Suc k) = a"
proof-
let ?r = "fps_radical r (Suc k) a"
from a0 r0 have r00: "r (Suc k) (a\$0) ≠ 0" by auto
{fix z have "?r ^ Suc k \$ z = a\$z"
proof(induct z rule: nat_less_induct)
fix n assume H: "∀m<n. ?r ^ Suc k \$ m = a\$m"
{assume "n = 0" then have "?r ^ Suc k \$ n = a \$n"
using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp}
moreover
{fix n1 assume n1: "n = Suc n1"
have fK: "finite {0..k}" by simp
have nz: "n ≠ 0" using n1 by arith
let ?Pnk = "natpermute n (k + 1)"
let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}"
let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}"
have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast
have d: "?Pnkn ∩ ?Pnknn = {}" by blast
have f: "finite ?Pnkn" "finite ?Pnknn"
using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
by (metis natpermute_finite)+
let ?f = "λv. ∏j∈{0..k}. ?r \$ v ! j"
have "sum ?f ?Pnkn = sum (λv. ?r \$ n * r (Suc k) (a \$ 0) ^ k) ?Pnkn"
proof(rule sum.cong2)
fix v assume v: "v ∈ {xs ∈ natpermute n (k + 1). n ∈ set xs}"
let ?ths = "(∏j∈{0..k}. fps_radical r (Suc k) a \$ v ! j) = fps_radical r (Suc k) a \$ n * r (Suc k) (a \$ 0) ^ k"
from v obtain i where i: "i ∈ {0..k}" "v = replicate (k+1) 0 [i:= n]"
unfolding natpermute_contain_maximal by auto
have "(∏j∈{0..k}. fps_radical r (Suc k) a \$ v ! j) = (∏j∈{0..k}. if j = i then fps_radical r (Suc k) a \$ n else r (Suc k) (a\$0))"
apply (rule prod.cong, simp)
using i r0 by (simp del: replicate.simps)
also have "… = (fps_radical r (Suc k) a \$ n) * r (Suc k) (a\$0) ^ k"
unfolding prod_gen_delta[OF fK] using i r0 by simp
finally show ?ths .
qed
then have "sum ?f ?Pnkn = of_nat (k+1) * ?r \$ n * r (Suc k) (a \$ 0) ^ k"
by (simp add: natpermute_max_card[OF nz, simplified])
also have "… = a\$n - sum ?f ?Pnknn"
unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc )
finally have fn: "sum ?f ?Pnkn = a\$n - sum ?f ?Pnknn" .
have "(?r ^ Suc k)\$n = sum ?f ?Pnkn + sum ?f ?Pnknn"
unfolding fps_power_nth_Suc sum.union_disjoint[OF f d, unfolded eq] ..
also have "… = a\$n" unfolding fn by simp
finally have "?r ^ Suc k \$ n = a \$n" .}
ultimately  show "?r ^ Suc k \$ n = a \$n" by (cases n, auto)
qed }
then show ?thesis by (simp add: fps_eq_iff)
qed

*)
lemma eq_divide_imp':
fixes c :: "'a::field"
shows "c ≠ 0 ⟹ a * c = b ⟹ a = b / c"

assumes r0: "(r (Suc k) (b\$0)) ^ Suc k = b\$0"
and a0: "r (Suc k) (b\$0 ::'a::field_char_0) = a\$0"
and b0: "b\$0 ≠ 0"
shows "a^(Suc k) = b ⟷ a = fps_radical r (Suc k) b"
(is "?lhs ⟷ ?rhs" is "_ ⟷ a = ?r")
proof
show ?lhs if ?rhs
using that using power_radical[OF b0, of r k, unfolded r0] by simp
show ?rhs if ?lhs
proof -
have r00: "r (Suc k) (b\$0) ≠ 0" using b0 r0 by auto
have ceq: "card {0..k} = Suc k" by simp
from a0 have a0r0: "a\$0 = ?r\$0" by simp
have "a \$ n = ?r \$ n" for n
proof (induct n rule: nat_less_induct)
fix n
assume h: "∀m<n. a\$m = ?r \$m"
show "a\$n = ?r \$ n"
proof (cases n)
case 0
then show ?thesis using a0 by simp
next
case (Suc n1)
have fK: "finite {0..k}" by simp
have nz: "n ≠ 0" using Suc by simp
let ?Pnk = "natpermute n (Suc k)"
let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}"
let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}"
have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast
have d: "?Pnkn ∩ ?Pnknn = {}" by blast
have f: "finite ?Pnkn" "finite ?Pnknn"
using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
by (metis natpermute_finite)+
let ?f = "λv. ∏j∈{0..k}. ?r \$ v ! j"
let ?g = "λv. ∏j∈{0..k}. a \$ v ! j"
have "sum ?g ?Pnkn = sum (λv. a \$ n * (?r\$0)^k) ?Pnkn"
proof (rule sum.cong)
fix v
assume v: "v ∈ {xs ∈ natpermute n (Suc k). n ∈ set xs}"
let ?ths = "(∏j∈{0..k}. a \$ v ! j) = a \$ n * (?r\$0)^k"
from v obtain i where i: "i ∈ {0..k}" "v = replicate (k+1) 0 [i:= n]"
unfolding Suc_eq_plus1 natpermute_contain_maximal
by (auto simp del: replicate.simps)
have "(∏j∈{0..k}. a \$ v ! j) = (∏j∈{0..k}. if j = i then a \$ n else r (Suc k) (b\$0))"
apply (rule prod.cong, simp)
using i a0
apply (simp del: replicate.simps)
done
also have "… = a \$ n * (?r \$ 0)^k"
using i by (simp add: prod_gen_delta)
finally show ?ths .
qed rule
then have th0: "sum ?g ?Pnkn = of_nat (k+1) * a \$ n * (?r \$ 0)^k"
by (simp add: natpermute_max_card[OF nz, simplified])
have th1: "sum ?g ?Pnknn = sum ?f ?Pnknn"
proof (rule sum.cong, rule refl, rule prod.cong, simp)
fix xs i
assume xs: "xs ∈ ?Pnknn" and i: "i ∈ {0..k}"
have False if c: "n ≤ xs ! i"
proof -
from xs i have "xs ! i ≠ n"
by (auto simp add: in_set_conv_nth natpermute_def)
with c have c': "n < xs!i" by arith
have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}"
by simp_all
have d: "{0 ..< i} ∩ ({i} ∪ {i+1 ..< Suc k}) = {}" "{i} ∩ {i+1..< Suc k} = {}"
by auto
have eqs: "{0..<Suc k} = {0 ..< i} ∪ ({i} ∪ {i+1 ..< Suc k})"
using i by auto
from xs have "n = sum_list xs"
also have "… = sum (nth xs) {0..<Suc k}"
using xs by (simp add: natpermute_def sum_list_sum_nth)
also have "… = xs!i + sum (nth xs) {0..<i} + sum (nth xs) {i+1..<Suc k}"
unfolding eqs  sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)]
by simp
finally show ?thesis using c' by simp
qed
then have thn: "xs!i < n" by presburger
from h[rule_format, OF thn] show "a\$(xs !i) = ?r\$(xs!i)" .
qed
have th00: "⋀x::'a. of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x"
by (simp add: field_simps del: of_nat_Suc)
from ‹?lhs› have "b\$n = a^Suc k \$ n"
also have "a ^ Suc k\$n = sum ?g ?Pnkn + sum ?g ?Pnknn"
unfolding fps_power_nth_Suc
using sum.union_disjoint[OF f d, unfolded Suc_eq_plus1[symmetric],
unfolded eq, of ?g] by simp
also have "… = of_nat (k+1) * a \$ n * (?r \$ 0)^k + sum ?f ?Pnknn"
unfolding th0 th1 ..
finally have "of_nat (k+1) * a \$ n * (?r \$ 0)^k = b\$n - sum ?f ?Pnknn"
by simp
then have "a\$n = (b\$n - sum ?f ?Pnknn) / (of_nat (k+1) * (?r \$ 0)^k)"
apply -
apply (rule eq_divide_imp')
using r00
apply (simp del: of_nat_Suc)
done
then show ?thesis
apply (simp del: of_nat_Suc)
apply (simp add: field_simps Suc th00 del: of_nat_Suc)
done
qed
qed
then show ?rhs by (simp add: fps_eq_iff)
qed
qed

assumes r0: "r (Suc k) ((a\$0) ^ Suc k) = a\$0"
and a0: "(a\$0 :: 'a::field_char_0) ≠ 0"
shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
proof -
let ?ak = "a^ Suc k"
have ak0: "?ak \$ 0 = (a\$0) ^ Suc k"
by (simp add: fps_nth_power_0 del: power_Suc)
from r0 have th0: "r (Suc k) (a ^ Suc k \$ 0) ^ Suc k = a ^ Suc k \$ 0"
using ak0 by auto
from r0 ak0 have th1: "r (Suc k) (a ^ Suc k \$ 0) = a \$ 0"
by auto
from ak0 a0 have ak00: "?ak \$ 0 ≠0 "
by auto
from radical_unique[of r k ?ak a, OF th0 th1 ak00] show ?thesis
by metis
qed

fixes a :: "'a::field_char_0 fps"
assumes r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0"
and a0: "a\$0 ≠ 0"
shows "fps_deriv (fps_radical r (Suc k) a) =
fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
proof -
let ?r = "fps_radical r (Suc k) a"
let ?w = "(fps_const (of_nat (Suc k)) * ?r ^ k)"
from a0 r0 have r0': "r (Suc k) (a\$0) ≠ 0"
by auto
from r0' have w0: "?w \$ 0 ≠ 0"
by (simp del: of_nat_Suc)
note th0 = inverse_mult_eq_1[OF w0]
let ?iw = "inverse ?w"
from iffD1[OF power_radical[of a r], OF a0 r0]
have "fps_deriv (?r ^ Suc k) = fps_deriv a"
by simp
then have "fps_deriv ?r * ?w = fps_deriv a"
by (simp add: fps_deriv_power ac_simps del: power_Suc)
then have "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a"
by simp
with a0 r0 have "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w"
by (subst fps_divide_unit) (auto simp del: of_nat_Suc)
then show ?thesis unfolding th0 by simp
qed

fixes a :: "'a::field_char_0 fps"
assumes k: "k > 0"
and ra0: "r k (a \$ 0) ^ k = a \$ 0"
and rb0: "r k (b \$ 0) ^ k = b \$ 0"
and a0: "a \$ 0 ≠ 0"
and b0: "b \$ 0 ≠ 0"
shows "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0) ⟷
(is "?lhs ⟷ ?rhs")
proof
show ?rhs if r0': ?lhs
proof -
from r0' have r0: "(r k ((a * b) \$ 0)) ^ k = (a * b) \$ 0"
by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
show ?thesis
proof (cases k)
case 0
then show ?thesis using r0' by simp
next
case (Suc h)
let ?ra = "fps_radical r (Suc h) a"
let ?rb = "fps_radical r (Suc h) b"
have th0: "r (Suc h) ((a * b) \$ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) \$ 0"
using r0' Suc by (simp add: fps_mult_nth)
have ab0: "(a*b) \$ 0 ≠ 0"
using a0 b0 by (simp add: fps_mult_nth)
from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded Suc] th0 ab0, symmetric]
iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded Suc]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded Suc]] Suc r0'
show ?thesis
by (auto simp add: power_mult_distrib simp del: power_Suc)
qed
qed
show ?lhs if ?rhs
proof -
from that have "(fps_radical r k (a * b)) \$ 0 = (fps_radical r k a * fps_radical r k b) \$ 0"
by simp
then show ?thesis
using k by (simp add: fps_mult_nth)
qed
qed

(*
fixes a:: "'a::field_char_0 fps"
assumes
ra0: "r k (a \$ 0) ^ k = a \$ 0"
and rb0: "r k (b \$ 0) ^ k = b \$ 0"
and r0': "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0)"
and a0: "a\$0 ≠ 0"
and b0: "b\$0 ≠ 0"
proof-
from r0' have r0: "(r (k) ((a*b)\$0)) ^ k = (a*b)\$0"
by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
{assume "k=0" then have ?thesis by simp}
moreover
{fix h assume k: "k = Suc h"
let ?ra = "fps_radical r (Suc h) a"
let ?rb = "fps_radical r (Suc h) b"
have th0: "r (Suc h) ((a * b) \$ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) \$ 0"
using r0' k by (simp add: fps_mult_nth)
have ab0: "(a*b) \$ 0 ≠ 0" using a0 b0 by (simp add: fps_mult_nth)
from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric]
power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k
have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)}
ultimately show ?thesis by (cases k, auto)
qed
*)

lemma fps_divide_1 [simp]: "(a :: 'a::field fps) / 1 = a"
by (fact div_by_1)

fixes a :: "'a::field_char_0 fps"
assumes kp: "k > 0"
and ra0: "(r k (a \$ 0)) ^ k = a \$ 0"
and rb0: "(r k (b \$ 0)) ^ k = b \$ 0"
and a0: "a\$0 ≠ 0"
and b0: "b\$0 ≠ 0"
shows "r k ((a \$ 0) / (b\$0)) = r k (a\$0) / r k (b \$ 0) ⟷
(is "?lhs = ?rhs")
proof
let ?r = "fps_radical r k"
from kp obtain h where k: "k = Suc h"
by (cases k) auto
have ra0': "r k (a\$0) ≠ 0" using a0 ra0 k by auto
have rb0': "r k (b\$0) ≠ 0" using b0 rb0 k by auto

show ?lhs if ?rhs
proof -
from that have "?r (a/b) \$ 0 = (?r a / ?r b)\$0"
by simp
then show ?thesis
using k a0 b0 rb0' by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse)
qed
show ?rhs if ?lhs
proof -
from a0 b0 have ab0[simp]: "(a/b)\$0 = a\$0 / b\$0"
by (simp add: fps_divide_def fps_mult_nth divide_inverse fps_inverse_def)
have th0: "r k ((a/b)\$0) ^ k = (a/b)\$0"
by (simp add: ‹?lhs› power_divide ra0 rb0)
from a0 b0 ra0' rb0' kp ‹?lhs›
have th1: "r k ((a / b) \$ 0) = (fps_radical r k a / fps_radical r k b) \$ 0"
by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse)
from a0 b0 ra0' rb0' kp have ab0': "(a / b) \$ 0 ≠ 0"
by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def nonzero_imp_inverse_nonzero)
note tha[simp] = iffD1[OF power_radical[where r=r and k=h], OF a0 ra0[unfolded k], unfolded k[symmetric]]
note thb[simp] = iffD1[OF power_radical[where r=r and k=h], OF b0 rb0[unfolded k], unfolded k[symmetric]]
from b0 rb0' have th2: "(?r a / ?r b)^k = a/b"
by (simp add: fps_divide_unit power_mult_distrib fps_inverse_power[symmetric])

from iffD1[OF radical_unique[where r=r and a="?r a / ?r b" and b="a/b" and k=h], symmetric, unfolded k[symmetric], OF th0 th1 ab0' th2]
show ?thesis .
qed
qed

fixes a :: "'a::field_char_0 fps"
assumes k: "k > 0"
and ra0: "r k (a \$ 0) ^ k = a \$ 0"
and r1: "(r k 1)^k = 1"
and a0: "a\$0 ≠ 0"
shows "r k (inverse (a \$ 0)) = r k 1 / (r k (a \$ 0)) ⟷
using radical_divide[where k=k and r=r and a=1 and b=a, OF k ] ra0 r1 a0

subsection ‹Derivative of composition›

lemma fps_compose_deriv:
fixes a :: "'a::idom fps"
assumes b0: "b\$0 = 0"
shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * fps_deriv b"
proof -
have "(fps_deriv (a oo b))\$n = (((fps_deriv a) oo b) * (fps_deriv b)) \$n" for n
proof -
have "(fps_deriv (a oo b))\$n = sum (λi. a \$ i * (fps_deriv (b^i))\$n) {0.. Suc n}"
by (simp add: fps_compose_def field_simps sum_distrib_left del: of_nat_Suc)
also have "… = sum (λi. a\$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))\$n) {0.. Suc n}"
by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
also have "… = sum (λi. of_nat i * a\$i * (((b^(i - 1)) * fps_deriv b))\$n) {0.. Suc n}"
unfolding fps_mult_left_const_nth  by (simp add: field_simps)
also have "… = sum (λi. of_nat i * a\$i * (sum (λj. (b^ (i - 1))\$j * (fps_deriv b)\$(n - j)) {0..n})) {0.. Suc n}"
unfolding fps_mult_nth ..
also have "… = sum (λi. of_nat i * a\$i * (sum (λj. (b^ (i - 1))\$j * (fps_deriv b)\$(n - j)) {0..n})) {1.. Suc n}"
apply (rule sum.mono_neutral_right)
apply (auto simp add: mult_delta_left sum.delta not_le)
done
also have "… = sum (λi. of_nat (i + 1) * a\$(i+1) * (sum (λj. (b^ i)\$j * of_nat (n - j + 1) * b\$(n - j + 1)) {0..n})) {0.. n}"
unfolding fps_deriv_nth
by (rule sum.reindex_cong [of Suc]) (auto simp add: mult.assoc)
finally have th0: "(fps_deriv (a oo b))\$n =
sum (λi. of_nat (i + 1) * a\$(i+1) * (sum (λj. (b^ i)\$j * of_nat (n - j + 1) * b\$(n - j + 1)) {0..n})) {0.. n}" .

have "(((fps_deriv a) oo b) * (fps_deriv b))\$n = sum (λi. (fps_deriv b)\$ (n - i) * ((fps_deriv a) oo b)\$i) {0..n}"
unfolding fps_mult_nth by (simp add: ac_simps)
also have "… = sum (λi. sum (λj. of_nat (n - i +1) * b\$(n - i + 1) * of_nat (j + 1) * a\$(j+1) * (b^j)\$i) {0..n}) {0..n}"
unfolding fps_deriv_nth fps_compose_nth sum_distrib_left mult.assoc
apply (rule sum.cong)
apply (rule refl)
apply (rule sum.mono_neutral_left)
apply clarify
apply (subgoal_tac "b^i\$x = 0")
apply simp
apply (rule startsby_zero_power_prefix[OF b0, rule_format])
apply simp
done
also have "… = sum (λi. of_nat (i + 1) * a\$(i+1) * (sum (λj. (b^ i)\$j * of_nat (n - j + 1) * b\$(n - j + 1)) {0..n})) {0.. n}"
unfolding sum_distrib_left
apply (subst sum.swap)
apply (rule sum.cong, rule refl)+
apply simp
done
finally show ?thesis
unfolding th0 by simp
qed
then show ?thesis by (simp add: fps_eq_iff)
qed

lemma fps_mult_fps_X_plus_1_nth:
"((1+fps_X)*a) \$n = (if n = 0 then (a\$n :: 'a::comm_ring_1) else a\$n + a\$(n - 1))"
proof (cases n)
case 0
then show ?thesis
next
case (Suc m)
have "((1 + fps_X)*a) \$ n = sum (λi. (1 + fps_X) \$ i * a \$ (n - i)) {0..n}"
also have "… = sum (λi. (1+fps_X)\$i * a\$(n-i)) {0.. 1}"
unfolding Suc by (rule sum.mono_neutral_right) auto
also have "… = (if n = 0 then (a\$n :: 'a::comm_ring_1) else a\$n + a\$(n - 1))"
finally show ?thesis .
qed

subsection ‹Finite FPS (i.e. polynomials) and fps_X›

lemma fps_poly_sum_fps_X:
assumes "∀i > n. a\$i = (0::'a::comm_ring_1)"
shows "a = sum (λi. fps_const (a\$i) * fps_X^i) {0..n}" (is "a = ?r")
proof -
have "a\$i = ?r\$i" for i
unfolding fps_sum_nth fps_mult_left_const_nth fps_X_power_nth
by (simp add: mult_delta_right sum.delta' assms)
then show ?thesis
unfolding fps_eq_iff by blast
qed

subsection ‹Compositional inverses›

fun compinv :: "'a fps ⇒ nat ⇒ 'a::field"
where
"compinv a 0 = fps_X\$0"
| "compinv a (Suc n) =
(fps_X\$ Suc n - sum (λi. (compinv a i) * (a^i)\$Suc n) {0 .. n}) / (a\$1) ^ Suc n"

definition "fps_inv a = Abs_fps (compinv a)"

lemma fps_inv:
assumes a0: "a\$0 = 0"
and a1: "a\$1 ≠ 0"
shows "fps_inv a oo a = fps_X"
proof -
let ?i = "fps_inv a oo a"
have "?i \$n = fps_X\$n" for n
proof (induct n rule: nat_less_induct)
fix n
assume h: "∀m<n. ?i\$m = fps_X\$m"
show "?i \$ n = fps_X\$n"
proof (cases n)
case 0
then show ?thesis using a0
next
case (Suc n1)
have "?i \$ n = sum (λi. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1} + fps_inv a \$ Suc n1 * (a \$ 1)^ Suc n1"
by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
also have "… = sum (λi. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1} +
(fps_X\$ Suc n1 - sum (λi. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1})"
using a0 a1 Suc by (simp add: fps_inv_def)
also have "… = fps_X\$n" using Suc by simp
finally show ?thesis .
qed
qed
then show ?thesis
qed

fun gcompinv :: "'a fps ⇒ 'a fps ⇒ nat ⇒ 'a::field"
where
"gcompinv b a 0 = b\$0"
| "gcompinv b a (Suc n) =
(b\$ Suc n - sum (λi. (gcompinv b a i) * (a^i)\$Suc n) {0 .. n}) / (a\$1) ^ Suc n"

definition "fps_ginv b a = Abs_fps (gcompinv b a)"

lemma fps_ginv:
assumes a0: "a\$0 = 0"
and a1: "a\$1 ≠ 0"
shows "fps_ginv b a oo a = b"
proof -
let ?i = "fps_ginv b a oo a"
have "?i \$n = b\$n" for n
proof (induct n rule: nat_less_induct)
fix n
assume h: "∀m<n. ?i\$m = b\$m"
show "?i \$ n = b\$n"
proof (cases n)
case 0
then show ?thesis using a0
next
case (Suc n1)
have "?i \$ n = sum (λi. (fps_ginv b a \$ i) * (a^i)\$n) {0 .. n1} + fps_ginv b a \$ Suc n1 * (a \$ 1)^ Suc n1"
by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
also have "… = sum (λi. (fps_ginv b a \$ i) * (a^i)\$n) {0 .. n1} +
(b\$ Suc n1 - sum (λi. (fps_ginv b a \$ i) * (a^i)\$n) {0 .. n1})"
using a0 a1 Suc by (simp add: fps_ginv_def)
also have "… = b\$n" using Suc by simp
finally show ?thesis .
qed
qed
then show ?thesis
qed

lemma fps_inv_ginv: "fps_inv = fps_ginv fps_X"
apply (auto simp add: fun_eq_iff fps_eq_iff fps_inv_def fps_ginv_def)
apply (induct_tac n rule: nat_less_induct)
apply auto
apply (case_tac na)
apply simp
apply simp
done

lemma fps_compose_1[simp]: "1 oo a = 1"
by (simp add: fps_eq_iff fps_compose_nth mult_delta_left sum.delta)

lemma fps_compose_0[simp]: "0 oo a = 0"

lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a \$ 0)"
by (auto simp add: fps_eq_iff fps_compose_nth power_0_left sum.neutral)

lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)"
by (simp add: fps_eq_iff fps_compose_nth field_simps sum.distrib)

lemma fps_compose_sum_distrib: "(sum f S) oo a = sum (λi. f i oo a) S"
proof (cases "finite S")
case True
show ?thesis
proof (rule finite_induct[OF True])
show "sum f {} oo a = (∑i∈{}. f i oo a)"
by simp
next
fix x F
assume fF: "finite F"
and xF: "x ∉ F"
and h: "sum f F oo a = sum (λi. f i oo a) F"
show "sum f (insert x F) oo a  = sum (λi. f i oo a) (insert x F)"
qed
next
case False
then show ?thesis by simp
qed

lemma convolution_eq:
"sum (λi. a (i :: nat) * b (n - i)) {0 .. n} =
sum (λ(i,j). a i * b j) {(i,j). i ≤ n ∧ j ≤ n ∧ i + j = n}"
by (rule sum.reindex_bij_witness[where i=fst and j="λi. (i, n - i)"]) auto

lemma product_composition_lemma:
assumes c0: "c\$0 = (0::'a::idom)"
and d0: "d\$0 = 0"
shows "((a oo c) * (b oo d))\$n =
sum (λ(k,m). a\$k * b\$m * (c^k * d^m) \$ n) {(k,m). k + m ≤ n}"  (is "?l = ?r")
proof -
let ?S = "{(k::nat, m::nat). k + m ≤ n}"
have s: "?S ⊆ {0..n} × {0..n}" by (auto simp add: subset_eq)
have f: "finite {(k::nat, m::nat). k + m ≤ n}"
apply (rule finite_subset[OF s])
apply auto
done
have "?r =  sum (λi. sum (λ(k,m). a\$k * (c^k)\$i * b\$m * (d^m) \$ (n - i)) {(k,m). k + m ≤ n}) {0..n}"
apply (subst sum.swap)
apply (rule sum.cong)
done
also have "… = ?l"
apply (simp add: fps_mult_nth fps_compose_nth sum_product)
apply (rule sum.cong)
apply (rule refl)
apply (rule sum.mono_neutral_right[OF f])
apply presburger
apply clarsimp
apply (rule ccontr)
apply (case_tac "x < aa")
apply simp
apply (frule_tac startsby_zero_power_prefix[rule_format, OF c0])
apply blast
apply simp
apply (frule_tac startsby_zero_power_prefix[rule_format, OF d0])
apply blast
done
finally show ?thesis by simp
qed

lemma product_composition_lemma':
assumes c0: "c\$0 = (0::'a::idom)"
and d0: "d\$0 = 0"
shows "((a oo c) * (b oo d))\$n =
sum (λk. sum (λm. a\$k * b\$m * (c^k * d^m) \$ n) {0..n}) {0..n}"  (is "?l = ?r")
unfolding product_composition_lemma[OF c0 d0]
unfolding sum.cartesian_product
apply (rule sum.mono_neutral_left)
apply simp
apply clarsimp
apply (rule ccontr)
apply (subgoal_tac "(c^aa * d^ba) \$ n = 0")
apply simp
unfolding fps_mult_nth
apply (rule sum.neutral)
apply (case_tac "x < aa")
apply (rule startsby_zero_power_prefix[OF c0, rule_format])
apply simp
apply (subgoal_tac "n - x < ba")
apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format])
apply simp
apply arith
done

lemma sum_pair_less_iff:
"sum (λ((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m ≤ n} =
sum (λs. sum (λi. a i * b (s - i) * c s) {0..s}) {0..n}"
(is "?l = ?r")
proof -
let ?KM = "{(k,m). k + m ≤ n}"
let ?f = "λs. UNION {(0::nat)..s} (λi. {(i,s - i)})"
have th0: "?KM = UNION {0..n} ?f"
by auto
show "?l = ?r "
unfolding th0
apply (subst sum.UNION_disjoint)
apply auto
apply (subst sum.UNION_disjoint)
apply auto
done
qed

lemma fps_compose_mult_distrib_lemma:
assumes c0: "c\$0 = (0::'a::idom)"
shows "((a oo c) * (b oo c))\$n = sum (λs. sum (λi. a\$i * b\$(s - i) * (c^s) \$ n) {0..s}) {0..n}"
unfolding sum_pair_less_iff[where a = "λk. a\$k" and b="λm. b\$m" and c="λs. (c ^ s)\$n" and n = n] ..

lemma fps_compose_mult_distrib:
assumes c0: "c \$ 0 = (0::'a::idom)"
shows "(a * b) oo c = (a oo c) * (b oo c)"
apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma [OF c0])
apply (simp add: fps_compose_nth fps_mult_nth sum_distrib_right)
done

lemma fps_compose_prod_distrib:
assumes c0: "c\$0 = (0::'a::idom)"
shows "prod a S oo c = prod (λk. a k oo c) S"
apply (cases "finite S")
apply simp_all
apply (induct S rule: finite_induct)
apply simp
done

lemma fps_compose_divide:
assumes [simp]: "g dvd f" "h \$ 0 = 0"
shows   "fps_compose f h = fps_compose (f / g :: 'a :: field fps) h * fps_compose g h"
proof -
have "f = (f / g) * g" by simp
also have "fps_compose … h = fps_compose (f / g) h * fps_compose g h"
by (subst fps_compose_mult_distrib) simp_all
finally show ?thesis .
qed

lemma fps_compose_divide_distrib:
assumes "g dvd f" "h \$ 0 = 0" "fps_compose g h ≠ 0"
shows   "fps_compose (f / g :: 'a :: field fps) h = fps_compose f h / fps_compose g h"
using fps_compose_divide[OF assms(1,2)] assms(3) by simp

lemma fps_compose_power:
assumes c0: "c\$0 = (0::'a::idom)"
shows "(a oo c)^n = a^n oo c"
proof (cases n)
case 0
then show ?thesis by simp
next
case (Suc m)
have "(∏n = 0..m. a) oo c = (∏n = 0..m. a oo c)"
using c0 fps_compose_prod_distrib by blast
moreover have th0: "a^n = prod (λk. a) {0..m}" "(a oo c) ^ n = prod (λk. a oo c) {0..m}"
ultimately show ?thesis
by presburger
qed

lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)"
by (simp add: fps_eq_iff fps_compose_nth field_simps sum_negf[symmetric])

lemma fps_compose_sub_distrib: "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)"

lemma fps_X_fps_compose: "fps_X oo a = Abs_fps (λn. if n = 0 then (0::'a::comm_ring_1) else a\$n)"
by (simp add: fps_eq_iff fps_compose_nth mult_delta_left sum.delta)

lemma fps_inverse_compose:
assumes b0: "(b\$0 :: 'a::field) = 0"
and a0: "a\$0 ≠ 0"
shows "inverse a oo b = inverse (a oo b)"
proof -
let ?ia = "inverse a"
let ?ab = "a oo b"
let ?iab = "inverse ?ab"

from a0 have ia0: "?ia \$ 0 ≠ 0" by simp
from a0 have ab0: "?ab \$ 0 ≠ 0" by (simp add: fps_compose_def)
have "(?ia oo b) *  (a oo b) = 1"
unfolding fps_compose_mult_distrib[OF b0, symmetric]
unfolding inverse_mult_eq_1[OF a0]
fps_compose_1 ..

then have "(?ia oo b) *  (a oo b) * ?iab  = 1 * ?iab" by simp
then have "(?ia oo b) *  (?iab * (a oo b))  = ?iab" by simp
then show ?thesis unfolding inverse_mult_eq_1[OF ab0] by simp
qed

lemma fps_divide_compose:
assumes c0: "(c\$0 :: 'a::field) = 0"
and b0: "b\$0 ≠ 0"
shows "(a/b) oo c = (a oo c) / (b oo c)"
using b0 c0 by (simp add: fps_divide_unit fps_inverse_compose fps_compose_mult_distrib)

lemma gp:
assumes a0: "a\$0 = (0::'a::field)"
shows "(Abs_fps (λn. 1)) oo a = 1/(1 - a)"
(is "?one oo a = _")
proof -
have o0: "?one \$ 0 ≠ 0" by simp
have th0: "(1 - fps_X) \$ 0 ≠ (0::'a)" by simp
from fps_inverse_gp[where ?'a = 'a]
have "inverse ?one = 1 - fps_X" by (simp add: fps_eq_iff)
then have "inverse (inverse ?one) = inverse (1 - fps_X)" by simp
then have th: "?one = 1/(1 - fps_X)" unfolding fps_inverse_idempotent[OF o0]
show ?thesis
unfolding th
unfolding fps_divide_compose[OF a0 th0]
fps_compose_1 fps_compose_sub_distrib fps_X_fps_compose_startby0[OF a0] ..
qed

lemma fps_const_power [simp]: "fps_const (c::'a::ring_1) ^ n = fps_const (c^n)"
by (induct n) auto

assumes b0: "b\$0 = (0::'a::field_char_0)"
and ra0: "r (Suc k) (a\$0) ^ Suc k = a\$0"
and a0: "a\$0 ≠ 0"
shows "fps_radical r (Suc k)  a oo b = fps_radical r (Suc k) (a oo b)"
proof -
let ?r = "fps_radical r (Suc k)"
let ?ab = "a oo b"
have ab0: "?ab \$ 0 = a\$0"
from ab0 a0 ra0 have rab0: "?ab \$ 0 ≠ 0" "r (Suc k) (?ab \$ 0) ^ Suc k = ?ab \$ 0"
by simp_all
have th00: "r (Suc k) ((a oo b) \$ 0) = (fps_radical r (Suc k) a oo b) \$ 0"
have th0: "(?r a oo b) ^ (Suc k) = a  oo b"
unfolding fps_compose_power[OF b0]
unfolding iffD1[OF power_radical[of a r k], OF a0 ra0]  ..
from iffD1[OF radical_unique[where r=r and k=k and b= ?ab and a = "?r a oo b", OF rab0(2) th00 rab0(1)], OF th0]
show ?thesis  .
qed

lemma fps_const_mult_apply_left: "fps_const c * (a oo b) = (fps_const c * a) oo b"
by (simp add: fps_eq_iff fps_compose_nth sum_distrib_left mult.assoc)

lemma fps_const_mult_apply_right:
"(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
by (auto simp add: fps_const_mult_apply_left mult.commute)

lemma fps_compose_assoc:
assumes c0: "c\$0 = (0::'a::idom)"
and b0: "b\$0 = 0"
shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
proof -
have "?l\$n = ?r\$n" for n
proof -
have "?l\$n = (sum (λi. (fps_const (a\$i) * b^i) oo c) {0..n})\$n"
by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left
sum_distrib_left mult.assoc fps_sum_nth)
also have "… = ((sum (λi. fps_const (a\$i) * b^i) {0..n}) oo c)\$n"
also have "… = ?r\$n"
apply (simp add: fps_compose_nth fps_sum_nth sum_distrib_right mult.assoc)
apply (rule sum.cong)
apply (rule refl)
apply (rule sum.mono_neutral_right)
apply (erule startsby_zero_power_prefix[OF b0, rule_format])
done
finally show ?thesis .
qed
then show ?thesis
qed

lemma fps_X_power_compose:
assumes a0: "a\$0=0"
shows "fps_X^k oo a = (a::'a::idom fps)^k"
(is "?l = ?r")
proof (cases k)
case 0
then show ?thesis by simp
next
case (Suc h)
have "?l \$ n = ?r \$n" for n
proof -
consider "k > n" | "k ≤ n" by arith
then show ?thesis
proof cases
case 1
then show ?thesis
using a0 startsby_zero_power_prefix[OF a0] Suc
by (simp add: fps_compose_nth del: power_Suc)
next
case 2
then show ?thesis
by (simp add: fps_compose_nth mult_delta_left sum.delta)
qed
qed
then show ?thesis
unfolding fps_eq_iff by blast
qed

lemma fps_inv_right:
assumes a0: "a\$0 = 0"
and a1: "a\$1 ≠ 0"
shows "a oo fps_inv a = fps_X"
proof -
let ?ia = "fps_inv a"
let ?iaa = "a oo fps_inv a"
have th0: "?ia \$ 0 = 0"
have th1: "?iaa \$ 0 = 0"
using a0 a1 by (simp add: fps_inv_def fps_compose_nth)
have th2: "fps_X\$0 = 0"
by simp
from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo fps_X"
by simp
then have "(a oo fps_inv a) oo a = fps_X oo a"
by (simp add: fps_compose_assoc[OF a0 th0] fps_X_fps_compose_startby0[OF a0])
with fps_compose_inj_right[OF a0 a1] show ?thesis
by simp
qed

lemma fps_inv_deriv:
assumes a0: "a\$0 = (0::'a::field)"
and a1: "a\$1 ≠ 0"
shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)"
proof -
let ?ia = "fps_inv a"
let ?d = "fps_deriv a oo ?ia"
let ?dia = "fps_deriv ?ia"
have ia0: "?ia\$0 = 0"
have th0: "?d\$0 ≠ 0"
using a1 by (simp add: fps_compose_nth)
from fps_inv_right[OF a0 a1] have "?d * ?dia = 1"
by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] )
then have "inverse ?d * ?d * ?dia = inverse ?d * 1"
by simp
with inverse_mult_eq_1 [OF th0] show "?dia = inverse ?d"
by simp
qed

lemma fps_inv_idempotent:
assumes a0: "a\$0 = 0"
and a1: "a\$1 ≠ 0"
shows "fps_inv (fps_inv a) = a"
proof -
let ?r = "fps_inv"
have ra0: "?r a \$ 0 = 0"
from a1 have ra1: "?r a \$ 1 ≠ 0"
have fps_X0: "fps_X\$0 = 0"
by simp
from fps_inv[OF ra0 ra1] have "?r (?r a) oo ?r a = fps_X" .
then have "?r (?r a) oo ?r a oo a = fps_X oo a"
by simp
then have "?r (?r a) oo (?r a oo a) = a"
unfolding fps_X_fps_compose_startby0[OF a0]
unfolding fps_compose_assoc[OF a0 ra0, symmetric] .
then show ?thesis
unfolding fps_inv[OF a0 a1] by simp
qed

lemma fps_ginv_ginv:
assumes a0: "a\$0 = 0"
and a1: "a\$1 ≠ 0"
and c0: "c\$0 = 0"
and  c1: "c\$1 ≠ 0"
shows "fps_ginv b (fps_ginv c a) = b oo a oo fps_inv c"
proof -
let ?r = "fps_ginv"
from c0 have rca0: "?r c a \$0 = 0"
from a1 c1 have rca1: "?r c a \$ 1 ≠ 0"
from fps_ginv[OF rca0 rca1]
have "?r b (?r c a) oo ?r c a = b" .
then have "?r b (?r c a) oo ?r c a oo a = b oo a"
by simp
then have "?r b (?r c a) oo (?r c a oo a) = b oo a"
apply (subst fps_compose_assoc)
using a0 c0
done
then have "?r b (?r c a) oo c = b oo a"
unfolding fps_ginv[OF a0 a1] .
then have "?r b (?r c a) oo c oo fps_inv c= b oo a oo fps_inv c"
by simp
then have "?r b (?r c a) oo (c oo fps_inv c) = b oo a oo fps_inv c"
apply (subst fps_compose_assoc)
using a0 c0
done
then show ?thesis
unfolding fps_inv_right[OF c0 c1] by simp
qed

lemma fps_ginv_deriv:
assumes a0:"a\$0 = (0::'a::field)"
and a1: "a\$1 ≠ 0"
shows "fps_deriv (fps_ginv b a) = (fps_deriv b / fps_deriv a) oo fps_ginv fps_X a"
proof -
let ?ia = "fps_ginv b a"
let ?ifps_Xa = "fps_ginv fps_X a"
let ?d = "fps_deriv"
let ?dia = "?d ?ia"
have ifps_Xa0: "?ifps_Xa \$ 0 = 0"
have da0: "?d a \$ 0 ≠ 0"
using a1 by simp
from fps_ginv[OF a0 a1, of b] have "?d (?ia oo a) = fps_deriv b"
by simp
then have "(?d ?ia oo a) * ?d a = ?d b"
unfolding fps_compose_deriv[OF a0] .
then have "(?d ?ia oo a) * ?d a * inverse (?d a) = ?d b * inverse (?d a)"
by simp
with a1 have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a"
then have "(?d ?ia oo a) oo ?ifps_Xa =  (?d b / ?d a) oo ?ifps_Xa"
unfolding inverse_mult_eq_1[OF da0] by simp
then have "?d ?ia oo (a oo ?ifps_Xa) =  (?d b / ?d a) oo ?ifps_Xa"
unfolding fps_compose_assoc[OF ifps_Xa0 a0] .
then show ?thesis unfolding fps_inv_ginv[symmetric]
unfolding fps_inv_right[OF a0 a1] by simp
qed

lemma fps_compose_linear:
"fps_compose (f :: 'a :: comm_ring_1 fps) (fps_const c * fps_X) = Abs_fps (λn. c^n * f \$ n)"
by (simp add: fps_eq_iff fps_compose_def power_mult_distrib
if_distrib sum.delta' cong: if_cong)

lemma fps_compose_uminus':
"fps_compose f (-fps_X :: 'a :: comm_ring_1 fps) = Abs_fps (λn. (-1)^n * f \$ n)"
using fps_compose_linear[of f "-1"]
by (simp only: fps_const_neg [symmetric] fps_const_1_eq_1) simp

subsection ‹Elementary series›

subsubsection ‹Exponential series›

definition "fps_exp x = Abs_fps (λn. x^n / of_nat (fact n))"

lemma fps_exp_deriv[simp]: "fps_deriv (fps_exp a) = fps_const (a::'a::field_char_0) * fps_exp a"
(is "?l = ?r")
proof -
have "?l\$n = ?r \$ n" for n
apply (auto simp add: fps_exp_def field_simps power_Suc[symmetric]
simp del: fact_Suc of_nat_Suc power_Suc)
done
then show ?thesis
qed

lemma fps_exp_unique_ODE:
"fps_deriv a = fps_const c * a ⟷ a = fps_const (a\$0) * fps_exp (c::'a::field_char_0)"
(is "?lhs ⟷ ?rhs")
proof
show ?rhs if ?lhs
proof -
from that have th: "⋀n. a \$ Suc n = c * a\$n / of_nat (Suc n)"
by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
have th': "a\$n = a\$0 * c ^ n/ (fact n)" for n
proof (induct n)
case 0
then show ?case by simp
next
case Suc
then show ?case
unfolding th
using fact_gt_zero
apply (simp add: field_simps del: of_nat_Suc fact_Suc)
apply simp
done
qed
show ?thesis
by (auto simp add: fps_eq_iff fps_const_mult_left fps_exp_def intro: th')
qed
show ?lhs if ?rhs
using that by (metis fps_exp_deriv fps_deriv_mult_const_left mult.left_commute)
qed

lemma fps_exp_add_mult: "fps_exp (a + b) = fps_exp (a::'a::field_char_0) * fps_exp b" (is "?l = ?r")
proof -
have "fps_deriv ?r = fps_const (a + b) * ?r"
then have "?r = ?l"
by (simp only: fps_exp_unique_ODE) (simp add: fps_mult_nth fps_exp_def)
then show ?thesis ..
qed

lemma fps_exp_nth[simp]: "fps_exp a \$ n = a^n / of_nat (fact n)"

lemma fps_exp_0[simp]: "fps_exp (0::'a::field) = 1"

lemma fps_exp_neg: "fps_exp (- a) = inverse (fps_exp (a::'a::field_char_0))"
proof -
from fps_exp_add_mult[of a "- a"] have th0: "fps_exp a * fps_exp (- a) = 1" by simp
from fps_inverse_unique[OF th0] show ?thesis by simp
qed

lemma fps_exp_nth_deriv[simp]:
"fps_nth_deriv n (fps_exp (a::'a::field_char_0)) = (fps_const a)^n * (fps_exp a)"
by (induct n) auto

lemma fps_X_compose_fps_exp[simp]: "fps_X oo fps_exp (a::'a::field) = fps_exp a - 1"

lemma fps_inv_fps_exp_compose:
assumes a: "a ≠ 0"
shows "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X"
and "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X"
proof -
let ?b = "fps_exp a - 1"
have b0: "?b \$ 0 = 0"
by simp
have b1: "?b \$ 1 ≠ 0"
from fps_inv[OF b0 b1] show "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X" .
from fps_inv_right[OF b0 b1] show "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X" .
qed

lemma fps_exp_power_mult: "(fps_exp (c::'a::field_char_0))^n = fps_exp (of_nat n * c)"

assumes r: "r (Suc k) 1 = 1"
shows "fps_radical r (Suc k) (fps_exp (c::'a::field_char_0)) = fps_exp (c / of_nat (Suc k))"
proof -
let ?ck = "(c / of_nat (Suc k))"
let ?r = "fps_radical r (Suc k)"
have eq0[simp]: "?ck * of_nat (Suc k) = c" "of_nat (Suc k) * ?ck = c"
by (simp_all del: of_nat_Suc)
have th0: "fps_exp ?ck ^ (Suc k) = fps_exp c" unfolding fps_exp_power_mult eq0 ..
have th: "r (Suc k) (fps_exp c \$0) ^ Suc k = fps_exp c \$ 0"
"r (Suc k) (fps_exp c \$ 0) = fps_exp ?ck \$ 0" "fps_exp c \$ 0 ≠ 0" using r by simp_all
from th0 radical_unique[where r=r and k=k, OF th] show ?thesis
by auto
qed

lemma fps_exp_compose_linear [simp]:
"fps_exp (d::'a::field_char_0) oo (fps_const c * fps_X) = fps_exp (c * d)"
by (simp add: fps_compose_linear fps_exp_def fps_eq_iff power_mult_distrib)

lemma fps_fps_exp_compose_minus [simp]:
"fps_compose (fps_exp c) (-fps_X) = fps_exp (-c :: 'a :: field_char_0)"
using fps_exp_compose_linear[of c "-1 :: 'a"]
unfolding fps_const_neg [symmetric] fps_const_1_eq_1 by simp

lemma fps_exp_eq_iff [simp]: "fps_exp c = fps_exp d ⟷ c = (d :: 'a :: field_char_0)"
proof
assume "fps_exp c = fps_exp d"
from arg_cong[of _ _ "λF. F \$ 1", OF this] show "c = d" by simp
qed simp_all

lemma fps_exp_eq_fps_const_iff [simp]:
"fps_exp (c :: 'a :: field_char_0) = fps_const c' ⟷ c = 0 ∧ c' = 1"
proof
assume "c = 0 ∧ c' = 1"
thus "fps_exp c = fps_const c'" by (auto simp: fps_eq_iff)
next
assume "fps_exp c = fps_const c'"
from arg_cong[of _ _ "λF. F \$ 1", OF this] arg_cong[of _ _ "λF. F \$ 0", OF this]
show "c = 0 ∧ c' = 1" by simp_all
qed

lemma fps_exp_neq_0 [simp]: "¬fps_exp (c :: 'a :: field_char_0) = 0"
unfolding fps_const_0_eq_0 [symmetric] fps_exp_eq_fps_const_iff by simp

lemma fps_exp_eq_1_iff [simp]: "fps_exp (c :: 'a :: field_char_0) = 1 ⟷ c = 0"
unfolding fps_const_1_eq_1 [symmetric] fps_exp_eq_fps_const_iff by simp

lemma fps_exp_neq_numeral_iff [simp]:
"fps_exp (c :: 'a :: field_char_0) = numeral n ⟷ c = 0 ∧ n = Num.One"
unfolding numeral_fps_const fps_exp_eq_fps_const_iff by simp

subsubsection ‹Logarithmic series›

lemma Abs_fps_if_0:
"Abs_fps (λn. if n = 0 then (v::'a::ring_1) else f n) =
fps_const v + fps_X * Abs_fps (λn. f (Suc n))"

definition fps_ln :: "'a::field_char_0 ⇒ 'a fps"
where "fps_ln c = fps_const (1/c) * Abs_fps (λn. if n = 0 then 0 else (- 1) ^ (n - 1) / of_nat n)"

lemma fps_ln_deriv: "fps_deriv (fps_ln c) = fps_const (1/c) * inverse (1 + fps_X)"
unfolding fps_inverse_fps_X_plus1
by (simp add: fps_ln_def fps_eq_iff del: of_nat_Suc)

lemma fps_ln_nth: "fps_ln c \$ n = (if n = 0 then 0 else 1/c * ((- 1) ^ (n - 1) / of_nat n))"

lemma fps_ln_0 [simp]: "fps_ln c \$ 0 = 0" by (simp add: fps_ln_def)

lemma fps_ln_fps_exp_inv:
fixes a :: "'a::field_char_0"
assumes a: "a ≠ 0"
shows "fps_ln a = fps_inv (fps_exp a - 1)"  (is "?l = ?r")
proof -
let ?b = "fps_exp a - 1"
have b0: "?b \$ 0 = 0" by simp
have b1: "?b \$ 1 ≠ 0" by (simp add: a)
have "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) =
(fps_const a * (fps_exp a - 1) + fps_const a) oo fps_inv (fps_exp a - 1)"
also have "… = fps_const a * (fps_X + 1)"
done
finally have eq: "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_const a * (fps_X + 1)" .
from fps_inv_deriv[OF b0 b1, unfolded eq]
have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (fps_X + 1)"
using a by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
then have "fps_deriv ?l = fps_deriv ?r"
then show ?thesis unfolding fps_deriv_eq_iff
qed

assumes c0: "c≠0"
and d0: "d≠0"
shows "fps_ln c + fps_ln d = fps_const (c+d) * fps_ln (c*d)"
(is "?r = ?l")
proof-
from c0 d0 have eq: "1/c + 1/d = (c+d)/(c*d)" by (simp add: field_simps)
have "fps_deriv ?r = fps_const (1/c + 1/d) * inverse (1 + fps_X)"
also have "… = fps_deriv ?l"
done
finally show ?thesis
unfolding fps_deriv_eq_iff by simp
qed

lemma fps_X_dvd_fps_ln [simp]: "fps_X dvd fps_ln c"
proof -
have "fps_ln c = fps_X * Abs_fps (λn. (-1) ^ n / (of_nat (Suc n) * c))"
by (intro fps_ext) (auto simp: fps_ln_def of_nat_diff)
thus ?thesis by simp
qed

subsubsection ‹Binomial series›

definition "fps_binomial a = Abs_fps (λn. a gchoose n)"

lemma fps_binomial_nth[simp]: "fps_binomial a \$ n = a gchoose n"

lemma fps_binomial_ODE_unique:
fixes c :: "'a::field_char_0"
shows "fps_deriv a = (fps_const c * a) / (1 + fps_X) ⟷ a = fps_const (a\$0) * fps_binomial c"
(is "?lhs ⟷ ?rhs")
proof
let ?da = "fps_deriv a"
let ?x1 = "(1 + fps_X):: 'a fps"
let ?l = "?x1 * ?da"
let ?r = "fps_const c * a"

have eq: "?l = ?r ⟷ ?lhs"
proof -
have x10: "?x1 \$ 0 ≠ 0" by simp
have "?l = ?r ⟷ inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
also have "… ⟷ ?da = (fps_const c * a) / ?x1"
apply (simp only: fps_divide_def  mult.assoc[symmetric] inverse_mult_eq_1[OF x10])
done
finally show ?thesis .
qed

show ?rhs if ?lhs
proof -
from eq that have h: "?l = ?r" ..
have th0: "a\$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a \$n" for n
proof -
from h have "?l \$ n = ?r \$ n" by simp
then show ?thesis
apply (simp add: field_simps del: of_nat_Suc)
apply (cases n)
apply (simp_all add: field_simps del: of_nat_Suc)
done
qed
have th1: "a \$ n = (c gchoose n) * a \$ 0" for n
proof (induct n)
case 0
then show ?case by simp
next
case (Suc m)
then show ?case
unfolding th0
apply (simp add: field_simps del: of_nat_Suc)
unfolding mult.assoc[symmetric] gbinomial_mult_1
done
qed
show ?thesis
apply (subst th1)
done
qed

show ?lhs if ?rhs
proof -
have th00: "x * (a \$ 0 * y) = a \$ 0 * (x * y)" for x y
have "?l = ?r"
apply (subst ‹?rhs›)
apply (subst (2) ‹?rhs›)
apply (clarsimp simp add: fps_eq_iff field_simps)
unfolding mult.assoc[symmetric] th00 gbinomial_mult_1
done
with eq show ?thesis ..
qed
qed

lemma fps_binomial_ODE_unique':
"(fps_deriv a = fps_const c * a / (1 + fps_X) ∧ a \$ 0 = 1) ⟷ (a = fps_binomial c)"
by (subst fps_binomial_ODE_unique) auto

lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + fps_X)"
proof -
let ?a = "fps_binomial c"
have th0: "?a = fps_const (?a\$0) * ?a" by (simp)
from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis .
qed

lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r")
proof -
let ?P = "?r - ?l"
let ?b = "fps_binomial"
let ?db = "λx. fps_deriv (?b x)"
have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)"  by simp
also have "… = inverse (1 + fps_X) *
(fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))"
unfolding fps_binomial_deriv
also have "… = (fps_const (c + d)/ (1 + fps_X)) * ?P"
finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + fps_X)"
have "?P = fps_const (?P\$0) * ?b (c + d)"
unfolding fps_binomial_ODE_unique[symmetric]
using th0 by simp
then have "?P = 0" by (simp add: fps_mult_nth)
then show ?thesis by simp
qed

lemma fps_binomial_minus_one: "fps_binomial (- 1) = inverse (1 + fps_X)"
(is "?l = inverse ?r")
proof-
have th: "?r\$0 ≠ 0" by simp
have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + fps_X)"
by (simp add: fps_inverse_deriv[OF th] fps_divide_def
power2_eq_square mult.commute fps_const_neg[symmetric] del: fps_const_neg)
have eq: "inverse ?r \$ 0 = 1"
from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + fps_X)" "- 1"] th'] eq
show ?thesis by (simp add: fps_inverse_def)
qed

lemma fps_binomial_of_nat: "fps_binomial (of_nat n) = (1 + fps_X :: 'a :: field_char_0 fps) ^ n"
proof (cases "n = 0")
case [simp]: True
have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = 0" by simp
also have "… = fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)" by (simp add: fps_binomial_def)
finally show ?thesis by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) simp_all
next
case False
have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = fps_const (of_nat n) * (1 + fps_X) ^ (n - 1)"
also have "(1 + fps_X :: 'a fps) \$ 0 ≠ 0" by simp
hence "(1 + fps_X :: 'a fps) ≠ 0" by (intro notI) (simp only: , simp)
with False have "(1 + fps_X :: 'a fps) ^ (n - 1) = (1 + fps_X) ^ n / (1 + fps_X)"
by (cases n) (simp_all )
also have "fps_const (of_nat n :: 'a) * ((1 + fps_X) ^ n / (1 + fps_X)) =
fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)"
finally show ?thesis
by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) (simp_all add: fps_power_nth)
qed

lemma fps_binomial_0 [simp]: "fps_binomial 0 = 1"
using fps_binomial_of_nat[of 0] by simp

lemma fps_binomial_power: "fps_binomial a ^ n = fps_binomial (of_nat n * a)"

lemma fps_binomial_1: "fps_binomial 1 = 1 + fps_X"
using fps_binomial_of_nat[of 1] by simp

lemma fps_binomial_minus_of_nat:
"fps_binomial (- of_nat n) = inverse ((1 + fps_X :: 'a :: field_char_0 fps) ^ n)"
by (rule sym, rule fps_inverse_unique)

lemma one_minus_const_fps_X_power:
"c ≠ 0 ⟹ (1 - fps_const c * fps_X) ^ n =
fps_compose (fps_binomial (of_nat n)) (-fps_const c * fps_X)"
by (subst fps_binomial_of_nat)
del: fps_const_neg)

lemma one_minus_fps_X_const_neg_power:
"inverse ((1 - fps_const c * fps_X) ^ n) =
fps_compose (fps_binomial (-of_nat n)) (-fps_const c * fps_X)"
proof (cases "c = 0")
case False
thus ?thesis
by (subst fps_binomial_minus_of_nat)
fps_const_neg [symmetric] del: fps_const_neg)
qed simp

lemma fps_X_plus_const_power:
"c ≠ 0 ⟹ (fps_X + fps_const c) ^ n =
fps_const (c^n) * fps_compose (fps_binomial (of_nat n)) (fps_const (inverse c) * fps_X)"
by (subst fps_binomial_of_nat)
fps_const_power [symmetric] power_mult_distrib [symmetric]
algebra_simps inverse_mult_eq_1' del: fps_const_power)

lemma fps_X_plus_const_neg_power:
"c ≠ 0 ⟹ inverse ((fps_X + fps_const c) ^ n) =
fps_const (inverse c^n) * fps_compose (fps_binomial (-of_nat n)) (fps_const (inverse c) * fps_X)"
by (subst fps_binomial_minus_of_nat)
fps_const_power [symmetric] power_mult_distrib [symmetric] fps_inverse_compose
algebra_simps fps_const_inverse [symmetric] fps_inverse_mult [symmetric]
fps_inverse_power [symmetric] inverse_mult_eq_1'
del: fps_const_power)

lemma one_minus_const_fps_X_neg_power':
"n > 0 ⟹ inverse ((1 - fps_const (c :: 'a :: field_char_0) * fps_X) ^ n) =
Abs_fps (λk. of_nat ((n + k - 1) choose k) * c^k)"
apply (rule fps_ext)
apply (subst one_minus_fps_X_const_neg_power, subst fps_const_neg, subst fps_compose_linear)
apply (simp add: power_mult_distrib [symmetric] mult.assoc [symmetric]
gbinomial_minus binomial_gbinomial of_nat_diff)
done

text ‹Vandermonde's Identity as a consequence.›
lemma gbinomial_Vandermonde:
"sum (λk. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
proof -
let ?ba = "fps_binomial a"
let ?bb = "fps_binomial b"
let ?bab = "fps_binomial (a + b)"
from fps_binomial_add_mult[of a b] have "?bab \$ n = (?ba * ?bb)\$n" by simp
then show ?thesis by (simp add: fps_mult_nth)
qed

lemma binomial_Vandermonde:
"sum (λk. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n]
by (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric]

lemma binomial_Vandermonde_same: "sum (λk. (n choose k)⇧2) {0..n} = (2 * n) choose n"
using binomial_Vandermonde[of n n n, symmetric]
unfolding mult_2
apply (rule sum.cong)
apply (auto intro:  binomial_symmetric)
done

lemma Vandermonde_pochhammer_lemma:
fixes a :: "'a::field_char_0"
assumes b: "∀j∈{0 ..<n}. b ≠ of_nat j"
shows "sum (λk. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) /
(of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} =
pochhammer (- (a + b)) n / pochhammer (- b) n"
(is "?l = ?r")
proof -
let ?m1 = "λm. (- 1 :: 'a) ^ m"
let ?f = "λm. of_nat (fact m)"
let ?p = "λ(x::'a). pochhammer (- x)"
from b have bn0: "?p b n ≠ 0"
unfolding pochhammer_eq_0_iff by simp
have th00:
"b gchoose (n - k) =
(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
(is ?gchoose)
"pochhammer (1 + b - of_nat n) k ≠ 0"
(is ?pochhammer)
if kn: "k ∈ {0..n}" for k
proof -
from kn have "k ≤ n" by simp
have nz: "pochhammer (1 + b - of_nat n) n ≠ 0"
proof
assume "pochhammer (1 + b - of_nat n) n = 0"
then have c: "pochhammer (b - of_nat n + 1) n = 0"
then obtain j where j: "j < n" "b - of_nat n + 1 = - of_nat j"
unfolding pochhammer_eq_0_iff by blast
from j have "b = of_nat n - of_nat j - of_nat 1"
then have "b = of_nat (n - j - 1)"
using j kn by (simp add: of_nat_diff)
with b show False using j by auto
qed

from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k ≠ 0"
by (rule pochhammer_neq_0_mono)

consider "k = 0 ∨ n = 0" | "k ≠ 0" "n ≠ 0"
by blast
then have "b gchoose (n - k) =
(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
proof cases
case 1
then show ?thesis
using kn by (cases "k = 0") (simp_all add: gbinomial_pochhammer)
next
case neq: 2
then obtain m where m: "n = Suc m"
by (cases n) auto
from neq(1) obtain h where h: "k = Suc h"
by (cases k) auto
show ?thesis
proof (cases "k = n")
case True
then show ?thesis
using pochhammer_minus'[where k=k and b=b]
using bn0
done
next
case False
with kn have kn': "k < n"
by simp
have m1nk: "?m1 n = prod (λi. - 1) {..m}" "?m1 k = prod (λi. - 1) {0..h}"
by (simp_all add: prod_constant m h)
have bnz0: "pochhammer (b - of_nat n + 1) k ≠ 0"
using bn0 kn
unfolding pochhammer_eq_0_iff
apply auto
apply (erule_tac x= "n - ka - 1" in allE)
apply (auto simp add: algebra_simps of_nat_diff)
done
have eq1: "prod (λk. (1::'a) + of_nat m - of_nat k) {..h} =
prod of_nat {Suc (m - h) .. Suc m}"
using kn' h m
by (intro prod.reindex_bij_witness[where i="λk. Suc m - k" and j="λk. Suc m - k"])
(auto simp: of_nat_diff)
have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
using ‹k ≤ n› apply (simp add: fact_split [of k n])
using prod.atLeastLessThan_shift_bounds [where ?'a = 'a, of "λi. 1 + of_nat i" 0 "n - k" k]
apply (auto simp add: of_nat_diff field_simps)
done
have th20: "?m1 n * ?p b n = prod (λi. b - of_nat i) {0..m}"
apply (simp add: pochhammer_minus field_simps m)
apply (auto simp add: pochhammer_prod_rev of_nat_diff prod.atLeast_Suc_atMost_Suc_shift)
done
have th21:"pochhammer (b - of_nat n + 1) k = prod (λi. b - of_nat i) {n - k .. n - 1}"
using kn apply (simp add: pochhammer_prod_rev m h prod.atLeast_Suc_atMost_Suc_shift)
using prod.atLeastAtMost_shift_0 [of "m - h" m, where ?'a = 'a]
apply (auto simp add: of_nat_diff field_simps)
done
have "?m1 n * ?p b n =
prod (λi. b - of_nat i) {0.. n - k - 1} * pochhammer (b - of_nat n + 1) k"
using kn' m h unfolding th20 th21 apply simp
```