Theory Formal_Power_Series

theory Formal_Power_Series
imports Binomial
(*  Title:      HOL/Library/Formal_Power_Series.thy
Author: Amine Chaieb, University of Cambridge
*)


header{* A formalization of formal power series *}

theory Formal_Power_Series
imports Binomial
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"
by (simp add: expand_fps_eq)

lemma fps_nth_Abs_fps [simp]: "Abs_fps f $ n = f n"
by (simp add: Abs_fps_inverse)

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:
"op + = (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:
"op - = (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:
"op * = (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

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

lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
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*}


instance fps :: (semigroup_add) semigroup_add
proof
fix a b c :: "'a fps"
show "a + b + c = a + (b + c)"
by (simp add: fps_ext add_assoc)
qed

instance fps :: (ab_semigroup_add) ab_semigroup_add
proof
fix a b :: "'a fps"
show "a + b = b + a"
by (simp add: fps_ext add_commute)
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))"

by (induct k) (simp_all add: Suc_diff_le setsum_addf add_assoc)

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 setsum_right_distrib setsum_left_distrib 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)"
proof (rule setsum_reindex_cong)
show "inj_on (λi. n - i) {0..n}"
by (rule inj_onI) simp
show "{0..n} = (λi. n - i) ` {0..n}"
apply auto
apply (rule_tac x = "n - x" in image_eqI)
apply simp_all
done
next
fix i
assume "i ∈ {0..n}"
then have "n - (n - i) = i" by simp
then show "f (n - i) i = f (n - i) (n - (n - i))" by simp
qed

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"
by (simp add: fps_mult_nth mult_commute)
qed
qed

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

instance fps :: (comm_monoid_add) comm_monoid_add
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 setsum_delta)
show "a * 1 = a" by (simp add: fps_ext fps_mult_nth mult_delta_right setsum_delta')
qed

instance fps :: (cancel_semigroup_add) cancel_semigroup_add
proof
fix a b c :: "'a fps"
{ assume "a + b = a + c" then show "b = c" by (simp add: expand_fps_eq) }
{ assume "b + a = c + a" then show "b = c" by (simp add: expand_fps_eq) }
qed

instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
proof
fix a b c :: "'a fps"
assume "a + b = a + c"
then show "b = c" by (simp add: expand_fps_eq)
qed

instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..

instance fps :: (group_add) group_add
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 diff_minus)
qed

instance fps :: (ab_group_add) ab_group_add
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
by default (simp add: expand_fps_eq)

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 setsum_addf)
show "a * (b + c) = a * b + a * c"
by (simp add: expand_fps_eq fps_mult_nth distrib_left setsum_addf)
qed

instance fps :: (semiring_0) semiring_0
proof
fix a :: "'a fps"
show "0 * a = 0" by (simp add: fps_ext fps_mult_nth)
show "a * 0 = 0" by (simp add: fps_ext fps_mult_nth)
qed

instance fps :: (semiring_0_cancel) semiring_0_cancel ..

subsection {* Selection of the nth power of the implicit variable in the infinite sum*}

lemma fps_nonzero_nth: "f ≠ 0 <-> (∃ n. f $n ≠ 0)"
by (simp add: expand_fps_eq)

lemma fps_nonzero_nth_minimal: "f ≠ 0 <-> (∃n. f $ n ≠ 0 ∧ (∀m < n. f $ m = 0))"
proof
let ?n = "LEAST n. f $ n ≠ 0"
assume "f ≠ 0"
then have "∃n. f $ n ≠ 0"
by (simp add: fps_nonzero_nth)
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 "∃n. f $ n ≠ 0 ∧ (∀m<n. f $ m = 0)" ..
next
assume "∃n. f $ n ≠ 0 ∧ (∀m<n. f $ m = 0)"
then show "f ≠ 0" 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_setsum_nth: "setsum f S $ n = setsum (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"
by (simp add: fps_ext)

lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1"
by (simp add: fps_ext)

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

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

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

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 setsum_0')

lemma fps_const_add_left: "fps_const (c::'a::monoid_add) + f =
Abs_fps (λn. if n = 0 then c + f$0 else f$n)"

by (simp add: fps_ext)

lemma fps_const_add_right: "f + fps_const (c::'a::monoid_add) =
Abs_fps (λn. if n = 0 then f$0 + c else f$n)"

by (simp add: fps_ext)

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 setsum_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 setsum_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 setsum_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 setsum_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: diff_minus distrib_right)

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

instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors
proof
fix a b :: "'a fps"
assume a0: "a ≠ 0" and b0: "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 setsum_diff1') simp_all
also have "(∑k∈{0..i+j}-{i}. a$k * b$(i+j-k)) = 0"
proof (rule setsum_0' [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
fps_const_add [symmetric])

lemma neg_numeral_fps_const: "neg_numeral k = fps_const (neg_numeral k)"
by (simp only: neg_numeral_def numeral_fps_const fps_const_neg)

subsection{* The eXtractor series X*}

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

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

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

lemma X_mult_right_nth[simp]:
"((f :: ('a::comm_semiring_1) fps) * X) $n = (if n = 0 then 0 else f $ (n - 1))"
by (metis X_mult_nth mult_commute)

lemma X_power_iff: "X^k = Abs_fps (λn. if n = k then (1::'a::comm_ring_1) else 0)"
proof (induct k)
case 0
thus ?case by (simp add: X_def fps_eq_iff)
next
case (Suc k)
{
fix m
have "(X^Suc k) $ m = (if m = 0 then (0::'a) else (X^k) $ (m - 1))"
by (simp del: One_nat_def)
then have "(X^Suc k) $ m = (if m = Suc k then (1::'a) else 0)"
using Suc.hyps by (auto cong del: if_weak_cong)
}
then show ?case by (simp add: fps_eq_iff)
qed

lemma X_power_mult_nth:
"(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 X_power_mult_right_nth:
"((f :: ('a::comm_ring_1) fps) * X^k) $n = (if n < k then 0 else f $ (n - k))"
by (metis X_power_mult_nth mult_commute)


subsection{* Formal Power series form a metric space *}

definition (in dist) "ball x r = {y. dist y x < r}"

instantiation fps :: (comm_ring_1) dist
begin

definition
dist_fps_def: "dist (a::'a fps) b =
(if (∃n. a$n ≠ b$n) then inverse (2 ^ The (leastP (λn. a$n ≠ b$n))) else 0)"


lemma dist_fps_ge0: "dist (a::'a fps) b ≥ 0"
by (simp add: dist_fps_def)

lemma dist_fps_sym: "dist (a::'a fps) b = dist b a"
apply (auto simp add: dist_fps_def)
apply (rule cong[OF refl, where x="(λn::nat. a $ n ≠ b $ n)"])
apply (rule ext)
apply auto
done

instance ..

end

lemma fps_nonzero_least_unique:
assumes a0: "a ≠ 0"
shows "∃!n. leastP (λn. a$n ≠ 0) n"
proof -
from fps_nonzero_nth_minimal [of a] a0
obtain n where "a$n ≠ 0" "∀m < n. a$m = 0" by blast
then have ln: "leastP (λn. a$n ≠ 0) n"
by (auto simp add: leastP_def setge_def not_le [symmetric])
moreover
{
fix m
assume "leastP (λn. a $ n ≠ 0) m"
then have "m = n" using ln
apply (auto simp add: leastP_def setge_def)
apply (erule allE[where x=n])
apply (erule allE[where x=m])
apply simp
done
}
ultimately show ?thesis by blast
qed

lemma fps_eq_least_unique:
assumes "(a::('a::ab_group_add) fps) ≠ b"
shows "∃! n. leastP (λn. a$n ≠ b$n) n"
using fps_nonzero_least_unique[of "a - b"] assms
by auto

instantiation fps :: (comm_ring_1) metric_space
begin

definition open_fps_def: "open (S :: 'a fps set) = (∀a ∈ S. ∃r. r >0 ∧ ball a r ⊆ S)"

instance
proof
fix S :: "'a fps set"
show "open S = (∀x∈S. ∃e>0. ∀y. dist y x < e --> y ∈ S)"
by (auto simp add: open_fps_def ball_def subset_eq)
next
{
fix a b :: "'a fps"
{
assume "a = b"
then have "¬ (∃n. a $ n ≠ b $ n)" by simp
then have "dist a b = 0" by (simp add: dist_fps_def)
}
moreover
{
assume d: "dist a b = 0"
then have "∀n. a$n = b$n"
by - (rule ccontr, simp add: dist_fps_def)
then have "a = b" by (simp add: fps_eq_iff)
}
ultimately show "dist a b =0 <-> a = b" by blast
}
note th = this
from th have th'[simp]: "!!a::'a fps. dist a a = 0" by simp
fix a b c :: "'a fps"
{
assume "a = b"
then have "dist a b = 0" unfolding th .
then have "dist a b ≤ dist a c + dist b c"
using dist_fps_ge0 [of a c] dist_fps_ge0 [of b c] by simp
}
moreover
{
assume "c = a ∨ c = b"
then have "dist a b ≤ dist a c + dist b c"
by (cases "c = a") (simp_all add: th dist_fps_sym)
}
moreover
{
assume ab: "a ≠ b" and ac: "a ≠ c" and bc: "b ≠ c"
let ?P = "λa b n. a$n ≠ b$n"
from fps_eq_least_unique[OF ab] fps_eq_least_unique[OF ac]
fps_eq_least_unique[OF bc]
obtain nab nac nbc where nab: "leastP (?P a b) nab"
and nac: "leastP (?P a c) nac"
and nbc: "leastP (?P b c) nbc" by blast
from nab have nab': "!!m. m < nab ==> a$m = b$m" "a$nab ≠ b$nab"
by (auto simp add: leastP_def setge_def)
from nac have nac': "!!m. m < nac ==> a$m = c$m" "a$nac ≠ c$nac"
by (auto simp add: leastP_def setge_def)
from nbc have nbc': "!!m. m < nbc ==> b$m = c$m" "b$nbc ≠ c$nbc"
by (auto simp add: leastP_def setge_def)

have th0: "!!(a::'a fps) b. a ≠ b <-> (∃n. a$n ≠ b$n)"
by (simp add: fps_eq_iff)
from ab ac bc nab nac nbc
have dab: "dist a b = inverse (2 ^ nab)"
and dac: "dist a c = inverse (2 ^ nac)"
and dbc: "dist b c = inverse (2 ^ nbc)"
unfolding th0
apply (simp_all add: dist_fps_def)
apply (erule the1_equality[OF fps_eq_least_unique[OF ab]])
apply (erule the1_equality[OF fps_eq_least_unique[OF ac]])
apply (erule the1_equality[OF fps_eq_least_unique[OF bc]])
done
from ab ac bc have nz: "dist a b ≠ 0" "dist a c ≠ 0" "dist b c ≠ 0"
unfolding th by simp_all
from nz have pos: "dist a b > 0" "dist a c > 0" "dist b c > 0"
using dist_fps_ge0[of a b] dist_fps_ge0[of a c] dist_fps_ge0[of b c]
by auto
have th1: "!!n. (2::real)^n >0" by auto
{
assume h: "dist a b > dist a c + dist b c"
then have gt: "dist a b > dist a c" "dist a b > dist b c"
using pos by auto
from gt have gtn: "nab < nbc" "nab < nac"
unfolding dab dbc dac by (auto simp add: th1)
from nac'(1)[OF gtn(2)] nbc'(1)[OF gtn(1)]
have "a $ nab = b $ nab" by simp
with nab'(2) have False by simp
}
then have "dist a b ≤ dist a c + dist b c"
by (auto simp add: not_le[symmetric])
}
ultimately show "dist a b ≤ dist a c + dist b c" by blast
qed

end

text{* The infinite sums and justification of the notation in textbooks*}

lemma reals_power_lt_ex:
assumes xp: "x > 0" and y1: "(y::real) > 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"
by (simp add: mult_ac)
then have "y ^ k * x > 1"
unfolding exp_zero exp_add exp_real_of_nat_mult exp_ln [OF xp] exp_ln [OF yp]
by simp
then have "x > (1 / y)^k" using yp
by (simp add: field_simps nonzero_power_divide)
then show ?thesis using kp by blast
qed

lemma X_nth[simp]: "X$n = (if n = 1 then 1 else 0)" by (simp add: X_def)

lemma X_power_nth[simp]: "(X^k) $n = (if n = k then 1 else (0::'a::comm_ring_1))"
by (simp add: X_power_iff)


lemma fps_sum_rep_nth: "(setsum (%i. fps_const(a$i)*X^i) {0..m})$n =
(if n ≤ m then a$n else (0::'a::comm_ring_1))"

apply (auto simp add: fps_setsum_nth cond_value_iff cong del: if_weak_cong)
apply (simp add: setsum_delta')
done

lemma fps_notation: "(%n. setsum (%i. fps_const(a$i) * X^i) {0..n}) ----> a"
(is "?s ----> a")
proof -
{
fix r:: real
assume rp: "r > 0"
have th0: "(2::real) > 1" by simp
from reals_power_lt_ex[OF rp th0]
obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0" by blast
{
fix n::nat
assume nn0: "n ≥ n0"
then have thnn0: "(1/2)^n <= (1/2 :: real)^n0"
by (auto intro: power_decreasing)
{
assume "?s n = a"
then have "dist (?s n) a < r"
unfolding dist_eq_0_iff[of "?s n" a, symmetric]
using rp by (simp del: dist_eq_0_iff)
}
moreover
{
assume neq: "?s n ≠ a"
from fps_eq_least_unique[OF neq]
obtain k where k: "leastP (λi. ?s n $ i ≠ a$i) k" by blast
have th0: "!!(a::'a fps) b. a ≠ b <-> (∃n. a$n ≠ b$n)"
by (simp add: fps_eq_iff)
from neq have dth: "dist (?s n) a = (1/2)^k"
unfolding th0 dist_fps_def
unfolding the1_equality[OF fps_eq_least_unique[OF neq], OF k]
by (auto simp add: inverse_eq_divide power_divide)

from k have kn: "k > n"
by (simp add: leastP_def setge_def fps_sum_rep_nth split:split_if_asm)
then have "dist (?s n) a < (1/2)^n" unfolding dth
by (auto intro: power_strict_decreasing)
also have "… <= (1/2)^n0" using nn0
by (auto intro: power_decreasing)
also have "… < r" using n0 by simp
finally have "dist (?s n) a < r" .
}
ultimately have "dist (?s n) a < r" by blast
}
then have "∃n0. ∀ n ≥ n0. dist (?s n) a < r " by blast
}
then show ?thesis unfolding LIMSEQ_def by blast
qed

subsection{* Inverses of formal power series *}

declare setsum_cong[fundef_cong]

instantiation fps :: ("{comm_monoid_add, inverse, times, uminus}") inverse
begin

fun natfun_inverse:: "'a fps => nat => 'a"
where
"natfun_inverse f 0 = inverse (f$0)"
| "natfun_inverse f n = - inverse (f$0) * setsum (λi. f$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: "divide = (λ(f::'a fps) g. f * inverse g)"

instance ..

end

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

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" by (simp add: mult_commute)
from f0 have ifn: "!!n. inverse f $ n = natfun_inverse f n"
by (simp add: fps_inverse_def)
from f0 have th0: "(inverse f * f) $ 0 = 1"
by (simp add: fps_mult_nth fps_inverse_def)
{
fix n :: nat
assume np: "n > 0"
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) =
(setsum (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: "setsum (λi. f$i * natfun_inverse f (n - i)) {1..n} = - (f$0) * (inverse f)$n"
by (simp add: field_simps)
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))"
by (simp add: eq)
also have "… = 0" unfolding th1 ifn by simp
finally have "(inverse f * f)$n = 0" unfolding c .
}
with th0 show ?thesis by (simp add: fps_eq_iff)
qed

lemma fps_inverse_0_iff[simp]: "(inverse f)$0 = (0::'a::division_ring) <-> f$0 = 0"
by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero)

lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) <-> f $0 = 0"
proof -
{
assume "f$0 = 0"
then have "inverse f = 0" by (simp add: fps_inverse_def)
}
moreover
{
assume h: "inverse f = 0" and c: "f $0 ≠ 0"
from inverse_mult_eq_1[OF c] h have False by simp
}
ultimately show ?thesis by blast
qed

lemma fps_inverse_idempotent[intro]:
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)"
by (simp add: mult_ac)
then show ?thesis using f0 unfolding mult_cancel_left by simp
qed

lemma fps_inverse_unique:
assumes f0: "f$0 ≠ (0::'a::field)"
and fg: "f*g = 1"
shows "inverse f = g"
proof -
from inverse_mult_eq_1[OF f0] fg
have th0: "inverse f * f = g * f" by (simp add: mult_ac)
then show ?thesis using f0 unfolding mult_cancel_right
by (auto simp add: expand_fps_eq)
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
apply (simp add: fps_eq_iff fps_mult_nth)
proof clarsimp
fix n :: nat
assume n: "n > 0"
let ?f = "λi. if n = i then (1::'a) 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: "setsum ?f {0..n} = setsum ?g {0..n}"
by (rule setsum_cong2) auto
have th2: "setsum ?g {0..n - 1} = setsum ?h {0..n - 1}"
using n apply - by (rule setsum_cong2) auto
have eq: "{0 .. n} = {0.. n - 1} ∪ {n}" by auto
from n have d: "{0.. n - 1} ∩ {n} = {}" by auto
have f: "finite {0.. n - 1}" "finite {n}" by auto
show "setsum ?f {0..n} = 0"
unfolding th1
apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
unfolding th2
apply (simp add: setsum_delta)
done
qed

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)"
by (simp add: fps_deriv_def)

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"

unfolding fps_eq_iff fps_add_nth fps_const_mult_left fps_deriv_nth by (simp add: field_simps)

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"
{ fix n::nat
let ?Zn = "{0 ..n}"
let ?Zn1 = "{0 .. n + 1}"
let ?f = "λi. i + 1"
have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def)
have eq: "{1.. n+1} = ?f ` {0..n}" by auto
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)"

{
fix k
assume k: "k ∈ {0..n}"
have "?h (k + 1) = ?g k" using k by auto
}
note th0 = this
have eq': "{0..n +1}- {1 .. n+1} = {0}" by auto
have s0: "setsum (λi. of_nat i * f $ i * g $ (n + 1 - i)) ?Zn1 =
setsum (λi. of_nat (n + 1 - i) * f $ (n + 1 - i) * g $ i) ?Zn1"

apply (rule setsum_reindex_cong[where f="λi. n + 1 - i"])
apply (simp add: inj_on_def Ball_def)
apply presburger
apply (rule set_eqI)
apply (presburger add: image_iff)
apply simp
done
have s1: "setsum (λi. f $ i * g $ (n + 1 - i)) ?Zn1 =
setsum (λi. f $ (n + 1 - i) * g $ i) ?Zn1"

apply (rule setsum_reindex_cong[where f="λi. n + 1 - i"])
apply (simp add: inj_on_def Ball_def)
apply presburger
apply (rule set_eqI)
apply (presburger add: image_iff)
apply simp
done
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)"
by (simp add: fps_mult_nth setsum_addf[symmetric])
also have "… = setsum ?h {1..n+1}"
using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto
also have "… = setsum ?h {0..n+1}"
apply (rule setsum_mono_zero_left)
apply simp
apply (simp add: subset_eq)
unfolding eq'
apply simp
done
also have "… = (fps_deriv (f * g)) $ n"
apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf)
unfolding s0 s1
unfolding setsum_addf[symmetric] setsum_right_distrib
apply (rule setsum_cong2)
apply (auto simp add: of_nat_diff field_simps)
done
finally have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" .
}
then show ?thesis unfolding fps_eq_iff by auto
qed

lemma fps_deriv_X[simp]: "fps_deriv X = 1"
by (simp add: fps_deriv_def X_def fps_eq_iff)

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

lemma fps_deriv_add[simp]: "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"
unfolding diff_minus 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_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"
by (simp add: fps_deriv_def fps_eq_iff)

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_setsum:
"fps_deriv (setsum f S) = setsum (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}))"
proof -
{
assume "f = fps_const (f$0)"
then have "fps_deriv f = fps_deriv (fps_const (f$0))" by simp
then have "fps_deriv f = 0" by simp
}
moreover
{
assume z: "fps_deriv f = 0"
then have "∀n. (fps_deriv f)$n = 0" by simp
then have "∀n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)
then have "f = fps_const (f$0)"
apply (clarsimp simp add: fps_eq_iff fps_const_def)
apply (erule_tac x="n - 1" in allE)
apply simp
done
}
ultimately show ?thesis by blast
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 by (simp add: field_simps)
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

lemma fps_nth_deriv_add[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_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"
unfolding diff_minus fps_nth_deriv_add 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_setsum:
"fps_nth_deriv n (setsum f S) = setsum (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 of_nat_mult)

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)
note h = Suc.hyps[OF `a$0 = 1`]
show ?case unfolding power_Suc fps_mult_nth
using h `a$0 = 1` fps_power_zeroth_eq_one[OF `a$0=1`]
by (simp add: field_simps)
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 (auto simp add: fps_mult_nth)
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::'a)"
let ?ths = "∀m<k. a ^ k $ m = 0"
{ assume "k = 0" then have ?ths by simp }
moreover
{
fix l
assume k: "k = Suc l"
{
fix m
assume mk: "m < k"
{
assume "m = 0"
then have "a^k $ m = 0"
using startsby_zero_power[of a k] k a0 by simp
}
moreover
{
assume m0: "m ≠ 0"
have "a ^k $ m = (a^l * a) $m" by (simp add: k mult_commute)
also have "… = (∑i = 0..m. a ^ l $ i * a $ (m - i))" by (simp add: fps_mult_nth)
also have "… = 0"
apply (rule setsum_0')
apply auto
apply (case_tac "x = m")
using a0 apply simp
apply (rule H[rule_format])
using a0 k mk apply auto
done
finally have "a^k $ m = 0" .
}
ultimately have "a^k $ m = 0" by blast
}
then have ?ths by blast
}
ultimately show ?ths by (cases k) auto
qed

lemma startsby_zero_setsum_depends:
assumes a0: "a $0 = (0::'a::idom)" and kn: "n ≥ k"
shows "setsum (λi. (a ^ i)$k) {0 .. n} = setsum (λi. (a ^ i)$k) {0 .. k}"
apply (rule setsum_mono_zero_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)$(Suc n)" by (simp add: field_simps)
also have "… = setsum (λi. a^n$i * a $ (Suc n - i)) {0.. Suc n}"
by (simp add: fps_mult_nth)
also have "… = setsum (λi. a^n$i * a $ (Suc n - i)) {n .. Suc n}"
apply (rule setsum_mono_zero_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"
proof -
{
assume a0: "a$0 = 0"
then have eq: "inverse a = 0" by (simp add: fps_inverse_def)
{ assume "n = 0" hence ?thesis by simp }
moreover
{
assume n: "n > 0"
from startsby_zero_power[OF a0 n] eq a0 n have ?thesis
by (simp add: fps_inverse_def)
}
ultimately have ?thesis by blast
}
moreover
{
assume a0: "a$0 ≠ 0"
have ?thesis
apply (rule fps_inverse_unique)
apply (simp add: a0)
unfolding power_mult_distrib[symmetric]
apply (rule ssubst[where t = "a * inverse a" and s= 1])
apply simp_all
apply (subst mult_commute)
apply (rule inverse_mult_eq_1[OF a0])
done
}
ultimately show ?thesis by blast
qed

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 (auto simp add: field_simps fps_const_add[symmetric] simp del: fps_const_add)
apply (case_tac n)
apply (auto simp add: field_simps)
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
hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp
hence "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
apply (simp add: field_simps)
apply (simp add: mult_assoc[symmetric])
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"
by (simp add: field_simps)
qed

lemma fps_inverse_mult:
fixes a::"('a :: field) fps"
shows "inverse (a * b) = inverse a * inverse b"
proof -
{
assume a0: "a$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all
have ?thesis unfolding th by simp
}
moreover
{
assume b0: "b$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all
have ?thesis unfolding th by simp
}
moreover
{
assume a0: "a$0 ≠ 0" and b0: "b$0 ≠ 0"
from a0 b0 have ab0:"(a*b) $ 0 ≠ 0" by (simp add: fps_mult_nth)
from inverse_mult_eq_1[OF ab0]
have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp
then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b"
by (simp add: field_simps)
then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp
}
ultimately show ?thesis by blast
qed

lemma fps_inverse_deriv':
fixes a:: "('a :: field) fps"
assumes a0: "a$0 ≠ 0"
shows "fps_deriv (inverse a) = - fps_deriv a / a2"
using fps_inverse_deriv[OF a0]
unfolding power2_eq_square fps_divide_def fps_inverse_mult
by simp

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_divide_deriv:
fixes a:: "('a :: field) fps"
assumes a0: "b$0 ≠ 0"
shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b2"
using fps_inverse_deriv[OF a0]
by (simp add: fps_divide_def field_simps
power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])


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

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


lemma fps_inverse_X_plus1:
"inverse (1 + X) = Abs_fps (λn. (- (1::'a::{field})) ^ n)" (is "_ = ?r")
proof-
have eq: "(1 + 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 simp del: minus_one)
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" by (simp add: fps_deriv_fps_integral)
moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def)
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
fps_compose_def: "a oo b = Abs_fps (λn. setsum (λi. a$i * (b^i$n)) {0..n})"

lemma fps_compose_nth: "(a oo b)$n = setsum (λi. a$i * (b^i$n)) {0..n}"
by (simp add: fps_compose_def)

lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)"
by (simp add: fps_ext fps_compose_def mult_delta_right setsum_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 setsum_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]: "(neg_numeral k::('a::{comm_ring_1}) fps) oo b = neg_numeral k"
unfolding neg_numeral_fps_const by simp

lemma X_fps_compose_startby0[simp]: "a$0 = 0 ==> X oo a = (a :: ('a :: comm_ring_1) fps)"
by (simp add: fps_eq_iff fps_compose_def mult_delta_left setsum_delta not_le)


subsection {* Rules from Herbert Wilf's Generatingfunctionology*}

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

lemma fps_power_mult_eq_shift:
"X^Suc k * Abs_fps (λn. a (n + Suc k)) =
Abs_fps a - setsum (λi. fps_const (a i :: 'a:: comm_ring_1) * X^i) {0 .. k}"

(is "?lhs = ?rhs")
proof -
{ fix n:: nat
have "?lhs $ n = (if n < Suc k then 0 else a n)"
unfolding X_power_mult_nth by auto
also have "… = ?rhs $ n"
proof (induct k)
case 0
thus ?case by (simp add: fps_setsum_nth)
next
case (Suc k)
note th = Suc.hyps[symmetric]
have "(Abs_fps a - setsum (λi. fps_const (a i :: 'a) * X^i) {0 .. Suc k})$n =
(Abs_fps a - setsum (λi. fps_const (a i :: 'a) * X^i) {0 .. k} -
fps_const (a (Suc k)) * X^ Suc k) $ n"

by (simp add: field_simps)
also have "… = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)$n"
using th unfolding fps_sub_nth by simp
also have "… = (if n < Suc (Suc k) then 0 else a n)"
unfolding 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 have "?lhs $ n = ?rhs $ n" .
}
then show ?thesis by (simp add: fps_eq_iff)
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 "XD = op * X o fps_deriv"

lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)"
by (simp add: XD_def field_simps)

lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a"
by (simp add: XD_def field_simps)

lemma XD_linear[simp]: "XD (fps_const c * a + fps_const d * b) =
fps_const c * XD a + fps_const d * XD (b :: ('a::comm_ring_1) fps)"

by simp

lemma XDN_linear:
"(XD ^^ n) (fps_const c * a + fps_const d * b) =
fps_const c * (XD ^^ n) a + fps_const d * (XD ^^ n) (b :: ('a::comm_ring_1) fps)"

by (induct n) simp_all

lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (λn. of_nat n* a$n)"
by (simp add: fps_eq_iff)


lemma fps_mult_XD_shift:
"(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: XD_def fps_eq_iff field_simps del: One_nat_def)


subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}

subsubsection{* Rule 5 --- summation and "division" by (1 - X)*}

lemma fps_divide_X_minus1_setsum_lemma:
"a = ((1::('a::comm_ring_1) fps) - X) * Abs_fps (λn. setsum (λi. a $ i) {0..n})"
proof -
let ?X = "X::('a::comm_ring_1) fps"
let ?sa = "Abs_fps (λn. setsum (λi. a $ i) {0..n})"
have th0: "!!i. (1 - (X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)"
by simp
{
fix n:: nat
{
assume "n=0"
hence "a$n = ((1 - ?X) * ?sa) $ n"
by (simp add: fps_mult_nth)
}
moreover
{
assume n0: "n ≠ 0"
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 n0 by simp_all
have f: "finite {0}" "finite {1}" "finite {2 .. n}"
"finite {0 .. n - 1}" "finite {n}" by simp_all
have "((1 - ?X) * ?sa) $ n = setsum (λi. (1 - ?X)$ i * ?sa $ (n - i)) {0 .. n}"
by (simp add: fps_mult_nth)
also have "… = a$n"
unfolding th0
unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)]
apply (simp)
unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)]
apply simp
done
finally have "a$n = ((1 - ?X) * ?sa) $ n" by simp
}
ultimately have "a$n = ((1 - ?X) * ?sa) $ n" by blast
}
then show ?thesis unfolding fps_eq_iff by blast
qed

lemma fps_divide_X_minus1_setsum:
"a /((1::('a::field) fps) - X) = Abs_fps (λn. setsum (λi. a $ i) {0..n})"
proof -
let ?X = "1 - (X::('a::field) fps)"
have th0: "?X $ 0 ≠ 0" by simp
have "a /?X = ?X * Abs_fps (λn::nat. setsum (op $ a) {0..n}) * inverse ?X"
using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0
by (simp add: fps_divide_def mult_assoc)
also have "… = (inverse ?X * ?X) * Abs_fps (λn::nat. setsum (op $ a) {0..n}) "
by (simp add: mult_ac)
finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0])
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 ∧ listsum l = n}"

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

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

lemma natpermute_split:
assumes mn: "h ≤ k"
shows "natpermute n k =
(\<Union>m ∈{0..n}. {l1 @ l2 |l1 l2. l1 ∈ natpermute m h ∧ l2 ∈ natpermute (n - m) (k - h)})"

(is "?L = ?R" is "?L = (\<Union>m ∈{0..n}. ?S m)")
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': "listsum xs = m"
by (simp add: natpermute_def)
from ys have ys': "listsum ys = n - m"
by (simp add: natpermute_def)
have "l ∈ ?L" using leq xs ys h
apply (clarsimp simp add: natpermute_def)
unfolding xs' ys'
using mn xs ys
unfolding natpermute_def
apply simp
done
}
moreover
{
fix l
assume l: "l ∈ natpermute n k"
let ?xs = "take h l"
let ?ys = "drop h l"
let ?m = "listsum ?xs"
from l have ls: "listsum (?xs @ ?ys) = n"
by (simp add: natpermute_def)
have xs: "?xs ∈ natpermute ?m h" using l mn
by (simp add: natpermute_def)
have l_take_drop: "listsum l = listsum (take h l @ drop h l)"
by simp
then have ys: "?ys ∈ natpermute (n - ?m) (k - h)"
using l mn 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 have "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
}
ultimately show ?thesis by blast
qed

lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"
by (auto simp add: natpermute_def)

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])
apply (simp add: natpermute_0)
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} = UNION {0 .. k} (λi. {(replicate (k+1) 0) [i:=n]})"
(is "?A = ?B")
proof -
{
fix xs
assume H: "xs ∈ natpermute n (k+1)" and n: "n ∈ set xs"
from n obtain i where i: "i ∈ {0.. k}" "xs!i = n" using H
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 = setsum (nth xs) {0..k}"
apply (simp add: natpermute_def)
apply (auto simp add: atLeastLessThanSuc_atLeastAtMost listsum_setsum_nth)
done
also have "… = n + setsum (nth xs) ({0..k} - {i})"
unfolding setsum_Un_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"
by (simp add: natpermute_def)
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 have "xs ∈ ?B" using i by blast
}
moreover
{
fix i
assume i: "i ∈ {0..k}"
let ?xs = "replicate (k+1) 0 [i:=n]"
have nxs: "n ∈ set ?xs"
apply (rule set_update_memI)
using i apply simp
done
have xsl: "length ?xs = k+1"
by (simp only: length_replicate length_list_update)
have "listsum ?xs = setsum (nth ?xs) {0..<k+1}"
unfolding listsum_setsum_nth xsl ..
also have "… = setsum (λj. if j = i then n else 0) {0..< k+1}"
by (rule setsum_cong2) (simp del: replicate.simps)
also have "… = n" using i by (simp add: setsum_delta)
finally have "?xs ∈ natpermute n (k+1)"
using xsl unfolding natpermute_def mem_Collect_eq by blast
then have "?xs ∈ ?A"
using nxs by blast
}
ultimately show ?thesis by auto
qed

(* The general form *)
lemma fps_setprod_nth:
fixes m :: nat
and a :: "nat => ('a::comm_ring_1) fps"
shows "(setprod a {0 .. m})$n =
setsum (λv. setprod (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 apply (simp add: set_eq_iff)
apply presburger
done
have f0: "finite {0 .. k}" "finite {m}" by auto
have d0: "{0 .. k} ∩ {m} = {}" using Suc by auto
have "(setprod a {0 .. m}) $ n = (setprod a {0 .. k} * a m) $ n"
unfolding setprod_Un_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)"
apply (simp add: Suc)
unfolding natpermute_split[of m "m + 1", simplified, of n,
unfolded natlist_trivial_1[unfolded One_nat_def] Suc]
apply (subst setsum_UN_disjoint)
apply simp
apply simp
unfolding image_Collect[symmetric]
apply clarsimp
apply (rule finite_imageI)
apply (rule natpermute_finite)
apply (clarsimp simp add: set_eq_iff)
apply auto
apply (rule setsum_cong2)
unfolding setsum_left_distrib
apply (rule sym)
apply (rule_tac f="λxs. xs @[n - x]" in setsum_reindex_cong)
apply (simp add: inj_on_def)
apply auto
unfolding setprod_Un_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 = setsum (λv. setprod (λj. a $ (v!j)) {0..m}) (natpermute n (m+1))"
proof -
have th0: "a^Suc m = setprod (λi. a) {0..m}" by (simp add: setprod_constant)
show ?thesis unfolding th0 fps_setprod_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 setsum (λv. setprod (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 = setprod (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 setprod_constant) simp
finally show ?thesis .
qed

lemma fps_compose_inj_right:
assumes a0: "a$0 = (0::'a::{idom})"
and a1: "a$1 ≠ 0"
shows "(b oo a = c oo a) <-> b = c" (is "?lhs <->?rhs")
proof
assume ?rhs
then show "?lhs" by simp
next
assume h: ?lhs
{
fix n
have "b$n = c$n"
proof (induct n rule: nat_less_induct)
fix n
assume H: "∀m<n. b$m = c$m"
{
assume n0: "n=0"
from h have "(b oo a)$n = (c oo a)$n" by simp
hence "b$n = c$n" using n0 by (simp add: fps_compose_nth)
}
moreover
{
fix n1 assume n1: "n = Suc n1"
have f: "finite {0 .. n1}" "finite {n}" by simp_all
have eq: "{0 .. n1} ∪ {n} = {0 .. n}" using n1 by auto
have d: "{0 .. n1} ∩ {n} = {}" using n1 by auto
have seq: "(∑i = 0..n1. b $ i * a ^ i $ n) = (∑i = 0..n1. c $ i * a ^ i $ n)"
apply (rule setsum_cong2)
using H n1
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 setsum_Un_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$1)^n"
unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq]
using startsby_zero_power_nth_same[OF a0]
by simp
from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
have "b$n = c$n" by auto
}
ultimately show "b$n = c$n" by (cases n) auto
qed}
then show ?rhs by (simp add: fps_eq_iff)
qed


subsection {* Radicals *}

declare setprod_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 - setsum (λxs. setprod (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

termination radical
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}"
{
assume c: "Suc n ≤ xs ! i"
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 = listsum xs"
by (simp add: natpermute_def)
also have "… = setsum (nth xs) {0..<Suc k}" using xs
by (simp add: natpermute_def listsum_setsum_nth)
also have "… = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
unfolding eqs setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
by simp
finally have False using c' by simp
}
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

definition "fps_radical r n a = Abs_fps (radical r n a)"

lemma fps_radical0[simp]: "fps_radical r 0 a = 1"
apply (auto simp add: fps_eq_iff fps_radical_def)
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))"
by (cases n) (simp_all add: fps_radical_def)

lemma fps_radical_power_nth[simp]:
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 setprod_cong)
apply simp
using Suc
apply (subgoal_tac "replicate k (0::nat) ! x = 0")
apply (auto intro: nth_replicate simp del: replicate.simps)
done
also have "… = a$0" using r Suc by (simp add: setprod_constant)
finally show ?thesis using Suc by simp
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"
{
assume eq: "replicate (k+1) 0 [i:=n] = replicate (k+1) 0 [j:= n]"
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 have False
using eq n0 by (simp del: replicate.simps)
}
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 (\<Union>i∈{0..k}. {replicate (k + 1) 0[i := n]}) = k+1"
by simp
qed

lemma power_radical:
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"
proof-
let ?r = "fps_radical r (Suc k) a"
{
assume r0: "(r (Suc k) (a$0)) ^ Suc k = a$0"
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"
hence "?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 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 "setsum ?f ?Pnkn = setsum (λv. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn"
proof (rule setsum_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 setprod_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: setprod_gen_delta)
finally show ?ths .
qed
then have "setsum ?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$n - setsum ?f ?Pnknn"
unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc)
finally have fn: "setsum ?f ?Pnkn = a$n - setsum ?f ?Pnknn" .
have "(?r ^ Suc k)$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
unfolding fps_power_nth_Suc setsum_Un_disjoint[OF f d, unfolded eq] ..
also have "… = a$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 have ?thesis using r0 by (simp add: fps_eq_iff)
}
moreover
{
assume h: "(fps_radical r (Suc k) a) ^ (Suc k) = a"
hence "((fps_radical r (Suc k) a) ^ (Suc k))$0 = a$0" by simp
then have "(r (Suc k) (a$0)) ^ Suc k = a$0"
unfolding fps_power_nth_Suc
by (simp add: setprod_constant del: replicate.simps)
}
ultimately show ?thesis by blast
qed

(*
lemma power_radical:
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" hence "?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 "setsum ?f ?Pnkn = setsum (λv. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn"
proof(rule setsum_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 setprod_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 setprod_gen_delta[OF fK] using i r0 by simp
finally show ?ths .
qed
then have "setsum ?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$n - setsum ?f ?Pnknn"
unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc )
finally have fn: "setsum ?f ?Pnkn = a$n - setsum ?f ?Pnknn" .
have "(?r ^ Suc k)$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
unfolding fps_power_nth_Suc setsum_Un_disjoint[OF f d, unfolded eq] ..
also have "… = a$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':
assumes c0: "(c::'a::field) ~= 0"
and eq: "a * c = b"
shows "a = b / c"
proof -
from eq have "a * c * inverse c = b * inverse c"
by simp
hence "a * (inverse c * c) = b/c"
by (simp only: field_simps divide_inverse)
then show "a = b/c"
unfolding field_inverse[OF c0] by simp
qed

lemma radical_unique:
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"
proof -
let ?r = "fps_radical r (Suc k) b"
have r00: "r (Suc k) (b$0) ≠ 0" using b0 r0 by auto
{
assume H: "a = ?r"
from H have "a^Suc k = b"
using power_radical[OF b0, of r k, unfolded r0] by simp
}
moreover
{
assume H: "a^Suc k = b"
have ceq: "card {0..k} = Suc k" by simp
from a0 have a0r0: "a$0 = ?r$0" by simp
{
fix n
have "a $ n = ?r $ n"
proof (induct n rule: nat_less_induct)
fix n
assume h: "∀m<n. a$m = ?r $m"
{
assume "n = 0"
hence "a$n = ?r $n" using a0 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 (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 "setsum ?g ?Pnkn = setsum (λv. a $ n * (?r$0)^k) ?Pnkn"
proof (rule setsum_cong2)
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 setprod_cong, simp)
using i a0 apply (simp del: replicate.simps)
done
also have "… = a $ n * (?r $ 0)^k"
using i by (simp add: setprod_gen_delta)
finally show ?ths .
qed
then have th0: "setsum ?g ?Pnkn = of_nat (k+1) * a $ n * (?r $ 0)^k"
by (simp add: natpermute_max_card[OF nz, simplified])
have th1: "setsum ?g ?Pnknn = setsum ?f ?Pnknn"
proof (rule setsum_cong2, rule setprod_cong, simp)
fix xs i
assume xs: "xs ∈ ?Pnknn" and i: "i ∈ {0..k}"
{
assume c: "n ≤ xs ! i"
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 = listsum xs"
by (simp add: natpermute_def)
also have "… = setsum (nth xs) {0..<Suc k}"
using xs by (simp add: natpermute_def listsum_setsum_nth)
also have "… = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
unfolding eqs setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
by simp
finally have False using c' by simp
}
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 H have "b$n = a^Suc k $ n"
by (simp add: fps_eq_iff)
also have "a ^ Suc k$n = setsum ?g ?Pnkn + setsum ?g ?Pnknn"
unfolding fps_power_nth_Suc
using setsum_Un_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 + setsum ?f ?Pnknn"
unfolding th0 th1 ..
finally have "of_nat (k+1) * a $ n * (?r $ 0)^k = b$n - setsum ?f ?Pnknn"
by simp
then have "a$n = (b$n - setsum ?f ?Pnknn) / (of_nat (k+1) * (?r $ 0)^k)"
apply -
apply (rule eq_divide_imp')
using r00
apply (simp del: of_nat_Suc)
apply (simp add: mult_ac)
done
then have "a$n = ?r $n"
apply (simp del: of_nat_Suc)
unfolding fps_radical_def n1
apply (simp add: field_simps n1 th00 del: of_nat_Suc)
done
}
ultimately show "a$n = ?r $ n" by (cases n) auto
qed
}
then have "a = ?r" by (simp add: fps_eq_iff)
}
ultimately show ?thesis by blast
qed


lemma radical_power:
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

lemma fps_deriv_radical:
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
hence "fps_deriv ?r * ?w = fps_deriv a"
by (simp add: fps_deriv_power mult_ac del: power_Suc)
hence "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a"
by simp
hence "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w"
by (simp add: fps_divide_def)
then show ?thesis unfolding th0 by simp
qed

lemma radical_mult_distrib:
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) <->
fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"

proof -
{
assume r0': "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0)"
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"
hence ?thesis using r0' 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]
iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded k]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded k]] k r0'
have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)
}
ultimately have ?thesis by (cases k) auto
}
moreover
{
assume h: "fps_radical r k (a*b) = fps_radical r k a * fps_radical r k b"
hence "(fps_radical r k (a*b))$0 = (fps_radical r k a * fps_radical r k b)$0"
by simp
then have "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0)"
using k by (simp add: fps_mult_nth)
}
ultimately show ?thesis by blast
qed

(*
lemma radical_mult_distrib:
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"
shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
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" hence ?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 (simp add: fps_divide_def)

lemma radical_divide:
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) <->
fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b"

(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

{
assume ?rhs
then have "?r (a/b) $ 0 = (?r a / ?r b)$0" by simp
then have ?lhs using k a0 b0 rb0'
by (simp add: fps_divide_def fps_mult_nth fps_inverse_def divide_inverse)
}
moreover
{
assume h: ?lhs
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: h nonzero_power_divide[OF rb0'] ra0 rb0)
from a0 b0 ra0' rb0' kp h
have th1: "r k ((a / b) $ 0) = (fps_radical r k a / fps_radical r k b) $ 0"
by (simp add: fps_divide_def 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_def 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]]
have th2: "(?r a / ?r b)^k = a/b"
by (simp add: fps_divide_def 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]
have ?rhs .
}
ultimately show ?thesis by blast
qed

lemma radical_inverse:
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)) <->
fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a"

using radical_divide[where k=k and r=r and a=1 and b=a, OF k ] ra0 r1 a0
by (simp add: divide_inverse fps_divide_def)

subsection{* Derivative of composition *}

lemma fps_compose_deriv:
fixes a:: "('a::idom) fps"
assumes b0: "b$0 = 0"
shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * (fps_deriv b)"
proof -
{
fix n
have "(fps_deriv (a oo b))$n = setsum (λi. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}"
by (simp add: fps_compose_def field_simps setsum_right_distrib del: of_nat_Suc)
also have "… = setsum (λi. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}"
by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
also have "… = setsum (λi. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}"
unfolding fps_mult_left_const_nth by (simp add: field_simps)
also have "… = setsum (λi. of_nat i * a$i * (setsum (λj. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {0.. Suc n}"
unfolding fps_mult_nth ..
also have "… = setsum (λi. of_nat i * a$i * (setsum (λj. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {1.. Suc n}"
apply (rule setsum_mono_zero_right)
apply (auto simp add: mult_delta_left setsum_delta not_le)
done
also have "… = setsum (λi. of_nat (i + 1) * a$(i+1) * (setsum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
unfolding fps_deriv_nth
by (rule setsum_reindex_cong [where f = Suc]) (auto simp add: mult_assoc)
finally have th0: "(fps_deriv (a oo b))$n =
setsum (λi. of_nat (i + 1) * a$(i+1) * (setsum (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))$n = setsum (λi. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}"
unfolding fps_mult_nth by (simp add: mult_ac)
also have "… = setsum (λi. setsum (λj. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}"
unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult_assoc
apply (rule setsum_cong2)
apply (rule setsum_mono_zero_left)
apply (simp_all add: subset_eq)
apply clarify
apply (subgoal_tac "b^i$x = 0")
apply simp
apply (rule startsby_zero_power_prefix[OF b0, rule_format])
apply simp
done
also have "… = setsum (λi. of_nat (i + 1) * a$(i+1) * (setsum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
unfolding setsum_right_distrib
apply (subst setsum_commute)
apply (rule setsum_cong2)+
apply simp
done
finally have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n"
unfolding th0 by simp
}
then show ?thesis by (simp add: fps_eq_iff)
qed

lemma fps_mult_X_plus_1_nth:
"((1+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 by (simp add: fps_mult_nth )
next
case (Suc m)
have "((1+X)*a) $n = setsum (λi. (1+X)$i * a$(n-i)) {0..n}"
by (simp add: fps_mult_nth)
also have "… = setsum (λi. (1+X)$i * a$(n-i)) {0.. 1}"
unfolding Suc by (rule setsum_mono_zero_right) auto
also have "… = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))"
by (simp add: Suc)
finally show ?thesis .
qed

subsection{* Finite FPS (i.e. polynomials) and X *}

lemma fps_poly_sum_X:
assumes z: "∀i > n. a$i = (0::'a::comm_ring_1)"
shows "a = setsum (λi. fps_const (a$i) * X^i) {0..n}" (is "a = ?r")
proof -
{
fix i
have "a$i = ?r$i"
unfolding fps_setsum_nth fps_mult_left_const_nth X_power_nth
by (simp add: mult_delta_right setsum_delta' z)
}
then show ?thesis unfolding fps_eq_iff by blast
qed


subsection{* Compositional inverses *}

fun compinv :: "'a fps => nat => 'a::{field}"
where
"compinv a 0 = X$0"
| "compinv a (Suc n) =
(X$ Suc n - setsum (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 = X"
proof -
let ?i = "fps_inv a oo a"
{
fix n
have "?i $n = X$n"
proof (induct n rule: nat_less_induct)
fix n
assume h: "∀m<n. ?i$m = X$m"
show "?i $ n = X$n"
proof (cases n)
case 0
then show ?thesis using a0
by (simp add: fps_compose_nth fps_inv_def)
next
case (Suc n1)
have "?i $ n = setsum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1"
by (simp add: fps_compose_nth Suc startsby_zero_power_nth_same[OF a0] del: power_Suc)
also have "… = setsum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1} +
(X$ Suc n1 - setsum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1})"

using a0 a1 Suc by (simp add: fps_inv_def)
also have "… = X$n" using Suc by simp
finally show ?thesis .
qed
qed
}
then show ?thesis by (simp add: fps_eq_iff)
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 - setsum (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"
{
fix n
have "?i $n = b$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
by (simp add: fps_compose_nth fps_ginv_def)
next
case (Suc n1)
have "?i $ n = setsum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1"
by (simp add: fps_compose_nth Suc startsby_zero_power_nth_same[OF a0] del: power_Suc)
also have "… = setsum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} +
(b$ Suc n1 - setsum (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 by (simp add: fps_eq_iff)
qed

lemma fps_inv_ginv: "fps_inv = fps_ginv 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 setsum_delta)

lemma fps_compose_0[simp]: "0 oo a = 0"
by (simp add: fps_eq_iff fps_compose_nth)

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 setsum_0')

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 setsum_addf)

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

lemma convolution_eq:
"setsum (%i. a (i :: nat) * b (n - i)) {0 .. n} = setsum (%(i,j). a i * b j) {(i,j). i <= n ∧ j ≤ n ∧ i + j = n}"
apply (rule setsum_reindex_cong[where f=fst])
apply (clarsimp simp add: inj_on_def)
apply (auto simp add: set_eq_iff image_iff)
apply (rule_tac x= "x" in exI)
apply clarsimp
apply (rule_tac x="n - x" in exI)
apply arith
done

lemma product_composition_lemma:
assumes c0: "c$0 = (0::'a::idom)"
and d0: "d$0 = 0"
shows "((a oo c) * (b oo d))$n =
setsum (%(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 = setsum (%i. setsum (%(k,m). a$k * (c^k)$i * b$m * (d^m) $ (n - i)) {(k,m). k + m ≤ n}) {0..n}"
apply (simp add: fps_mult_nth setsum_right_distrib)
apply (subst setsum_commute)
apply (rule setsum_cong2)
apply (auto simp add: field_simps)
done
also have "… = ?l"
apply (simp add: fps_mult_nth fps_compose_nth setsum_product)
apply (rule setsum_cong2)
apply (simp add: setsum_cartesian_product mult_assoc)
apply (rule setsum_mono_zero_right[OF f])
apply (simp add: subset_eq) apply presburger
apply clarsimp
apply (rule ccontr)
apply (clarsimp simp add: not_le)
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 =
setsum (%k. setsum (%m. a$k * b$m * (c^k * d^m) $ n) {0..n}) {0..n}"
(is "?l = ?r")
unfolding product_composition_lemma[OF c0 d0]
unfolding setsum_cartesian_product
apply (rule setsum_mono_zero_left)
apply simp
apply (clarsimp simp add: subset_eq)
apply clarsimp
apply (rule ccontr)
apply (subgoal_tac "(c^aa * d^ba) $ n = 0")
apply simp
unfolding fps_mult_nth
apply (rule setsum_0')
apply (clarsimp simp add: not_le)
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 setsum_pair_less_iff:
"setsum (%((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m ≤ n} =
setsum (%s. setsum (%i. a i * b (s - i) * c s) {0..s}) {0..n}"

(is "?l = ?r")
proof -
let ?KM = "{(k,m). k + m ≤ n}"
let ?f = "%s. UNION {(0::nat)..s} (%i. {(i,s - i)})"
have th0: "?KM = UNION {0..n} ?f"
apply (simp add: set_eq_iff)
apply presburger (* FIXME: slow! *)
done
show "?l = ?r "
unfolding th0
apply (subst setsum_UN_disjoint)
apply auto
apply (subst setsum_UN_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 =
setsum (%s. setsum (%i. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}"

(is "?l = ?r")
unfolding product_composition_lemma[OF c0 c0] power_add[symmetric]
unfolding setsum_pair_less_iff[where a = "%k. a$k" and b="%m. b$m" and c="%s. (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)" (is "?l = ?r")
apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma[OF c0])
apply (simp add: fps_compose_nth fps_mult_nth setsum_left_distrib)
done

lemma fps_compose_setprod_distrib:
assumes c0: "c$0 = (0::'a::idom)"
shows "(setprod a S) oo c = setprod (%k. a k oo c) S" (is "?l = ?r")
apply (cases "finite S")
apply simp_all
apply (induct S rule: finite_induct)
apply simp
apply (simp add: fps_compose_mult_distrib[OF c0])
done

lemma fps_compose_power:
assumes c0: "c$0 = (0::'a::idom)"
shows "(a oo c)^n = a^n oo c"
(is "?l = ?r")
proof (cases n)
case 0
then show ?thesis by simp
next
case (Suc m)
have th0: "a^n = setprod (%k. a) {0..m}" "(a oo c) ^ n = setprod (%k. a oo c) {0..m}"
by (simp_all add: setprod_constant Suc)
then show ?thesis
by (simp add: fps_compose_setprod_distrib[OF c0])
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 setsum_negf[symmetric])

lemma fps_compose_sub_distrib: "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)"
unfolding diff_minus fps_compose_uminus fps_compose_add_distrib ..

lemma X_fps_compose: "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 setsum_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)"
unfolding fps_divide_def fps_compose_mult_distrib[OF c0]
fps_inverse_compose[OF c0 b0] ..

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 - X) $ 0 ≠ (0::'a)" by simp
from fps_inverse_gp[where ?'a = 'a]
have "inverse ?one = 1 - X" by (simp add: fps_eq_iff)
hence "inverse (inverse ?one) = inverse (1 - X)" by simp
hence th: "?one = 1/(1 - X)" unfolding fps_inverse_idempotent[OF o0]
by (simp add: fps_divide_def)
show ?thesis
unfolding th
unfolding fps_divide_compose[OF a0 th0]
fps_compose_1 fps_compose_sub_distrib 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

lemma fps_compose_radical:
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"
by (simp add: fps_compose_def)
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"
by (simp add: ab0 fps_compose_def)
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 setsum_right_distrib 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$0 = 0"
shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
proof -
{
fix n
have "?l$n = (setsum (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
setsum_right_distrib mult_assoc fps_setsum_nth)
also have "… = ((setsum (λi. fps_const (a$i) * b^i) {0..n}) oo c)$n"
by (simp add: fps_compose_setsum_distrib)
also have "… = ?r$n"
apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult_assoc)
apply (rule setsum_cong2)
apply (rule setsum_mono_zero_right)
apply (auto simp add: not_le)
apply (erule startsby_zero_power_prefix[OF b0, rule_format])
done
finally have "?l$n = ?r$n" .
}
then show ?thesis by (simp add: fps_eq_iff)
qed


lemma fps_X_power_compose:
assumes a0: "a$0=0"
shows "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)
{
fix n
{
assume kn: "k>n"
hence "?l $ n = ?r $n" using a0 startsby_zero_power_prefix[OF a0] Suc
by (simp add: fps_compose_nth del: power_Suc)
}
moreover
{
assume kn: "k ≤ n"
hence "?l$n = ?r$n"
by (simp add: fps_compose_nth mult_delta_left setsum_delta)
}
moreover have "k >n ∨ k≤ n" by arith
ultimately have "?l$n = ?r$n" by blast
}
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 = X"
proof -
let ?ia = "fps_inv a"
let ?iaa = "a oo fps_inv a"
have th0: "?ia $ 0 = 0" by (simp add: fps_inv_def)
have th1: "?iaa $ 0 = 0" using a0 a1
by (simp add: fps_inv_def fps_compose_nth)
have th2: "X$0 = 0" by simp
from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo X" by simp
then have "(a oo fps_inv a) oo a = X oo a"
by (simp add: fps_compose_assoc[OF a0 th0] 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" by (simp add: fps_inv_def)
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] )
hence "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" by (simp add: fps_inv_def)
from a1 have ra1: "?r a $ 1 ≠ 0" by (simp add: fps_inv_def field_simps)
have X0: "X$0 = 0" by simp
from fps_inv[OF ra0 ra1] have "?r (?r a) oo ?r a = X" .
then have "?r (?r a) oo ?r a oo a = X oo a" by simp
then have "?r (?r a) oo (?r a oo a) = a"
unfolding 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" by (simp add: fps_ginv_def)
from a1 c1 have rca1: "?r c a $ 1 ≠ 0" by (simp add: fps_ginv_def field_simps)
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
apply (auto simp add: fps_ginv_def)
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
apply (auto simp add: fps_inv_def)
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 X a"
proof -
let ?ia = "fps_ginv b a"
let ?iXa = "fps_ginv X a"
let ?d = "fps_deriv"
let ?dia = "?d ?ia"
have iXa0: "?iXa $ 0 = 0" by (simp add: fps_ginv_def)
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
then have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a"
by (simp add: fps_divide_def)
then have "(?d ?ia oo a) oo ?iXa = (?d b / ?d a) oo ?iXa "
unfolding inverse_mult_eq_1[OF da0] by simp
then have "?d ?ia oo (a oo ?iXa) = (?d b / ?d a) oo ?iXa"
unfolding fps_compose_assoc[OF iXa0 a0] .
then show ?thesis unfolding fps_inv_ginv[symmetric]
unfolding fps_inv_right[OF a0 a1] by simp
qed

subsection{* Elementary series *}

subsubsection{* Exponential series *}

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

lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::field_char_0) * E a" (is "?l = ?r")
proof -
{
fix n
have "?l$n = ?r $ n"
apply (auto simp add: E_def field_simps power_Suc[symmetric]
simp del: fact_Suc of_nat_Suc power_Suc)
apply (simp add: of_nat_mult field_simps)
done
}
then show ?thesis by (simp add: fps_eq_iff)
qed

lemma E_unique_ODE:
"fps_deriv a = fps_const c * a <-> a = fps_const (a$0) * E (c :: 'a::field_char_0)"
(is "?lhs <-> ?rhs")
proof
assume d: ?lhs
from d have th: "!!n. a $ Suc n = c * a$n / of_nat (Suc n)"
by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
{
fix n
have "a$n = a$0 * c ^ n/ (of_nat (fact n))"
apply (induct n)
apply simp
unfolding th
using fact_gt_zero_nat
apply (simp add: field_simps del: of_nat_Suc fact_Suc)
apply (drule sym)
apply (simp add: field_simps of_nat_mult)
done
}
note th' = this
show ?rhs by (auto simp add: fps_eq_iff fps_const_mult_left E_def intro: th')
next
assume h: ?rhs
show ?lhs
apply (subst h)
apply simp
apply (simp only: h[symmetric])
apply simp
done
qed

lemma E_add_mult: "E (a + b) = E (a::'a::field_char_0) * E b" (is "?l = ?r")
proof -
have "fps_deriv (?r) = fps_const (a+b) * ?r"
by (simp add: fps_const_add[symmetric] field_simps del: fps_const_add)
then have "?r = ?l" apply (simp only: E_unique_ODE)
by (simp add: fps_mult_nth E_def)
then show ?thesis ..
qed

lemma E_nth[simp]: "E a $ n = a^n / of_nat (fact n)"
by (simp add: E_def)

lemma E0[simp]: "E (0::'a::{field}) = 1"
by (simp add: fps_eq_iff power_0_left)

lemma E_neg: "E (- a) = inverse (E (a::'a::field_char_0))"
proof -
from E_add_mult[of a "- a"] have th0: "E a * E (- a) = 1"
by (simp )
have th1: "E a $ 0 ≠ 0" by simp
from fps_inverse_unique[OF th1 th0] show ?thesis by simp
qed

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

lemma X_compose_E[simp]: "X oo E (a::'a::{field}) = E a - 1"
by (simp add: fps_eq_iff X_fps_compose)

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

lemma fps_const_inverse:
"a ≠ 0 ==> inverse (fps_const (a::'a::field)) = fps_const (inverse a)"
apply (auto simp add: fps_eq_iff fps_inverse_def)
apply (case_tac n)
apply auto
done

lemma inverse_one_plus_X:
"inverse (1 + X) = Abs_fps (λn. (- 1 ::'a::{field})^n)"
(is "inverse ?l = ?r")
proof -
have th: "?l * ?r = 1"
by (auto simp add: field_simps fps_eq_iff minus_one_power_iff simp del: minus_one)
have th': "?l $ 0 ≠ 0" by (simp add: )
from fps_inverse_unique[OF th' th] show ?thesis .
qed

lemma E_power_mult: "(E (c::'a::field_char_0))^n = E (of_nat n * c)"
by (induct n) (auto simp add: field_simps E_add_mult)

lemma radical_E:
assumes r: "r (Suc k) 1 = 1"
shows "fps_radical r (Suc k) (E (c::'a::{field_char_0})) = E (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: "E ?ck ^ (Suc k) = E c" unfolding E_power_mult eq0 ..
have th: "r (Suc k) (E c $0) ^ Suc k = E c $ 0"
"r (Suc k) (E c $ 0) = E ?ck $ 0" "E 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 Ec_E1_eq: "E (1::'a::{field_char_0}) oo (fps_const c * X) = E c"
apply (auto simp add: fps_eq_iff E_def fps_compose_def power_mult_distrib)
apply (simp add: cond_value_iff cond_application_beta setsum_delta' cong del: if_weak_cong)
done

text{* The generalized binomial theorem as a consequence of @{thm E_add_mult} *}

lemma gbinomial_theorem:
"((a::'a::{field_char_0, field_inverse_zero})+b) ^ n =
(∑k=0..n. of_nat (n choose k) * a^k * b^(n-k))"

proof -
from E_add_mult[of a b]
have "(E (a + b)) $ n = (E a * E b)$n" by simp
then have "(a + b) ^ n =
(∑i::nat = 0::nat..n. a ^ i * b ^ (n - i) * (of_nat (fact n) / of_nat (fact i * fact (n - i))))"

by (simp add: field_simps fps_mult_nth of_nat_mult[symmetric] setsum_right_distrib)
then show ?thesis
apply simp
apply (rule setsum_cong2)
apply simp
apply (frule binomial_fact[where ?'a = 'a, symmetric])
apply (simp add: field_simps of_nat_mult)
done
qed

text{* And the nat-form -- also available from Binomial.thy *}
lemma binomial_theorem: "(a+b) ^ n = (∑k=0..n. (n choose k) * a^k * b^(n-k))"
using gbinomial_theorem[of "of_nat a" "of_nat b" n]
unfolding of_nat_add[symmetric] of_nat_power[symmetric] of_nat_mult[symmetric]
of_nat_setsum[symmetric]
by simp


subsubsection{* Logarithmic series *}

lemma Abs_fps_if_0:
"Abs_fps(%n. if n=0 then (v::'a::ring_1) else f n) = fps_const v + X * Abs_fps (%n. f (Suc n))"
by (auto simp add: fps_eq_iff)

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

lemma fps_deriv_L: "fps_deriv (L c) = fps_const (1/c) * inverse (1 + X)"
unfolding inverse_one_plus_X
by (simp add: L_def fps_eq_iff del: of_nat_Suc)

lemma L_nth: "L c $ n = (if n=0 then 0 else 1/c * ((- 1) ^ (n - 1) / of_nat n))"
by (simp add: L_def field_simps)

lemma L_0[simp]: "L c $ 0 = 0" by (simp add: L_def)

lemma L_E_inv:
assumes a: "a≠ (0::'a::{field_char_0})"
shows "L a = fps_inv (E a - 1)" (is "?l = ?r")
proof -
let ?b = "E a - 1"
have b0: "?b $ 0 = 0" by simp
have b1: "?b $ 1 ≠ 0" by (simp add: a)
have "fps_deriv (E a - 1) oo fps_inv (E a - 1) =
(fps_const a * (E a - 1) + fps_const a) oo fps_inv (E a - 1)"

by (simp add: field_simps)
also have "… = fps_const a * (X + 1)"
apply (simp add: fps_compose_add_distrib fps_const_mult_apply_left[symmetric] fps_inv_right[OF b0 b1])
apply (simp add: field_simps)
done
finally have eq: "fps_deriv (E a - 1) oo fps_inv (E a - 1) = fps_const a * (X + 1)" .
from fps_inv_deriv[OF b0 b1, unfolded eq]
have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (X + 1)"
using a
by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
hence "fps_deriv ?l = fps_deriv ?r"
by (simp add: fps_deriv_L add_commute fps_divide_def divide_inverse)
then show ?thesis unfolding fps_deriv_eq_iff
by (simp add: L_nth fps_inv_def)
qed

lemma L_mult_add:
assumes c0: "c≠0"
and d0: "d≠0"
shows "L c + L d = fps_const (c+d) * L (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 + X)"
by (simp add: fps_deriv_L fps_const_add[symmetric] algebra_simps del: fps_const_add)
also have "… = fps_deriv ?l"
apply (simp add: fps_deriv_L)
apply (simp add: fps_eq_iff eq)
done
finally show ?thesis
unfolding fps_deriv_eq_iff 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"
by (simp add: fps_binomial_def)

lemma fps_binomial_ODE_unique:
fixes c :: "'a::field_char_0"
shows "fps_deriv a = (fps_const c * a) / (1 + X) <-> a = fps_const (a$0) * fps_binomial c"
(is "?lhs <-> ?rhs")
proof -
let ?da = "fps_deriv a"
let ?x1 = "(1 + X):: 'a fps"
let ?l = "?x1 * ?da"
let ?r = "fps_const c * a"
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])
apply (simp add: field_simps)
done
finally have eq: "?l = ?r <-> ?lhs" by simp
moreover
{assume h: "?l = ?r"
{fix n
from h have lrn: "?l $ n = ?r$n" by simp

from lrn
have "a$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a $n"
apply (simp add: field_simps del: of_nat_Suc)
by (cases n, simp_all add: field_simps del: of_nat_Suc)
}
note th0 = this
{
fix n
have "a$n = (c gchoose n) * a$0"
proof (induct n)
case 0
thus ?case by simp
next
case (Suc m)
thus ?case unfolding th0
apply (simp add: field_simps del: of_nat_Suc)
unfolding mult_assoc[symmetric] gbinomial_mult_1
apply (simp add: field_simps)
done
qed
}
note th1 = this
have ?rhs
apply (simp add: fps_eq_iff)
apply (subst th1)
apply (simp add: field_simps)
done
}
moreover
{
assume h: ?rhs
have th00: "!!x y. x * (a$0 * y) = a$0 * (x*y)"
by (simp add: mult_commute)
have "?l = ?r"
apply (subst h)
apply (subst (2) h)
apply (clarsimp simp add: fps_eq_iff field_simps)
unfolding mult_assoc[symmetric] th00 gbinomial_mult_1
apply (simp add: field_simps gbinomial_mult_1)
done
}
ultimately show ?thesis by blast
qed

lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + 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 + 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
by (simp add: fps_divide_def field_simps)
also have "… = (fps_const (c + d)/ (1 + X)) * ?P"
by (simp add: field_simps fps_divide_def fps_const_add[symmetric] del: fps_const_add)
finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + X)"
by (simp add: fps_divide_def)
have "?P = fps_const (?P$0) * ?b (c + d)"
unfolding fps_binomial_ODE_unique[symmetric]
using th0 by simp
hence "?P = 0" by (simp add: fps_mult_nth)
then show ?thesis by simp
qed

lemma fps_minomial_minus_one: "fps_binomial (- 1) = inverse (1 + 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 + 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 minus_one)
have eq: "inverse ?r $ 0 = 1"
by (simp add: fps_inverse_def)
from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + X)" "- 1"] th'] eq
show ?thesis by (simp add: fps_inverse_def)
qed

text{* Vandermonde's Identity as a consequence *}
lemma gbinomial_Vandermonde:
"setsum (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:
"setsum (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]
apply (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric]
of_nat_setsum[symmetric] of_nat_add[symmetric])
apply simp
done

lemma binomial_Vandermonde_same: "setsum (λk. (n choose k)2) {0..n} = (2*n) choose n"
using binomial_Vandermonde[of n n n,symmetric]
unfolding mult_2
apply (simp add: power2_eq_square)
apply (rule setsum_cong2)
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 "setsum (%k. (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 = "%m. (- 1 :: 'a) ^ m"
let ?f = "%m. of_nat (fact m)"
let ?p = "%(x::'a). pochhammer (- x)"
from b have bn0: "?p b n ≠ 0" unfolding pochhammer_eq_0_iff by simp
{
fix k
assume kn: "k ∈ {0..n}"
{
assume 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"
by (simp add: algebra_simps)
then have "b = of_nat (n - j - 1)"
using j kn by (simp add: of_nat_diff)
with b have False using j by auto
}
then have nz: "pochhammer (1 + b - of_nat n) n ≠ 0"
by (auto simp add: algebra_simps)

from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k ≠ 0"
by (rule pochhammer_neq_0_mono)
{
assume k0: "k = 0 ∨ n =0"
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)"

using kn
by (cases "k = 0") (simp_all add: gbinomial_pochhammer)
}
moreover
{
assume n0: "n ≠ 0" and k0: "k ≠ 0"
then obtain m where m: "n = Suc m" by (cases n) auto
from k0 obtain h where h: "k = Suc h" by (cases k) auto
{
assume kn: "k = n"
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)"

using kn pochhammer_minus'[where k=k and n=n and b=b]
apply (simp add: pochhammer_same)
using bn0
apply (simp add: field_simps power_add[symmetric])
done
}
moreover
{
assume nk: "k ≠ n"
have m1nk: "?m1 n = setprod (%i. - 1) {0..m}" "?m1 k = setprod (%i. - 1) {0..h}"
by (simp_all add: setprod_constant m h)
from kn nk have kn': "k < n" by simp
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: "setprod (%k. (1::'a) + of_nat m - of_nat k) {0 .. h} =
setprod of_nat {Suc (m - h) .. Suc m}"

apply (rule strong_setprod_reindex_cong[where f="%k. Suc m - k "])
using kn' h m
apply (auto simp add: inj_on_def image_def)
apply (rule_tac x="Suc m - x" in bexI)
apply (simp_all add: of_nat_diff)
done

have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
unfolding m1nk
unfolding m h pochhammer_Suc_setprod
apply (simp add: field_simps del: fact_Suc minus_one)
unfolding fact_altdef_nat id_def
unfolding of_nat_setprod
unfolding setprod_timesf[symmetric]
apply auto
unfolding eq1
apply (subst setprod_Un_disjoint[symmetric])
apply (auto)
apply (rule setprod_cong)
apply auto
done
have th20: "?m1 n * ?p b n = setprod (%i. b - of_nat i) {0..m}"
unfolding m1nk
unfolding m h pochhammer_Suc_setprod
unfolding setprod_timesf[symmetric]
apply (rule setprod_cong)
apply auto
done
have th21:"pochhammer (b - of_nat n + 1) k = setprod (%i. b - of_nat i) {n - k .. n - 1}"
unfolding h m
unfolding pochhammer_Suc_setprod
apply (rule strong_setprod_reindex_cong[where f="%k. n - 1 - k"])
using kn
apply (auto simp add: inj_on_def m h image_def)
apply (rule_tac x= "m - x" in bexI)
apply (auto simp add: of_nat_diff)
done

have "?m1 n * ?p b n =
pochhammer (b - of_nat n + 1) k * setprod (%i. b - of_nat i) {0.. n - k - 1}"

unfolding th20 th21
unfolding h m
apply (subst setprod_Un_disjoint[symmetric])
using kn' h m
apply auto
apply (rule setprod_cong)
apply auto
done
then have th2: "(?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k =
setprod (%i. b - of_nat i) {0.. n - k - 1}"

using nz' by (simp add: field_simps)
have "(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k) =
((?m1 k * ?p (of_nat n) k) / ?f n) * ((?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k)"

using bnz0
by (simp add: field_simps)
also have "… = b gchoose (n - k)"
unfolding th1 th2
using kn' by (simp add: gbinomial_def)
finally 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)"

by simp
}
ultimately
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)"

by (cases "k = n") auto
}
ultimately 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)"

"pochhammer (1 + b - of_nat n) k ≠ 0 "
apply (cases "n = 0")
using nz'
apply auto
apply (cases k)
apply auto
done
}
note th00 = this
have "?r = ((a + b) gchoose n) * (of_nat (fact n)/ (?m1 n * pochhammer (- b) n))"
unfolding gbinomial_pochhammer
using bn0 by (auto simp add: field_simps)
also have "… = ?l"
unfolding gbinomial_Vandermonde[symmetric]
apply (simp add: th00)
unfolding gbinomial_pochhammer
using bn0
apply (simp add: setsum_left_distrib setsum_right_distrib field_simps)
apply (rule setsum_cong2)
apply (drule th00(2))
apply (simp add: field_simps power_add[symmetric])
done
finally show ?thesis by simp
qed

lemma Vandermonde_pochhammer:
fixes a :: "'a::field_char_0"
assumes c: "ALL i : {0..< n}. c ≠ - of_nat i"
shows "setsum (%k. (pochhammer a k * pochhammer (- (of_nat n)) k) /
(of_nat (fact k) * pochhammer c k)) {0..n} = pochhammer (c - a) n / pochhammer c n"

proof -
let ?a = "- a"
let ?b = "c + of_nat n - 1"
have h: "∀ j ∈{0..< n}. ?b ≠ of_nat j" using c
apply (auto simp add: algebra_simps of_nat_diff)
apply (erule_tac x= "n - j - 1" in ballE)
apply (auto simp add: of_nat_diff algebra_simps)
done
have th0: "pochhammer (- (?a + ?b)) n = (- 1)^n * pochhammer (c - a) n"
unfolding pochhammer_minus[OF le_refl]
by (simp add: algebra_simps)
have th1: "pochhammer (- ?b) n = (- 1)^n * pochhammer c n"
unfolding pochhammer_minus[OF le_refl]
by simp
have nz: "pochhammer c n ≠ 0" using c
by (simp add: pochhammer_eq_0_iff)
from Vandermonde_pochhammer_lemma[where a = "?a" and b="?b" and n=n, OF h, unfolded th0 th1]
show ?thesis using nz by (simp add: field_simps setsum_right_distrib)
qed


subsubsection{* Formal trigonometric functions *}

definition "fps_sin (c::'a::field_char_0) =
Abs_fps (λn. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"


definition "fps_cos (c::'a::field_char_0) =
Abs_fps (λn. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"


lemma fps_sin_deriv:
"fps_deriv (fps_sin c) = fps_const c * fps_cos c"
(is "?lhs = ?rhs")
proof (rule fps_ext)
fix n :: nat
{
assume en: "even n"
have "?lhs$n = of_nat (n+1) * (fps_sin c $ (n+1))" by simp
also have "… = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))"
using en by (simp add: fps_sin_def)
also have "… = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
unfolding fact_Suc of_nat_mult
by (simp add: field_simps del: of_nat_add of_nat_Suc)
also have "… = (- 1)^(n div 2) *c^Suc n / of_nat (fact n)"
by (simp add: field_simps del: of_nat_add of_nat_Suc)
finally have "?lhs $n = ?rhs$n" using en
by (simp add: fps_cos_def field_simps)
}
then show "?lhs $ n = ?rhs $ n"
by (cases "even n") (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
qed

lemma fps_cos_deriv: "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)"
(is "?lhs = ?rhs")
proof (rule fps_ext)
have th0: "!!n. - ((- 1::'a) ^ n) = (- 1)^Suc n" by simp
have th1: "!!n. odd n ==> Suc ((n - 1) div 2) = Suc n div 2"
by (case_tac n, simp_all)
fix n::nat
{
assume en: "odd n"
from en have n0: "n ≠0 " by presburger
have "?lhs$n = of_nat (n+1) * (fps_cos c $ (n+1))" by simp
also have "… = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))"
using en by (simp add: fps_cos_def)
also have "… = (- 1)^((n + 1) div 2)*c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
unfolding fact_Suc of_nat_mult
by (simp add: field_simps del: of_nat_add of_nat_Suc)
also have "… = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)"
by (simp add: field_simps del: of_nat_add of_nat_Suc)
also have "… = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)"
unfolding th0 unfolding th1[OF en] by simp
finally have "?lhs $n = ?rhs$n" using en
by (simp add: fps_sin_def field_simps)
}
then show "?lhs $ n = ?rhs $ n"
by (cases "even n") (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
qed

lemma fps_sin_cos_sum_of_squares:
"(fps_cos c)2 + (fps_sin c)2 = 1" (is "?lhs = 1")
proof -
have "fps_deriv ?lhs = 0"
apply (simp add: fps_deriv_power fps_sin_deriv fps_cos_deriv)
apply (simp add: field_simps fps_const_neg[symmetric] del: fps_const_neg)
done
then have "?lhs = fps_const (?lhs $ 0)"
unfolding fps_deriv_eq_0_iff .
also have "… = 1"
by (auto simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
finally show ?thesis .
qed

lemma divide_eq_iff: "a ≠ (0::'a::field) ==> x / a = y <-> x = y * a"
by auto

lemma eq_divide_iff: "a ≠ (0::'a::field) ==> x = y / a <-> x * a = y"
by auto

lemma fps_sin_nth_0 [simp]: "fps_sin c $ 0 = 0"
unfolding fps_sin_def by simp

lemma fps_sin_nth_1 [simp]: "fps_sin c $ 1 = c"
unfolding fps_sin_def by simp

lemma fps_sin_nth_add_2:
"fps_sin c $ (n + 2) = - (c * c * fps_sin c $ n / (of_nat(n+1) * of_nat(n+2)))"
unfolding fps_sin_def
apply (cases n, simp)
apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
done

lemma fps_cos_nth_0 [simp]: "fps_cos c $ 0 = 1"
unfolding fps_cos_def by simp

lemma fps_cos_nth_1 [simp]: "fps_cos c $ 1 = 0"
unfolding fps_cos_def by simp

lemma fps_cos_nth_add_2:
"fps_cos c $ (n + 2) = - (c * c * fps_cos c $ n / (of_nat(n+1) * of_nat(n+2)))"
unfolding fps_cos_def
apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
done

lemma nat_induct2: "P 0 ==> P 1 ==> (!!n. P n ==> P (n + 2)) ==> P (n::nat)"
unfolding One_nat_def numeral_2_eq_2
apply (induct n rule: nat_less_induct)
apply (case_tac n)
apply simp
apply (rename_tac m)
apply (case_tac m)
apply simp
apply (rename_tac k)
apply (case_tac k)
apply simp_all
done

lemma nat_add_1_add_1: "(n::nat) + 1 + 1 = n + 2"
by simp

lemma eq_fps_sin:
assumes 0: "a $ 0 = 0"
and 1: "a $ 1 = c"
and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
shows "a = fps_sin c"
apply (rule fps_ext)
apply (induct_tac n rule: nat_induct2)
apply (simp add: 0)
apply (simp add: 1 del: One_nat_def)
apply (rename_tac m, cut_tac f="λa. a $ m" in arg_cong [OF 2])
apply (simp add: nat_add_1_add_1 fps_sin_nth_add_2
del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
apply (subst minus_divide_left)
apply (subst eq_divide_iff)
apply (simp del: of_nat_add of_nat_Suc)
apply (simp only: mult_ac)
done

lemma eq_fps_cos:
assumes 0: "a $ 0 = 1"
and 1: "a $ 1 = 0"
and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
shows "a = fps_cos c"
apply (rule fps_ext)
apply (induct_tac n rule: nat_induct2)
apply (simp add: 0)
apply (simp add: 1 del: One_nat_def)
apply (rename_tac m, cut_tac f="λa. a $ m" in arg_cong [OF 2])
apply (simp add: nat_add_1_add_1 fps_cos_nth_add_2
del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
apply (subst minus_divide_left)
apply (subst eq_divide_iff)
apply (simp del: of_nat_add of_nat_Suc)
apply (simp only: mult_ac)
done

lemma mult_nth_0 [simp]: "(a * b) $ 0 = a $ 0 * b $ 0"
by (simp add: fps_mult_nth)

lemma mult_nth_1 [simp]: "(a * b) $ 1 = a $ 0 * b $ 1 + a $ 1 * b $ 0"
by (simp add: fps_mult_nth)

lemma fps_sin_add: "fps_sin (a + b) = fps_sin a * fps_cos b + fps_cos a * fps_sin b"
apply (rule eq_fps_sin [symmetric], simp, simp del: One_nat_def)
apply (simp del: fps_const_neg fps_const_add fps_const_mult
add: fps_const_add [symmetric] fps_const_neg [symmetric]
fps_sin_deriv fps_cos_deriv algebra_simps)
done

lemma fps_cos_add: "fps_cos (a + b) = fps_cos a * fps_cos b - fps_sin a * fps_sin b"
apply (rule eq_fps_cos [symmetric], simp, simp del: One_nat_def)
apply (simp del: fps_const_neg fps_const_add fps_const_mult
add: fps_const_add [symmetric] fps_const_neg [symmetric]
fps_sin_deriv fps_cos_deriv algebra_simps)
done

lemma fps_sin_even: "fps_sin (- c) = - fps_sin c"
by (auto simp add: fps_eq_iff fps_sin_def)

lemma fps_cos_odd: "fps_cos (- c) = fps_cos c"
by (auto simp add: fps_eq_iff fps_cos_def)

definition "fps_tan c = fps_sin c / fps_cos c"

lemma fps_tan_deriv: "fps_deriv (fps_tan c) = fps_const c / (fps_cos c)2"
proof -
have th0: "fps_cos c $ 0 ≠ 0" by (simp add: fps_cos_def)
show ?thesis
using fps_sin_cos_sum_of_squares[of c]
apply (simp add: fps_tan_def fps_divide_deriv[OF th0] fps_sin_deriv fps_cos_deriv
fps_const_neg[symmetric] field_simps power2_eq_square del: fps_const_neg)
unfolding distrib_left[symmetric]
apply simp
done
qed

text {* Connection to E c over the complex numbers --- Euler and De Moivre*}
lemma Eii_sin_cos: "E (ii * c) = fps_cos c + fps_const ii * fps_sin c "
(is "?l = ?r")
proof -
{ fix n :: nat
{
assume en: "even n"
from en obtain m where m: "n = 2 * m"
unfolding even_mult_two_ex by blast

have "?l $n = ?r$n"
by (simp add: m fps_sin_def fps_cos_def power_mult_distrib power_mult power_minus)
}
moreover
{
assume on: "odd n"
from on obtain m where m: "n = 2*m + 1"
unfolding odd_nat_equiv_def2 by (auto simp add: mult_2)
have "?l $n = ?r$n"
by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
power_mult power_minus)
}
ultimately have "?l $n = ?r$n" by blast
} then show ?thesis by (simp add: fps_eq_iff)
qed

lemma E_minus_ii_sin_cos: "E (- (ii * c)) = fps_cos c - fps_const ii * fps_sin c"
unfolding minus_mult_right Eii_sin_cos by (simp add: fps_sin_even fps_cos_odd)

lemma fps_const_minus: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"
by (simp add: fps_eq_iff fps_const_def)

lemma fps_numeral_fps_const: "numeral i = fps_const (numeral i :: 'a:: {comm_ring_1})"
by (fact numeral_fps_const) (* FIXME: duplicate *)

lemma fps_cos_Eii: "fps_cos c = (E (ii * c) + E (- ii * c)) / fps_const 2"
proof -
have th: "fps_cos c + fps_cos c = fps_cos c * fps_const 2"
by (simp add: numeral_fps_const)
show ?thesis
unfolding Eii_sin_cos minus_mult_commute
by (simp add: fps_sin_even fps_cos_odd numeral_fps_const fps_divide_def fps_const_inverse th)
qed

lemma fps_sin_Eii: "fps_sin c = (E (ii * c) - E (- ii * c)) / fps_const (2*ii)"
proof -
have th: "fps_const \<i> * fps_sin c + fps_const \<i> * fps_sin c = fps_sin c * fps_const (2 * ii)"
by (simp add: fps_eq_iff numeral_fps_const)
show ?thesis
unfolding Eii_sin_cos minus_mult_commute
by (simp add: fps_sin_even fps_cos_odd fps_divide_def fps_const_inverse th)
qed

lemma fps_tan_Eii:
"fps_tan c = (E (ii * c) - E (- ii * c)) / (fps_const ii * (E (ii * c) + E (- ii * c)))"
unfolding fps_tan_def fps_sin_Eii fps_cos_Eii mult_minus_left E_neg
apply (simp add: fps_divide_def fps_inverse_mult fps_const_mult[symmetric] fps_const_inverse del: fps_const_mult)
apply simp
done

lemma fps_demoivre: "(fps_cos a + fps_const ii * fps_sin a)^n = fps_cos (of_nat n * a) + fps_const ii * fps_sin (of_nat n * a)"
unfolding Eii_sin_cos[symmetric] E_power_mult
by (simp add: mult_ac)


subsection {* Hypergeometric series *}

definition "F as bs (c::'a::{field_char_0, field_inverse_zero}) =
Abs_fps (%n. (foldl (%r a. r* pochhammer a n) 1 as * c^n) /
(foldl (%r b. r * pochhammer b n) 1 bs * of_nat (fact n)))"


lemma F_nth[simp]: "F as bs c $ n =
(foldl (λr a. r* pochhammer a n) 1 as * c^n) /
(foldl (λr b. r * pochhammer b n) 1 bs * of_nat (fact n))"

by (simp add: F_def)

lemma foldl_mult_start:
"foldl (%r x. r * f x) (v::'a::comm_ring_1) as * x = foldl (%r x. r * f x) (v * x) as "
by (induct as arbitrary: x v) (auto simp add: algebra_simps)

lemma foldr_mult_foldl:
"foldr (%x r. r * f x) as v = foldl (%r x. r * f x) (v :: 'a::comm_ring_1) as"
by (induct as arbitrary: v) (auto simp add: foldl_mult_start)

lemma F_nth_alt:
"F as bs c $ n = foldr (λa r. r * pochhammer a n) as (c ^ n) /
foldr (λb r. r * pochhammer b n) bs (of_nat (fact n))"

by (simp add: foldl_mult_start foldr_mult_foldl)

lemma F_E[simp]: "F [] [] c = E c"
by (simp add: fps_eq_iff)

lemma F_1_0[simp]: "F [1] [] c = 1/(1 - fps_const c * X)"
proof -
let ?a = "(Abs_fps (λn. 1)) oo (fps_const c * X)"
have th0: "(fps_const c * X) $ 0 = 0" by simp
show ?thesis unfolding gp[OF th0, symmetric]
by (auto simp add: fps_eq_iff pochhammer_fact[symmetric]
fps_compose_nth power_mult_distrib cond_value_iff setsum_delta' cong del: if_weak_cong)
qed

lemma F_B[simp]: "F [-a] [] (- 1) = fps_binomial a"
by (simp add: fps_eq_iff gbinomial_pochhammer algebra_simps)

lemma F_0[simp]: "F as bs c $0 = 1"
apply simp
apply (subgoal_tac "ALL as. foldl (%(r::'a) (a::'a). r) 1 as = 1")
apply auto
apply (induct_tac as)
apply auto
done

lemma foldl_prod_prod:
"foldl (%(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (%r x. r * g x) w as =
foldl (%r x. r * f x * g x) (v*w) as"

by (induct as arbitrary: v w) (auto simp add: algebra_simps)


lemma F_rec:
"F as bs c $ Suc n = ((foldl (%r a. r* (a + of_nat n)) c as) /
(foldl (%r b. r * (b + of_nat n)) (of_nat (Suc n)) bs )) * F as bs c $ n"

apply (simp del: of_nat_Suc of_nat_add fact_Suc)
apply (simp add: foldl_mult_start del: fact_Suc of_nat_Suc)
unfolding foldl_prod_prod[unfolded foldl_mult_start] pochhammer_Suc
apply (simp add: algebra_simps of_nat_mult)
done

lemma XD_nth[simp]: "XD a $ n = (if n=0 then 0 else of_nat n * a$n)"
by (simp add: XD_def)

lemma XD_0th[simp]: "XD a $ 0 = 0" by simp
lemma XD_Suc[simp]:" XD a $ Suc n = of_nat (Suc n) * a $ Suc n" by simp

definition "XDp c a = XD a + fps_const c * a"

lemma XDp_nth[simp]: "XDp c a $ n = (c + of_nat n) * a$n"
by (simp add: XDp_def algebra_simps)

lemma XDp_commute: "XDp b o XDp (c::'a::comm_ring_1) = XDp c o XDp b"
by (auto simp add: XDp_def fun_eq_iff fps_eq_iff algebra_simps)

lemma XDp0 [simp]: "XDp 0 = XD"
by (simp add: fun_eq_iff fps_eq_iff)

lemma XDp_fps_integral [simp]: "XDp 0 (fps_integral a c) = X * a"
by (simp add: fps_eq_iff fps_integral_def)

lemma F_minus_nat:
"F [- of_nat n] [- of_nat (n + m)] (c::'a::{field_char_0, field_inverse_zero}) $ k =
(if k <= n then
pochhammer (- of_nat n) k * c ^ k / (pochhammer (- of_nat (n + m)) k * of_nat (fact k))
else 0)"

"F [- of_nat m] [- of_nat (m + n)] (c::'a::{field_char_0, field_inverse_zero}) $ k =
(if k <= m then
pochhammer (- of_nat m) k * c ^ k / (pochhammer (- of_nat (m + n)) k * of_nat (fact k))
else 0)"

by (auto simp add: pochhammer_eq_0_iff)

lemma setsum_eq_if: "setsum f {(n::nat) .. m} = (if m < n then 0 else f n + setsum f {n+1 .. m})"
apply simp
apply (subst setsum_insert[symmetric])
apply (auto simp add: not_less setsum_head_Suc)
done

lemma pochhammer_rec_if: "pochhammer a n = (if n = 0 then 1 else a * pochhammer (a + 1) (n - 1))"
by (cases n) (simp_all add: pochhammer_rec)

lemma XDp_foldr_nth [simp]: "foldr (%c r. XDp c o r) cs (%c. XDp c a) c0 $ n =
foldr (%c r. (c + of_nat n) * r) cs (c0 + of_nat n) * a$n"

by (induct cs arbitrary: c0) (auto simp add: algebra_simps)

lemma genric_XDp_foldr_nth:
assumes f: "ALL n c a. f c a $ n = (of_nat n + k c) * a$n"
shows "foldr (%c r. f c o r) cs (%c. g c a) c0 $ n =
foldr (%c r. (k c + of_nat n) * r) cs (g c0 a $ n)"

by (induct cs arbitrary: c0) (auto simp add: algebra_simps f)

lemma dist_less_imp_nth_equal:
assumes "dist f g < inverse (2 ^ i)"
and"j ≤ i"
shows "f $ j = g $ j"
proof (cases "f = g")
case False
hence "∃n. f $ n ≠ g $ n" by (simp add: fps_eq_iff)
with assms have "i < The (leastP (λn. f $ n ≠ g $ n))"
by (simp add: split_if_asm dist_fps_def)
moreover
from fps_eq_least_unique[OF `f ≠ g`]
obtain n where n: "leastP (λn. f$n ≠ g$n) n" "The (leastP (λn. f $ n ≠ g $ n)) = n" by blast
moreover from n have "!!m. m < n ==> f$m = g$m" "f$n ≠ g$n"
by (auto simp add: leastP_def setge_def)
ultimately show ?thesis using `j ≤ i` by simp
next
case True
then show ?thesis by simp
qed

lemma nth_equal_imp_dist_less:
assumes "!!j. j ≤ i ==> f $ j = g $ j"
shows "dist f g < inverse (2 ^ i)"
proof (cases "f = g")
case False
hence "∃n. f $ n ≠ g $ n" by (simp add: fps_eq_iff)
with assms have "dist f g = inverse (2 ^ (The (leastP (λn. f $ n ≠ g $ n))))"
by (simp add: split_if_asm dist_fps_def)
moreover
from fps_eq_least_unique[OF `f ≠ g`]
obtain n where "leastP (λn. f$n ≠ g$n) n" "The (leastP (λn. f $ n ≠ g $ n)) = n" by blast
with assms have "i < The (leastP (λn. f $ n ≠ g $ n))"
by (metis (full_types) leastPD1 not_le)
ultimately show ?thesis by simp
next
case True
then show ?thesis by simp
qed

lemma dist_less_eq_nth_equal: "dist f g < inverse (2 ^ i) <-> (∀j ≤ i. f $ j = g $ j)"
using dist_less_imp_nth_equal nth_equal_imp_dist_less by blast

instance fps :: (comm_ring_1) complete_space
proof
fix X::"nat => 'a fps"
assume "Cauchy X"
{
fix i
have "0 < inverse ((2::real)^i)" by simp
from metric_CauchyD[OF `Cauchy X` this] dist_less_imp_nth_equal
have "∃M. ∀m ≥ M. ∀j≤i. X M $ j = X m $ j" by blast
}
then obtain M where M: "∀i. ∀m ≥ M i. ∀j ≤ i. X (M i) $ j = X m $ j" by metis
hence "∀i. ∀m ≥ M i. ∀j ≤ i. X (M i) $ j = X m $ j" by metis
show "convergent X"
proof (rule convergentI)
show "X ----> Abs_fps (λi. X (M i) $ i)"
unfolding tendsto_iff
proof safe
fix e::real assume "0 < e"
with LIMSEQ_inverse_realpow_zero[of 2, simplified, simplified filterlim_iff,
THEN spec, of "λx. x < e"]
have "eventually (λi. inverse (2 ^ i) < e) sequentially"
apply safe
apply (auto simp: eventually_nhds)
done
then obtain i where "inverse (2 ^ i) < e" by (auto simp: eventually_sequentially)
have "eventually (λx. M i ≤ x) sequentially" by (auto simp: eventually_sequentially)
thus "eventually (λx. dist (X x) (Abs_fps (λi. X (M i) $ i)) < e) sequentially"
proof eventually_elim
fix x
assume "M i ≤ x"
moreover
have "!!j. j ≤ i ==> X (M i) $ j = X (M j) $ j"
using M by (metis nat_le_linear)
ultimately have "dist (X x) (Abs_fps (λj. X (M j) $ j)) < inverse (2 ^ i)"
using M by (force simp: dist_less_eq_nth_equal)
also note `inverse (2 ^ i) < e`
finally show "dist (X x) (Abs_fps (λj. X (M j) $ j)) < e" .
qed
qed
qed
qed

end