isabelle: comparison src/HOL/Library/Formal_Power

equal deleted inserted replaced

-:903bb1495239
+:839169c70e92
 (*  Title:      HOL/Library/Formal_Power_Series.thy
 Author:     Amine Chaieb, University of Cambridge
 *)
-section\<open>A formalization of formal power series\<close>
+section \<open>A formalization of formal power series\<close>
 theory Formal_Power_Series
 imports Complex_Main
 begin
 subsection \<open>The type of formal power series\<close>
 typedef 'a fps = "{f :: nat \<Rightarrow> 'a. True}"
 morphisms fps_nth Abs_fps
 by simp
 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\<open>Definition of the basic elements 0 and 1 and the basic operations of addition,
+text \<open>Definition of the basic elements 0 and 1 and the basic operations of addition,
-negation and multiplication\<close>
+negation and multiplication.\<close>
 instantiation fps :: (zero) zero
 begin
+definition fps_zero_def: "0 = Abs_fps (\<lambda>n. 0)"
-definition fps_zero_def:
+instance ..
-"0 = Abs_fps (\<lambda>n. 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 (\<lambda>n. if n = 0 then 1 else 0)"
-definition fps_one_def:
+instance ..
-"1 = Abs_fps (\<lambda>n. 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 + = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n + g $ n))"
-definition fps_plus_def:
+instance ..
-"op + = (\<lambda>f g. Abs_fps (\<lambda>n. 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 - = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n - g $ n))"
-definition fps_minus_def:
+instance ..
-"op - = (\<lambda>f g. Abs_fps (\<lambda>n. 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 = (\<lambda>f. Abs_fps (\<lambda>n. - (f $ n)))"
-definition fps_uminus_def:
+instance ..
-"uminus = (\<lambda>f. Abs_fps (\<lambda>n. - (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 * = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f $ i * g $ (n - i)))"
-definition fps_times_def:
+instance ..
-"op * = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f $ i * g $ (n - i)))"
-instance ..
 end
 lemma fps_mult_nth: "(f * g) $ n = (\<Sum>i=0..n. f$i * g$(n - i))"
 unfolding fps_times_def by simp
 by auto
 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
 by auto
-subsection\<open>Formal power series form a commutative ring with unity, if the range of sequences
+subsection \<open>Formal power series form a commutative ring with unity, if the range of sequences
 they represent is a commutative ring with unity\<close>
 instance fps :: (semigroup_add) semigroup_add
 proof
 fix a b c :: "'a fps"
 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 "1 * a = a"
-show "a * 1 = a" by (simp add: fps_ext fps_mult_nth mult_delta_right setsum.delta')
+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) }
+show "b = c" if "a + b = a + c"
-{ assume "b + a = c + a" then show "b = c" by (simp add: expand_fps_eq) }
+using that by (simp add: expand_fps_eq)
+show "b = c" if "b + a = c + a"
+using that by (simp add: expand_fps_eq)
 qed
 instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
 proof
 fix a b c :: "'a fps"
-show "a + b - a = b" by (simp add: expand_fps_eq)
+show "a + b - a = b"
-show "a - b - c = a - (b + c)" by (simp add: expand_fps_eq diff_diff_eq)
+by (simp add: expand_fps_eq)
+show "a - b - c = a - (b + c)"
+by (simp add: expand_fps_eq diff_diff_eq)
 qed
 instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..
 instance fps :: (group_add) group_add
 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 "0 * a = 0"
-show "a * 0 = 0" by (simp add: fps_ext fps_mult_nth)
+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 \<open>Selection of the nth power of the implicit variable in the infinite sum\<close>
 lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0)"
 by (simp add: expand_fps_eq)
 lemma fps_nonzero_nth_minimal: "f \<noteq> 0 \<longleftrightarrow> (\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m < n. f $ m = 0))"
+(is "?lhs \<longleftrightarrow> ?rhs")
 proof
 let ?n = "LEAST n. f $ n \<noteq> 0"
-assume "f \<noteq> 0"
+show ?rhs if ?lhs
-then have "\<exists>n. f $ n \<noteq> 0"
+proof -
-by (simp add: fps_nonzero_nth)
+from that have "\<exists>n. f $ n \<noteq> 0"
-then have "f $ ?n \<noteq> 0"
+by (simp add: fps_nonzero_nth)
-by (rule LeastI_ex)
+then have "f $ ?n \<noteq> 0"
-moreover have "\<forall>m<?n. f $ m = 0"
+by (rule LeastI_ex)
-by (auto dest: not_less_Least)
+moreover have "\<forall>m<?n. f $ m = 0"
-ultimately have "f $ ?n \<noteq> 0 \<and> (\<forall>m<?n. f $ m = 0)" ..
+by (auto dest: not_less_Least)
-then show "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)" ..
+ultimately have "f $ ?n \<noteq> 0 \<and> (\<forall>m<?n. f $ m = 0)" ..
-next
+then show ?thesis ..
-assume "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)"
+qed
-then show "f \<noteq> 0" by (auto simp add: expand_fps_eq)
+show ?lhs if ?rhs
+using that by (auto simp add: expand_fps_eq)
 qed
 lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f $ n = g $n)"
 by (rule expand_fps_eq)
 next
 case False
 then show ?thesis by simp
 qed
-subsection\<open>Injection of the basic ring elements and multiplication by scalars\<close>
+subsection \<open>Injection of the basic ring elements and multiplication by scalars\<close>
 definition "fps_const c = Abs_fps (\<lambda>n. 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
 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 \<open>Formal power series form an integral domain\<close>
 instance fps :: (ring) ring ..
 instance fps :: (ring_1) ring_1
 by (intro_classes, auto simp add: distrib_right)
 instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors
 proof
 fix a b :: "'a fps"
-assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0"
+assume "a \<noteq> 0" and "b \<noteq> 0"
-then obtain i j where i: "a$i\<noteq>0" "\<forall>k<i. a$k=0" and j: "b$j \<noteq>0" "\<forall>k<j. b$k =0"
+then obtain i j where i: "a $ i \<noteq> 0" "\<forall>k<i. a $ k = 0" and j: "b $ j \<noteq> 0" "\<forall>k<j. b $ k =0"
 unfolding fps_nonzero_nth_minimal
 by blast+
-have "(a * b) $ (i+j) = (\<Sum>k=0..i+j. a$k * b$(i+j-k))"
+have "(a * b) $ (i + j) = (\<Sum>k=0..i+j. a $ k * b $ (i + j - k))"
 by (rule fps_mult_nth)
-also have "\<dots> = (a$i * b$(i+j-i)) + (\<Sum>k\<in>{0..i+j}-{i}. a$k * b$(i+j-k))"
+also have "\<dots> = (a $ i * b $ (i + j - i)) + (\<Sum>k\<in>{0..i+j} - {i}. a $ k * b $ (i + j - k))"
 by (rule setsum.remove) simp_all
-also have "(\<Sum>k\<in>{0..i+j}-{i}. a$k * b$(i+j-k)) = 0"
+also have "(\<Sum>k\<in>{0..i+j}-{i}. a $ k * b $ (i + j - k)) = 0"
 proof (rule setsum.neutral [rule_format])
 fix k assume "k \<in> {0..i+j} - {i}"
-then have "k < i \<or> i+j-k < j" by auto
+then have "k < i \<or> i+j-k < j"
-then show "a$k * b$(i+j-k) = 0" using i j by auto
+by auto
-qed
+then show "a $ k * b $ (i + j - k) = 0"
-also have "a$i * b$(i+j-i) + 0 = a$i * b$j" by simp
+using i j by auto
-also have "a$i * b$j \<noteq> 0" using i j by simp
+qed
+also have "a $ i * b $ (i + j - i) + 0 = a $ i * b $ j"
+by simp
+also have "a $ i * b $ j \<noteq> 0"
+using i j by simp
 finally have "(a*b) $ (i+j) \<noteq> 0" .
-then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast
+then show "a * b \<noteq> 0"
+unfolding fps_nonzero_nth by blast
 qed
 instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors ..
 instance fps :: (idom) idom ..
 fps_const_add [symmetric])
 lemma neg_numeral_fps_const: "- numeral k = fps_const (- numeral k)"
 by (simp only: numeral_fps_const fps_const_neg)
-subsection\<open>The eXtractor series X\<close>
+subsection \<open>The eXtractor series X\<close>
 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 (\<lambda>n. if n = 1 then 1 else 0)"
 case False
 have "(X * f) $n = (\<Sum>i = 0..n. X $ i * f $ (n - i))"
 by (simp add: fps_mult_nth)
 also have "\<dots> = f $ (n - 1)"
 using False by (simp add: X_def mult_delta_left setsum.delta)
-finally show ?thesis using False by simp
+finally show ?thesis
+using False by simp
 next
 case True
-then show ?thesis by (simp add: fps_mult_nth X_def)
+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)
 proof (induct k)
 case 0
 then show ?case by (simp add: X_def fps_eq_iff)
 next
 case (Suc k)
-{
+have "(X^Suc k) $ m = (if m = Suc k then 1::'a else 0)" for m
-fix m
+proof -
-have "(X^Suc k) $ m = (if m = 0 then 0::'a else (X^k) $ (m - 1))"
+have "(X^Suc k) $ m = (if m = 0 then 0 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)"
+then show ?thesis
 using Suc.hyps by (auto cong del: if_weak_cong)
-}
+qed
-then show ?case by (simp add: fps_eq_iff)
+then show ?case
-qed
+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))"
+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
 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\<open>Formal Power series form a metric space\<close>
+subsection \<open>Formal Power series form a metric space\<close>
 definition (in dist) "ball x r = {y. dist y x < r}"
 instantiation fps :: (comm_ring_1) dist
 begin
 definition open_fps_def: "open (S :: 'a fps set) = (\<forall>a \<in> S. \<exists>r. r >0 \<and> ball a r \<subseteq> S)"
 instance
 proof
-fix S :: "'a fps set"
+show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" for S :: "'a fps set"
-show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
 by (auto simp add: open_fps_def ball_def subset_eq)
-next
+show th: "dist a b = 0 \<longleftrightarrow> a = b" for a b :: "'a fps"
-{
+proof
-fix a b :: "'a fps"
+assume "a = b"
-{
+then have "\<not> (\<exists>n. a $ n \<noteq> b $ n)" by simp
-assume "a = b"
+then show "dist a b = 0" by (simp add: dist_fps_def)
-then have "\<not> (\<exists>n. a $ n \<noteq> b $ n)" by simp
+next
-then have "dist a b = 0" by (simp add: dist_fps_def)
+assume d: "dist a b = 0"
-}
+then have "\<forall>n. a$n = b$n"
-moreover
+by - (rule ccontr, simp add: dist_fps_def)
-{
+then show "a = b" by (simp add: fps_eq_iff)
-assume d: "dist a b = 0"
+qed
-then have "\<forall>n. a$n = b$n"
+then have th'[simp]: "dist a a = 0" for a :: "'a fps"
-by - (rule ccontr, simp add: dist_fps_def)
+by simp
-then have "a = b" by (simp add: fps_eq_iff)
-}
-ultimately show "dist a b =0 \<longleftrightarrow> a = b" by blast
-}
-note th = this
-from th have th'[simp]: "\<And>a::'a fps. dist a a = 0" by simp
 fix a b c :: "'a fps"
-{
+consider "a = b" | "c = a \<or> c = b" | "a \<noteq> b" "a \<noteq> c" "b \<noteq> c" by blast
-assume "a = b"
+then show "dist a b \<le> dist a c + dist b c"
+proof cases
+case 1
 then have "dist a b = 0" unfolding th .
-then have "dist a b \<le> dist a c + dist b c"
+then show ?thesis
 using dist_fps_ge0 [of a c] dist_fps_ge0 [of b c] by simp
-}
+next
-moreover
+case 2
-{
+then show ?thesis
-assume "c = a \<or> c = b"
-then have "dist a b \<le> dist a c + dist b c"
 by (cases "c = a") (simp_all add: th dist_fps_sym)
-}
+next
-moreover
+case 3
-{
-assume ab: "a \<noteq> b" and ac: "a \<noteq> c" and bc: "b \<noteq> c"
 def n \<equiv> "\<lambda>a b::'a fps. LEAST n. a$n \<noteq> b$n"
 then have n': "\<And>m a b. m < n a b \<Longrightarrow> a$m = b$m"
 by (auto dest: not_less_Least)
+from 3 have dab: "dist a b = inverse (2 ^ n a b)"
-from ab ac bc
-have dab: "dist a b = inverse (2 ^ n a b)"
 and dac: "dist a c = inverse (2 ^ n a c)"
 and dbc: "dist b c = inverse (2 ^ n b c)"
 by (simp_all add: dist_fps_def n_def fps_eq_iff)
-from ab ac bc have nz: "dist a b \<noteq> 0" "dist a c \<noteq> 0" "dist b c \<noteq> 0"
+from 3 have nz: "dist a b \<noteq> 0" "dist a c \<noteq> 0" "dist b c \<noteq> 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: "\<And>n. (2::real)^n >0" by auto
+have th1: "\<And>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: "n a b < n b c" "n a b < n a c"
 unfolding dab dbc dac by (auto simp add: th1)
 from n'[OF gtn(2)] n'(1)[OF gtn(1)]
 have "a $ n a b = b $ n a b" by simp
 moreover have "a $ n a b \<noteq> b $ n a b"
-unfolding n_def by (rule LeastI_ex) (insert ab, simp add: fps_eq_iff)
+unfolding n_def by (rule LeastI_ex) (insert \<open>a \<noteq> b\<close>, simp add: fps_eq_iff)
 ultimately have False by contradiction
 }
-then have "dist a b \<le> dist a c + dist b c"
+then show ?thesis
 by (auto simp add: not_le[symmetric])
-}
+qed
-ultimately show "dist a b \<le> dist a c + dist b c" by blast
 qed
 end
-text\<open>The infinite sums and justification of the notation in textbooks\<close>
+text \<open>The infinite sums and justification of the notation in textbooks\<close>
 lemma reals_power_lt_ex:
 fixes x y :: real
 assumes xp: "x > 0"
 and y1: "y > 1"
 lemma fps_notation: "(\<lambda>n. setsum (\<lambda>i. fps_const(a$i) * X^i) {0..n}) ----> a"
 (is "?s ----> a")
 proof -
 {
 fix r :: real
-assume rp: "r > 0"
+assume "r > 0"
-have th0: "(2::real) > 1" by simp
+obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0"
-from reals_power_lt_ex[OF rp th0]
+using reals_power_lt_ex[OF \<open>r > 0\<close>, of 2] by auto
-obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0" by blast
+have "\<exists>n0. \<forall>n \<ge> n0. dist (?s n) a < r"
-{
+proof -
-fix n :: nat
-assume nn0: "n \<ge> n0"
-then have thnn0: "(1/2)^n \<le> (1/2 :: real)^n0"
-by (simp add: divide_simps)
 {
-assume "?s n = a"
+fix n :: nat
-then have "dist (?s n) a < r"
+assume nn0: "n \<ge> n0"
-unfolding dist_eq_0_iff[of "?s n" a, symmetric]
+then have thnn0: "(1/2)^n \<le> (1/2 :: real)^n0"
-using rp by (simp del: dist_eq_0_iff)
+by (simp add: divide_simps)
+have "dist (?s n) a < r"
+proof (cases "?s n = a")
+case True
+then show ?thesis
+unfolding dist_eq_0_iff[of "?s n" a, symmetric]
+using \<open>r > 0\<close> by (simp del: dist_eq_0_iff)
+next
+case False
+def k \<equiv> "LEAST i. ?s n $ i \<noteq> a $ i"
+from False have dth: "dist (?s n) a = (1/2)^k"
+by (auto simp add: dist_fps_def inverse_eq_divide power_divide k_def fps_eq_iff)
+from False have kn: "k > n"
+by (auto simp: fps_sum_rep_nth not_le k_def fps_eq_iff
+split: split_if_asm intro: LeastI2_ex)
+then have "dist (?s n) a < (1/2)^n"
+unfolding dth by (simp add: divide_simps)
+also have "\<dots> \<le> (1/2)^n0"
+using nn0 by (simp add: divide_simps)
+also have "\<dots> < r"
+using n0 by simp
+finally show ?thesis .
+qed
 }
-moreover
+then show ?thesis by blast
-{
+qed
-assume neq: "?s n \<noteq> a"
-def k \<equiv> "LEAST i. ?s n $ i \<noteq> a $ i"
-from neq have dth: "dist (?s n) a = (1/2)^k"
-by (auto simp add: dist_fps_def inverse_eq_divide power_divide k_def fps_eq_iff)
-from neq have kn: "k > n"
-by (auto simp: fps_sum_rep_nth not_le k_def fps_eq_iff
-split: split_if_asm intro: LeastI2_ex)
-then have "dist (?s n) a < (1/2)^n"
-unfolding dth by (simp add: divide_simps)
-also have "\<dots> \<le> (1/2)^n0"
-using nn0 by (simp add: divide_simps)
-also have "\<dots> < r"
-using n0 by simp
-finally have "dist (?s n) a < r" .
-}
-ultimately have "dist (?s n) a < r"
-by blast
-}
-then have "\<exists>n0. \<forall> n \<ge> n0. dist (?s n) a < r"
-by blast
 }
 then show ?thesis
 unfolding lim_sequentially by blast
 qed
-subsection\<open>Inverses of formal power series\<close>
+subsection \<open>Inverses of formal power series\<close>
 declare setsum.cong[fundef_cong]
 instantiation fps :: ("{comm_monoid_add, inverse, times, uminus}") inverse
 begin
 fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a"
 where
 "natfun_inverse f 0 = inverse (f$0)"
 | "natfun_inverse f n = - inverse (f$0) * setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}"
-definition
+definition fps_inverse_def: "inverse f = (if f $ 0 = 0 then 0 else Abs_fps (natfun_inverse f))"
-fps_inverse_def: "inverse f = (if f $ 0 = 0 then 0 else Abs_fps (natfun_inverse f))"
+definition fps_divide_def: "divide = (\<lambda>(f::'a fps) g. f * inverse g)"
-definition
-fps_divide_def: "divide = (\<lambda>(f::'a fps) g. f * inverse g)"
 instance ..
 end
 by (simp add: mult.commute)
 from f0 have ifn: "\<And>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)
-{
+have "(inverse f * f)$n = 0" if np: "n > 0" for n
-fix n :: nat
+proof -
-assume np: "n > 0"
 from np have eq: "{0..n} = {0} \<union> {1 .. n}"
 by auto
 have d: "{0} \<inter> {1 .. n} = {}"
 by auto
 from f0 np have th0: "- (inverse f $ n) =
 unfolding fps_mult_nth ifn ..
 also have "\<dots> = f$0 * natfun_inverse f n + (\<Sum>i = 1..n. f$i * natfun_inverse f (n-i))"
 by (simp add: eq)
 also have "\<dots> = 0"
 unfolding th1 ifn by simp
-finally have "(inverse f * f)$n = 0"
+finally show ?thesis unfolding c .
-unfolding c .
+qed
-}
 with th0 show ?thesis
 by (simp add: fps_eq_iff)
 qed
-lemma fps_inverse_0_iff[simp]: "(inverse f)$0 = (0::'a::division_ring) \<longleftrightarrow> f$0 = 0"
+lemma fps_inverse_0_iff[simp]: "(inverse f) $ 0 = (0::'a::division_ring) \<longleftrightarrow> 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) \<longleftrightarrow> f $0 = 0"
+lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f $ 0 = 0"
-proof -
+(is "?lhs \<longleftrightarrow> ?rhs")
-{
+proof
-assume "f $ 0 = 0"
+show ?lhs if ?rhs
-then have "inverse f = 0"
+using that by (simp add: fps_inverse_def)
-by (simp add: fps_inverse_def)
+show ?rhs if h: ?lhs
-}
+proof (rule ccontr)
-moreover
-{
-assume h: "inverse f = 0"
 assume c: "f $0 \<noteq> 0"
-from inverse_mult_eq_1[OF c] h have False
+from inverse_mult_eq_1[OF c] h show False
 by simp
-}
+qed
-ultimately show ?thesis by blast
 qed
 lemma fps_inverse_idempotent[intro]:
 assumes f0: "f$0 \<noteq> (0::'a::field)"
 shows "inverse (inverse f) = f"
 unfolding th2
 apply (simp add: setsum.delta)
 done
 qed
-lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
+lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field))) =
-= Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
+Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
 apply (rule fps_inverse_unique)
 apply (simp_all add: fps_eq_iff fps_mult_nth setsum_zero_lemma)
 done
 show ?thesis by (induct rule: finite_induct [OF True]) simp_all
 qed
 lemma fps_deriv_eq_0_iff [simp]:
 "fps_deriv f = 0 \<longleftrightarrow> f = fps_const (f$0 :: 'a::{idom,semiring_char_0})"
-proof -
+(is "?lhs \<longleftrightarrow> ?rhs")
-{
+proof
-assume "f = fps_const (f$0)"
+show ?lhs if ?rhs
-then have "fps_deriv f = fps_deriv (fps_const (f$0))" by simp
+proof -
-then have "fps_deriv f = 0" by simp
+from that have "fps_deriv f = fps_deriv (fps_const (f$0))"
-}
+by simp
-moreover
+then show ?thesis
-{
+by simp
-assume z: "fps_deriv f = 0"
+qed
-then have "\<forall>n. (fps_deriv f)$n = 0" by simp
+show ?rhs if ?lhs
-then have "\<forall>n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)
+proof -
-then have "f = fps_const (f$0)"
+from that have "\<forall>n. (fps_deriv f)$n = 0"
+by simp
+then have "\<forall>n. f$(n+1) = 0"
+by (simp del: of_nat_Suc of_nat_add One_nat_def)
+then show ?thesis
 apply (clarsimp simp add: fps_eq_iff fps_const_def)
 apply (erule_tac x="n - 1" in allE)
 apply simp
 done
-}
+qed
-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 \<longleftrightarrow> (f = fps_const(f$0 - g$0) + g)"
 proof -
 have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0"
 by simp
 also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f - g) $ 0)"
 unfolding fps_deriv_eq_0_iff ..
-finally show ?thesis by (simp add: field_simps)
+finally show ?thesis
+by (simp add: field_simps)
 qed
 lemma fps_deriv_eq_iff_ex:
 "(fps_deriv f = fps_deriv g) \<longleftrightarrow> (\<exists>c::'a::{idom,semiring_char_0}. f = fps_const c + g)"
 by (auto simp: fps_deriv_eq_iff)
 proof (induct n)
 case 0
 then show ?case by simp
 next
 case (Suc n)
-note h = Suc.hyps[OF \<open>a$0 = 1\<close>]
 show ?case unfolding power_Suc fps_mult_nth
-using h \<open>a$0 = 1\<close> fps_power_zeroth_eq_one[OF \<open>a$0=1\<close>]
+using Suc.hyps[OF \<open>a$0 = 1\<close>] \<open>a$0 = 1\<close> fps_power_zeroth_eq_one[OF \<open>a$0=1\<close>]
 by (simp add: field_simps)
 qed
 lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1"
 by (induct n) (auto simp add: fps_mult_nth)
 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)"
+assumes a0: "a $ 0 = (0::'a::idom)"
 shows "\<forall>n < k. a ^ k $ n = 0"
 using a0
 proof (induct k rule: nat_less_induct)
 fix k
 assume H: "\<forall>m<k. a $0 =  0 \<longrightarrow> (\<forall>n<m. a ^ m $ n = 0)" and a0: "a $ 0 = 0"
-let ?ths = "\<forall>m<k. a ^ k $ m = 0"
+show "\<forall>m<k. a ^ k $ m = 0"
-{
+proof (cases k)
-assume "k = 0"
+case 0
-then have ?ths by simp
+then show ?thesis by simp
-}
+next
-moreover
+case (Suc l)
-{
+have "a^k $ m = 0" if mk: "m < k" for m
-fix l
+proof (cases "m = 0")
-assume k: "k = Suc l"
+case True
-{
+then show ?thesis
-fix m
+using startsby_zero_power[of a k] Suc a0 by simp
-assume mk: "m < k"
+next
-{
+case False
-assume "m = 0"
+have "a ^k $ m = (a^l * a) $m"
-then have "a^k $ m = 0"
+by (simp add: Suc mult.commute)
-using startsby_zero_power[of a k] k a0 by simp
+also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m - i))"
-}
+by (simp add: fps_mult_nth)
-moreover
+also have "\<dots> = 0"
-{
+apply (rule setsum.neutral)
-assume m0: "m \<noteq> 0"
+apply auto
-have "a ^k $ m = (a^l * a) $m"
+apply (case_tac "x = m")
-by (simp add: k mult.commute)
+using a0 apply simp
-also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m - i))"
+apply (rule H[rule_format])
-by (simp add: fps_mult_nth)
+using a0 Suc mk apply auto
-also have "\<dots> = 0"
+done
-apply (rule setsum.neutral)
+finally show ?thesis .
-apply auto
+qed
-apply (case_tac "x = m")
+then show ?thesis by blast
-using a0 apply simp
+qed
-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 \<ge> k"
 qed
 lemma fps_inverse_power:
 fixes a :: "'a::field fps"
 shows "inverse (a^n) = inverse a ^ n"
-proof -
+proof (cases "a$0 = 0")
-{
+case True
-assume a0: "a$0 = 0"
+then have eq: "inverse a = 0"
-then have eq: "inverse a = 0"
+by (simp add: fps_inverse_def)
+consider "n = 0" | "n > 0" by blast
+then show ?thesis
+proof cases
+case 1
+then show ?thesis by simp
+next
+case 2
+from startsby_zero_power[OF True 2] eq show ?thesis
 by (simp add: fps_inverse_def)
-{
+qed
-assume "n = 0"
+next
-then have ?thesis by simp
+case False
-}
+show ?thesis
-moreover
+apply (rule fps_inverse_unique)
-{
+apply (simp add: False)
-assume n: "n > 0"
+unfolding power_mult_distrib[symmetric]
-from startsby_zero_power[OF a0 n] eq a0 n have ?thesis
+apply (rule ssubst[where t = "a * inverse a" and s= 1])
-by (simp add: fps_inverse_def)
+apply simp_all
-}
+apply (subst mult.commute)
-ultimately have ?thesis by blast
+apply (rule inverse_mult_eq_1[OF False])
-}
+done
-moreover
-{
-assume a0: "a$0 \<noteq> 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)
 lemma fps_inverse_mult:
 fixes a :: "'a::field fps"
 shows "inverse (a * b) = inverse a * inverse b"
 proof -
-{
+consider "a $ 0 = 0" | "b $ 0 = 0" | "a $ 0 \<noteq> 0" "b $ 0 \<noteq> 0"
-assume a0: "a$0 = 0"
+by blast
-then have ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
+then show ?thesis
-from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all
+proof cases
-have ?thesis unfolding th by simp
+case 1
-}
+then have "(a * b) $ 0 = 0"
-moreover
+by (simp add: fps_mult_nth)
-{
+with 1 have th: "inverse a = 0" "inverse (a * b) = 0"
-assume b0: "b$0 = 0"
+by simp_all
-then have ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
+show ?thesis
-from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all
+unfolding th by simp
-have ?thesis unfolding th by simp
+next
-}
+case 2
-moreover
+then have "(a * b) $ 0 = 0"
-{
+by (simp add: fps_mult_nth)
-assume a0: "a$0 \<noteq> 0" and b0: "b$0 \<noteq> 0"
+with 2 have th: "inverse b = 0" "inverse (a * b) = 0"
-from a0 b0 have ab0:"(a*b) $ 0 \<noteq> 0" by (simp  add: fps_mult_nth)
+by simp_all
+show ?thesis
+unfolding th by simp
+next
+case 3
+then have ab0:"(a * b) $ 0 \<noteq> 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
+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
+then show ?thesis
-}
+using inverse_mult_eq_1[OF \<open>a $ 0 \<noteq> 0\<close>] inverse_mult_eq_1[OF \<open>b $ 0 \<noteq> 0\<close>] by simp
-ultimately show ?thesis by blast
+qed
 qed
 lemma fps_inverse_deriv':
 fixes a :: "'a::field fps"
-assumes a0: "a$0 \<noteq> 0"
+assumes a0: "a $ 0 \<noteq> 0"
 shows "fps_deriv (inverse a) = - fps_deriv a / a\<^sup>2"
 using fps_inverse_deriv[OF a0]
 unfolding power2_eq_square fps_divide_def fps_inverse_mult
 by simp
 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 (\<lambda>n. (- (1::'a::field)) ^ n)"
-lemma fps_inverse_X_plus1:
+(is "_ = ?r")
-"inverse (1 + X) = Abs_fps (\<lambda>n. (- (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)
 qed
-subsection\<open>Integration\<close>
+subsection \<open>Integration\<close>
 definition fps_integral :: "'a::field_char_0 fps \<Rightarrow> 'a \<Rightarrow> 'a fps"
 where "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n - 1) / of_nat n))"
 lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a"
 qed
 subsection \<open>Composition of FPSs\<close>
-definition fps_compose :: "'a::semiring_1 fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55)
+definition fps_compose :: "'a::semiring_1 fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps"  (infixl "oo" 55)
 where "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a$i * (b^i$n)) {0..n})"
 lemma fps_compose_nth: "(a oo b)$n = setsum (\<lambda>i. 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]:
+lemma fps_const_compose[simp]: "fps_const (a::'a::comm_ring_1) oo b = fps_const a"
-"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 fps_power_mult_eq_shift:
 "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) =
 Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a::comm_ring_1) * X^i) {0 .. k}"
 (is "?lhs = ?rhs")
 proof -
-{ fix n :: nat
+have "?lhs $ n = ?rhs $ n" for n :: nat
+proof -
 have "?lhs $ n = (if n < Suc k then 0 else a n)"
 unfolding X_power_mult_nth by auto
 also have "\<dots> = ?rhs $ n"
 proof (induct k)
 case 0
-then show ?case by (simp add: fps_setsum_nth)
+then show ?case
+by (simp add: fps_setsum_nth)
 next
 case (Suc k)
-note th = Suc.hyps[symmetric]
 have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. Suc k})$n =
 (Abs_fps a - setsum (\<lambda>i. 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 "\<dots> = (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
+using Suc.hyps[symmetric] unfolding fps_sub_nth by simp
 also have "\<dots> = (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
+finally show ?case
+by simp
 qed
-finally have "?lhs $ n = ?rhs $ n" .
+finally show ?thesis .
-}
+qed
-then show ?thesis by (simp add: fps_eq_iff)
+then show ?thesis
+by (simp add: fps_eq_iff)
 qed
 subsubsection \<open>Rule 2\<close>
 by (induct n) simp_all
 lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)"
 by (simp add: fps_eq_iff)
 lemma fps_mult_XD_shift:
 "(XD ^^ k) (a :: 'a::comm_ring_1 fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
 by (induct k arbitrary: a) (simp_all add: XD_def fps_eq_iff field_simps del: One_nat_def)
-subsubsection \<open>Rule 3 is trivial and is given by @{text fps_times_def}\<close>
+subsubsection \<open>Rule 3\<close>
+text \<open>Rule 3 is trivial and is given by @{text fps_times_def}.\<close>
 subsubsection \<open>Rule 5 --- summation and "division" by (1 - X)\<close>
 lemma fps_divide_X_minus1_setsum_lemma:
 "a = ((1::'a::comm_ring_1 fps) - X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
 proof -
 let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
 have th0: "\<And>i. (1 - (X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)"
 by simp
-{
+have "a$n = ((1 - X) * ?sa) $ n" for n
-fix n :: nat
+proof (cases "n = 0")
-{
+case True
-assume "n = 0"
+then show ?thesis
-then have "a $ n = ((1 - X) * ?sa) $ n"
+by (simp add: fps_mult_nth)
-by (simp add: fps_mult_nth)
+next
-}
+case False
-moreover
+then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1} \<union> {2..n} = {1..n}"
-{
+"{0..n - 1} \<union> {n} = {0..n}"
-assume n0: "n \<noteq> 0"
+by (auto simp: set_eq_iff)
-then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1} \<union> {2..n} = {1..n}"
+have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}" "{0..n - 1} \<inter> {n} = {}"
-"{0..n - 1} \<union> {n} = {0..n}"
+using False by simp_all
-by (auto simp: set_eq_iff)
+have f: "finite {0}" "finite {1}" "finite {2 .. n}"
-have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}" "{0..n - 1} \<inter> {n} = {}"
+"finite {0 .. n - 1}" "finite {n}" by simp_all
-using n0 by simp_all
+have "((1 - X) * ?sa) $ n = setsum (\<lambda>i. (1 - X)$ i * ?sa $ (n - i)) {0 .. n}"
-have f: "finite {0}" "finite {1}" "finite {2 .. n}"
+by (simp add: fps_mult_nth)
-"finite {0 .. n - 1}" "finite {n}" by simp_all
+also have "\<dots> = a$n"
-have "((1 - X) * ?sa) $ n = setsum (\<lambda>i. (1 - X)$ i * ?sa $ (n - i)) {0 .. n}"
+unfolding th0
-by (simp add: fps_mult_nth)
+unfolding setsum.union_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
-also have "\<dots> = a$n"
+unfolding setsum.union_disjoint[OF f(2) f(3) d(2)]
-unfolding th0
+apply (simp)
-unfolding setsum.union_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
+unfolding setsum.union_disjoint[OF f(4,5) d(3), unfolded u(3)]
-unfolding setsum.union_disjoint[OF f(2) f(3) d(2)]
+apply simp
-apply (simp)
+done
-unfolding setsum.union_disjoint[OF f(4,5) d(3), unfolded u(3)]
+finally show ?thesis
-apply simp
+by simp
-done
+qed
-finally have "a$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:
 finally show ?thesis
 by (simp add: inverse_mult_eq_1[OF th0])
 qed
-subsubsection\<open>Rule 4 in its more general form: generalizes Rule 3 for an arbitrary
+subsubsection \<open>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\<close>
 definition "natpermute n k = {l :: nat list. length l = k \<and> listsum l = n}"
 lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
 done
 finally show ?thesis .
 qed
 qed
-text\<open>The special form for powers\<close>
+text \<open>The special form for powers\<close>
 lemma fps_power_nth_Suc:
 fixes m :: nat
 and a :: "'a::comm_ring_1 fps"
 shows "(a ^ Suc m)$n = setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m}) (natpermute n (m+1))"
 proof -
 assumes a0: "a$0 = (0::'a::idom)"
 and a1: "a$1 \<noteq> 0"
 shows "(b oo a = c oo a) \<longleftrightarrow> b = c"
 (is "?lhs \<longleftrightarrow>?rhs")
 proof
-assume ?rhs
+show ?lhs if ?rhs using that by simp
-then show "?lhs" by simp
+show ?rhs if ?lhs
-next
+proof -
-assume h: ?lhs
+have "b$n = c$n" for n
-{
-fix n
-have "b$n = c$n"
 proof (induct n rule: nat_less_induct)
 fix n
 assume H: "\<forall>m<n. b$m = c$m"
-{
+show "b$n = c$n"
-assume n0: "n=0"
+proof (cases n)
-from h have "(b oo a)$n = (c oo a)$n" by simp
+case 0
-then have "b$n = c$n" using n0 by (simp add: fps_compose_nth)
+from \<open>?lhs\<close> have "(b oo a)$n = (c oo a)$n"
-}
+by simp
-moreover
+then show ?thesis
-{
+using 0 by (simp add: fps_compose_nth)
-fix n1 assume n1: "n = Suc n1"
+next
+case (Suc n1)
 have f: "finite {0 .. n1}" "finite {n}" by simp_all
-have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto
+have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using Suc by auto
-have d: "{0 .. n1} \<inter> {n} = {}" using n1 by auto
+have d: "{0 .. n1} \<inter> {n} = {}" using Suc by auto
 have seq: "(\<Sum>i = 0..n1. b $ i * a ^ i $ n) = (\<Sum>i = 0..n1. c $ i * a ^ i $ n)"
 apply (rule setsum.cong)
-using H n1
+using H Suc
 apply auto
 done
 have th0: "(b oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n"
 unfolding fps_compose_nth setsum.union_disjoint[OF f d, unfolded eq] seq
 using startsby_zero_power_nth_same[OF a0]
 by simp
 have th1: "(c oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n"
 unfolding fps_compose_nth setsum.union_disjoint[OF f d, unfolded eq]
 using startsby_zero_power_nth_same[OF a0]
 by simp
-from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
+from \<open>?lhs\<close>[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
-have "b$n = c$n" by auto
+show ?thesis by auto
-}
+qed
-ultimately show "b$n = c$n" by (cases n) auto
+qed
-qed}
+then show ?rhs by (simp add: fps_eq_iff)
-then show ?rhs by (simp add: fps_eq_iff)
+qed
 qed
 subsection \<open>Radicals\<close>
 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))"
+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)"
 apply simp
 using Suc
 apply (subgoal_tac "replicate k 0 ! x = 0")
 apply (auto intro: nth_replicate simp del: replicate.simps)
 done
-also have "\<dots> = a$0" using r Suc by (simp add: setprod_constant)
+also have "\<dots> = a$0"
-finally show ?thesis using Suc by simp
+using r Suc by (simp add: setprod_constant)
+finally show ?thesis
+using Suc by simp
 qed
 lemma natpermute_max_card:
 assumes n0: "n \<noteq> 0"
 shows "card {xs \<in> natpermute n (k+1). n \<in> set xs} = k + 1"
 unfolding natpermute_contain_maximal
 proof -
-let ?A= "\<lambda>i. {replicate (k + 1) 0[i := n]}"
+let ?A = "\<lambda>i. {replicate (k + 1) 0[i := n]}"
 let ?K = "{0 ..k}"
-have fK: "finite ?K" by simp
+have fK: "finite ?K"
-have fAK: "\<forall>i\<in>?K. finite (?A i)" by auto
+by simp
+have fAK: "\<forall>i\<in>?K. finite (?A i)"
+by auto
 have d: "\<forall>i\<in> ?K. \<forall>j\<in> ?K. i \<noteq> j \<longrightarrow>
 {replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
 proof clarify
 fix i j
-assume i: "i \<in> ?K" and j: "j\<in> ?K" and ij: "i\<noteq>j"
+assume i: "i \<in> ?K" and j: "j \<in> ?K" and ij: "i \<noteq> 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
 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) \<noteq> 0" by auto
-{
+have "?r ^ Suc k $ z = a$z" for z
-fix z
+proof (induct z rule: nat_less_induct)
-have "?r ^ Suc k $ z = a$z"
+fix n
-proof (induct z rule: nat_less_induct)
+assume H: "\<forall>m<n. ?r ^ Suc k $ m = a$m"
-fix n
+show "?r ^ Suc k $ n = a $n"
-assume H: "\<forall>m<n. ?r ^ Suc k $ m = a$m"
+proof (cases n)
-{
+case 0
-assume "n = 0"
+then show ?thesis
-then have "?r ^ Suc k $ n = a $n"
+using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp
-using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp
+next
-}
+case (Suc n1)
-moreover
+then have "n \<noteq> 0" by simp
-{
+let ?Pnk = "natpermute n (k + 1)"
-fix n1 assume n1: "n = Suc n1"
+let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
-have nz: "n \<noteq> 0" using n1 by arith
+let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
-let ?Pnk = "natpermute n (k + 1)"
+have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
-let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
+have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
-let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
+have f: "finite ?Pnkn" "finite ?Pnknn"
-have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
+using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
-have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
+by (metis natpermute_finite)+
-have f: "finite ?Pnkn" "finite ?Pnknn"
+let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
-using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
+have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn"
-by (metis natpermute_finite)+
+proof (rule setsum.cong)
-let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
+fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
-have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn"
+let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) =
-proof (rule setsum.cong)
+fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k"
-fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
+from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
-let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) =
+unfolding natpermute_contain_maximal by auto
-fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k"
+have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) =
-from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
+(\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))"
-unfolding natpermute_contain_maximal by auto
+apply (rule setprod.cong, simp)
-have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) =
+using i r0
-(\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))"
+apply (simp del: replicate.simps)
-apply (rule setprod.cong, simp)
+done
-using i r0
+also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k"
-apply (simp del: replicate.simps)
+using i r0 by (simp add: setprod_gen_delta)
-done
+finally show ?ths .
-also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k"
+qed rule
-using i r0 by (simp add: setprod_gen_delta)
+then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k"
-finally show ?ths .
+by (simp add: natpermute_max_card[OF \<open>n \<noteq> 0\<close>, simplified])
-qed rule
+also have "\<dots> = a$n - setsum ?f ?Pnknn"
-then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k"
+unfolding Suc using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc)
-by (simp add: natpermute_max_card[OF nz, simplified])
+finally have fn: "setsum ?f ?Pnkn = a$n - setsum ?f ?Pnknn" .
-also have "\<dots> = a$n - setsum ?f ?Pnknn"
+have "(?r ^ Suc k)$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
-unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc)
+unfolding fps_power_nth_Suc setsum.union_disjoint[OF f d, unfolded eq] ..
-finally have fn: "setsum ?f ?Pnkn = a$n - setsum ?f ?Pnknn" .
+also have "\<dots> = a$n" unfolding fn by simp
-have "(?r ^ Suc k)$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
+finally show ?thesis .
-unfolding fps_power_nth_Suc setsum.union_disjoint[OF f d, unfolded eq] ..
-also have "\<dots> = 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
-}
+qed
 then have ?thesis using r0 by (simp add: fps_eq_iff)
 }
 moreover
 {
 assume h: "(fps_radical r (Suc k) a) ^ (Suc k) = a"
 then show ?thesis by (simp add: fps_eq_iff)
 qed
 *)
 lemma eq_divide_imp':
-fixes c :: "'a::field" shows "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
+fixes c :: "'a::field"
+shows "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
 by (simp add: field_simps)
 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 \<noteq> 0"
 shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b"
-proof -
+(is "?lhs \<longleftrightarrow> ?rhs" is "_ \<longleftrightarrow> a = ?r")
-let ?r = "fps_radical r (Suc k) b"
+proof
-have r00: "r (Suc k) (b$0) \<noteq> 0" using b0 r0 by auto
+show ?lhs if ?rhs
-{
+using that using power_radical[OF b0, of r k, unfolded r0] by simp
-assume H: "a = ?r"
+show ?rhs if ?lhs
-from H have "a^Suc k = b"
+proof -
-using power_radical[OF b0, of r k, unfolded r0] by simp
+have r00: "r (Suc k) (b$0) \<noteq> 0" using b0 r0 by auto
-}
-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
-{
+have "a $ n = ?r $ n" for n
+proof (induct n rule: nat_less_induct)
 fix n
-have "a $ n = ?r $ n"
+assume h: "\<forall>m<n. a$m = ?r $m"
-proof (induct n rule: nat_less_induct)
+show "a$n = ?r $ n"
-fix n
+proof (cases n)
-assume h: "\<forall>m<n. a$m = ?r $m"
+case 0
-{
+then show ?thesis using a0 by simp
-assume "n = 0"
+next
-then have "a$n = ?r $n" using a0 by simp
+case (Suc n1)
-}
+have fK: "finite {0..k}" by simp
-moreover
+have nz: "n \<noteq> 0" using Suc by simp
-{
-fix n1
-assume n1: "n = Suc n1"
-have fK: "finite {0..k}" by simp
-have nz: "n \<noteq> 0" using n1 by arith
 let ?Pnk = "natpermute n (Suc k)"
 let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
 let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
 have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
 have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
 by (simp add: natpermute_max_card[OF nz, simplified])
 have th1: "setsum ?g ?Pnknn = setsum ?f ?Pnknn"
 proof (rule setsum.cong, rule refl, rule setprod.cong, simp)
 fix xs i
 assume xs: "xs \<in> ?Pnknn" and i: "i \<in> {0..k}"
-{
+have False if c: "n \<le> xs ! i"
-assume c: "n \<le> xs ! i"
+proof -
-from xs i have "xs !i \<noteq> n"
+from xs i have "xs ! i \<noteq> n"
 by (auto simp add: in_set_conv_nth natpermute_def)
 with c have c': "n < xs!i" by arith
 have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}"
 by simp_all
 have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}"
 using xs by (simp add: natpermute_def listsum_setsum_nth)
 also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
 unfolding eqs  setsum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
 unfolding setsum.union_disjoint[OF fths(2) fths(3) d(2)]
 by simp
-finally have False using c' by simp
+finally show ?thesis using c' by simp
-}
+qed
 then have thn: "xs!i < n" by presburger
 from h[rule_format, OF thn] show "a$(xs !i) = ?r$(xs!i)" .
 qed
 have th00: "\<And>x::'a. of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x"
 by (simp add: field_simps del: of_nat_Suc)
-from H have "b$n = a^Suc k $ n"
+from \<open>?lhs\<close> have "b$n = a^Suc k $ n"
 by (simp add: fps_eq_iff)
 also have "a ^ Suc k$n = setsum ?g ?Pnkn + setsum ?g ?Pnknn"
 unfolding fps_power_nth_Suc
 using setsum.union_disjoint[OF f d, unfolded Suc_eq_plus1[symmetric],
 unfolded eq, of ?g] by simp
 apply (rule eq_divide_imp')
 using r00
 apply (simp del: of_nat_Suc)
 apply (simp add: ac_simps)
 done
-then have "a$n = ?r $n"
+then show ?thesis
 apply (simp del: of_nat_Suc)
-unfolding fps_radical_def n1
+unfolding fps_radical_def Suc
-apply (simp add: field_simps n1 th00 del: of_nat_Suc)
+apply (simp add: field_simps Suc th00 del: of_nat_Suc)
 done
-}
-ultimately show "a$n = ?r $ n" by (cases n) auto
 qed
-}
+qed
-then have "a = ?r" by (simp add: fps_eq_iff)
+then show ?rhs by (simp add: fps_eq_iff)
-}
+qed
-ultimately show ?thesis by blast
 qed
 lemma radical_power:
 assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0"
 and b0: "b$0 \<noteq> 0"
 shows "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0) \<longleftrightarrow>
 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)"
+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"
+then have r0: "(r (k) ((a*b)$0)) ^ k = (a*b)$0"
 by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
-{
+have ?thesis
-assume "k = 0"
+proof (cases k)
-then have ?thesis using r0' by simp
+case 0
-}
+then show ?thesis using r0' by simp
-moreover
+next
-{
+case (Suc h)
-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)
+using r0' Suc by (simp add: fps_mult_nth)
 have ab0: "(a*b) $ 0 \<noteq> 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]
+from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded Suc] th0 ab0, symmetric]
-iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded k]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded k]] k r0'
+iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded Suc]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded Suc]] Suc r0'
-have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)
+show ?thesis
-}
+by (auto simp add: power_mult_distrib simp del: power_Suc)
-ultimately have ?thesis by (cases k) auto
+qed
 }
 moreover
 {
 assume h: "fps_radical r k (a*b) = fps_radical r k a * fps_radical r k b"
 then have "(fps_radical r k (a*b))$0 = (fps_radical r k a * fps_radical r k b)$0"
 and a0: "a$0 \<noteq> 0"
 and b0: "b$0 \<noteq> 0"
 shows "r k ((a $ 0) / (b$0)) = r k (a$0) / r k (b $ 0) \<longleftrightarrow>
 fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b"
 (is "?lhs = ?rhs")
-proof -
+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) \<noteq> 0" using a0 ra0 k by auto
 have rb0': "r k (b$0) \<noteq> 0" using b0 rb0 k by auto
-{
+show ?lhs if ?rhs
-assume ?rhs
+proof -
-then have "?r (a/b) $ 0 = (?r a / ?r b)$0" by simp
+from that have "?r (a/b) $ 0 = (?r a / ?r b)$0"
-then have ?lhs using k a0 b0 rb0'
+by simp
-by (simp add: fps_divide_def fps_mult_nth fps_inverse_def divide_inverse)
+then show ?thesis
-}
+using k a0 b0 rb0' by (simp add: fps_divide_def fps_mult_nth fps_inverse_def divide_inverse)
-moreover
+qed
-{
+show ?rhs if ?lhs
-assume h: ?lhs
+proof -
 from a0 b0 have ab0[simp]: "(a/b)$0 = a$0 / b$0"
 by (simp add: fps_divide_def fps_mult_nth divide_inverse fps_inverse_def)
 have th0: "r k ((a/b)$0) ^ k = (a/b)$0"
-by (simp add: h nonzero_power_divide[OF rb0'] ra0 rb0)
+by (simp add: \<open>?lhs\<close> nonzero_power_divide[OF rb0'] ra0 rb0)
-from a0 b0 ra0' rb0' kp h
+from a0 b0 ra0' rb0' kp \<open>?lhs\<close>
 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 \<noteq> 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 .
+show ?thesis .
-}
+qed
-ultimately show ?thesis by blast
 qed
 lemma radical_inverse:
 fixes a :: "'a::field_char_0 fps"
 assumes k: "k > 0"
 shows "r k (inverse (a $ 0)) = r k 1 / (r k (a $ 0)) \<longleftrightarrow>
 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\<open>Derivative of composition\<close>
+subsection \<open>Derivative of composition\<close>
 lemma fps_compose_deriv:
 fixes a :: "'a::idom fps"
 assumes b0: "b$0 = 0"
 shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * fps_deriv b"
 proof -
-{
+have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n" for n
-fix n
+proof -
 have "(fps_deriv (a oo b))$n = setsum (\<lambda>i. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}"
 by (simp add: fps_compose_def field_simps setsum_right_distrib del: of_nat_Suc)
 also have "\<dots> = setsum (\<lambda>i. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}"
 by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
 also have "\<dots> = setsum (\<lambda>i. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}"
 unfolding setsum_right_distrib
 apply (subst setsum.commute)
 apply (rule setsum.cong, rule refl)+
 apply simp
 done
-finally have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n"
+finally show ?thesis
 unfolding th0 by simp
-}
+qed
 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 )
+by (simp add: fps_mult_nth)
 next
 case (Suc m)
-have "((1+X)*a) $n = setsum (\<lambda>i. (1+X)$i * a$(n-i)) {0..n}"
+have "((1 + X)*a) $ n = setsum (\<lambda>i. (1 + X) $ i * a $ (n - i)) {0..n}"
 by (simp add: fps_mult_nth)
 also have "\<dots> = setsum (\<lambda>i. (1+X)$i * a$(n-i)) {0.. 1}"
 unfolding Suc by (rule setsum.mono_neutral_right) auto
 also have "\<dots> = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))"
 by (simp add: Suc)
 subsection \<open>Finite FPS (i.e. polynomials) and X\<close>
 lemma fps_poly_sum_X:
-assumes z: "\<forall>i > n. a$i = (0::'a::comm_ring_1)"
+assumes "\<forall>i > n. a$i = (0::'a::comm_ring_1)"
 shows "a = setsum (\<lambda>i. fps_const (a$i) * X^i) {0..n}" (is "a = ?r")
 proof -
-{
+have "a$i = ?r$i" for i
-fix i
+unfolding fps_setsum_nth fps_mult_left_const_nth X_power_nth
-have "a$i = ?r$i"
+by (simp add: mult_delta_right setsum.delta' assms)
-unfolding fps_setsum_nth fps_mult_left_const_nth X_power_nth
+then show ?thesis
-by (simp add: mult_delta_right setsum.delta' z)
+unfolding fps_eq_iff by blast
-}
+qed
-then show ?thesis unfolding fps_eq_iff by blast
-qed
+subsection \<open>Compositional inverses\<close>
-subsection\<open>Compositional inverses\<close>
 fun compinv :: "'a fps \<Rightarrow> nat \<Rightarrow> 'a::field"
 where
 "compinv a 0 = X$0"
 | "compinv a (Suc n) =
 assumes a0: "a$0 = 0"
 and a1: "a$1 \<noteq> 0"
 shows "fps_inv a oo a = X"
 proof -
 let ?i = "fps_inv a oo a"
-{
+have "?i $n = X$n" for n
+proof (induct n rule: nat_less_induct)
 fix n
-have "?i $n = X$n"
+assume h: "\<forall>m<n. ?i$m = X$m"
-proof (induct n rule: nat_less_induct)
+show "?i $ n = X$n"
-fix n
+proof (cases n)
-assume h: "\<forall>m<n. ?i$m = X$m"
+case 0
-show "?i $ n = X$n"
+then show ?thesis using a0
-proof (cases n)
+by (simp add: fps_compose_nth fps_inv_def)
-case 0
+next
-then show ?thesis using a0
+case (Suc n1)
-by (simp add: fps_compose_nth fps_inv_def)
+have "?i $ n = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1"
-next
+by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
-case (Suc n1)
+also have "\<dots> = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} +
-have "?i $ n = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1"
+(X$ Suc n1 - setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1})"
-by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
+using a0 a1 Suc by (simp add: fps_inv_def)
-also have "\<dots> = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} +
+also have "\<dots> = X$n" using Suc by simp
-(X$ Suc n1 - setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1})"
+finally show ?thesis .
-using a0 a1 Suc by (simp add: fps_inv_def)
-also have "\<dots> = X$n" using Suc by simp
-finally show ?thesis .
-qed
 qed
-}
+qed
-then show ?thesis by (simp add: fps_eq_iff)
+then show ?thesis
+by (simp add: fps_eq_iff)
 qed
 fun gcompinv :: "'a fps \<Rightarrow> 'a fps \<Rightarrow> nat \<Rightarrow> 'a::field"
 where
 assumes a0: "a$0 = 0"
 and a1: "a$1 \<noteq> 0"
 shows "fps_ginv b a oo a = b"
 proof -
 let ?i = "fps_ginv b a oo a"
-{
+have "?i $n = b$n" for n
+proof (induct n rule: nat_less_induct)
 fix n
-have "?i $n = b$n"
+assume h: "\<forall>m<n. ?i$m = b$m"
-proof (induct n rule: nat_less_induct)
+show "?i $ n = b$n"
-fix n
+proof (cases n)
-assume h: "\<forall>m<n. ?i$m = b$m"
+case 0
-show "?i $ n = b$n"
+then show ?thesis using a0
-proof (cases n)
+by (simp add: fps_compose_nth fps_ginv_def)
-case 0
+next
-then show ?thesis using a0
+case (Suc n1)
-by (simp add: fps_compose_nth fps_ginv_def)
+have "?i $ n = setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1"
-next
+by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
-case (Suc n1)
+also have "\<dots> = setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} +
-have "?i $ n = setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1"
+(b$ Suc n1 - setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})"
-by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
+using a0 a1 Suc by (simp add: fps_ginv_def)
-also have "\<dots> = setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} +
+also have "\<dots> = b$n" using Suc by simp
-(b$ Suc n1 - setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})"
+finally show ?thesis .
-using a0 a1 Suc by (simp add: fps_ginv_def)
-also have "\<dots> = b$n" using Suc by simp
-finally show ?thesis .
-qed
 qed
-}
+qed
-then show ?thesis by (simp add: fps_eq_iff)
+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)
 lemma fps_compose_setsum_distrib: "(setsum f S) oo a = setsum (\<lambda>i. f i oo a) S"
 proof (cases "finite S")
 case True
 show ?thesis
 proof (rule finite_induct[OF True])
-show "setsum f {} oo a = (\<Sum>i\<in>{}. f i oo a)" by simp
+show "setsum f {} oo a = (\<Sum>i\<in>{}. f i oo a)"
+by simp
 next
 fix x F
 assume fF: "finite F"
 and xF: "x \<notin> F"
 and h: "setsum f F oo a = setsum (\<lambda>i. f i oo a) F"
 done
 qed
 lemma fps_compose_mult_distrib_lemma:
 assumes c0: "c$0 = (0::'a::idom)"
-shows "((a oo c) * (b oo c))$n =
+shows "((a oo c) * (b oo c))$n = setsum (\<lambda>s. setsum (\<lambda>i. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}"
-setsum (\<lambda>s. setsum (\<lambda>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 = "\<lambda>k. a$k" and b="\<lambda>m. b$m" and c="\<lambda>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)"
 apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma [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)
 lemma fps_compose_assoc:
 assumes c0: "c$0 = (0::'a::idom)"
 and b0: "b$0 = 0"
 shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
 proof -
-{
+have "?l$n = ?r$n" for n
-fix n
+proof -
 have "?l$n = (setsum (\<lambda>i. (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 "\<dots> = ((setsum (\<lambda>i. fps_const (a$i) * b^i) {0..n}) oo c)$n"
 by (simp add: fps_compose_setsum_distrib)
 apply (rule refl)
 apply (rule setsum.mono_neutral_right)
 apply (auto simp add: not_le)
 apply (erule startsby_zero_power_prefix[OF b0, rule_format])
 done
-finally have "?l$n = ?r$n" .
+finally show ?thesis .
-}
+qed
-then show ?thesis by (simp add: fps_eq_iff)
+then show ?thesis
+by (simp add: fps_eq_iff)
 qed
 lemma fps_X_power_compose:
 assumes a0: "a$0=0"
 proof (cases k)
 case 0
 then show ?thesis by simp
 next
 case (Suc h)
-{
+have "?l $ n = ?r $n" for n
-fix n
+proof -
-{
+consider "k > n" | "k \<le> n" by arith
-assume kn: "k>n"
+then show ?thesis
-then have "?l $ n = ?r $n" using a0 startsby_zero_power_prefix[OF a0] Suc
+proof cases
+case 1
+then show ?thesis
+using a0 startsby_zero_power_prefix[OF a0] Suc
 by (simp add: fps_compose_nth del: power_Suc)
-}
+next
-moreover
+case 2
-{
+then show ?thesis
-assume kn: "k \<le> n"
-then have "?l$n = ?r$n"
 by (simp add: fps_compose_nth mult_delta_left setsum.delta)
-}
+qed
-moreover have "k >n \<or> k\<le> n"  by arith
+qed
-ultimately have "?l$n = ?r$n"  by blast
+then show ?thesis
-}
+unfolding fps_eq_iff 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 \<noteq> 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 th0: "?ia $ 0 = 0"
-have th1: "?iaa $ 0 = 0" using a0 a1
+by (simp add: fps_inv_def)
-by (simp add: fps_inv_def fps_compose_nth)
+have th1: "?iaa $ 0 = 0"
-have th2: "X$0 = 0" by simp
+using a0 a1 by (simp add: fps_inv_def fps_compose_nth)
-from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo X" by simp
+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]
+with fps_compose_inj_right[OF a0 a1] show ?thesis
-show ?thesis by simp
+by simp
 qed
 lemma fps_inv_deriv:
-assumes a0:"a$0 = (0::'a::field)"
+assumes a0: "a$0 = (0::'a::field)"
 and a1: "a$1 \<noteq> 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 ia0: "?ia$0 = 0"
-have th0: "?d$0 \<noteq> 0" using a1 by (simp add: fps_compose_nth)
+by (simp add: fps_inv_def)
+have th0: "?d$0 \<noteq> 0"
+using a1 by (simp add: fps_compose_nth)
 from fps_inv_right[OF a0 a1] have "?d * ?dia = 1"
 by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] )
-then have "inverse ?d * ?d * ?dia = inverse ?d * 1" by simp
+then have "inverse ?d * ?d * ?dia = inverse ?d * 1"
-with inverse_mult_eq_1 [OF th0]
+by simp
-show "?dia = inverse ?d" 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 \<noteq> 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)
+have ra0: "?r a $ 0 = 0"
-from a1 have ra1: "?r a $ 1 \<noteq> 0" by (simp add: fps_inv_def field_simps)
+by (simp add: fps_inv_def)
-have X0: "X$0 = 0" by simp
+from a1 have ra1: "?r a $ 1 \<noteq> 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 = 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
+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 \<noteq> 0"
 and c0: "c$0 = 0"
 and  c1: "c$1 \<noteq> 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 c0 have rca0: "?r c a $0 = 0"
-from a1 c1 have rca1: "?r c a $ 1 \<noteq> 0" by (simp add: fps_ginv_def field_simps)
+by (simp add: fps_ginv_def)
+from a1 c1 have rca1: "?r c a $ 1 \<noteq> 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"
+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"
+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
+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 \<noteq> 0"
 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 iXa0: "?iXa $ 0 = 0"
-have da0: "?d a $ 0 \<noteq> 0" using a1 by simp
+by (simp add: fps_ginv_def)
-from fps_ginv[OF a0 a1, of b] have "?d (?ia oo a) = fps_deriv b" by simp
+have da0: "?d a $ 0 \<noteq> 0"
-then have "(?d ?ia oo a) * ?d a = ?d b" unfolding fps_compose_deriv[OF a0] .
+using a1 by simp
-then have "(?d ?ia oo a) * ?d a * inverse (?d a) = ?d b * inverse (?d a)" 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 "
+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\<open>Elementary series\<close>
+subsection \<open>Elementary series\<close>
-subsubsection\<open>Exponential series\<close>
+subsubsection \<open>Exponential series\<close>
 definition "E x = Abs_fps (\<lambda>n. 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 -
-{
+have "?l$n = ?r $ n" for n
-fix n
+apply (auto simp add: E_def field_simps power_Suc[symmetric]
-have "?l$n = ?r $ n"
+simp del: fact.simps of_nat_Suc power_Suc)
-apply (auto simp add: E_def field_simps power_Suc[symmetric]
+apply (simp add: of_nat_mult field_simps)
-simp del: fact.simps of_nat_Suc power_Suc)
+done
-apply (simp add: of_nat_mult field_simps)
+then show ?thesis
-done
+by (simp add: fps_eq_iff)
-}
-then show ?thesis by (simp add: fps_eq_iff)
 qed
 lemma E_unique_ODE:
 "fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a$0) * E (c::'a::field_char_0)"
 (is "?lhs \<longleftrightarrow> ?rhs")
 proof
-assume d: ?lhs
+show ?rhs if ?lhs
-from d have th: "\<And>n. a $ Suc n = c * a$n / of_nat (Suc n)"
+proof -
-by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
+from that have th: "\<And>n. a $ Suc n = c * a$n / of_nat (Suc n)"
-{
+by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
-fix n
+have th': "a$n = a$0 * c ^ n/ (fact n)" for n
-have "a$n = a$0 * c ^ n/ (fact n)"
+proof (induct n)
-apply (induct n)
+case 0
-apply simp
+then show ?case by simp
-unfolding th
+next
-using fact_gt_zero
+case Suc
-apply (simp add: field_simps del: of_nat_Suc fact_Suc)
+then show ?case
-apply simp
+unfolding th
-done
+using fact_gt_zero
-}
+apply (simp add: field_simps del: of_nat_Suc fact_Suc)
-note th' = this
+apply simp
-show ?rhs by (auto simp add: fps_eq_iff fps_const_mult_left E_def intro: th')
+done
-next
+qed
-assume h: ?rhs
+show ?thesis
-show ?lhs
+by (auto simp add: fps_eq_iff fps_const_mult_left E_def intro: th')
-by (metis E_deriv fps_deriv_mult_const_left h mult.left_commute)
+qed
+show ?lhs if ?rhs
+using that by (metis E_deriv fps_deriv_mult_const_left mult.left_commute)
 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"
+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)
+then have "?r = ?l"
-by (simp add: fps_mult_nth E_def)
+by (simp only: E_unique_ODE) (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 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\<noteq>0"
+assumes a: "a \<noteq> 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 b0: "?b $ 0 = 0"
-have b1: "?b $ 1 \<noteq> 0" by (simp add: a)
+by simp
+have b1: "?b $ 1 \<noteq> 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:
+lemma fps_const_inverse: "a \<noteq> 0 \<Longrightarrow> inverse (fps_const (a::'a::field)) = fps_const (inverse a)"
-"a \<noteq> 0 \<Longrightarrow> 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
 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 \<noteq> 0" using r by simp_all
-from th0 radical_unique[where r=r and k=k, OF th]
+from th0 radical_unique[where r=r and k=k, OF th] show ?thesis
-show ?thesis by auto
+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
-subsubsection\<open>Logarithmic series\<close>
+subsubsection \<open>Logarithmic series\<close>
 lemma Abs_fps_if_0:
-"Abs_fps(\<lambda>n. if n=0 then (v::'a::ring_1) else f n) = fps_const v + X * Abs_fps (\<lambda>n. f (Suc n))"
+"Abs_fps (\<lambda>n. if n = 0 then (v::'a::ring_1) else f n) =
+fps_const v + X * Abs_fps (\<lambda>n. f (Suc n))"
 by (auto simp add: fps_eq_iff)
 definition L :: "'a::field_char_0 \<Rightarrow> 'a fps"
 where "L c = fps_const (1/c) * Abs_fps (\<lambda>n. 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 fps_inverse_X_plus1
 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))"
+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:
 finally show ?thesis
 unfolding fps_deriv_eq_iff by simp
 qed
-subsubsection\<open>Binomial series\<close>
+subsubsection \<open>Binomial series\<close>
 definition "fps_binomial a = Abs_fps (\<lambda>n. 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) \<longleftrightarrow> a = fps_const (a$0) * fps_binomial c"
 (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
+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 \<noteq> 0" by simp
-have "?l = ?r \<longleftrightarrow> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
+have eq: "?l = ?r \<longleftrightarrow> ?lhs"
-also have "\<dots> \<longleftrightarrow> ?da = (fps_const c * a) / ?x1"
+proof -
-apply (simp only: fps_divide_def  mult.assoc[symmetric] inverse_mult_eq_1[OF x10])
+have x10: "?x1 $ 0 \<noteq> 0" by simp
-apply (simp add: field_simps)
+have "?l = ?r \<longleftrightarrow> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
-done
+also have "\<dots> \<longleftrightarrow> ?da = (fps_const c * a) / ?x1"
-finally have eq: "?l = ?r \<longleftrightarrow> ?lhs" by simp
+apply (simp only: fps_divide_def  mult.assoc[symmetric] inverse_mult_eq_1[OF x10])
-moreover
+apply (simp add: field_simps)
-{assume h: "?l = ?r"
+done
-{fix n
+finally show ?thesis .
-from h have lrn: "?l $ n = ?r$n" by simp
+qed
-from lrn
+show ?rhs if ?lhs
-have "a$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a $n"
+proof -
+from eq that have h: "?l = ?r" ..
+have th0: "a$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a $n" for n
+proof -
+from h have "?l $ n = ?r $ n" by simp
+then show ?thesis
 apply (simp add: field_simps del: of_nat_Suc)
-by (cases n, simp_all add: field_simps del: of_nat_Suc)
+apply (cases n)
-}
+apply (simp_all add: field_simps del: of_nat_Suc)
-note th0 = this
+done
-{
+qed
-fix n
+have th1: "a $ n = (c gchoose n) * a $ 0" for n
-have "a$n = (c gchoose n) * a$0"
+proof (induct n)
-proof (induct n)
+case 0
-case 0
+then show ?case by simp
-then show ?case by simp
+next
-next
+case (Suc m)
-case (Suc m)
+then show ?case
-then show ?case unfolding th0
+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
-}
+show ?thesis
-note th1 = this
-have ?rhs
 apply (simp add: fps_eq_iff)
 apply (subst th1)
 apply (simp add: field_simps)
 done
-}
+qed
-moreover
-{
+show ?lhs if ?rhs
-assume h: ?rhs
+proof -
-have th00: "\<And>x y. x * (a$0 * y) = a$0 * (x*y)"
+have th00: "x * (a $ 0 * y) = a $ 0 * (x * y)" for x y
 by (simp add: mult.commute)
 have "?l = ?r"
-apply (subst h)
+apply (subst \<open>?rhs\<close>)
-apply (subst (2) h)
+apply (subst (2) \<open>?rhs\<close>)
 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
-}
+with eq show ?thesis ..
-ultimately show ?thesis by blast
+qed
 qed
 lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + X)"
 proof -
 let ?a = "fps_binomial c"
 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\<open>Vandermonde's Identity as a consequence\<close>
+text \<open>Vandermonde's Identity as a consequence\<close>
 lemma gbinomial_Vandermonde:
 "setsum (\<lambda>k. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
 proof -
 let ?ba = "fps_binomial a"
 let ?bb = "fps_binomial b"
 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 (\<lambda>k. (n choose k)\<^sup>2) {0..n} = (2*n) choose n"
+lemma binomial_Vandermonde_same: "setsum (\<lambda>k. (n choose k)\<^sup>2) {0..n} = (2 * n) choose n"
-using binomial_Vandermonde[of n n n,symmetric]
+using binomial_Vandermonde[of n n n, symmetric]
 unfolding mult_2
 apply (simp add: power2_eq_square)
 apply (rule setsum.cong)
 apply (auto intro:  binomial_symmetric)
 done
 (is "?l = ?r")
 proof -
 let ?m1 = "\<lambda>m. (- 1 :: 'a) ^ m"
 let ?f = "\<lambda>m. of_nat (fact m)"
 let ?p = "\<lambda>(x::'a). pochhammer (- x)"
-from b have bn0: "?p b n \<noteq> 0" unfolding pochhammer_eq_0_iff by simp
+from b have bn0: "?p b n \<noteq> 0"
+unfolding pochhammer_eq_0_iff by simp
 {
 fix k
 assume kn: "k \<in> {0..n}"
-{
+have nz: "pochhammer (1 + b - of_nat n) n \<noteq> 0"
-assume c:"pochhammer (b - of_nat n + 1) n = 0"
+proof
+assume "pochhammer (1 + b - of_nat n) n = 0"
+then have c: "pochhammer (b - of_nat n + 1) n = 0"
+by (simp add: algebra_simps)
 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
+with b show False using j by auto
-}
+qed
-then have nz: "pochhammer (1 + b - of_nat n) n \<noteq> 0"
-by (auto simp add: algebra_simps)
 from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k \<noteq> 0"
 by (rule pochhammer_neq_0_mono)
 {
 assume k0: "k = 0 \<or> n =0"
 by (cases "k = 0") (simp_all add: gbinomial_pochhammer)
 }
 moreover
 {
 assume n0: "n \<noteq> 0" and k0: "k \<noteq> 0"
-then obtain m where m: "n = Suc m" by (cases n) auto
+then obtain m where m: "n = Suc m"
-from k0 obtain h where h: "k = Suc h" by (cases k) auto
+by (cases n) auto
-{
+from k0 obtain h where h: "k = Suc h"
-assume "k = n"
+by (cases k) auto
-then have "b gchoose (n - k) =
+have "b gchoose (n - k) =
 (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
+proof (cases "k = n")
+case True
+then show ?thesis
 using pochhammer_minus'[where k=k and b=b]
 apply (simp add: pochhammer_same)
 using bn0
 apply (simp add: field_simps power_add[symmetric])
 done
-}
+next
-moreover
+case False
-{
+with kn have kn': "k < n"
-assume nk: "k \<noteq> n"
+by simp
 have m1nk: "?m1 n = setprod (\<lambda>i. - 1) {0..m}" "?m1 k = setprod (\<lambda>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 \<noteq> 0"
 using bn0 kn
 unfolding pochhammer_eq_0_iff
 apply auto
 apply (erule_tac x= "n - ka - 1" in allE)
 using bnz0
 by (simp add: field_simps)
 also have "\<dots> = b gchoose (n - k)"
 unfolding th1 th2
 using kn' by (simp add: gbinomial_def)
-finally have "b gchoose (n - k) =
+finally show ?thesis by simp
-(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
+qed
-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 \<noteq> 0 "
 apply (cases "n = 0")
 proof -
 let ?a = "- a"
 let ?b = "c + of_nat n - 1"
 have h: "\<forall> j \<in>{0..< n}. ?b \<noteq> of_nat j" using c
 apply (auto simp add: algebra_simps of_nat_diff)
-apply (erule_tac x= "n - j - 1" in ballE)
+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
 by (simp add: algebra_simps)
 unfolding pochhammer_minus
 by simp
 have nz: "pochhammer c n \<noteq> 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)
+show ?thesis
-qed
+using nz by (simp add: field_simps setsum_right_distrib)
+qed
-subsubsection\<open>Formal trigonometric functions\<close>
+subsubsection \<open>Formal trigonometric functions\<close>
 definition "fps_sin (c::'a::field_char_0) =
 Abs_fps (\<lambda>n. 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) =
 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
-{
+show "?lhs $ n = ?rhs $ n"
-assume en: "even n"
+proof (cases "even n")
+case True
 have "?lhs$n = of_nat (n+1) * (fps_sin c $ (n+1))" by simp
 also have "\<dots> = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))"
-using en by (simp add: fps_sin_def)
+using True by (simp add: fps_sin_def)
 also have "\<dots> = (- 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 "\<dots> = (- 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
+finally show ?thesis
-by (simp add: fps_cos_def field_simps)
+using True by (simp add: fps_cos_def field_simps)
-}
+next
-then show "?lhs $ n = ?rhs $ n"
+case False
-by (cases "even n") (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
+then show ?thesis
+by (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
+qed
 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: "\<And>n. - ((- 1::'a) ^ n) = (- 1)^Suc n" by simp
+have th0: "- ((- 1::'a) ^ n) = (- 1)^Suc n" for n
-have th1: "\<And>n. odd n \<Longrightarrow> Suc ((n - 1) div 2) = Suc n div 2"
+by simp
-by (case_tac n, simp_all)
+show "?lhs $ n = ?rhs $ n" for n
-fix n::nat
+proof (cases "even n")
-{
+case False
-assume en: "odd n"
+then have n0: "n \<noteq> 0" by presburger
-from en have n0: "n \<noteq>0 " by presburger
+from False have th1: "Suc ((n - 1) div 2) = Suc n div 2"
+by (cases n) simp_all
 have "?lhs$n = of_nat (n+1) * (fps_cos c $ (n+1))" by simp
 also have "\<dots> = 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)
+using False by (simp add: fps_cos_def)
 also have "\<dots> = (- 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 "\<dots> = (- 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 "\<dots> = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)"
-unfolding th0 unfolding th1[OF en] by simp
+unfolding th0 unfolding th1 by simp
-finally have "?lhs $n = ?rhs$n" using en
+finally show ?thesis
-by (simp add: fps_sin_def field_simps)
+using False by (simp add: fps_sin_def field_simps)
-}
+next
-then show "?lhs $ n = ?rhs $ n"
+case True
-by (cases "even n") (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
+then show ?thesis
-qed
+by (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
+qed
-lemma fps_sin_cos_sum_of_squares:
+qed
-"(fps_cos c)\<^sup>2 + (fps_sin c)\<^sup>2 = 1" (is "?lhs = 1")
+lemma fps_sin_cos_sum_of_squares: "(fps_cos c)\<^sup>2 + (fps_sin c)\<^sup>2 = 1"
+(is "?lhs = _")
 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
 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)))"
+"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 (cases n)
+apply simp
 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq 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"
 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)))"
+"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: nonzero_divide_eq_eq nonzero_eq_divide_eq del: of_nat_Suc fact_Suc)
 apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
 done
 apply simp
 done
 qed
 text \<open>Connection to E c over the complex numbers --- Euler and De Moivre\<close>
-lemma Eii_sin_cos: "E (ii * c) = fps_cos c + fps_const ii * fps_sin c "
+lemma Eii_sin_cos: "E (ii * c) = fps_cos c + fps_const ii * fps_sin c"
 (is "?l = ?r")
 proof -
-{ fix n :: nat
+have "?l $ n = ?r $ n" for n
-{
+proof (cases "even n")
-assume en: "even n"
+case True
-from en obtain m where m: "n = 2 * m" ..
+then obtain m where m: "n = 2 * m" ..
+show ?thesis
-have "?l $n = ?r$n"
+by (simp add: m fps_sin_def fps_cos_def power_mult_distrib power_mult power_minus [of "c ^ 2"])
-by (simp add: m fps_sin_def fps_cos_def power_mult_distrib power_mult power_minus [of "c ^ 2"])
+next
-}
+case False
-moreover
+then obtain m where m: "n = 2 * m + 1" ..
-{
+show ?thesis
-assume "odd n"
+by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
-then obtain m where m: "n = 2 * m + 1" ..
+power_mult power_minus [of "c ^ 2"])
-have "?l $n = ?r$n"
+qed
-by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
+then show ?thesis
-power_mult power_minus [of "c ^ 2"])
+by (simp add: fps_eq_iff)
-}
-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_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)"
 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)"
+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: ac_simps)
 subsection \<open>Hypergeometric series\<close>
 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"
+lemma F_0[simp]: "F as bs c $ 0 = 1"
 apply simp
 apply (subgoal_tac "\<forall>as. foldl (\<lambda>(r::'a) (a::'a). r) 1 as = 1")
 apply auto
 apply (induct_tac as)
 apply auto
 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_0th[simp]: "XD a $ 0 = 0"
-lemma XD_Suc[simp]:" XD a $ Suc n = of_nat (Suc n) * a $ Suc n" by simp
+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 nth_equal_imp_dist_less:
 assumes "\<And>j. j \<le> i \<Longrightarrow> f $ j = g $ j"
 shows "dist f g < inverse (2 ^ i)"
 proof (cases "f = g")
+case True
+then show ?thesis by simp
+next
 case False
 then have "\<exists>n. f $ n \<noteq> g $ n" by (simp add: fps_eq_iff)
 with assms have "dist f g = inverse (2 ^ (LEAST n. f $ n \<noteq> g $ n))"
 by (simp add: split_if_asm dist_fps_def)
 moreover
 from assms \<open>\<exists>n. f $ n \<noteq> g $ n\<close> have "i < (LEAST n. f $ n \<noteq> g $ n)"
 by (metis (mono_tags) LeastI not_less)
 ultimately show ?thesis by simp
-qed simp
+qed
 lemma dist_less_eq_nth_equal: "dist f g < inverse (2 ^ i) \<longleftrightarrow> (\<forall>j \<le> 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 \<Rightarrow> 'a fps"
 assume "Cauchy X"
-{
+obtain M where M: "\<forall>i. \<forall>m \<ge> M i. \<forall>j \<le> i. X (M i) $ j = X m $ j"
-fix i
+proof -
-have "0 < inverse ((2::real)^i)" by simp
+have "\<exists>M. \<forall>m \<ge> M. \<forall>j\<le>i. X M $ j = X m $ j" for i
-from metric_CauchyD[OF \<open>Cauchy X\<close> this] dist_less_imp_nth_equal
+proof -
-have "\<exists>M. \<forall>m \<ge> M. \<forall>j\<le>i. X M $ j = X m $ j" by blast
+have "0 < inverse ((2::real)^i)" by simp
-}
+from metric_CauchyD[OF \<open>Cauchy X\<close> this] dist_less_imp_nth_equal
-then obtain M where M: "\<forall>i. \<forall>m \<ge> M i. \<forall>j \<le> i. X (M i) $ j = X m $ j" by metis
+show ?thesis by blast
-then have "\<forall>i. \<forall>m \<ge> M i. \<forall>j \<le> i. X (M i) $ j = X m $ j" by metis
+qed
+then show ?thesis using that by metis
+qed
 show "convergent X"
 proof (rule convergentI)
 show "X ----> Abs_fps (\<lambda>i. X (M i) $ i)"
 unfolding tendsto_iff
 proof safe
 THEN spec, of "\<lambda>x. x < e"]
 have "eventually (\<lambda>i. inverse (2 ^ i) < e) sequentially"
 unfolding eventually_nhds
 apply clarsimp
 apply (rule FalseE)
-apply auto --\<open>slow\<close>
+apply auto -- \<open>slow\<close>
 done
-then obtain i where "inverse (2 ^ i) < e" by (auto simp: eventually_sequentially)
+then obtain i where "inverse (2 ^ i) < e"
-have "eventually (\<lambda>x. M i \<le> x) sequentially" by (auto simp: eventually_sequentially)
+by (auto simp: eventually_sequentially)
+have "eventually (\<lambda>x. M i \<le> x) sequentially"
+by (auto simp: eventually_sequentially)
 then show "eventually (\<lambda>x. dist (X x) (Abs_fps (\<lambda>i. X (M i) $ i)) < e) sequentially"
 proof eventually_elim
 fix x
-assume "M i \<le> x"
+assume x: "M i \<le> x"
-moreover
+have "X (M i) $ j = X (M j) $ j" if "j \<le> i" for j
-have "\<And>j. j \<le> i \<Longrightarrow> X (M i) $ j = X (M j) $ j"
+using M that by (metis nat_le_linear)
-using M by (metis nat_le_linear)
+with x have "dist (X x) (Abs_fps (\<lambda>j. X (M j) $ j)) < inverse (2 ^ i)"
-ultimately have "dist (X x) (Abs_fps (\<lambda>j. X (M j) $ j)) < inverse (2 ^ i)"
 using M by (force simp: dist_less_eq_nth_equal)
 also note \<open>inverse (2 ^ i) < e\<close>
 finally show "dist (X x) (Abs_fps (\<lambda>j. X (M j) $ j)) < e" .
 qed
 qed

changeset 60501	839169c70e92
parent 60500	903bb1495239
child 60504	17741c71a731