(* Title: HOL/Computational_Algebra/Formal_Power_Series.thy Author: Amine Chaieb, University of Cambridge *) section ‹A formalization of formal power series› theory Formal_Power_Series imports Complex_Main Euclidean_Algorithm begin subsection ‹The type of formal power series› typedef 'a fps = "{f :: nat ⇒ 'a. True}" morphisms fps_nth Abs_fps by simp notation fps_nth (infixl "$" 75) lemma expand_fps_eq: "p = q ⟷ (∀n. p $ n = q $ n)" by (simp add: fps_nth_inject [symmetric] fun_eq_iff) lemma fps_ext: "(⋀n. p $ n = q $ n) ⟹ p = q" 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: "(+) = (λf g. Abs_fps (λn. f $ n + g $ n))" instance .. end lemma fps_add_nth [simp]: "(f + g) $ n = f $ n + g $ n" unfolding fps_plus_def by simp instantiation fps :: (minus) minus begin definition fps_minus_def: "(-) = (λf g. Abs_fps (λn. f $ n - g $ n))" instance .. end lemma fps_sub_nth [simp]: "(f - g) $ n = f $ n - g $ n" unfolding fps_minus_def by simp instantiation fps :: (uminus) uminus begin definition fps_uminus_def: "uminus = (λf. Abs_fps (λn. - (f $ n)))" instance .. end lemma fps_neg_nth [simp]: "(- f) $ n = - (f $ n)" unfolding fps_uminus_def by simp instantiation fps :: ("{comm_monoid_add, times}") times begin definition fps_times_def: "( * ) = (λf g. Abs_fps (λn. ∑i=0..n. f $ i * g $ (n - i)))" instance .. end lemma fps_mult_nth: "(f * g) $ n = (∑i=0..n. f$i * g$(n - i))" unfolding fps_times_def by simp lemma fps_mult_nth_0 [simp]: "(f * g) $ 0 = f $ 0 * g $ 0" unfolding fps_times_def by simp declare atLeastAtMost_iff [presburger] declare Bex_def [presburger] declare Ball_def [presburger] lemma mult_delta_left: fixes x y :: "'a::mult_zero" shows "(if b then x else 0) * y = (if b then x * y else 0)" by simp lemma mult_delta_right: fixes x y :: "'a::mult_zero" shows "x * (if b then y else 0) = (if b then x * y else 0)" by simp lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)" by auto subsection ‹Formal power series form a commutative ring with unity, if the range of sequences they represent is a commutative ring with unity› 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 sum.distrib 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 sum_distrib_left sum_distrib_right mult.assoc) qed qed lemma fps_mult_commute_lemma: fixes n :: nat and f :: "nat ⇒ nat ⇒ 'a::comm_monoid_add" shows "(∑i=0..n. f i (n - i)) = (∑i=0..n. f (n - i) i)" by (rule sum.reindex_bij_witness[where i="(-) n" and j="(-) n"]) auto instance fps :: (comm_semiring_0) ab_semigroup_mult proof fix a b :: "'a fps" show "a * b = b * a" proof (rule fps_ext) fix n :: nat have "(∑i=0..n. a$i * b$(n - i)) = (∑i=0..n. a$(n - i) * b$i)" by (rule fps_mult_commute_lemma) then show "(a * b) $ n = (b * a) $ n" 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 sum.delta) show "a * 1 = a" by (simp add: fps_ext fps_mult_nth mult_delta_right sum.delta') qed instance fps :: (cancel_semigroup_add) cancel_semigroup_add proof fix a b c :: "'a fps" show "b = c" if "a + b = a + c" using that by (simp add: expand_fps_eq) show "b = c" if "b + a = c + a" using that by (simp add: expand_fps_eq) qed 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 - 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 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 :: (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 standard (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 sum.distrib) show "a * (b + c) = a * b + a * c" by (simp add: expand_fps_eq fps_mult_nth distrib_left sum.distrib) qed instance fps :: (semiring_0) semiring_0 proof fix a :: "'a fps" show "0 * a = 0" 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 .. instance fps :: (semiring_1) semiring_1 .. subsection ‹Selection of the nth power of the implicit variable in the infinite sum› lemma fps_square_nth: "(f^2) $ n = (∑k≤n. f $ k * f $ (n - k))" by (simp add: power2_eq_square fps_mult_nth atLeast0AtMost) lemma fps_nonzero_nth: "f ≠ 0 ⟷ (∃ n. f $n ≠ 0)" by (simp add: expand_fps_eq) lemma fps_nonzero_nth_minimal: "f ≠ 0 ⟷ (∃n. f $ n ≠ 0 ∧ (∀m < n. f $ m = 0))" (is "?lhs ⟷ ?rhs") proof let ?n = "LEAST n. f $ n ≠ 0" show ?rhs if ?lhs proof - from that have "∃n. f $ n ≠ 0" 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 ?thesis .. qed show ?lhs if ?rhs using that by (auto simp add: expand_fps_eq) qed lemma fps_eq_iff: "f = g ⟷ (∀n. f $ n = g $n)" by (rule expand_fps_eq) lemma fps_sum_nth: "sum f S $ n = sum (λk. (f k) $ n) S" proof (cases "finite S") case True then show ?thesis by (induct set: finite) auto next case False then show ?thesis by simp qed subsection ‹Injection of the basic ring elements and multiplication by scalars› definition "fps_const c = Abs_fps (λn. if n = 0 then c else 0)" lemma fps_nth_fps_const [simp]: "fps_const c $ n = (if n = 0 then c else 0)" unfolding fps_const_def by simp lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0" 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 sum.neutral) 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$n)" unfolding fps_eq_iff fps_mult_nth by (simp add: fps_const_def mult_delta_left sum.delta) lemma fps_const_mult_right: "f * fps_const (c::'a::semiring_0) = Abs_fps (λn. f$n * c)" unfolding fps_eq_iff fps_mult_nth by (simp add: fps_const_def mult_delta_right sum.delta') lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n" by (simp add: fps_mult_nth mult_delta_left sum.delta) lemma fps_mult_right_const_nth [simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c" by (simp add: fps_mult_nth mult_delta_right sum.delta') subsection ‹Formal power series form an integral domain› instance fps :: (ring) ring .. instance fps :: (ring_1) ring_1 by (intro_classes, auto simp add: distrib_right) instance fps :: (comm_ring_1) comm_ring_1 by (intro_classes, auto simp add: distrib_right) instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors proof fix a b :: "'a fps" assume "a ≠ 0" and "b ≠ 0" then obtain i j where i: "a $ i ≠ 0" "∀k<i. a $ k = 0" and j: "b $ j ≠ 0" "∀k<j. b $ k =0" unfolding fps_nonzero_nth_minimal by blast+ have "(a * b) $ (i + j) = (∑k=0..i+j. a $ k * b $ (i + j - k))" by (rule fps_mult_nth) also have "… = (a $ i * b $ (i + j - i)) + (∑k∈{0..i+j} - {i}. a $ k * b $ (i + j - k))" by (rule sum.remove) simp_all also have "(∑k∈{0..i+j}-{i}. a $ k * b $ (i + j - k)) = 0" proof (rule sum.neutral [rule_format]) fix k assume "k ∈ {0..i+j} - {i}" then have "k < i ∨ i+j-k < j" by auto then show "a $ k * b $ (i + j - k) = 0" using i j by auto qed also have "a $ i * b $ (i + j - i) + 0 = a $ i * b $ j" by simp also have "a $ i * b $ j ≠ 0" using i j by simp finally have "(a*b) $ (i+j) ≠ 0" . then show "a * b ≠ 0" unfolding fps_nonzero_nth by blast qed instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors .. instance fps :: (idom) idom .. lemma numeral_fps_const: "numeral k = fps_const (numeral k)" by (induct k) (simp_all only: numeral.simps fps_const_1_eq_1 fps_const_add [symmetric]) lemma neg_numeral_fps_const: "(- numeral k :: 'a :: ring_1 fps) = fps_const (- numeral k)" by (simp add: numeral_fps_const) lemma fps_numeral_nth: "numeral n $ i = (if i = 0 then numeral n else 0)" by (simp add: numeral_fps_const) lemma fps_numeral_nth_0 [simp]: "numeral n $ 0 = numeral n" by (simp add: numeral_fps_const) lemma fps_of_nat: "fps_const (of_nat c) = of_nat c" by (induction c) (simp_all add: fps_const_add [symmetric] del: fps_const_add) lemma numeral_neq_fps_zero [simp]: "(numeral f :: 'a :: field_char_0 fps) ≠ 0" proof assume "numeral f = (0 :: 'a fps)" from arg_cong[of _ _ "λF. F $ 0", OF this] show False by simp qed subsection ‹The efps_Xtractor series fps_X› lemma minus_one_power_iff: "(- (1::'a::comm_ring_1)) ^ n = (if even n then 1 else - 1)" by (induct n) auto definition "fps_X = Abs_fps (λn. if n = 1 then 1 else 0)" lemma fps_X_mult_nth [simp]: "(fps_X * (f :: 'a::semiring_1 fps)) $n = (if n = 0 then 0 else f $ (n - 1))" proof (cases "n = 0") case False have "(fps_X * f) $n = (∑i = 0..n. fps_X $ i * f $ (n - i))" by (simp add: fps_mult_nth) also have "… = f $ (n - 1)" using False by (simp add: fps_X_def mult_delta_left sum.delta) finally show ?thesis using False by simp next case True then show ?thesis by (simp add: fps_mult_nth fps_X_def) qed lemma fps_X_mult_right_nth[simp]: "((a::'a::semiring_1 fps) * fps_X) $ n = (if n = 0 then 0 else a $ (n - 1))" proof - have "(a * fps_X) $ n = (∑i = 0..n. a $ i * (if n - i = Suc 0 then 1 else 0))" by (simp add: fps_times_def fps_X_def) also have "… = (∑i = 0..n. if i = n - 1 then if n = 0 then 0 else a $ i else 0)" by (intro sum.cong) auto also have "… = (if n = 0 then 0 else a $ (n - 1))" by (simp add: sum.delta) finally show ?thesis . qed lemma fps_mult_fps_X_commute: "fps_X * (a :: 'a :: semiring_1 fps) = a * fps_X" by (simp add: fps_eq_iff) lemma fps_X_power_iff: "fps_X ^ n = Abs_fps (λm. if m = n then 1 else 0)" by (induction n) (auto simp: fps_eq_iff) lemma fps_X_nth[simp]: "fps_X$n = (if n = 1 then 1 else 0)" by (simp add: fps_X_def) lemma fps_X_power_nth[simp]: "(fps_X^k) $n = (if n = k then 1 else 0::'a::comm_ring_1)" by (simp add: fps_X_power_iff) lemma fps_X_power_mult_nth: "(fps_X^k * (f :: 'a::comm_ring_1 fps)) $n = (if n < k then 0 else f $ (n - k))" apply (induct k arbitrary: n) apply simp unfolding power_Suc mult.assoc apply (case_tac n) apply auto done lemma fps_X_power_mult_right_nth: "((f :: 'a::comm_ring_1 fps) * fps_X^k) $n = (if n < k then 0 else f $ (n - k))" by (metis fps_X_power_mult_nth mult.commute) lemma fps_X_neq_fps_const [simp]: "(fps_X :: 'a :: zero_neq_one fps) ≠ fps_const c" proof assume "(fps_X::'a fps) = fps_const (c::'a)" hence "fps_X$1 = (fps_const (c::'a))$1" by (simp only:) thus False by auto qed lemma fps_X_neq_zero [simp]: "(fps_X :: 'a :: zero_neq_one fps) ≠ 0" by (simp only: fps_const_0_eq_0[symmetric] fps_X_neq_fps_const) simp lemma fps_X_neq_one [simp]: "(fps_X :: 'a :: zero_neq_one fps) ≠ 1" by (simp only: fps_const_1_eq_1[symmetric] fps_X_neq_fps_const) simp lemma fps_X_neq_numeral [simp]: "(fps_X :: 'a :: {semiring_1,zero_neq_one} fps) ≠ numeral c" by (simp only: numeral_fps_const fps_X_neq_fps_const) simp lemma fps_X_pow_eq_fps_X_pow_iff [simp]: "(fps_X :: ('a :: {comm_ring_1}) fps) ^ m = fps_X ^ n ⟷ m = n" proof assume "(fps_X :: 'a fps) ^ m = fps_X ^ n" hence "(fps_X :: 'a fps) ^ m $ m = fps_X ^ n $ m" by (simp only:) thus "m = n" by (simp split: if_split_asm) qed simp_all subsection ‹Subdegrees› definition subdegree :: "('a::zero) fps ⇒ nat" where "subdegree f = (if f = 0 then 0 else LEAST n. f$n ≠ 0)" lemma subdegreeI: assumes "f $ d ≠ 0" and "⋀i. i < d ⟹ f $ i = 0" shows "subdegree f = d" proof- from assms(1) have "f ≠ 0" by auto moreover from assms(1) have "(LEAST i. f $ i ≠ 0) = d" proof (rule Least_equality) fix e assume "f $ e ≠ 0" with assms(2) have "¬(e < d)" by blast thus "e ≥ d" by simp qed ultimately show ?thesis unfolding subdegree_def by simp qed lemma nth_subdegree_nonzero [simp,intro]: "f ≠ 0 ⟹ f $ subdegree f ≠ 0" proof- assume "f ≠ 0" hence "subdegree f = (LEAST n. f $ n ≠ 0)" by (simp add: subdegree_def) also from ‹f ≠ 0› have "∃n. f$n ≠ 0" using fps_nonzero_nth by blast from LeastI_ex[OF this] have "f $ (LEAST n. f $ n ≠ 0) ≠ 0" . finally show ?thesis . qed lemma nth_less_subdegree_zero [dest]: "n < subdegree f ⟹ f $ n = 0" proof (cases "f = 0") assume "f ≠ 0" and less: "n < subdegree f" note less also from ‹f ≠ 0› have "subdegree f = (LEAST n. f $ n ≠ 0)" by (simp add: subdegree_def) finally show "f $ n = 0" using not_less_Least by blast qed simp_all lemma subdegree_geI: assumes "f ≠ 0" "⋀i. i < n ⟹ f$i = 0" shows "subdegree f ≥ n" proof (rule ccontr) assume "¬(subdegree f ≥ n)" with assms(2) have "f $ subdegree f = 0" by simp moreover from assms(1) have "f $ subdegree f ≠ 0" by simp ultimately show False by contradiction qed lemma subdegree_greaterI: assumes "f ≠ 0" "⋀i. i ≤ n ⟹ f$i = 0" shows "subdegree f > n" proof (rule ccontr) assume "¬(subdegree f > n)" with assms(2) have "f $ subdegree f = 0" by simp moreover from assms(1) have "f $ subdegree f ≠ 0" by simp ultimately show False by contradiction qed lemma subdegree_leI: "f $ n ≠ 0 ⟹ subdegree f ≤ n" by (rule leI) auto lemma subdegree_0 [simp]: "subdegree 0 = 0" by (simp add: subdegree_def) lemma subdegree_1 [simp]: "subdegree (1 :: ('a :: zero_neq_one) fps) = 0" by (auto intro!: subdegreeI) lemma subdegree_fps_X [simp]: "subdegree (fps_X :: ('a :: zero_neq_one) fps) = 1" by (auto intro!: subdegreeI simp: fps_X_def) lemma subdegree_fps_const [simp]: "subdegree (fps_const c) = 0" by (cases "c = 0") (auto intro!: subdegreeI) lemma subdegree_numeral [simp]: "subdegree (numeral n) = 0" by (simp add: numeral_fps_const) lemma subdegree_eq_0_iff: "subdegree f = 0 ⟷ f = 0 ∨ f $ 0 ≠ 0" proof (cases "f = 0") assume "f ≠ 0" thus ?thesis using nth_subdegree_nonzero[OF ‹f ≠ 0›] by (fastforce intro!: subdegreeI) qed simp_all lemma subdegree_eq_0 [simp]: "f $ 0 ≠ 0 ⟹ subdegree f = 0" by (simp add: subdegree_eq_0_iff) lemma nth_subdegree_mult [simp]: fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps" shows "(f * g) $ (subdegree f + subdegree g) = f $ subdegree f * g $ subdegree g" proof- let ?n = "subdegree f + subdegree g" have "(f * g) $ ?n = (∑i=0..?n. f$i * g$(?n-i))" by (simp add: fps_mult_nth) also have "... = (∑i=0..?n. if i = subdegree f then f$i * g$(?n-i) else 0)" proof (intro sum.cong) fix x assume x: "x ∈ {0..?n}" hence "x = subdegree f ∨ x < subdegree f ∨ ?n - x < subdegree g" by auto thus "f $ x * g $ (?n - x) = (if x = subdegree f then f $ x * g $ (?n - x) else 0)" by (elim disjE conjE) auto qed auto also have "... = f $ subdegree f * g $ subdegree g" by (simp add: sum.delta) finally show ?thesis . qed lemma subdegree_mult [simp]: assumes "f ≠ 0" "g ≠ 0" shows "subdegree ((f :: ('a :: {ring_no_zero_divisors}) fps) * g) = subdegree f + subdegree g" proof (rule subdegreeI) let ?n = "subdegree f + subdegree g" have "(f * g) $ ?n = (∑i=0..?n. f$i * g$(?n-i))" by (simp add: fps_mult_nth) also have "... = (∑i=0..?n. if i = subdegree f then f$i * g$(?n-i) else 0)" proof (intro sum.cong) fix x assume x: "x ∈ {0..?n}" hence "x = subdegree f ∨ x < subdegree f ∨ ?n - x < subdegree g" by auto thus "f $ x * g $ (?n - x) = (if x = subdegree f then f $ x * g $ (?n - x) else 0)" by (elim disjE conjE) auto qed auto also have "... = f $ subdegree f * g $ subdegree g" by (simp add: sum.delta) also from assms have "... ≠ 0" by auto finally show "(f * g) $ (subdegree f + subdegree g) ≠ 0" . next fix m assume m: "m < subdegree f + subdegree g" have "(f * g) $ m = (∑i=0..m. f$i * g$(m-i))" by (simp add: fps_mult_nth) also have "... = (∑i=0..m. 0)" proof (rule sum.cong) fix i assume "i ∈ {0..m}" with m have "i < subdegree f ∨ m - i < subdegree g" by auto thus "f$i * g$(m-i) = 0" by (elim disjE) auto qed auto finally show "(f * g) $ m = 0" by simp qed lemma subdegree_power [simp]: "subdegree ((f :: ('a :: ring_1_no_zero_divisors) fps) ^ n) = n * subdegree f" by (cases "f = 0"; induction n) simp_all lemma subdegree_uminus [simp]: "subdegree (-(f::('a::group_add) fps)) = subdegree f" by (simp add: subdegree_def) lemma subdegree_minus_commute [simp]: "subdegree (f-(g::('a::group_add) fps)) = subdegree (g - f)" proof - have "f - g = -(g - f)" by simp also have "subdegree ... = subdegree (g - f)" by (simp only: subdegree_uminus) finally show ?thesis . qed lemma subdegree_add_ge: assumes "f ≠ -(g :: ('a :: {group_add}) fps)" shows "subdegree (f + g) ≥ min (subdegree f) (subdegree g)" proof (rule subdegree_geI) from assms show "f + g ≠ 0" by (subst (asm) eq_neg_iff_add_eq_0) next fix i assume "i < min (subdegree f) (subdegree g)" hence "f $ i = 0" and "g $ i = 0" by auto thus "(f + g) $ i = 0" by force qed lemma subdegree_add_eq1: assumes "f ≠ 0" assumes "subdegree f < subdegree (g :: ('a :: {group_add}) fps)" shows "subdegree (f + g) = subdegree f" proof (rule antisym[OF subdegree_leI]) from assms show "subdegree (f + g) ≥ subdegree f" by (intro order.trans[OF min.boundedI subdegree_add_ge]) auto from assms have "f $ subdegree f ≠ 0" "g $ subdegree f = 0" by auto thus "(f + g) $ subdegree f ≠ 0" by simp qed lemma subdegree_add_eq2: assumes "g ≠ 0" assumes "subdegree g < subdegree (f :: ('a :: {ab_group_add}) fps)" shows "subdegree (f + g) = subdegree g" using subdegree_add_eq1[OF assms] by (simp add: add.commute) lemma subdegree_diff_eq1: assumes "f ≠ 0" assumes "subdegree f < subdegree (g :: ('a :: {ab_group_add}) fps)" shows "subdegree (f - g) = subdegree f" using subdegree_add_eq1[of f "-g"] assms by (simp add: add.commute) lemma subdegree_diff_eq2: assumes "g ≠ 0" assumes "subdegree g < subdegree (f :: ('a :: {ab_group_add}) fps)" shows "subdegree (f - g) = subdegree g" using subdegree_add_eq2[of "-g" f] assms by (simp add: add.commute) lemma subdegree_diff_ge [simp]: assumes "f ≠ (g :: ('a :: {group_add}) fps)" shows "subdegree (f - g) ≥ min (subdegree f) (subdegree g)" using assms subdegree_add_ge[of f "-g"] by simp subsection ‹Shifting and slicing› definition fps_shift :: "nat ⇒ 'a fps ⇒ 'a fps" where "fps_shift n f = Abs_fps (λi. f $ (i + n))" lemma fps_shift_nth [simp]: "fps_shift n f $ i = f $ (i + n)" by (simp add: fps_shift_def) lemma fps_shift_0 [simp]: "fps_shift 0 f = f" by (intro fps_ext) (simp add: fps_shift_def) lemma fps_shift_zero [simp]: "fps_shift n 0 = 0" by (intro fps_ext) (simp add: fps_shift_def) lemma fps_shift_one: "fps_shift n 1 = (if n = 0 then 1 else 0)" by (intro fps_ext) (simp add: fps_shift_def) lemma fps_shift_fps_const: "fps_shift n (fps_const c) = (if n = 0 then fps_const c else 0)" by (intro fps_ext) (simp add: fps_shift_def) lemma fps_shift_numeral: "fps_shift n (numeral c) = (if n = 0 then numeral c else 0)" by (simp add: numeral_fps_const fps_shift_fps_const) lemma fps_shift_fps_X_power [simp]: "n ≤ m ⟹ fps_shift n (fps_X ^ m) = (fps_X ^ (m - n) ::'a::comm_ring_1 fps)" by (intro fps_ext) (auto simp: fps_shift_def ) lemma fps_shift_times_fps_X_power: "n ≤ subdegree f ⟹ fps_shift n f * fps_X ^ n = (f :: 'a :: comm_ring_1 fps)" by (intro fps_ext) (auto simp: fps_X_power_mult_right_nth nth_less_subdegree_zero) lemma fps_shift_times_fps_X_power' [simp]: "fps_shift n (f * fps_X^n) = (f :: 'a :: comm_ring_1 fps)" by (intro fps_ext) (auto simp: fps_X_power_mult_right_nth nth_less_subdegree_zero) lemma fps_shift_times_fps_X_power'': "m ≤ n ⟹ fps_shift n (f * fps_X^m) = fps_shift (n - m) (f :: 'a :: comm_ring_1 fps)" by (intro fps_ext) (auto simp: fps_X_power_mult_right_nth nth_less_subdegree_zero) lemma fps_shift_subdegree [simp]: "n ≤ subdegree f ⟹ subdegree (fps_shift n f) = subdegree (f :: 'a :: comm_ring_1 fps) - n" by (cases "f = 0") (force intro: nth_less_subdegree_zero subdegreeI)+ lemma subdegree_decompose: "f = fps_shift (subdegree f) f * fps_X ^ subdegree (f :: ('a :: comm_ring_1) fps)" by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth) lemma subdegree_decompose': "n ≤ subdegree (f :: ('a :: comm_ring_1) fps) ⟹ f = fps_shift n f * fps_X^n" by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth intro!: nth_less_subdegree_zero) lemma fps_shift_fps_shift: "fps_shift (m + n) f = fps_shift m (fps_shift n f)" by (rule fps_ext) (simp add: add_ac) lemma fps_shift_add: "fps_shift n (f + g) = fps_shift n f + fps_shift n g" by (simp add: fps_eq_iff) lemma fps_shift_mult: assumes "n ≤ subdegree (g :: 'b :: {comm_ring_1} fps)" shows "fps_shift n (h*g) = h * fps_shift n g" proof - from assms have "g = fps_shift n g * fps_X^n" by (rule subdegree_decompose') also have "h * ... = (h * fps_shift n g) * fps_X^n" by simp also have "fps_shift n ... = h * fps_shift n g" by simp finally show ?thesis . qed lemma fps_shift_mult_right: assumes "n ≤ subdegree (g :: 'b :: {comm_ring_1} fps)" shows "fps_shift n (g*h) = h * fps_shift n g" by (subst mult.commute, subst fps_shift_mult) (simp_all add: assms) lemma nth_subdegree_zero_iff [simp]: "f $ subdegree f = 0 ⟷ f = 0" by (cases "f = 0") auto lemma fps_shift_subdegree_zero_iff [simp]: "fps_shift (subdegree f) f = 0 ⟷ f = 0" by (subst (1) nth_subdegree_zero_iff[symmetric], cases "f = 0") (simp_all del: nth_subdegree_zero_iff) definition "fps_cutoff n f = Abs_fps (λi. if i < n then f$i else 0)" lemma fps_cutoff_nth [simp]: "fps_cutoff n f $ i = (if i < n then f$i else 0)" unfolding fps_cutoff_def by simp lemma fps_cutoff_zero_iff: "fps_cutoff n f = 0 ⟷ (f = 0 ∨ n ≤ subdegree f)" proof assume A: "fps_cutoff n f = 0" thus "f = 0 ∨ n ≤ subdegree f" proof (cases "f = 0") assume "f ≠ 0" with A have "n ≤ subdegree f" by (intro subdegree_geI) (auto simp: fps_eq_iff split: if_split_asm) thus ?thesis .. qed simp qed (auto simp: fps_eq_iff intro: nth_less_subdegree_zero) lemma fps_cutoff_0 [simp]: "fps_cutoff 0 f = 0" by (simp add: fps_eq_iff) lemma fps_cutoff_zero [simp]: "fps_cutoff n 0 = 0" by (simp add: fps_eq_iff) lemma fps_cutoff_one: "fps_cutoff n 1 = (if n = 0 then 0 else 1)" by (simp add: fps_eq_iff) lemma fps_cutoff_fps_const: "fps_cutoff n (fps_const c) = (if n = 0 then 0 else fps_const c)" by (simp add: fps_eq_iff) lemma fps_cutoff_numeral: "fps_cutoff n (numeral c) = (if n = 0 then 0 else numeral c)" by (simp add: numeral_fps_const fps_cutoff_fps_const) lemma fps_shift_cutoff: "fps_shift n (f :: ('a :: comm_ring_1) fps) * fps_X^n + fps_cutoff n f = f" by (simp add: fps_eq_iff fps_X_power_mult_right_nth) subsection ‹Formal Power series form a metric space› instantiation fps :: (comm_ring_1) dist begin definition dist_fps_def: "dist (a :: 'a fps) b = (if a = b then 0 else inverse (2 ^ subdegree (a - b)))" lemma dist_fps_ge0: "dist (a :: 'a fps) b ≥ 0" by (simp add: dist_fps_def) lemma dist_fps_sym: "dist (a :: 'a fps) b = dist b a" by (simp add: dist_fps_def) instance .. end instantiation fps :: (comm_ring_1) metric_space begin definition uniformity_fps_def [code del]: "(uniformity :: ('a fps × 'a fps) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})" definition open_fps_def' [code del]: "open (U :: 'a fps set) ⟷ (∀x∈U. eventually (λ(x', y). x' = x ⟶ y ∈ U) uniformity)" instance proof show th: "dist a b = 0 ⟷ a = b" for a b :: "'a fps" by (simp add: dist_fps_def split: if_split_asm) then have th'[simp]: "dist a a = 0" for a :: "'a fps" by simp fix a b c :: "'a fps" consider "a = b" | "c = a ∨ c = b" | "a ≠ b" "a ≠ c" "b ≠ c" by blast then show "dist a b ≤ dist a c + dist b c" proof cases case 1 then show ?thesis by (simp add: dist_fps_def) next case 2 then show ?thesis by (cases "c = a") (simp_all add: th dist_fps_sym) next case neq: 3 have False if "dist a b > dist a c + dist b c" proof - let ?n = "subdegree (a - b)" from neq have "dist a b > 0" "dist b c > 0" and "dist a c > 0" by (simp_all add: dist_fps_def) with that have "dist a b > dist a c" and "dist a b > dist b c" by simp_all with neq have "?n < subdegree (a - c)" and "?n < subdegree (b - c)" by (simp_all add: dist_fps_def field_simps) hence "(a - c) $ ?n = 0" and "(b - c) $ ?n = 0" by (simp_all only: nth_less_subdegree_zero) hence "(a - b) $ ?n = 0" by simp moreover from neq have "(a - b) $ ?n ≠ 0" by (intro nth_subdegree_nonzero) simp_all ultimately show False by contradiction qed thus ?thesis by (auto simp add: not_le[symmetric]) qed qed (rule open_fps_def' uniformity_fps_def)+ end declare uniformity_Abort[where 'a="'a :: comm_ring_1 fps", code] lemma open_fps_def: "open (S :: 'a::comm_ring_1 fps set) = (∀a ∈ S. ∃r. r >0 ∧ {y. dist y a < r} ⊆ S)" unfolding open_dist subset_eq by simp text ‹The infinite sums and justification of the notation in textbooks.› lemma reals_power_lt_ex: fixes x y :: real assumes xp: "x > 0" and y1: "y > 1" shows "∃k>0. (1/y)^k < x" proof - have yp: "y > 0" using y1 by simp from reals_Archimedean2[of "max 0 (- log y x) + 1"] obtain k :: nat where k: "real k > max 0 (- log y x) + 1" by blast from k have kp: "k > 0" by simp from k have "real k > - log y x" by simp then have "ln y * real k > - ln x" unfolding log_def using ln_gt_zero_iff[OF yp] y1 by (simp add: minus_divide_left field_simps del: minus_divide_left[symmetric]) then have "ln y * real k + ln x > 0" by simp then have "exp (real k * ln y + ln x) > exp 0" by (simp add: ac_simps) then have "y ^ k * x > 1" unfolding exp_zero exp_add exp_of_nat_mult exp_ln [OF xp] exp_ln [OF yp] by simp then have "x > (1 / y)^k" using yp by (simp add: field_simps) then show ?thesis using kp by blast qed lemma fps_sum_rep_nth: "(sum (λi. fps_const(a$i)*fps_X^i) {0..m})$n = (if n ≤ m then a$n else 0::'a::comm_ring_1)" by (auto simp add: fps_sum_nth cond_value_iff cong del: if_weak_cong) lemma fps_notation: "(λn. sum (λi. fps_const(a$i) * fps_X^i) {0..n}) ⇢ a" (is "?s ⇢ a") proof - have "∃n0. ∀n ≥ n0. dist (?s n) a < r" if "r > 0" for r proof - obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0" using reals_power_lt_ex[OF ‹r > 0›, of 2] by auto show ?thesis proof - have "dist (?s n) a < r" if nn0: "n ≥ n0" for n proof - from that have thnn0: "(1/2)^n ≤ (1/2 :: real)^n0" by (simp add: divide_simps) show ?thesis proof (cases "?s n = a") case True then show ?thesis unfolding dist_eq_0_iff[of "?s n" a, symmetric] using ‹r > 0› by (simp del: dist_eq_0_iff) next case False from False have dth: "dist (?s n) a = (1/2)^subdegree (?s n - a)" by (simp add: dist_fps_def field_simps) from False have kn: "subdegree (?s n - a) > n" by (intro subdegree_greaterI) (simp_all add: fps_sum_rep_nth) then have "dist (?s n) a < (1/2)^n" by (simp add: field_simps dist_fps_def) also have "… ≤ (1/2)^n0" using nn0 by (simp add: divide_simps) also have "… < r" using n0 by simp finally show ?thesis . qed qed then show ?thesis by blast qed qed then show ?thesis unfolding lim_sequentially by blast qed subsection ‹Inverses of formal power series› declare sum.cong[fundef_cong] 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$0) * sum (λi. f$i * natfun_inverse f (n - i)) {1..n}" definition fps_inverse_def: "inverse f = (if f $ 0 = 0 then 0 else Abs_fps (natfun_inverse f))" definition fps_divide_def: "f div g = (if g = 0 then 0 else let n = subdegree g; h = fps_shift n g in fps_shift n (f * inverse h))" instance .. end lemma fps_inverse_zero [simp]: "inverse (0 :: 'a::{comm_monoid_add,inverse,times,uminus} fps) = 0" 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) have "(inverse f * f)$n = 0" if np: "n > 0" for n proof - from np have eq: "{0..n} = {0} ∪ {1 .. n}" by auto have d: "{0} ∩ {1 .. n} = {}" by auto from f0 np have th0: "- (inverse f $ n) = (sum (λi. f$i * natfun_inverse f (n - i)) {1..n}) / (f$0)" by (cases n) (simp_all add: divide_inverse fps_inverse_def) from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]] have th1: "sum (λi. f$i * natfun_inverse f (n - i)) {1..n} = - (f$0) * (inverse f)$n" 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 show ?thesis 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) ⟷ f $ 0 = 0" by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero) lemma fps_inverse_nth_0 [simp]: "inverse f $ 0 = inverse (f $ 0 :: 'a :: division_ring)" by (simp add: fps_inverse_def) lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::division_ring) fps) ⟷ f $ 0 = 0" proof assume A: "inverse f = 0" have "0 = inverse f $ 0" by (subst A) simp thus "f $ 0 = 0" by simp qed (simp add: fps_inverse_def) lemma fps_inverse_idempotent[intro, simp]: assumes f0: "f$0 ≠ (0::'a::field)" shows "inverse (inverse f) = f" proof - from f0 have if0: "inverse f $ 0 ≠ 0" by simp from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0] have "inverse f * f = inverse f * inverse (inverse f)" by (simp add: ac_simps) then show ?thesis using f0 unfolding mult_cancel_left by simp qed lemma fps_inverse_unique: assumes fg: "(f :: 'a :: field fps) * g = 1" shows "inverse f = g" proof - have f0: "f $ 0 ≠ 0" proof assume "f $ 0 = 0" hence "0 = (f * g) $ 0" by simp also from fg have "(f * g) $ 0 = 1" by simp finally show False by simp qed from inverse_mult_eq_1[OF this] fg have th0: "inverse f * f = g * f" by (simp add: ac_simps) then show ?thesis using f0 unfolding mult_cancel_right by (auto simp add: expand_fps_eq) qed lemma fps_inverse_eq_0: "f$0 = 0 ⟹ inverse (f :: 'a :: division_ring fps) = 0" by simp lemma sum_zero_lemma: fixes n::nat assumes "0 < n" shows "(∑i = 0..n. if n = i then 1 else if n - i = 1 then - 1 else 0) = (0::'a::field)" proof - let ?f = "λi. if n = i then 1 else if n - i = 1 then - 1 else 0" let ?g = "λi. if i = n then 1 else if i = n - 1 then - 1 else 0" let ?h = "λi. if i=n - 1 then - 1 else 0" have th1: "sum ?f {0..n} = sum ?g {0..n}" by (rule sum.cong) auto have th2: "sum ?g {0..n - 1} = sum ?h {0..n - 1}" apply (rule sum.cong) using assms apply auto done have eq: "{0 .. n} = {0.. n - 1} ∪ {n}" by auto from assms have d: "{0.. n - 1} ∩ {n} = {}" by auto have f: "finite {0.. n - 1}" "finite {n}" by auto show ?thesis unfolding th1 apply (simp add: sum.union_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def) unfolding th2 apply (simp add: sum.delta) done qed lemma fps_inverse_mult: "inverse (f * g :: 'a::field fps) = inverse f * inverse g" proof (cases "f$0 = 0 ∨ g$0 = 0") assume "¬(f$0 = 0 ∨ g$0 = 0)" hence [simp]: "f$0 ≠ 0" "g$0 ≠ 0" by simp_all show ?thesis proof (rule fps_inverse_unique) have "f * g * (inverse f * inverse g) = (inverse f * f) * (inverse g * g)" by simp also have "... = 1" by (subst (1 2) inverse_mult_eq_1) simp_all finally show "f * g * (inverse f * inverse g) = 1" . qed next assume A: "f$0 = 0 ∨ g$0 = 0" hence "inverse (f * g) = 0" by simp also from A have "... = inverse f * inverse g" by auto finally show "inverse (f * g) = inverse f * inverse g" . qed lemma fps_inverse_gp: "inverse (Abs_fps(λn. (1::'a::field))) = Abs_fps (λn. if n= 0 then 1 else if n=1 then - 1 else 0)" apply (rule fps_inverse_unique) apply (simp_all add: fps_eq_iff fps_mult_nth sum_zero_lemma) done lemma subdegree_inverse [simp]: "subdegree (inverse (f::'a::field fps)) = 0" proof (cases "f$0 = 0") assume nz: "f$0 ≠ 0" hence "subdegree (inverse f) + subdegree f = subdegree (inverse f * f)" by (subst subdegree_mult) auto also from nz have "subdegree f = 0" by (simp add: subdegree_eq_0_iff) also from nz have "inverse f * f = 1" by (rule inverse_mult_eq_1) finally show "subdegree (inverse f) = 0" by simp qed (simp_all add: fps_inverse_def) lemma fps_is_unit_iff [simp]: "(f :: 'a :: field fps) dvd 1 ⟷ f $ 0 ≠ 0" proof assume "f dvd 1" then obtain g where "1 = f * g" by (elim dvdE) from this[symmetric] have "(f*g) $ 0 = 1" by simp thus "f $ 0 ≠ 0" by auto next assume A: "f $ 0 ≠ 0" thus "f dvd 1" by (simp add: inverse_mult_eq_1[OF A, symmetric]) qed lemma subdegree_eq_0' [simp]: "(f :: 'a :: field fps) dvd 1 ⟹ subdegree f = 0" by simp lemma fps_unit_dvd [simp]: "(f $ 0 :: 'a :: field) ≠ 0 ⟹ f dvd g" by (rule dvd_trans, subst fps_is_unit_iff) simp_all instantiation fps :: (field) normalization_semidom begin definition fps_unit_factor_def [simp]: "unit_factor f = fps_shift (subdegree f) f" definition fps_normalize_def [simp]: "normalize f = (if f = 0 then 0 else fps_X ^ subdegree f)" instance proof fix f :: "'a fps" show "unit_factor f * normalize f = f" by (simp add: fps_shift_times_fps_X_power) next fix f g :: "'a fps" show "unit_factor (f * g) = unit_factor f * unit_factor g" proof (cases "f = 0 ∨ g = 0") assume "¬(f = 0 ∨ g = 0)" thus "unit_factor (f * g) = unit_factor f * unit_factor g" unfolding fps_unit_factor_def by (auto simp: fps_shift_fps_shift fps_shift_mult fps_shift_mult_right) qed auto next fix f g :: "'a fps" assume "g ≠ 0" then have "f * (fps_shift (subdegree g) g * inverse (fps_shift (subdegree g) g)) = f" by (metis add_cancel_right_left fps_shift_nth inverse_mult_eq_1 mult.commute mult_cancel_left2 nth_subdegree_nonzero) then have "fps_shift (subdegree g) (g * (f * inverse (fps_shift (subdegree g) g))) = f" by (simp add: fps_shift_mult_right mult.commute) with ‹g ≠ 0› show "f * g / g = f" by (simp add: fps_divide_def Let_def ac_simps) qed (auto simp add: fps_divide_def Let_def) end instantiation fps :: (field) idom_modulo begin definition fps_mod_def: "f mod g = (if g = 0 then f else let n = subdegree g; h = fps_shift n g in fps_cutoff n (f * inverse h) * h)" lemma fps_mod_eq_zero: assumes "g ≠ 0" and "subdegree f ≥ subdegree g" shows "f mod g = 0" using assms by (cases "f = 0") (auto simp: fps_cutoff_zero_iff fps_mod_def Let_def) lemma fps_times_divide_eq: assumes "g ≠ 0" and "subdegree f ≥ subdegree (g :: 'a fps)" shows "f div g * g = f" proof (cases "f = 0") assume nz: "f ≠ 0" define n where "n = subdegree g" define h where "h = fps_shift n g" from assms have [simp]: "h $ 0 ≠ 0" unfolding h_def by (simp add: n_def) from assms nz have "f div g * g = fps_shift n (f * inverse h) * g" by (simp add: fps_divide_def Let_def h_def n_def) also have "... = fps_shift n (f * inverse h) * fps_X^n * h" unfolding h_def n_def by (subst subdegree_decompose[of g]) simp also have "fps_shift n (f * inverse h) * fps_X^n = f * inverse h" by (rule fps_shift_times_fps_X_power) (simp_all add: nz assms n_def) also have "... * h = f * (inverse h * h)" by simp also have "inverse h * h = 1" by (rule inverse_mult_eq_1) simp finally show ?thesis by simp qed (simp_all add: fps_divide_def Let_def) lemma assumes "g$0 ≠ 0" shows fps_divide_unit: "f div g = f * inverse g" and fps_mod_unit [simp]: "f mod g = 0" proof - from assms have [simp]: "subdegree g = 0" by (simp add: subdegree_eq_0_iff) from assms show "f div g = f * inverse g" by (auto simp: fps_divide_def Let_def subdegree_eq_0_iff) from assms show "f mod g = 0" by (intro fps_mod_eq_zero) auto qed instance proof fix f g :: "'a fps" define n where "n = subdegree g" define h where "h = fps_shift n g" show "f div g * g + f mod g = f" proof (cases "g = 0 ∨ f = 0") assume "¬(g = 0 ∨ f = 0)" hence nz [simp]: "f ≠ 0" "g ≠ 0" by simp_all show ?thesis proof (rule disjE[OF le_less_linear]) assume "subdegree f ≥ subdegree g" with nz show ?thesis by (simp add: fps_mod_eq_zero fps_times_divide_eq) next assume "subdegree f < subdegree g" have g_decomp: "g = h * fps_X^n" unfolding h_def n_def by (rule subdegree_decompose) have "f div g * g + f mod g = fps_shift n (f * inverse h) * g + fps_cutoff n (f * inverse h) * h" by (simp add: fps_mod_def fps_divide_def Let_def n_def h_def) also have "... = h * (fps_shift n (f * inverse h) * fps_X^n + fps_cutoff n (f * inverse h))" by (subst g_decomp) (simp add: algebra_simps) also have "... = f * (inverse h * h)" by (subst fps_shift_cutoff) simp also have "inverse h * h = 1" by (rule inverse_mult_eq_1) (simp add: h_def n_def) finally show ?thesis by simp qed qed (auto simp: fps_mod_def fps_divide_def Let_def) qed end lemma subdegree_mod: assumes "f ≠ 0" "subdegree f < subdegree g" shows "subdegree (f mod g) = subdegree f" proof (cases "f div g * g = 0") assume "f div g * g ≠ 0" hence [simp]: "f div g ≠ 0" "g ≠ 0" by auto from div_mult_mod_eq[of f g] have "f mod g = f - f div g * g" by (simp add: algebra_simps) also from assms have "subdegree ... = subdegree f" by (intro subdegree_diff_eq1) simp_all finally show ?thesis . next assume zero: "f div g * g = 0" from div_mult_mod_eq[of f g] have "f mod g = f - f div g * g" by (simp add: algebra_simps) also note zero finally show ?thesis by simp qed lemma fps_divide_nth_0 [simp]: "g $ 0 ≠ 0 ⟹ (f div g) $ 0 = f $ 0 / (g $ 0 :: _ :: field)" by (simp add: fps_divide_unit divide_inverse) lemma dvd_imp_subdegree_le: "(f :: 'a :: idom fps) dvd g ⟹ g ≠ 0 ⟹ subdegree f ≤ subdegree g" by (auto elim: dvdE) lemma fps_dvd_iff: assumes "(f :: 'a :: field fps) ≠ 0" "g ≠ 0" shows "f dvd g ⟷ subdegree f ≤ subdegree g" proof assume "subdegree f ≤ subdegree g" with assms have "g mod f = 0" by (simp add: fps_mod_def Let_def fps_cutoff_zero_iff) thus "f dvd g" by (simp add: dvd_eq_mod_eq_0) qed (simp add: assms dvd_imp_subdegree_le) lemma fps_shift_altdef: "fps_shift n f = (f :: 'a :: field fps) div fps_X^n" by (simp add: fps_divide_def) lemma fps_div_fps_X_power_nth: "((f :: 'a :: field fps) div fps_X^n) $ k = f $ (k + n)" by (simp add: fps_shift_altdef [symmetric]) lemma fps_div_fps_X_nth: "((f :: 'a :: field fps) div fps_X) $ k = f $ Suc k" using fps_div_fps_X_power_nth[of f 1] by simp lemma fps_const_inverse: "inverse (fps_const (a::'a::field)) = fps_const (inverse a)" by (cases "a ≠ 0", rule fps_inverse_unique) (auto simp: fps_eq_iff) lemma fps_const_divide: "fps_const (x :: _ :: field) / fps_const y = fps_const (x / y)" by (cases "y = 0") (simp_all add: fps_divide_unit fps_const_inverse divide_inverse) lemma inverse_fps_numeral: "inverse (numeral n :: ('a :: field_char_0) fps) = fps_const (inverse (numeral n))" by (intro fps_inverse_unique fps_ext) (simp_all add: fps_numeral_nth) lemma fps_numeral_divide_divide: "x / numeral b / numeral c = (x / numeral (b * c) :: 'a :: field fps)" by (cases "numeral b = (0::'a)"; cases "numeral c = (0::'a)") (simp_all add: fps_divide_unit fps_inverse_mult [symmetric] numeral_fps_const numeral_mult del: numeral_mult [symmetric]) lemma fps_numeral_mult_divide: "numeral b * x / numeral c = (numeral b / numeral c * x :: 'a :: field fps)" by (cases "numeral c = (0::'a)") (simp_all add: fps_divide_unit numeral_fps_const) lemmas fps_numeral_simps = fps_numeral_divide_divide fps_numeral_mult_divide inverse_fps_numeral neg_numeral_fps_const lemma subdegree_div: assumes "q dvd p" shows "subdegree ((p :: 'a :: field fps) div q) = subdegree p - subdegree q" proof (cases "p = 0") case False from assms have "p = p div q * q" by simp also from assms False have "subdegree … = subdegree (p div q) + subdegree q" by (intro subdegree_mult) (auto simp: dvd_div_eq_0_iff) finally show ?thesis by simp qed simp_all lemma subdegree_div_unit: assumes "q $ 0 ≠ 0" shows "subdegree ((p :: 'a :: field fps) div q) = subdegree p" using assms by (subst subdegree_div) simp_all subsection ‹Formal power series form a Euclidean ring› instantiation fps :: (field) euclidean_ring_cancel begin definition fps_euclidean_size_def: "euclidean_size f = (if f = 0 then 0 else 2 ^ subdegree f)" context begin private lemma fps_divide_cancel_aux1: assumes "h$0 ≠ (0 :: 'a :: field)" shows "(h * f) div (h * g) = f div g" proof (cases "g = 0") assume "g ≠ 0" from assms have "h ≠ 0" by auto note nz [simp] = ‹g ≠ 0› ‹h ≠ 0› from assms have [simp]: "subdegree h = 0" by (simp add: subdegree_eq_0_iff) have "(h * f) div (h * g) = fps_shift (subdegree g) (h * f * inverse (fps_shift (subdegree g) (h*g)))" by (simp add: fps_divide_def Let_def) also have "h * f * inverse (fps_shift (subdegree g) (h*g)) = (inverse h * h) * f * inverse (fps_shift (subdegree g) g)" by (subst fps_shift_mult) (simp_all add: algebra_simps fps_inverse_mult) also from assms have "inverse h * h = 1" by (rule inverse_mult_eq_1) finally show "(h * f) div (h * g) = f div g" by (simp_all add: fps_divide_def Let_def) qed (simp_all add: fps_divide_def) private lemma fps_divide_cancel_aux2: "(f * fps_X^m) div (g * fps_X^m) = f div (g :: 'a :: field fps)" proof (cases "g = 0") assume [simp]: "g ≠ 0" have "(f * fps_X^m) div (g * fps_X^m) = fps_shift (subdegree g + m) (f*inverse (fps_shift (subdegree g + m) (g*fps_X^m))*fps_X^m)" by (simp add: fps_divide_def Let_def algebra_simps) also have "... = f div g" by (simp add: fps_shift_times_fps_X_power'' fps_divide_def Let_def) finally show ?thesis . qed (simp_all add: fps_divide_def) instance proof fix f g :: "'a fps" assume [simp]: "g ≠ 0" show "euclidean_size f ≤ euclidean_size (f * g)" by (cases "f = 0") (auto simp: fps_euclidean_size_def) show "euclidean_size (f mod g) < euclidean_size g" apply (cases "f = 0", simp add: fps_euclidean_size_def) apply (rule disjE[OF le_less_linear[of "subdegree g" "subdegree f"]]) apply (simp_all add: fps_mod_eq_zero fps_euclidean_size_def subdegree_mod) done show "(h * f) div (h * g) = f div g" if "h ≠ 0" for f g h :: "'a fps" proof - define m where "m = subdegree h" define h' where "h' = fps_shift m h" have h_decomp: "h = h' * fps_X ^ m" unfolding h'_def m_def by (rule subdegree_decompose) from ‹h ≠ 0› have [simp]: "h'$0 ≠ 0" by (simp add: h'_def m_def) have "(h * f) div (h * g) = (h' * f * fps_X^m) div (h' * g * fps_X^m)" by (simp add: h_decomp algebra_simps) also have "... = f div g" by (simp add: fps_divide_cancel_aux1 fps_divide_cancel_aux2) finally show ?thesis . qed show "(f + g * h) div h = g + f div h" if "h ≠ 0" for f g h :: "'a fps" proof - define n h' where dfs: "n = subdegree h" "h' = fps_shift n h" have "(f + g * h) div h = fps_shift n (f * inverse h') + fps_shift n (g * (h * inverse h'))" by (simp add: fps_divide_def Let_def dfs [symmetric] algebra_simps fps_shift_add that) also have "h * inverse h' = (inverse h' * h') * fps_X^n" by (subst subdegree_decompose) (simp_all add: dfs) also have "... = fps_X^n" by (subst inverse_mult_eq_1) (simp_all add: dfs that) also have "fps_shift n (g * fps_X^n) = g" by simp also have "fps_shift n (f * inverse h') = f div h" by (simp add: fps_divide_def Let_def dfs) finally show ?thesis by simp qed qed (simp_all add: fps_euclidean_size_def) end end instance fps :: (field) normalization_euclidean_semiring .. instantiation fps :: (field) euclidean_ring_gcd begin definition fps_gcd_def: "(gcd :: 'a fps ⇒ _) = Euclidean_Algorithm.gcd" definition fps_lcm_def: "(lcm :: 'a fps ⇒ _) = Euclidean_Algorithm.lcm" definition fps_Gcd_def: "(Gcd :: 'a fps set ⇒ _) = Euclidean_Algorithm.Gcd" definition fps_Lcm_def: "(Lcm :: 'a fps set ⇒ _) = Euclidean_Algorithm.Lcm" instance by standard (simp_all add: fps_gcd_def fps_lcm_def fps_Gcd_def fps_Lcm_def) end lemma fps_gcd: assumes [simp]: "f ≠ 0" "g ≠ 0" shows "gcd f g = fps_X ^ min (subdegree f) (subdegree g)" proof - let ?m = "min (subdegree f) (subdegree g)" show "gcd f g = fps_X ^ ?m" proof (rule sym, rule gcdI) fix d assume "d dvd f" "d dvd g" thus "d dvd fps_X ^ ?m" by (cases "d = 0") (auto simp: fps_dvd_iff) qed (simp_all add: fps_dvd_iff) qed lemma fps_gcd_altdef: "gcd (f :: 'a :: field fps) g = (if f = 0 ∧ g = 0 then 0 else if f = 0 then fps_X ^ subdegree g else if g = 0 then fps_X ^ subdegree f else fps_X ^ min (subdegree f) (subdegree g))" by (simp add: fps_gcd) lemma fps_lcm: assumes [simp]: "f ≠ 0" "g ≠ 0" shows "lcm f g = fps_X ^ max (subdegree f) (subdegree g)" proof - let ?m = "max (subdegree f) (subdegree g)" show "lcm f g = fps_X ^ ?m" proof (rule sym, rule lcmI) fix d assume "f dvd d" "g dvd d" thus "fps_X ^ ?m dvd d" by (cases "d = 0") (auto simp: fps_dvd_iff) qed (simp_all add: fps_dvd_iff) qed lemma fps_lcm_altdef: "lcm (f :: 'a :: field fps) g = (if f = 0 ∨ g = 0 then 0 else fps_X ^ max (subdegree f) (subdegree g))" by (simp add: fps_lcm) lemma fps_Gcd: assumes "A - {0} ≠ {}" shows "Gcd A = fps_X ^ (INF f:A-{0}. subdegree f)" proof (rule sym, rule GcdI) fix f assume "f ∈ A" thus "fps_X ^ (INF f:A - {0}. subdegree f) dvd f" by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cINF_lower) next fix d assume d: "⋀f. f ∈ A ⟹ d dvd f" from assms obtain f where "f ∈ A - {0}" by auto with d[of f] have [simp]: "d ≠ 0" by auto from d assms have "subdegree d ≤ (INF f:A-{0}. subdegree f)" by (intro cINF_greatest) (auto simp: fps_dvd_iff[symmetric]) with d assms show "d dvd fps_X ^ (INF f:A-{0}. subdegree f)" by (simp add: fps_dvd_iff) qed simp_all lemma fps_Gcd_altdef: "Gcd (A :: 'a :: field fps set) = (if A ⊆ {0} then 0 else fps_X ^ (INF f:A-{0}. subdegree f))" using fps_Gcd by auto lemma fps_Lcm: assumes "A ≠ {}" "0 ∉ A" "bdd_above (subdegree`A)" shows "Lcm A = fps_X ^ (SUP f:A. subdegree f)" proof (rule sym, rule LcmI) fix f assume "f ∈ A" moreover from assms(3) have "bdd_above (subdegree ` A)" by auto ultimately show "f dvd fps_X ^ (SUP f:A. subdegree f)" using assms(2) by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cSUP_upper) next fix d assume d: "⋀f. f ∈ A ⟹ f dvd d" from assms obtain f where f: "f ∈ A" "f ≠ 0" by auto show "fps_X ^ (SUP f:A. subdegree f) dvd d" proof (cases "d = 0") assume "d ≠ 0" moreover from d have "⋀f. f ∈ A ⟹ f ≠ 0 ⟹ f dvd d" by blast ultimately have "subdegree d ≥ (SUP f:A. subdegree f)" using assms by (intro cSUP_least) (auto simp: fps_dvd_iff) with ‹d ≠ 0› show ?thesis by (simp add: fps_dvd_iff) qed simp_all qed simp_all lemma fps_Lcm_altdef: "Lcm (A :: 'a :: field fps set) = (if 0 ∈ A ∨ ¬bdd_above (subdegree`A) then 0 else if A = {} then 1 else fps_X ^ (SUP f:A. subdegree f))" proof (cases "bdd_above (subdegree`A)") assume unbounded: "¬bdd_above (subdegree`A)" have "Lcm A = 0" proof (rule ccontr) assume "Lcm A ≠ 0" from unbounded obtain f where f: "f ∈ A" "subdegree (Lcm A) < subdegree f" unfolding bdd_above_def by (auto simp: not_le) moreover from f and ‹Lcm A ≠ 0› have "subdegree f ≤ subdegree (Lcm A)" by (intro dvd_imp_subdegree_le dvd_Lcm) simp_all ultimately show False by simp qed with unbounded show ?thesis by simp qed (simp_all add: fps_Lcm Lcm_eq_0_I) 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_0th_higher_deriv: "(fps_deriv ^^ n) f $ 0 = (fact n * f $ n :: 'a :: {comm_ring_1, semiring_char_0})" by (induction n arbitrary: f) (simp_all del: funpow.simps add: funpow_Suc_right algebra_simps) lemma fps_deriv_linear[simp]: "fps_deriv (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = fps_const a * fps_deriv f + fps_const b * fps_deriv g" 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" have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" for n proof - let ?Zn = "{0 ..n}" let ?Zn1 = "{0 .. n + 1}" let ?g = "λi. of_nat (i+1) * g $ (i+1) * f $ (n - i) + of_nat (i+1)* f $ (i+1) * g $ (n - i)" let ?h = "λi. of_nat i * g $ i * f $ ((n+1) - i) + of_nat i* f $ i * g $ ((n + 1) - i)" have s0: "sum (λi. of_nat i * f $ i * g $ (n + 1 - i)) ?Zn1 = sum (λi. of_nat (n + 1 - i) * f $ (n + 1 - i) * g $ i) ?Zn1" by (rule sum.reindex_bij_witness[where i="(-) (n + 1)" and j="(-) (n + 1)"]) auto have s1: "sum (λi. f $ i * g $ (n + 1 - i)) ?Zn1 = sum (λi. f $ (n + 1 - i) * g $ i) ?Zn1" by (rule sum.reindex_bij_witness[where i="(-) (n + 1)" and j="(-) (n + 1)"]) auto have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n" by (simp only: mult.commute) also have "… = (∑i = 0..n. ?g i)" by (simp add: fps_mult_nth sum.distrib[symmetric]) also have "… = sum ?h {0..n+1}" by (rule sum.reindex_bij_witness_not_neutral [where S'="{}" and T'="{0}" and j="Suc" and i="λi. i - 1"]) auto also have "… = (fps_deriv (f * g)) $ n" apply (simp only: fps_deriv_nth fps_mult_nth sum.distrib) unfolding s0 s1 unfolding sum.distrib[symmetric] sum_distrib_left apply (rule sum.cong) apply (auto simp add: of_nat_diff field_simps) done finally show ?thesis . qed then show ?thesis unfolding fps_eq_iff by auto qed lemma fps_deriv_fps_X[simp]: "fps_deriv fps_X = 1" by (simp add: fps_deriv_def fps_X_def fps_eq_iff) lemma fps_deriv_neg[simp]: "fps_deriv (- (f:: 'a::comm_ring_1 fps)) = - (fps_deriv f)" 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" using fps_deriv_add [of f "- g"] by simp lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0" by (simp add: fps_ext fps_deriv_def fps_const_def) lemma fps_deriv_of_nat [simp]: "fps_deriv (of_nat n) = 0" by (simp add: fps_of_nat [symmetric]) lemma fps_deriv_numeral [simp]: "fps_deriv (numeral n) = 0" by (simp add: numeral_fps_const) 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_sum: "fps_deriv (sum f S) = sum (λi. fps_deriv (f i :: 'a::comm_ring_1 fps)) S" proof (cases "finite S") case False then show ?thesis by simp next case True show ?thesis by (induct rule: finite_induct [OF True]) simp_all qed lemma fps_deriv_eq_0_iff [simp]: "fps_deriv f = 0 ⟷ f = fps_const (f$0 :: 'a::{idom,semiring_char_0})" (is "?lhs ⟷ ?rhs") proof show ?lhs if ?rhs proof - from that have "fps_deriv f = fps_deriv (fps_const (f$0))" by simp then show ?thesis by simp qed show ?rhs if ?lhs proof - from that have "∀n. (fps_deriv f)$n = 0" by simp then have "∀n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def) then show ?thesis apply (clarsimp simp add: fps_eq_iff fps_const_def) apply (erule_tac x="n - 1" in allE) apply simp done qed qed lemma fps_deriv_eq_iff: fixes f :: "'a::{idom,semiring_char_0} fps" shows "fps_deriv f = fps_deriv g ⟷ (f = fps_const(f$0 - g$0) + g)" proof - have "fps_deriv f = fps_deriv g ⟷ fps_deriv (f - g) = 0" by simp also have "… ⟷ f - g = fps_const ((f - g) $ 0)" unfolding fps_deriv_eq_0_iff .. finally show ?thesis 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" using fps_nth_deriv_add [of n f "- g"] by simp lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0" by (induct n) simp_all lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)" by (induct n) simp_all lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)" by (cases n) simp_all lemma fps_nth_deriv_mult_const_left[simp]: "fps_nth_deriv n (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_nth_deriv n f" using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp lemma fps_nth_deriv_mult_const_right[simp]: "fps_nth_deriv n (f * fps_const (c::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c" using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult.commute) lemma fps_nth_deriv_sum: "fps_nth_deriv n (sum f S) = sum (λi. fps_nth_deriv n (f i :: 'a::comm_ring_1 fps)) S" proof (cases "finite S") case True show ?thesis by (induct rule: finite_induct [OF True]) simp_all next case False then show ?thesis by simp qed lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f :: 'a::comm_semiring_1 fps)) $ 0 = of_nat (fact k) * f $ k" by (induct k arbitrary: f) (auto simp add: field_simps) subsection ‹Powers› lemma fps_power_zeroth_eq_one: "a$0 =1 ⟹ a^n $ 0 = (1::'a::semiring_1)" by (induct n) (auto simp add: expand_fps_eq fps_mult_nth) lemma fps_power_first_eq: "(a :: 'a::comm_ring_1 fps) $ 0 =1 ⟹ a^n $ 1 = of_nat n * a$1" proof (induct n) case 0 then show ?case by simp next case (Suc n) show ?case unfolding power_Suc fps_mult_nth using Suc.hyps[OF ‹a$0 = 1›] ‹a$0 = 1› fps_power_zeroth_eq_one[OF ‹a$0=1›] 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" show "∀m<k. a ^ k $ m = 0" proof (cases k) case 0 then show ?thesis by simp next case (Suc l) have "a^k $ m = 0" if mk: "m < k" for m proof (cases "m = 0") case True then show ?thesis using startsby_zero_power[of a k] Suc a0 by simp next case False have "a ^k $ m = (a^l * a) $m" by (simp add: Suc mult.commute) also have "… = (∑i = 0..m. a ^ l $ i * a $ (m - i))" by (simp add: fps_mult_nth) also have "… = 0" apply (rule sum.neutral) apply auto apply (case_tac "x = m") using a0 apply simp apply (rule H[rule_format]) using a0 Suc mk apply auto done finally show ?thesis . qed then show ?thesis by blast qed qed lemma startsby_zero_sum_depends: assumes a0: "a $0 = (0::'a::idom)" and kn: "n ≥ k" shows "sum (λi. (a ^ i)$k) {0 .. n} = sum (λi. (a ^ i)$k) {0 .. k}" apply (rule sum.mono_neutral_right) using kn apply auto apply (rule startsby_zero_power_prefix[rule_format, OF a0]) apply arith done lemma startsby_zero_power_nth_same: assumes a0: "a$0 = (0::'a::idom)" shows "a^n $ n = (a$1) ^ n" proof (induct n) case 0 then show ?case by simp next case (Suc n) have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)" by (simp add: field_simps) also have "… = sum (λi. a^n$i * a $ (Suc n - i)) {0.. Suc n}" by (simp add: fps_mult_nth) also have "… = sum (λi. a^n$i * a $ (Suc n - i)) {n .. Suc n}" apply (rule sum.mono_neutral_right) apply simp apply clarsimp apply clarsimp apply (rule startsby_zero_power_prefix[rule_format, OF a0]) apply arith done also have "… = a^n $ n * a$1" using a0 by simp finally show ?case using Suc.hyps by simp qed lemma fps_inverse_power: fixes a :: "'a::field fps" shows "inverse (a^n) = inverse a ^ n" by (induction n) (simp_all add: fps_inverse_mult) lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a::comm_ring_1) * fps_deriv a * a ^ (n - 1)" apply (induct n) apply (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 then have "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp then have "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0" by simp with inverse_mult_eq_1[OF a0] have "(inverse a)⇧^{2}* fps_deriv a + fps_deriv (inverse a) = 0" unfolding power2_eq_square 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_deriv': fixes a :: "'a::field fps" assumes a0: "a $ 0 ≠ 0" shows "fps_deriv (inverse a) = - fps_deriv a / a⇧^{2}" using fps_inverse_deriv[OF a0] a0 by (simp add: fps_divide_unit power2_eq_square fps_inverse_mult) lemma inverse_mult_eq_1': assumes f0: "f$0 ≠ (0::'a::field)" shows "f * inverse f = 1" by (metis mult.commute inverse_mult_eq_1 f0) lemma fps_inverse_minus [simp]: "inverse (-f) = -inverse (f :: 'a :: field fps)" by (cases "f$0 = 0") (auto intro: fps_inverse_unique simp: inverse_mult_eq_1' fps_inverse_eq_0) lemma divide_fps_const [simp]: "f / fps_const (c :: 'a :: field) = fps_const (inverse c) * f" by (cases "c = 0") (simp_all add: fps_divide_unit fps_const_inverse) (* FIfps_XME: The last part of this proof should go through by simp once we have a proper theorem collection for simplifying division on rings *) lemma fps_divide_deriv: assumes "b dvd (a :: 'a :: field fps)" shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b^2" proof - have eq_divide_imp: "c ≠ 0 ⟹ a * c = b ⟹ a = b div c" for a b c :: "'a :: field fps" by (drule sym) (simp add: mult.assoc) from assms have "a = a / b * b" by simp also have "fps_deriv (a / b * b) = fps_deriv (a / b) * b + a / b * fps_deriv b" by simp finally have "fps_deriv (a / b) * b^2 = fps_deriv a * b - a * fps_deriv b" using assms by (simp add: power2_eq_square algebra_simps) thus ?thesis by (cases "b = 0") (auto simp: eq_divide_imp) qed lemma fps_inverse_gp': "inverse (Abs_fps (λn. 1::'a::field)) = 1 - fps_X" by (simp add: fps_inverse_gp fps_eq_iff fps_X_def) lemma fps_one_over_one_minus_fps_X_squared: "inverse ((1 - fps_X)^2 :: 'a :: field fps) = Abs_fps (λn. of_nat (n+1))" proof - have "inverse ((1 - fps_X)^2 :: 'a fps) = fps_deriv (inverse (1 - fps_X))" by (subst fps_inverse_deriv) (simp_all add: fps_inverse_power) also have "inverse (1 - fps_X :: 'a fps) = Abs_fps (λ_. 1)" by (subst fps_inverse_gp' [symmetric]) simp also have "fps_deriv … = Abs_fps (λn. of_nat (n + 1))" by (simp add: fps_deriv_def) finally show ?thesis . qed lemma fps_nth_deriv_fps_X[simp]: "fps_nth_deriv n fps_X = (if n = 0 then fps_X else if n=1 then 1 else 0)" by (cases n) simp_all lemma fps_inverse_fps_X_plus1: "inverse (1 + fps_X) = Abs_fps (λn. (- (1::'a::field)) ^ n)" (is "_ = ?r") proof - have eq: "(1 + fps_X) * ?r = 1" unfolding minus_one_power_iff by (auto simp add: field_simps fps_eq_iff) show ?thesis by (auto simp add: eq intro: fps_inverse_unique) qed subsection ‹Integration› definition fps_integral :: "'a::field_char_0 fps ⇒ 'a ⇒ 'a fps" where "fps_integral a a0 = Abs_fps (λn. if n = 0 then a0 else (a$(n - 1) / of_nat n))" lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a" unfolding fps_integral_def fps_deriv_def by (simp add: fps_eq_iff del: of_nat_Suc) lemma fps_integral_linear: "fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) = fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0" (is "?l = ?r") proof - have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral) moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def) ultimately show ?thesis unfolding fps_deriv_eq_iff by auto qed subsection ‹Composition of FPSs› definition fps_compose :: "'a::semiring_1 fps ⇒ 'a fps ⇒ 'a fps" (infixl "oo" 55) where "a oo b = Abs_fps (λn. sum (λi. a$i * (b^i$n)) {0..n})" lemma fps_compose_nth: "(a oo b)$n = sum (λi. a$i * (b^i$n)) {0..n}" by (simp add: fps_compose_def) lemma fps_compose_nth_0 [simp]: "(f oo g) $ 0 = f $ 0" by (simp add: fps_compose_nth) lemma fps_compose_fps_X[simp]: "a oo fps_X = (a :: 'a::comm_ring_1 fps)" by (simp add: fps_ext fps_compose_def mult_delta_right sum.delta') lemma fps_const_compose[simp]: "fps_const (a::'a::comm_ring_1) oo b = fps_const a" by (simp add: fps_eq_iff fps_compose_nth mult_delta_left sum.delta) lemma numeral_compose[simp]: "(numeral k :: 'a::comm_ring_1 fps) oo b = numeral k" unfolding numeral_fps_const by simp lemma neg_numeral_compose[simp]: "(- numeral k :: 'a::comm_ring_1 fps) oo b = - numeral k" unfolding neg_numeral_fps_const by simp lemma fps_X_fps_compose_startby0[simp]: "a$0 = 0 ⟹ fps_X oo a = (a :: 'a::comm_ring_1 fps)" by (simp add: fps_eq_iff fps_compose_def mult_delta_left sum.delta not_le) subsection ‹Rules from Herbert Wilf's Generatingfunctionology› subsubsection ‹Rule 1› (* {a_{n+k}}_0^infty Corresponds to (f - sum (λi. a_i * x^i))/x^h, for h>0*) lemma fps_power_mult_eq_shift: "fps_X^Suc k * Abs_fps (λn. a (n + Suc k)) = Abs_fps a - sum (λi. fps_const (a i :: 'a::comm_ring_1) * fps_X^i) {0 .. k}" (is "?lhs = ?rhs") proof - have "?lhs $ n = ?rhs $ n" for n :: nat proof - have "?lhs $ n = (if n < Suc k then 0 else a n)" unfolding fps_X_power_mult_nth by auto also have "… = ?rhs $ n" proof (induct k) case 0 then show ?case by (simp add: fps_sum_nth) next case (Suc k) have "(Abs_fps a - sum (λi. fps_const (a i :: 'a) * fps_X^i) {0 .. Suc k})$n = (Abs_fps a - sum (λi. fps_const (a i :: 'a) * fps_X^i) {0 .. k} - fps_const (a (Suc k)) * fps_X^ Suc k) $ n" by (simp add: field_simps) also have "… = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * fps_X^ Suc k)$n" using Suc.hyps[symmetric] unfolding fps_sub_nth by simp also have "… = (if n < Suc (Suc k) then 0 else a n)" unfolding fps_X_power_mult_right_nth apply (auto simp add: not_less fps_const_def) apply (rule cong[of a a, OF refl]) apply arith done finally show ?case by simp qed finally show ?thesis . qed then show ?thesis by (simp add: fps_eq_iff) qed subsubsection ‹Rule 2› (* We can not reach the form of Wilf, but still near to it using rewrite rules*) (* If f reprents {a_n} and P is a polynomial, then P(xD) f represents {P(n) a_n}*) definition "fps_XD = ( * ) fps_X ∘ fps_deriv" lemma fps_XD_add[simp]:"fps_XD (a + b) = fps_XD a + fps_XD (b :: 'a::comm_ring_1 fps)" by (simp add: fps_XD_def field_simps) lemma fps_XD_mult_const[simp]:"fps_XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * fps_XD a" by (simp add: fps_XD_def field_simps) lemma fps_XD_linear[simp]: "fps_XD (fps_const c * a + fps_const d * b) = fps_const c * fps_XD a + fps_const d * fps_XD (b :: 'a::comm_ring_1 fps)" by simp lemma fps_XDN_linear: "(fps_XD ^^ n) (fps_const c * a + fps_const d * b) = fps_const c * (fps_XD ^^ n) a + fps_const d * (fps_XD ^^ n) (b :: 'a::comm_ring_1 fps)" by (induct n) simp_all lemma fps_mult_fps_X_deriv_shift: "fps_X* fps_deriv a = Abs_fps (λn. of_nat n* a$n)" by (simp add: fps_eq_iff) lemma fps_mult_fps_XD_shift: "(fps_XD ^^ k) (a :: 'a::comm_ring_1 fps) = Abs_fps (λn. (of_nat n ^ k) * a$n)" by (induct k arbitrary: a) (simp_all add: fps_XD_def fps_eq_iff field_simps del: One_nat_def) subsubsection ‹Rule 3› text ‹Rule 3 is trivial and is given by ‹fps_times_def›.› subsubsection ‹Rule 5 --- summation and "division" by (1 - fps_X)› lemma fps_divide_fps_X_minus1_sum_lemma: "a = ((1::'a::comm_ring_1 fps) - fps_X) * Abs_fps (λn. sum (λi. a $ i) {0..n})" proof - let ?sa = "Abs_fps (λn. sum (λi. a $ i) {0..n})" have th0: "⋀i. (1 - (fps_X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)" by simp have "a$n = ((1 - fps_X) * ?sa) $ n" for n proof (cases "n = 0") case True then show ?thesis by (simp add: fps_mult_nth) next case False then have u: "{0} ∪ ({1} ∪ {2..n}) = {0..n}" "{1} ∪ {2..n} = {1..n}" "{0..n - 1} ∪ {n} = {0..n}" by (auto simp: set_eq_iff) have d: "{0} ∩ ({1} ∪ {2..n}) = {}" "{1} ∩ {2..n} = {}" "{0..n - 1} ∩ {n} = {}" using False by simp_all have f: "finite {0}" "finite {1}" "finite {2 .. n}" "finite {0 .. n - 1}" "finite {n}" by simp_all have "((1 - fps_X) * ?sa) $ n = sum (λi. (1 - fps_X)$ i * ?sa $ (n - i)) {0 .. n}" by (simp add: fps_mult_nth) also have "… = a$n" unfolding th0 unfolding sum.union_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)] unfolding sum.union_disjoint[OF f(2) f(3) d(2)] apply (simp) unfolding sum.union_disjoint[OF f(4,5) d(3), unfolded u(3)] apply simp done finally show ?thesis by simp qed then show ?thesis unfolding fps_eq_iff by blast qed lemma fps_divide_fps_X_minus1_sum: "a /((1::'a::field fps) - fps_X) = Abs_fps (λn. sum (λi. a $ i) {0..n})" proof - let ?fps_X = "1 - (fps_X::'a fps)" have th0: "?fps_X $ 0 ≠ 0" by simp have "a /?fps_X = ?fps_X * Abs_fps (λn::nat. sum (($) a) {0..n}) * inverse ?fps_X" using fps_divide_fps_X_minus1_sum_lemma[of a, symmetric] th0 by (simp add: fps_divide_def mult.assoc) also have "… = (inverse ?fps_X * ?fps_X) * Abs_fps (λn::nat. sum (($) a) {0..n}) " by (simp add: ac_simps) 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 ∧ sum_list 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 "xs @ ys ∈ natpermute n k" shows "sum_list xs ≤ n" and "sum_list ys ≤ n" proof - from assms have "sum_list (xs @ ys) = n" by (simp add: natpermute_def) then have "sum_list xs + sum_list ys = n" by simp then show "sum_list xs ≤ n" and "sum_list ys ≤ n" by simp_all qed lemma natpermute_split: assumes "h ≤ k" shows "natpermute n k = (⋃m ∈{0..n}. {l1 @ l2 |l1 l2. l1 ∈ natpermute m h ∧ l2 ∈ natpermute (n - m) (k - h)})" (is "?L = ?R" is "_ = (⋃m ∈{0..n}. ?S m)") proof show "?R ⊆ ?L" proof fix l assume l: "l ∈ ?R" from l obtain m xs ys where h: "m ∈ {0..n}" and xs: "xs ∈ natpermute m h" and ys: "ys ∈ natpermute (n - m) (k - h)" and leq: "l = xs@ys" by blast from xs have xs': "sum_list xs = m" by (simp add: natpermute_def) from ys have ys': "sum_list ys = n - m" by (simp add: natpermute_def) show "l ∈ ?L" using leq xs ys h apply (clarsimp simp add: natpermute_def) unfolding xs' ys' using assms xs ys unfolding natpermute_def apply simp done qed show "?L ⊆ ?R" proof fix l assume l: "l ∈ natpermute n k" let ?xs = "take h l" let ?ys = "drop h l" let ?m = "sum_list ?xs" from l have ls: "sum_list (?xs @ ?ys) = n" by (simp add: natpermute_def) have xs: "?xs ∈ natpermute ?m h" using l assms by (simp add: natpermute_def) have l_take_drop: "sum_list l = sum_list (take h l @ drop h l)" by simp then have ys: "?ys ∈ natpermute (n - ?m) (k - h)" using l assms ls by (auto simp add: natpermute_def simp del: append_take_drop_id) from ls have m: "?m ∈ {0..n}" by (simp add: l_take_drop del: append_take_drop_id) from xs ys ls show "l ∈ ?R" apply auto apply (rule bexI [where x = "?m"]) apply (rule exI [where x = "?xs"]) apply (rule exI [where x = "?ys"]) using ls l apply (auto simp add: natpermute_def l_take_drop simp del: append_take_drop_id) apply simp done qed qed lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})" 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} = (⋃i∈{0 .. k}. {(replicate (k + 1) 0) [i:=n]})" (is "?A = ?B") proof show "?A ⊆ ?B" proof fix xs assume "xs ∈ ?A" then have H: "xs ∈ natpermute n (k + 1)" and n: "n ∈ set xs" by blast+ then obtain i where i: "i ∈ {0.. k}" "xs!i = n" unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def) have eqs: "({0..k} - {i}) ∪ {i} = {0..k}" using i by auto have f: "finite({0..k} - {i})" "finite {i}" by auto have d: "({0..k} - {i}) ∩ {i} = {}" using i by auto from H have "n = sum (nth xs) {0..k}" apply (simp add: natpermute_def) apply (auto simp add: atLeastLessThanSuc_atLeastAtMost sum_list_sum_nth) done also have "… = n + sum (nth xs) ({0..k} - {i})" unfolding sum.union_disjoint[OF f d, unfolded eqs] using i by simp finally have zxs: "∀ j∈ {0..k} - {i}. xs!j = 0" by auto from H have xsl: "length xs = k+1" 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 show "xs ∈ ?B" using i by blast qed show "?B ⊆ ?A" proof fix xs assume "xs ∈ ?B" then obtain i where i: "i ∈ {0..k}" and xs: "xs = replicate (k + 1) 0 [i:=n]" by auto have nxs: "n ∈ set xs" unfolding xs apply (rule set_update_memI) using i apply simp done have xsl: "length xs = k + 1" by (simp only: xs length_replicate length_list_update) have "sum_list xs = sum (nth xs) {0..<k+1}" unfolding sum_list_sum_nth xsl .. also have "… = sum (λj. if j = i then n else 0) {0..< k+1}" by (rule sum.cong) (simp_all add: xs del: replicate.simps) also have "… = n" using i by (simp add: sum.delta) finally have "xs ∈ natpermute n (k + 1)" using xsl unfolding natpermute_def mem_Collect_eq by blast then show "xs ∈ ?A" using nxs by blast qed qed text ‹The general form.› lemma fps_prod_nth: fixes m :: nat and a :: "nat ⇒ 'a::comm_ring_1 fps" shows "(prod a {0 .. m}) $ n = sum (λv. prod (λj. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))" (is "?P m n") proof (induct m arbitrary: n rule: nat_less_induct) fix m n assume H: "∀m' < m. ∀n. ?P m' n" show "?P m n" proof (cases m) case 0 then show ?thesis apply simp unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] apply simp done next case (Suc k) then have km: "k < m" by arith have u0: "{0 .. k} ∪ {m} = {0..m}" using Suc by (simp add: set_eq_iff) presburger have f0: "finite {0 .. k}" "finite {m}" by auto have d0: "{0 .. k} ∩ {m} = {}" using Suc by auto have "(prod a {0 .. m}) $ n = (prod a {0 .. k} * a m) $ n" unfolding prod.union_disjoint[OF f0 d0, unfolded u0] by simp also have "… = (∑i = 0..n. (∑v∈natpermute i (k + 1). ∏j∈{0..k}. a j $ v ! j) * a m $ (n - i))" unfolding fps_mult_nth H[rule_format, OF km] .. also have "… = (∑v∈natpermute n (m + 1). ∏j∈{0..m}. a j $ v ! j)" 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 sum.UNION_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 sum.cong) apply (rule refl) unfolding sum_distrib_right apply (rule sym) apply (rule_tac l = "λxs. xs @ [n - x]" in sum.reindex_cong) apply (simp add: inj_on_def) apply auto unfolding prod.union_disjoint[OF f0 d0, unfolded u0, unfolded Suc] apply (clarsimp simp add: natpermute_def nth_append) done finally show ?thesis . qed qed text ‹The special form for powers.› lemma fps_power_nth_Suc: fixes m :: nat and a :: "'a::comm_ring_1 fps" shows "(a ^ Suc m)$n = sum (λv. prod (λj. a $ (v!j)) {0..m}) (natpermute n (m+1))" proof - have th0: "a^Suc m = prod (λi. a) {0..m}" by (simp add: prod_constant) show ?thesis unfolding th0 fps_prod_nth .. qed lemma fps_power_nth: fixes m :: nat and a :: "'a::comm_ring_1 fps" shows "(a ^m)$n = (if m=0 then 1$n else sum (λv. prod (λj. a $ (v!j)) {0..m - 1}) (natpermute n m))" by (cases m) (simp_all add: fps_power_nth_Suc del: power_Suc) lemma fps_nth_power_0: fixes m :: nat and a :: "'a::comm_ring_1 fps" shows "(a ^m)$0 = (a$0) ^ m" proof (cases m) case 0 then show ?thesis by simp next case (Suc n) then have c: "m = card {0..n}" by simp have "(a ^m)$0 = prod (λi. a$0) {0..n}" by (simp add: Suc fps_power_nth del: replicate.simps power_Suc) also have "… = (a$0) ^ m" unfolding c by (rule prod_constant) finally show ?thesis . qed lemma natpermute_max_card: assumes n0: "n ≠ 0" shows "card {xs ∈ natpermute n (k + 1). n ∈ set xs} = k + 1" unfolding natpermute_contain_maximal proof - let ?A = "λi. {replicate (k + 1) 0[i := n]}" let ?K = "{0 ..k}" have fK: "finite ?K" by simp have fAK: "∀i∈?K. finite (?A i)" by auto have d: "∀i∈ ?K. ∀j∈ ?K. i ≠ j ⟶ {replicate (k + 1) 0[i := n]} ∩ {replicate (k + 1) 0[j := n]} = {}" proof clarify fix i j assume i: "i ∈ ?K" and j: "j ∈ ?K" and ij: "i ≠ j" have False if eq: "replicate (k+1) 0 [i:=n] = replicate (k+1) 0 [j:= n]" proof - have "(replicate (k+1) 0 [i:=n] ! i) = n" using i by (simp del: replicate.simps) moreover have "(replicate (k+1) 0 [j:=n] ! i) = 0" using i ij by (simp del: replicate.simps) ultimately show ?thesis using eq n0 by (simp del: replicate.simps) qed then show "{replicate (k + 1) 0[i := n]} ∩ {replicate (k + 1) 0[j := n]} = {}" by auto qed from card_UN_disjoint[OF fK fAK d] show "card (⋃i∈{0..k}. {replicate (k + 1) 0[i := n]}) = k + 1" by simp qed lemma fps_power_Suc_nth: fixes f :: "'a :: comm_ring_1 fps" assumes k: "k > 0" shows "(f ^ Suc m) $ k = of_nat (Suc m) * (f $ k * (f $ 0) ^ m) + (∑v∈{v∈natpermute k (m+1). k ∉ set v}. ∏j = 0..m. f $ v ! j)" proof - define A B where "A = {v∈natpermute k (m+1). k ∈ set v}" and "B = {v∈natpermute k (m+1). k ∉ set v}" have [simp]: "finite A" "finite B" "A ∩ B = {}" by (auto simp: A_def B_def natpermute_finite) from natpermute_max_card[of k m] k have card_A: "card A = m + 1" by (simp add: A_def) { fix v assume v: "v ∈ A" from v have [simp]: "length v = Suc m" by (simp add: A_def natpermute_def) from v have "∃j. j ≤ m ∧ v ! j = k" by (auto simp: set_conv_nth A_def natpermute_def less_Suc_eq_le) then guess j by (elim exE conjE) note j = this from v have "k = sum_list v" by (simp add: A_def natpermute_def) also have "… = (∑i=0..m. v ! i)" by (simp add: sum_list_sum_nth atLeastLessThanSuc_atLeastAtMost del: sum_op_ivl_Suc) also from j have "{0..m} = insert j ({0..m}-{j})" by auto also from j have "(∑i∈…. v ! i) = k + (∑i∈{0..m}-{j}. v ! i)" by (subst sum.insert) simp_all finally have "(∑i∈{0..m}-{j}. v ! i) = 0" by simp hence zero: "v ! i = 0" if "i ∈ {0..m}-{j}" for i using that by (subst (asm) sum_eq_0_iff) auto from j have "{0..m} = insert j ({0..m} - {j})" by auto also from j have "(∏i∈…. f $ (v ! i)) = f $ k * (∏i∈{0..m} - {j}. f $ (v ! i))" by (subst prod.insert) auto also have "(∏i∈{0..m} - {j}. f $ (v ! i)) = (∏i∈{0..m} - {j}. f $ 0)" by (intro prod.cong) (simp_all add: zero) also from j have "… = (f $ 0) ^ m" by (subst prod_constant) simp_all finally have "(∏j = 0..m. f $ (v ! j)) = f $ k * (f $ 0) ^ m" . } note A = this have "(f ^ Suc m) $ k = (∑v∈natpermute k (m + 1). ∏j = 0..m. f $ v ! j)" by (rule fps_power_nth_Suc) also have "natpermute k (m+1) = A ∪ B" unfolding A_def B_def by blast also have "(∑v∈…. ∏j = 0..m. f $ (v ! j)) = (∑v∈A. ∏j = 0..m. f $ (v ! j)) + (∑v∈B. ∏j = 0..m. f $ (v ! j))" by (intro sum.union_disjoint) simp_all also have "(∑v∈A. ∏j = 0..m. f $ (v ! j)) = of_nat (Suc m) * (f $ k * (f $ 0) ^ m)" by (simp add: A card_A) finally show ?thesis by (simp add: B_def) qed lemma fps_power_Suc_eqD: fixes f g :: "'a :: {idom,semiring_char_0} fps" assumes "f ^ Suc m = g ^ Suc m" "f $ 0 = g $ 0" "f $ 0 ≠ 0" shows "f = g" proof (rule fps_ext) fix k :: nat show "f $ k = g $ k" proof (induction k rule: less_induct) case (less k) show ?case proof (cases "k = 0") case False let ?h = "λf. (∑v | v ∈ natpermute k (m + 1) ∧ k ∉ set v. ∏j = 0..m. f $ v ! j)" from False fps_power_Suc_nth[of k f m] fps_power_Suc_nth[of k g m] have "f $ k * (of_nat (Suc m) * (f $ 0) ^ m) + ?h f = g $ k * (of_nat (Suc m) * (f $ 0) ^ m) + ?h g" using assms by (simp add: mult_ac del: power_Suc of_nat_Suc) also have "v ! i < k" if "v ∈ {v∈natpermute k (m+1). k ∉ set v}" "i ≤ m" for v i using that elem_le_sum_list[of i v] unfolding natpermute_def by (auto simp: set_conv_nth dest!: spec[of _ i]) hence "?h f = ?h g" by (intro sum.cong refl prod.cong less lessI) (auto simp: natpermute_def) finally have "f $ k * (of_nat (Suc m) * (f $ 0) ^ m) = g $ k * (of_nat (Suc m) * (f $ 0) ^ m)" by simp with assms show "f $ k = g $ k" by (subst (asm) mult_right_cancel) (auto simp del: of_nat_Suc) qed (simp_all add: assms) qed qed lemma fps_power_Suc_eqD': fixes f g :: "'a :: {idom,semiring_char_0} fps" assumes "f ^ Suc m = g ^ Suc m" "f $ subdegree f = g $ subdegree g" shows "f = g" proof (cases "f = 0") case False have "Suc m * subdegree f = subdegree (f ^ Suc m)" by (rule subdegree_power [symmetric]) also have "f ^ Suc m = g ^ Suc m" by fact also have "subdegree … = Suc m * subdegree g" by (rule subdegree_power) finally have [simp]: "subdegree f = subdegree g" by (subst (asm) Suc_mult_cancel1) have "fps_shift (subdegree f) f * fps_X ^ subdegree f = f" by (rule subdegree_decompose [symmetric]) also have "… ^ Suc m = g ^ Suc m" by fact also have "g = fps_shift (subdegree g) g * fps_X ^ subdegree g" by (rule subdegree_decompose) also have "subdegree f = subdegree g" by fact finally have "fps_shift (subdegree g) f ^ Suc m = fps_shift (subdegree g) g ^ Suc m" by (simp add: algebra_simps power_mult_distrib del: power_Suc) hence "fps_shift (subdegree g) f = fps_shift (subdegree g) g" by (rule fps_power_Suc_eqD) (insert assms False, auto) with subdegree_decompose[of f] subdegree_decompose[of g] show ?thesis by simp qed (insert assms, simp_all) lemma fps_power_eqD': fixes f g :: "'a :: {idom,semiring_char_0} fps" assumes "f ^ m = g ^ m" "f $ subdegree f = g $ subdegree g" "m > 0" shows "f = g" using fps_power_Suc_eqD'[of f "m-1" g] assms by simp lemma fps_power_eqD: fixes f g :: "'a :: {idom,semiring_char_0} fps" assumes "f ^ m = g ^ m" "f $ 0 = g $ 0" "f $ 0 ≠ 0" "m > 0" shows "f = g" by (rule fps_power_eqD'[of f m g]) (insert assms, simp_all) lemma fps_compose_inj_right: assumes a0: "a$0 = (0::'a::idom)" and a1: "a$1 ≠ 0" shows "(b oo a = c oo a) ⟷ b = c" (is "?lhs ⟷?rhs") proof show ?lhs if ?rhs using that by simp show ?rhs if ?lhs proof - have "b$n = c$n" for n proof (induct n rule: nat_less_induct) fix n assume H: "∀m<n. b$m = c$m" show "b$n = c$n" proof (cases n) case 0 from ‹?lhs› have "(b oo a)$n = (c oo a)$n" by simp then show ?thesis using 0 by (simp add: fps_compose_nth) next case (Suc n1) have f: "finite {0 .. n1}" "finite {n}" by simp_all have eq: "{0 .. n1} ∪ {n} = {0 .. n}" using Suc by auto have d: "{0 .. n1} ∩ {n} = {}" using Suc by auto have seq: "(∑i = 0..n1. b $ i * a ^ i $ n) = (∑i = 0..n1. c $ i * a ^ i $ n)" apply (rule sum.cong) using H Suc apply auto done have th0: "(b oo a) $n = (∑i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n" unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq] seq using startsby_zero_power_nth_same[OF a0] by simp have th1: "(c oo a) $n = (∑i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n" unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq] using startsby_zero_power_nth_same[OF a0] by simp from ‹?lhs›[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1 show ?thesis by auto qed qed then show ?rhs by (simp add: fps_eq_iff) qed qed subsection ‹Radicals› declare prod.cong [fundef_cong] function radical :: "(nat ⇒ 'a ⇒ 'a) ⇒ nat ⇒ 'a::field fps ⇒ nat ⇒ 'a" where "radical r 0 a 0 = 1" | "radical r 0 a (Suc n) = 0" | "radical r (Suc k) a 0 = r (Suc k) (a$0)" | "radical r (Suc k) a (Suc n) = (a$ Suc n - sum (λxs. prod (λj. radical r (Suc k) a (xs ! j)) {0..k}) {xs. xs ∈ natpermute (Suc n) (Suc k) ∧ Suc n ∉ set xs}) / (of_nat (Suc k) * (radical r (Suc k) a 0)^k)" by pat_completeness auto 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}" have False if c: "Suc n ≤ xs ! i" proof - from xs i have "xs !i ≠ Suc n" by (auto simp add: in_set_conv_nth natpermute_def) with c have c': "Suc n < xs!i" by arith have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all have d: "{0 ..< i} ∩ ({i} ∪ {i+1 ..< Suc k}) = {}" "{i} ∩ {i+1..< Suc k} = {}" by auto have eqs: "{0..<Suc k} = {0 ..< i} ∪ ({i} ∪ {i+1 ..< Suc k})" using i by auto from xs have "Suc n = sum_list xs" by (simp add: natpermute_def) also have "… = sum (nth xs) {0..<Suc k}" using xs by (simp add: natpermute_def sum_list_sum_nth) also have "… = xs!i + sum (nth xs) {0..<i} + sum (nth xs) {i+1..<Suc k}" unfolding eqs sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)] unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)] by simp finally show ?thesis using c' by simp qed then 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 prod.cong) apply simp using Suc apply (subgoal_tac "replicate k 0 ! x = 0") apply (auto intro: nth_replicate simp del: replicate.simps) done also have "… = a$0" using r Suc by (simp add: prod_constant) finally show ?thesis using Suc by simp qed 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" (is "?lhs ⟷ ?rhs") proof let ?r = "fps_radical r (Suc k) a" show ?rhs if r0: ?lhs proof - from a0 r0 have r00: "r (Suc k) (a$0) ≠ 0" by auto have "?r ^ Suc k $ z = a$z" for z proof (induct z rule: nat_less_induct) fix n assume H: "∀m<n. ?r ^ Suc k $ m = a$m" show "?r ^ Suc k $ n = a $n" proof (cases n) case 0 then show ?thesis using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp next case (Suc n1) then have "n ≠ 0" by simp let ?Pnk = "natpermute n (k + 1)" let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}" let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}" have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast have d: "?Pnkn ∩ ?Pnknn = {}" by blast have f: "finite ?Pnkn" "finite ?Pnknn" using finite_Un[of ?Pnkn ?Pnknn, unfolded eq] by (metis natpermute_finite)+ let ?f = "λv. ∏j∈{0..k}. ?r $ v ! j" have "sum ?f ?Pnkn = sum (λv. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn" proof (rule sum.cong) fix v assume v: "v ∈ {xs ∈ natpermute n (k + 1). n ∈ set xs}" let ?ths = "(∏j∈{0..k}. fps_radical r (Suc k) a $ v ! j) = fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k" from v obtain i where i: "i ∈ {0..k}" "v = replicate (k+1) 0 [i:= n]" unfolding natpermute_contain_maximal by auto have "(∏j∈{0..k}. fps_radical r (Suc k) a $ v ! j) = (∏j∈{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))" apply (rule prod.cong, simp) using i r0 apply (simp del: replicate.simps) done also have "… = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k" using i r0 by (simp add: prod_gen_delta) finally show ?ths . qed rule then have "sum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k" by (simp add: natpermute_max_card[OF ‹n ≠ 0›, simplified]) also have "… = a$n - sum ?f ?Pnknn" unfolding Suc using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc) finally have fn: "sum ?f ?Pnkn = a$n - sum ?f ?Pnknn" . have "(?r ^ Suc k)$n = sum ?f ?Pnkn + sum ?f ?Pnknn" unfolding fps_power_nth_Suc sum.union_disjoint[OF f d, unfolded eq] .. also have "… = a$n" unfolding fn by simp finally show ?thesis . qed qed then show ?thesis using r0 by (simp add: fps_eq_iff) qed show ?lhs if ?rhs proof - from that have "((fps_radical r (Suc k) a) ^ (Suc k))$0 = a$0" by simp then show ?thesis unfolding fps_power_nth_Suc by (simp add: prod_constant del: replicate.simps) qed qed (* 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" then have "?r ^ Suc k $ n = a $n" using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp} moreover {fix n1 assume n1: "n = Suc n1" have fK: "finite {0..k}" by simp have nz: "n ≠ 0" using n1 by arith let ?Pnk = "natpermute n (k + 1)" let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}" let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}" have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast have d: "?Pnkn ∩ ?Pnknn = {}" by blast have f: "finite ?Pnkn" "finite ?Pnknn" using finite_Un[of ?Pnkn ?Pnknn, unfolded eq] by (metis natpermute_finite)+ let ?f = "λv. ∏j∈{0..k}. ?r $ v ! j" have "sum ?f ?Pnkn = sum (λv. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn" proof(rule sum.cong2) fix v assume v: "v ∈ {xs ∈ natpermute n (k + 1). n ∈ set xs}" let ?ths = "(∏j∈{0..k}. fps_radical r (Suc k) a $ v ! j) = fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k" from v obtain i where i: "i ∈ {0..k}" "v = replicate (k+1) 0 [i:= n]" unfolding natpermute_contain_maximal by auto have "(∏j∈{0..k}. fps_radical r (Suc k) a $ v ! j) = (∏j∈{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))" apply (rule prod.cong, simp) using i r0 by (simp del: replicate.simps) also have "… = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k" unfolding prod_gen_delta[OF fK] using i r0 by simp finally show ?ths . qed then have "sum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k" by (simp add: natpermute_max_card[OF nz, simplified]) also have "… = a$n - sum ?f ?Pnknn" unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc ) finally have fn: "sum ?f ?Pnkn = a$n - sum ?f ?Pnknn" . have "(?r ^ Suc k)$n = sum ?f ?Pnkn + sum ?f ?Pnknn" unfolding fps_power_nth_Suc sum.union_disjoint[OF f d, unfolded eq] .. also have "… = a$n" unfolding fn by simp finally have "?r ^ Suc k $ n = a $n" .} ultimately show "?r ^ Suc k $ n = a $n" by (cases n, auto) qed } then show ?thesis by (simp add: fps_eq_iff) qed *) lemma eq_divide_imp': fixes c :: "'a::field" shows "c ≠ 0 ⟹ a * c = b ⟹ a = b / c" 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 ≠ 0" shows "a^(Suc k) = b ⟷ a = fps_radical r (Suc k) b" (is "?lhs ⟷ ?rhs" is "_ ⟷ a = ?r") proof show ?lhs if ?rhs using that using power_radical[OF b0, of r k, unfolded r0] by simp show ?rhs if ?lhs proof - have r00: "r (Suc k) (b$0) ≠ 0" using b0 r0 by auto have ceq: "card {0..k} = Suc k" by simp from a0 have a0r0: "a$0 = ?r$0" by simp have "a $ n = ?r $ n" for n proof (induct n rule: nat_less_induct) fix n assume h: "∀m<n. a$m = ?r $m" show "a$n = ?r $ n" proof (cases n) case 0 then show ?thesis using a0 by simp next case (Suc n1) have fK: "finite {0..k}" by simp have nz: "n ≠ 0" using Suc by simp let ?Pnk = "natpermute n (Suc k)" let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}" let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}" have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast have d: "?Pnkn ∩ ?Pnknn = {}" by blast have f: "finite ?Pnkn" "finite ?Pnknn" using finite_Un[of ?Pnkn ?Pnknn, unfolded eq] by (metis natpermute_finite)+ let ?f = "λv. ∏j∈{0..k}. ?r $ v ! j" let ?g = "λv. ∏j∈{0..k}. a $ v ! j" have "sum ?g ?Pnkn = sum (λv. a $ n * (?r$0)^k) ?Pnkn" proof (rule sum.cong) fix v assume v: "v ∈ {xs ∈ natpermute n (Suc k). n ∈ set xs}" let ?ths = "(∏j∈{0..k}. a $ v ! j) = a $ n * (?r$0)^k" from v obtain i where i: "i ∈ {0..k}" "v = replicate (k+1) 0 [i:= n]" unfolding Suc_eq_plus1 natpermute_contain_maximal by (auto simp del: replicate.simps) have "(∏j∈{0..k}. a $ v ! j) = (∏j∈{0..k}. if j = i then a $ n else r (Suc k) (b$0))" apply (rule prod.cong, simp) using i a0 apply (simp del: replicate.simps) done also have "… = a $ n * (?r $ 0)^k" using i by (simp add: prod_gen_delta) finally show ?ths . qed rule then have th0: "sum ?g ?Pnkn = of_nat (k+1) * a $ n * (?r $ 0)^k" by (simp add: natpermute_max_card[OF nz, simplified]) have th1: "sum ?g ?Pnknn = sum ?f ?Pnknn" proof (rule sum.cong, rule refl, rule prod.cong, simp) fix xs i assume xs: "xs ∈ ?Pnknn" and i: "i ∈ {0..k}" have False if c: "n ≤ xs ! i" proof - from xs i have "xs ! i ≠ n" by (auto simp add: in_set_conv_nth natpermute_def) with c have c': "n < xs!i" by arith have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all have d: "{0 ..< i} ∩ ({i} ∪ {i+1 ..< Suc k}) = {}" "{i} ∩ {i+1..< Suc k} = {}" by auto have eqs: "{0..<Suc k} = {0 ..< i} ∪ ({i} ∪ {i+1 ..< Suc k})" using i by auto from xs have "n = sum_list xs" by (simp add: natpermute_def) also have "… = sum (nth xs) {0..<Suc k}" using xs by (simp add: natpermute_def sum_list_sum_nth) also have "… = xs!i + sum (nth xs) {0..<i} + sum (nth xs) {i+1..<Suc k}" unfolding eqs sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)] unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)] by simp finally show ?thesis using c' by simp qed then have thn: "xs!i < n" by presburger from h[rule_format, OF thn] show "a$(xs !i) = ?r$(xs!i)" . qed have th00: "⋀x::'a. of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x" by (simp add: field_simps del: of_nat_Suc) from ‹?lhs› have "b$n = a^Suc k $ n" by (simp add: fps_eq_iff) also have "a ^ Suc k$n = sum ?g ?Pnkn + sum ?g ?Pnknn" unfolding fps_power_nth_Suc using sum.union_disjoint[OF f d, unfolded Suc_eq_plus1[symmetric], unfolded eq, of ?g] by simp also have "… = of_nat (k+1) * a $ n * (?r $ 0)^k + sum ?f ?Pnknn" unfolding th0 th1 .. finally have "of_nat (k+1) * a $ n * (?r $ 0)^k = b$n - sum ?f ?Pnknn" by simp then have "a$n = (b$n - sum ?f ?Pnknn) / (of_nat (k+1) * (?r $ 0)^k)" apply - apply (rule eq_divide_imp') using r00 apply (simp del: of_nat_Suc) apply (simp add: ac_simps) done then show ?thesis apply (simp del: of_nat_Suc) unfolding fps_radical_def Suc apply (simp add: field_simps Suc th00 del: of_nat_Suc) done qed qed then show ?rhs by (simp add: fps_eq_iff) qed qed 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 then have "fps_deriv ?r * ?w = fps_deriv a" by (simp add: fps_deriv_power ac_simps del: power_Suc) then have "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a" by simp with a0 r0 have "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w" by (subst fps_divide_unit) (auto simp del: of_nat_Suc) then show ?thesis unfolding th0 by simp qed 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" (is "?lhs ⟷ ?rhs") proof show ?rhs if r0': ?lhs proof - from r0' have r0: "(r k ((a * b) $ 0)) ^ k = (a * b) $ 0" by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib) show ?thesis proof (cases k) case 0 then show ?thesis using r0' by simp next case (Suc h) let ?ra = "fps_radical r (Suc h) a" let ?rb = "fps_radical r (Suc h) b" have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0" using r0' Suc by (simp add: fps_mult_nth) have ab0: "(a*b) $ 0 ≠ 0" using a0 b0 by (simp add: fps_mult_nth) from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded Suc] th0 ab0, symmetric] iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded Suc]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded Suc]] Suc r0' show ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc) qed qed show ?lhs if ?rhs proof - from that have "(fps_radical r k (a * b)) $ 0 = (fps_radical r k a * fps_radical r k b) $ 0" by simp then show ?thesis using k by (simp add: fps_mult_nth) qed qed (* 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" then have ?thesis by simp} moreover {fix h assume k: "k = Suc h" let ?ra = "fps_radical r (Suc h) a" let ?rb = "fps_radical r (Suc h) b" have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0" using r0' k by (simp add: fps_mult_nth) have ab0: "(a*b) $ 0 ≠ 0" using a0 b0 by (simp add: fps_mult_nth) from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric] power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)} ultimately show ?thesis by (cases k, auto) qed *) lemma fps_divide_1 [simp]: "(a :: 'a::field fps) / 1 = a" by (fact div_by_1) 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 show ?lhs if ?rhs proof - from that have "?r (a/b) $ 0 = (?r a / ?r b)$0" by simp then show ?thesis using k a0 b0 rb0' by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse) qed show ?rhs if ?lhs proof - from a0 b0 have ab0[simp]: "(a/b)$0 = a$0 / b$0" by (simp add: fps_divide_def fps_mult_nth divide_inverse fps_inverse_def) have th0: "r k ((a/b)$0) ^ k = (a/b)$0" by (simp add: ‹?lhs› power_divide ra0 rb0) from a0 b0 ra0' rb0' kp ‹?lhs› have th1: "r k ((a / b) $ 0) = (fps_radical r k a / fps_radical r k b) $ 0" by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse) from a0 b0 ra0' rb0' kp have ab0': "(a / b) $ 0 ≠ 0" by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def nonzero_imp_inverse_nonzero) note tha[simp] = iffD1[OF power_radical[where r=r and k=h], OF a0 ra0[unfolded k], unfolded k[symmetric]] note thb[simp] = iffD1[OF power_radical[where r=r and k=h], OF b0 rb0[unfolded k], unfolded k[symmetric]] from b0 rb0' have th2: "(?r a / ?r b)^k = a/b" by (simp add: fps_divide_unit power_mult_distrib fps_inverse_power[symmetric]) from iffD1[OF radical_unique[where r=r and a="?r a / ?r b" and b="a/b" and k=h], symmetric, unfolded k[symmetric], OF th0 th1 ab0' th2] show ?thesis . qed qed 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$0 = 0" shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * fps_deriv b" proof - have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n" for n proof - have "(fps_deriv (a oo b))$n = sum (λi. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}" by (simp add: fps_compose_def field_simps sum_distrib_left del: of_nat_Suc) also have "… = sum (λi. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}" by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc) also have "… = sum (λi. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}" unfolding fps_mult_left_const_nth by (simp add: field_simps) also have "… = sum (λi. of_nat i * a$i * (sum (λj. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {0.. Suc n}" unfolding fps_mult_nth .. also have "… = sum (λi. of_nat i * a$i * (sum (λj. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {1.. Suc n}" apply (rule sum.mono_neutral_right) apply (auto simp add: mult_delta_left sum.delta not_le) done also have "… = sum (λi. of_nat (i + 1) * a$(i+1) * (sum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" unfolding fps_deriv_nth by (rule sum.reindex_cong [of Suc]) (auto simp add: mult.assoc) finally have th0: "(fps_deriv (a oo b))$n = sum (λi. of_nat (i + 1) * a$(i+1) * (sum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" . have "(((fps_deriv a) oo b) * (fps_deriv b))$n = sum (λi. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}" unfolding fps_mult_nth by (simp add: ac_simps) also have "… = sum (λi. sum (λj. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}" unfolding fps_deriv_nth fps_compose_nth sum_distrib_left mult.assoc apply (rule sum.cong) apply (rule refl) apply (rule sum.mono_neutral_left) apply (simp_all add: subset_eq) apply clarify apply (subgoal_tac "b^i$x = 0") apply simp apply (rule startsby_zero_power_prefix[OF b0, rule_format]) apply simp done also have "… = sum (λi. of_nat (i + 1) * a$(i+1) * (sum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" unfolding sum_distrib_left apply (subst sum.swap) apply (rule sum.cong, rule refl)+ apply simp done finally show ?thesis unfolding th0 by simp qed then show ?thesis by (simp add: fps_eq_iff) qed lemma fps_mult_fps_X_plus_1_nth: "((1+fps_X)*a) $n = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))" proof (cases n) case 0 then show ?thesis by (simp add: fps_mult_nth) next case (Suc m) have "((1 + fps_X)*a) $ n = sum (λi. (1 + fps_X) $ i * a $ (n - i)) {0..n}" by (simp add: fps_mult_nth) also have "… = sum (λi. (1+fps_X)$i * a$(n-i)) {0.. 1}" unfolding Suc by (rule sum.mono_neutral_right) auto also have "… = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))" by (simp add: Suc) finally show ?thesis . qed subsection ‹Finite FPS (i.e. polynomials) and fps_X› lemma fps_poly_sum_fps_X: assumes "∀i > n. a$i = (0::'a::comm_ring_1)" shows "a = sum (λi. fps_const (a$i) * fps_X^i) {0..n}" (is "a = ?r") proof - have "a$i = ?r$i" for i unfolding fps_sum_nth fps_mult_left_const_nth fps_X_power_nth by (simp add: mult_delta_right sum.delta' assms) then show ?thesis unfolding fps_eq_iff by blast qed subsection ‹Compositional inverses› fun compinv :: "'a fps ⇒ nat ⇒ 'a::field" where "compinv a 0 = fps_X$0" | "compinv a (Suc n) = (fps_X$ Suc n - sum (λi. (compinv a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n" definition "fps_inv a = Abs_fps (compinv a)" lemma fps_inv: assumes a0: "a$0 = 0" and a1: "a$1 ≠ 0" shows "fps_inv a oo a = fps_X" proof - let ?i = "fps_inv a oo a" have "?i $n = fps_X$n" for n proof (induct n rule: nat_less_induct) fix n assume h: "∀m<n. ?i$m = fps_X$m" show "?i $ n = fps_X$n" proof (cases n) case 0 then show ?thesis using a0 by (simp add: fps_compose_nth fps_inv_def) next case (Suc n1) have "?i $ n = sum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1" by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc) also have "… = sum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + (fps_X$ Suc n1 - sum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1})" using a0 a1 Suc by (simp add: fps_inv_def) also have "… = fps_X$n" using Suc by simp finally show ?thesis . qed qed then show ?thesis 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 - sum (λi. (gcompinv b a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n" definition "fps_ginv b a = Abs_fps (gcompinv b a)" lemma fps_ginv: assumes a0: "a$0 = 0" and a1: "a$1 ≠ 0" shows "fps_ginv b a oo a = b" proof - let ?i = "fps_ginv b a oo a" have "?i $n = b$n" for n proof (induct n rule: nat_less_induct) fix n assume h: "∀m<n. ?i$m = b$m" show "?i $ n = b$n" proof (cases n) case 0 then show ?thesis using a0 by (simp add: fps_compose_nth fps_ginv_def) next case (Suc n1) have "?i $ n = sum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1" by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc) also have "… = sum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + (b$ Suc n1 - sum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})" using a0 a1 Suc by (simp add: fps_ginv_def) also have "… = b$n" using Suc by simp finally show ?thesis . qed qed then show ?thesis by (simp add: fps_eq_iff) qed lemma fps_inv_ginv: "fps_inv = fps_ginv fps_X" apply (auto simp add: fun_eq_iff fps_eq_iff fps_inv_def fps_ginv_def) apply (induct_tac n rule: nat_less_induct) apply auto apply (case_tac na) apply simp apply simp done lemma fps_compose_1[simp]: "1 oo a = 1" by (simp add: fps_eq_iff fps_compose_nth mult_delta_left sum.delta) lemma fps_compose_0[simp]: "0 oo a = 0" 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 sum.neutral) lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)" by (simp add: fps_eq_iff fps_compose_nth field_simps sum.distrib) lemma fps_compose_sum_distrib: "(sum f S) oo a = sum (λi. f i oo a) S" proof (cases "finite S") case True show ?thesis proof (rule finite_induct[OF True]) show "sum f {} oo a = (∑i∈{}. f i oo a)" by simp next fix x F assume fF: "finite F" and xF: "x ∉ F" and h: "sum f F oo a = sum (λi. f i oo a) F" show "sum f (insert x F) oo a = sum (λi. f i oo a) (insert x F)" using fF xF h by (simp add: fps_compose_add_distrib) qed next case False then show ?thesis by simp qed lemma convolution_eq: "sum (λi. a (i :: nat) * b (n - i)) {0 .. n} = sum (λ(i,j). a i * b j) {(i,j). i ≤ n ∧ j ≤ n ∧ i + j = n}" by (rule sum.reindex_bij_witness[where i=fst and j="λi. (i, n - i)"]) auto lemma product_composition_lemma: assumes c0: "c$0 = (0::'a::idom)" and d0: "d$0 = 0" shows "((a oo c) * (b oo d))$n = sum (λ(k,m). a$k * b$m * (c^k * d^m) $ n) {(k,m). k + m ≤ n}" (is "?l = ?r") proof - let ?S = "{(k::nat, m::nat). k + m ≤ n}" have s: "?S ⊆ {0..n} × {0..n}" by (auto simp add: subset_eq) have f: "finite {(k::nat, m::nat). k + m ≤ n}" apply (rule finite_subset[OF s]) apply auto done have "?r = sum (λi. sum (λ(k,m). a$k * (c^k)$i * b$m * (d^m) $ (n - i)) {(k,m). k + m ≤ n}) {0..n}" apply (simp add: fps_mult_nth sum_distrib_left) apply (subst sum.swap) apply (rule sum.cong) apply (auto simp add: field_simps) done also have "… = ?l" apply (simp add: fps_mult_nth fps_compose_nth sum_product) apply (rule sum.cong) apply (rule refl) apply (simp add: sum.cartesian_product mult.assoc) apply (rule sum.mono_neutral_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 = sum (λk. sum (λm. a$k * b$m * (c^k * d^m) $ n) {0..n}) {0..n}" (is "?l = ?r") unfolding product_composition_lemma[OF c0 d0] unfolding sum.cartesian_product apply (rule sum.mono_neutral_left) apply simp apply (clarsimp 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 sum.neutral) 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 sum_pair_less_iff: "sum (λ((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m ≤ n} = sum (λs. sum (λi. a i * b (s - i) * c s) {0..s}) {0..n}" (is "?l = ?r") proof - let ?KM = "{(k,m). k + m ≤ n}" let ?f = "λs. UNION {(0::nat)..s} (λi. {(i,s - i)})" have th0: "?KM = UNION {0..n} ?f" by auto show "?l = ?r " unfolding th0 apply (subst sum.UNION_disjoint) apply auto apply (subst sum.UNION_disjoint) apply auto done qed lemma fps_compose_mult_distrib_lemma: assumes c0: "c$0 = (0::'a::idom)" shows "((a oo c) * (b oo c))$n = sum (λs. sum (λi. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}" unfolding product_composition_lemma[OF c0 c0] power_add[symmetric] unfolding sum_pair_less_iff[where a = "λk. a$k" and b="λm. b$m" and c="λs. (c ^ s)$n" and n = n] .. lemma fps_compose_mult_distrib: assumes c0: "c $ 0 = (0::'a::idom)" shows "(a * b) oo c = (a oo c) * (b oo c)" apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma [OF c0]) apply (simp add: fps_compose_nth fps_mult_nth sum_distrib_right) done lemma fps_compose_prod_distrib: assumes c0: "c$0 = (0::'a::idom)" shows "prod a S oo c = prod (λk. a k oo c) S" apply (cases "finite S") apply simp_all apply (induct S rule: finite_induct) apply simp apply (simp add: fps_compose_mult_distrib[OF c0]) done lemma fps_compose_divide: assumes [simp]: "g dvd f" "h $ 0 = 0" shows "fps_compose f h = fps_compose (f / g :: 'a :: field fps) h * fps_compose g h" proof - have "f = (f / g) * g" by simp also have "fps_compose … h = fps_compose (f / g) h * fps_compose g h" by (subst fps_compose_mult_distrib) simp_all finally show ?thesis . qed lemma fps_compose_divide_distrib: assumes "g dvd f" "h $ 0 = 0" "fps_compose g h ≠ 0" shows "fps_compose (f / g :: 'a :: field fps) h = fps_compose f h / fps_compose g h" using fps_compose_divide[OF assms(1,2)] assms(3) by simp lemma fps_compose_power: assumes c0: "c$0 = (0::'a::idom)" shows "(a oo c)^n = a^n oo c" proof (cases n) case 0 then show ?thesis by simp next case (Suc m) have "(∏n = 0..m. a) oo c = (∏n = 0..m. a oo c)" using c0 fps_compose_prod_distrib by blast moreover have th0: "a^n = prod (λk. a) {0..m}" "(a oo c) ^ n = prod (λk. a oo c) {0..m}" by (simp_all add: prod_constant Suc) ultimately show ?thesis by presburger qed lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)" by (simp add: fps_eq_iff fps_compose_nth field_simps sum_negf[symmetric]) lemma fps_compose_sub_distrib: "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)" using fps_compose_add_distrib [of a "- b" c] by (simp add: fps_compose_uminus) lemma fps_X_fps_compose: "fps_X oo a = Abs_fps (λn. if n = 0 then (0::'a::comm_ring_1) else a$n)" by (simp add: fps_eq_iff fps_compose_nth mult_delta_left sum.delta) lemma fps_inverse_compose: assumes b0: "(b$0 :: 'a::field) = 0" and a0: "a$0 ≠ 0" shows "inverse a oo b = inverse (a oo b)" proof - let ?ia = "inverse a" let ?ab = "a oo b" let ?iab = "inverse ?ab" from a0 have ia0: "?ia $ 0 ≠ 0" by simp from a0 have ab0: "?ab $ 0 ≠ 0" by (simp add: fps_compose_def) have "(?ia oo b) * (a oo b) = 1" unfolding fps_compose_mult_distrib[OF b0, symmetric] unfolding inverse_mult_eq_1[OF a0] fps_compose_1 .. then have "(?ia oo b) * (a oo b) * ?iab = 1 * ?iab" by simp then have "(?ia oo b) * (?iab * (a oo b)) = ?iab" by simp then show ?thesis unfolding inverse_mult_eq_1[OF ab0] by simp qed lemma fps_divide_compose: assumes c0: "(c$0 :: 'a::field) = 0" and b0: "b$0 ≠ 0" shows "(a/b) oo c = (a oo c) / (b oo c)" using b0 c0 by (simp add: fps_divide_unit fps_inverse_compose fps_compose_mult_distrib) lemma gp: assumes a0: "a$0 = (0::'a::field)" shows "(Abs_fps (λn. 1)) oo a = 1/(1 - a)" (is "?one oo a = _") proof - have o0: "?one $ 0 ≠ 0" by simp have th0: "(1 - fps_X) $ 0 ≠ (0::'a)" by simp from fps_inverse_gp[where ?'a = 'a] have "inverse ?one = 1 - fps_X" by (simp add: fps_eq_iff) then have "inverse (inverse ?one) = inverse (1 - fps_X)" by simp then have th: "?one = 1/(1 - fps_X)" unfolding fps_inverse_idempotent[OF o0] by (simp add: fps_divide_def) show ?thesis unfolding th unfolding fps_divide_compose[OF a0 th0] fps_compose_1 fps_compose_sub_distrib fps_X_fps_compose_startby0[OF a0] .. qed lemma fps_const_power [simp]: "fps_const (c::'a::ring_1) ^ n = fps_const (c^n)" by (induct n) auto 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 sum_distrib_left mult.assoc) lemma fps_const_mult_apply_right: "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b" by (auto simp add: fps_const_mult_apply_left mult.commute) lemma fps_compose_assoc: assumes c0: "c$0 = (0::'a::idom)" and b0: "b$0 = 0" shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r") proof - have "?l$n = ?r$n" for n proof - have "?l$n = (sum (λi. (fps_const (a$i) * b^i) oo c) {0..n})$n" by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left sum_distrib_left mult.assoc fps_sum_nth) also have "… = ((sum (λi. fps_const (a$i) * b^i) {0..n}) oo c)$n" by (simp add: fps_compose_sum_distrib) also have "… = ?r$n" apply (simp add: fps_compose_nth fps_sum_nth sum_distrib_right mult.assoc) apply (rule sum.cong) apply (rule refl) apply (rule sum.mono_neutral_right) apply (auto simp add: not_le) apply (erule startsby_zero_power_prefix[OF b0, rule_format]) done finally show ?thesis . qed then show ?thesis by (simp add: fps_eq_iff) qed lemma fps_X_power_compose: assumes a0: "a$0=0" shows "fps_X^k oo a = (a::'a::idom fps)^k" (is "?l = ?r") proof (cases k) case 0 then show ?thesis by simp next case (Suc h) have "?l $ n = ?r $n" for n proof - consider "k > n" | "k ≤ n" by arith then show ?thesis proof cases case 1 then show ?thesis using a0 startsby_zero_power_prefix[OF a0] Suc by (simp add: fps_compose_nth del: power_Suc) next case 2 then show ?thesis by (simp add: fps_compose_nth mult_delta_left sum.delta) qed qed then show ?thesis unfolding fps_eq_iff by blast qed lemma fps_inv_right: assumes a0: "a$0 = 0" and a1: "a$1 ≠ 0" shows "a oo fps_inv a = fps_X" proof - let ?ia = "fps_inv a" let ?iaa = "a oo fps_inv a" have th0: "?ia $ 0 = 0" 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: "fps_X$0 = 0" by simp from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo fps_X" by simp then have "(a oo fps_inv a) oo a = fps_X oo a" by (simp add: fps_compose_assoc[OF a0 th0] fps_X_fps_compose_startby0[OF a0]) with fps_compose_inj_right[OF a0 a1] show ?thesis by simp qed lemma fps_inv_deriv: assumes a0: "a$0 = (0::'a::field)" and a1: "a$1 ≠ 0" shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)" proof - let ?ia = "fps_inv a" let ?d = "fps_deriv a oo ?ia" let ?dia = "fps_deriv ?ia" have ia0: "?ia$0 = 0" 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] ) then have "inverse ?d * ?d * ?dia = inverse ?d * 1" by simp with inverse_mult_eq_1 [OF th0] show "?dia = inverse ?d" by simp qed lemma fps_inv_idempotent: assumes a0: "a$0 = 0" and a1: "a$1 ≠ 0" shows "fps_inv (fps_inv a) = a" proof - let ?r = "fps_inv" have ra0: "?r a $ 0 = 0" by (simp add: fps_inv_def) from a1 have ra1: "?r a $ 1 ≠ 0" by (simp add: fps_inv_def field_simps) have fps_X0: "fps_X$0 = 0" by simp from fps_inv[OF ra0 ra1] have "?r (?r a) oo ?r a = fps_X" . then have "?r (?r a) oo ?r a oo a = fps_X oo a" by simp then have "?r (?r a) oo (?r a oo a) = a" unfolding fps_X_fps_compose_startby0[OF a0] unfolding fps_compose_assoc[OF a0 ra0, symmetric] . then show ?thesis unfolding fps_inv[OF a0 a1] by simp qed lemma fps_ginv_ginv: assumes a0: "a$0 = 0" and a1: "a$1 ≠ 0" and c0: "c$0 = 0" and c1: "c$1 ≠ 0" shows "fps_ginv b (fps_ginv c a) = b oo a oo fps_inv c" proof - let ?r = "fps_ginv" from c0 have rca0: "?r c a $0 = 0" 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 fps_X a" proof - let ?ia = "fps_ginv b a" let ?ifps_Xa = "fps_ginv fps_X a" let ?d = "fps_deriv" let ?dia = "?d ?ia" have ifps_Xa0: "?ifps_Xa $ 0 = 0" 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 with a1 have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a" by (simp add: fps_divide_unit) then have "(?d ?ia oo a) oo ?ifps_Xa = (?d b / ?d a) oo ?ifps_Xa" unfolding inverse_mult_eq_1[OF da0] by simp then have "?d ?ia oo (a oo ?ifps_Xa) = (?d b / ?d a) oo ?ifps_Xa" unfolding fps_compose_assoc[OF ifps_Xa0 a0] . then show ?thesis unfolding fps_inv_ginv[symmetric] unfolding fps_inv_right[OF a0 a1] by simp qed lemma fps_compose_linear: "fps_compose (f :: 'a :: comm_ring_1 fps) (fps_const c * fps_X) = Abs_fps (λn. c^n * f $ n)" by (simp add: fps_eq_iff fps_compose_def power_mult_distrib if_distrib sum.delta' cong: if_cong) lemma fps_compose_uminus': "fps_compose f (-fps_X :: 'a :: comm_ring_1 fps) = Abs_fps (λn. (-1)^n * f $ n)" using fps_compose_linear[of f "-1"] by (simp only: fps_const_neg [symmetric] fps_const_1_eq_1) simp subsection ‹Elementary series› subsubsection ‹Exponential series› definition "fps_exp x = Abs_fps (λn. x^n / of_nat (fact n))" lemma fps_exp_deriv[simp]: "fps_deriv (fps_exp a) = fps_const (a::'a::field_char_0) * fps_exp a" (is "?l = ?r") proof - have "?l$n = ?r $ n" for n apply (auto simp add: fps_exp_def field_simps power_Suc[symmetric] simp del: fact_Suc of_nat_Suc power_Suc) apply (simp add: field_simps) done then show ?thesis by (simp add: fps_eq_iff) qed lemma fps_exp_unique_ODE: "fps_deriv a = fps_const c * a ⟷ a = fps_const (a$0) * fps_exp (c::'a::field_char_0)" (is "?lhs ⟷ ?rhs") proof show ?rhs if ?lhs proof - from that have th: "⋀n. a $ Suc n = c * a$n / of_nat (Suc n)" by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc) have th': "a$n = a$0 * c ^ n/ (fact n)" for n proof (induct n) case 0 then show ?case by simp next case Suc then show ?case unfolding th using fact_gt_zero apply (simp add: field_simps del: of_nat_Suc fact_Suc) apply simp done qed show ?thesis by (auto simp add: fps_eq_iff fps_const_mult_left fps_exp_def intro: th') qed show ?lhs if ?rhs using that by (metis fps_exp_deriv fps_deriv_mult_const_left mult.left_commute) qed lemma fps_exp_add_mult: "fps_exp (a + b) = fps_exp (a::'a::field_char_0) * fps_exp b" (is "?l = ?r") proof - have "fps_deriv ?r = fps_const (a + b) * ?r" by (simp add: fps_const_add[symmetric] field_simps del: fps_const_add) then have "?r = ?l" by (simp only: fps_exp_unique_ODE) (simp add: fps_mult_nth fps_exp_def) then show ?thesis .. qed lemma fps_exp_nth[simp]: "fps_exp a $ n = a^n / of_nat (fact n)" by (simp add: fps_exp_def) lemma fps_exp_0[simp]: "fps_exp (0::'a::field) = 1" by (simp add: fps_eq_iff power_0_left) lemma fps_exp_neg: "fps_exp (- a) = inverse (fps_exp (a::'a::field_char_0))" proof - from fps_exp_add_mult[of a "- a"] have th0: "fps_exp a * fps_exp (- a) = 1" by simp from fps_inverse_unique[OF th0] show ?thesis by simp qed lemma fps_exp_nth_deriv[simp]: "fps_nth_deriv n (fps_exp (a::'a::field_char_0)) = (fps_const a)^n * (fps_exp a)" by (induct n) auto lemma fps_X_compose_fps_exp[simp]: "fps_X oo fps_exp (a::'a::field) = fps_exp a - 1" by (simp add: fps_eq_iff fps_X_fps_compose) lemma fps_inv_fps_exp_compose: assumes a: "a ≠ 0" shows "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X" and "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X" proof - let ?b = "fps_exp a - 1" have b0: "?b $ 0 = 0" by simp have b1: "?b $ 1 ≠ 0" by (simp add: a) from fps_inv[OF b0 b1] show "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X" . from fps_inv_right[OF b0 b1] show "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X" . qed lemma fps_exp_power_mult: "(fps_exp (c::'a::field_char_0))^n = fps_exp (of_nat n * c)" by (induct n) (auto simp add: field_simps fps_exp_add_mult) lemma radical_fps_exp: assumes r: "r (Suc k) 1 = 1" shows "fps_radical r (Suc k) (fps_exp (c::'a::field_char_0)) = fps_exp (c / of_nat (Suc k))" proof - let ?ck = "(c / of_nat (Suc k))" let ?r = "fps_radical r (Suc k)" have eq0[simp]: "?ck * of_nat (Suc k) = c" "of_nat (Suc k) * ?ck = c" by (simp_all del: of_nat_Suc) have th0: "fps_exp ?ck ^ (Suc k) = fps_exp c" unfolding fps_exp_power_mult eq0 .. have th: "r (Suc k) (fps_exp c $0) ^ Suc k = fps_exp c $ 0" "r (Suc k) (fps_exp c $ 0) = fps_exp ?ck $ 0" "fps_exp c $ 0 ≠ 0" using r by simp_all from th0 radical_unique[where r=r and k=k, OF th] show ?thesis by auto qed lemma fps_exp_compose_linear [simp]: "fps_exp (d::'a::field_char_0) oo (fps_const c * fps_X) = fps_exp (c * d)" by (simp add: fps_compose_linear fps_exp_def fps_eq_iff power_mult_distrib) lemma fps_fps_exp_compose_minus [simp]: "fps_compose (fps_exp c) (-fps_X) = fps_exp (-c :: 'a :: field_char_0)" using fps_exp_compose_linear[of c "-1 :: 'a"] unfolding fps_const_neg [symmetric] fps_const_1_eq_1 by simp lemma fps_exp_eq_iff [simp]: "fps_exp c = fps_exp d ⟷ c = (d :: 'a :: field_char_0)" proof assume "fps_exp c = fps_exp d" from arg_cong[of _ _ "λF. F $ 1", OF this] show "c = d" by simp qed simp_all lemma fps_exp_eq_fps_const_iff [simp]: "fps_exp (c :: 'a :: field_char_0) = fps_const c' ⟷ c = 0 ∧ c' = 1" proof assume "c = 0 ∧ c' = 1" thus "fps_exp c = fps_const c'" by (auto simp: fps_eq_iff) next assume "fps_exp c = fps_const c'" from arg_cong[of _ _ "λF. F $ 1", OF this] arg_cong[of _ _ "λF. F $ 0", OF this] show "c = 0 ∧ c' = 1" by simp_all qed lemma fps_exp_neq_0 [simp]: "¬fps_exp (c :: 'a :: field_char_0) = 0" unfolding fps_const_0_eq_0 [symmetric] fps_exp_eq_fps_const_iff by simp lemma fps_exp_eq_1_iff [simp]: "fps_exp (c :: 'a :: field_char_0) = 1 ⟷ c = 0" unfolding fps_const_1_eq_1 [symmetric] fps_exp_eq_fps_const_iff by simp lemma fps_exp_neq_numeral_iff [simp]: "fps_exp (c :: 'a :: field_char_0) = numeral n ⟷ c = 0 ∧ n = Num.One" unfolding numeral_fps_const fps_exp_eq_fps_const_iff by simp subsubsection ‹Logarithmic series› lemma Abs_fps_if_0: "Abs_fps (λn. if n = 0 then (v::'a::ring_1) else f n) = fps_const v + fps_X * Abs_fps (λn. f (Suc n))" by (auto simp add: fps_eq_iff) definition fps_ln :: "'a::field_char_0 ⇒ 'a fps" where "fps_ln c = fps_const (1/c) * Abs_fps (λn. if n = 0 then 0 else (- 1) ^ (n - 1) / of_nat n)" lemma fps_ln_deriv: "fps_deriv (fps_ln c) = fps_const (1/c) * inverse (1 + fps_X)" unfolding fps_inverse_fps_X_plus1 by (simp add: fps_ln_def fps_eq_iff del: of_nat_Suc) lemma fps_ln_nth: "fps_ln c $ n = (if n = 0 then 0 else 1/c * ((- 1) ^ (n - 1) / of_nat n))" by (simp add: fps_ln_def field_simps) lemma fps_ln_0 [simp]: "fps_ln c $ 0 = 0" by (simp add: fps_ln_def) lemma fps_ln_fps_exp_inv: fixes a :: "'a::field_char_0" assumes a: "a ≠ 0" shows "fps_ln a = fps_inv (fps_exp a - 1)" (is "?l = ?r") proof - let ?b = "fps_exp a - 1" have b0: "?b $ 0 = 0" by simp have b1: "?b $ 1 ≠ 0" by (simp add: a) have "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) = (fps_const a * (fps_exp a - 1) + fps_const a) oo fps_inv (fps_exp a - 1)" by (simp add: field_simps) also have "… = fps_const a * (fps_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 (fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_const a * (fps_X + 1)" . from fps_inv_deriv[OF b0 b1, unfolded eq] have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (fps_X + 1)" using a by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult) then have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_ln_deriv add.commute fps_divide_def divide_inverse) then show ?thesis unfolding fps_deriv_eq_iff by (simp add: fps_ln_nth fps_inv_def) qed lemma fps_ln_mult_add: assumes c0: "c≠0" and d0: "d≠0" shows "fps_ln c + fps_ln d = fps_const (c+d) * fps_ln (c*d)" (is "?r = ?l") proof- from c0 d0 have eq: "1/c + 1/d = (c+d)/(c*d)" by (simp add: field_simps) have "fps_deriv ?r = fps_const (1/c + 1/d) * inverse (1 + fps_X)" by (simp add: fps_ln_deriv fps_const_add[symmetric] algebra_simps del: fps_const_add) also have "… = fps_deriv ?l" apply (simp add: fps_ln_deriv) apply (simp add: fps_eq_iff eq) done finally show ?thesis unfolding fps_deriv_eq_iff by simp qed lemma fps_X_dvd_fps_ln [simp]: "fps_X dvd fps_ln c" proof - have "fps_ln c = fps_X * Abs_fps (λn. (-1) ^ n / (of_nat (Suc n) * c))" by (intro fps_ext) (auto simp: fps_ln_def of_nat_diff) thus ?thesis by simp qed subsubsection ‹Binomial series› definition "fps_binomial a = Abs_fps (λn. a gchoose n)" lemma fps_binomial_nth[simp]: "fps_binomial a $ n = a gchoose n" 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 + fps_X) ⟷ a = fps_const (a$0) * fps_binomial c" (is "?lhs ⟷ ?rhs") proof let ?da = "fps_deriv a" let ?x1 = "(1 + fps_X):: 'a fps" let ?l = "?x1 * ?da" let ?r = "fps_const c * a" have eq: "?l = ?r ⟷ ?lhs" proof - have x10: "?x1 $ 0 ≠ 0" by simp have "?l = ?r ⟷ inverse ?x1 * ?l = inverse ?x1 * ?r" by simp also have "… ⟷ ?da = (fps_const c * a) / ?x1" apply (simp only: fps_divide_def mult.assoc[symmetric] inverse_mult_eq_1[OF x10]) apply (simp add: field_simps) done finally show ?thesis . qed show ?rhs if ?lhs proof - from eq that have h: "?l = ?r" .. have th0: "a$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a $n" for n proof - from h have "?l $ n = ?r $ n" by simp then show ?thesis apply (simp add: field_simps del: of_nat_Suc) apply (cases n) apply (simp_all add: field_simps del: of_nat_Suc) done qed have th1: "a $ n = (c gchoose n) * a $ 0" for n proof (induct n) case 0 then show ?case by simp next case (Suc m) then show ?case unfolding th0 apply (simp add: field_simps del: of_nat_Suc) unfolding mult.assoc[symmetric] gbinomial_mult_1 apply (simp add: field_simps) done qed show ?thesis apply (simp add: fps_eq_iff) apply (subst th1) apply (simp add: field_simps) done qed show ?lhs if ?rhs proof - have th00: "x * (a $ 0 * y) = a $ 0 * (x * y)" for x y by (simp add: mult.commute) have "?l = ?r" apply (subst ‹?rhs›) apply (subst (2) ‹?rhs›) apply (clarsimp simp add: fps_eq_iff field_simps) unfolding mult.assoc[symmetric] th00 gbinomial_mult_1 apply (simp add: field_simps gbinomial_mult_1) done with eq show ?thesis .. qed qed lemma fps_binomial_ODE_unique': "(fps_deriv a = fps_const c * a / (1 + fps_X) ∧ a $ 0 = 1) ⟷ (a = fps_binomial c)" by (subst fps_binomial_ODE_unique) auto lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + fps_X)" proof - let ?a = "fps_binomial c" have th0: "?a = fps_const (?a$0) * ?a" by (simp) from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis . qed lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r") proof - let ?P = "?r - ?l" let ?b = "fps_binomial" let ?db = "λx. fps_deriv (?b x)" have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)" by simp also have "… = inverse (1 + fps_X) * (fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))" unfolding fps_binomial_deriv by (simp add: fps_divide_def field_simps) also have "… = (fps_const (c + d)/ (1 + fps_X)) * ?P" by (simp add: field_simps fps_divide_unit fps_const_add[symmetric] del: fps_const_add) finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + fps_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 then have "?P = 0" by (simp add: fps_mult_nth) then show ?thesis by simp qed lemma fps_binomial_minus_one: "fps_binomial (- 1) = inverse (1 + fps_X)" (is "?l = inverse ?r") proof- have th: "?r$0 ≠ 0" by simp have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + fps_X)" by (simp add: fps_inverse_deriv[OF th] fps_divide_def power2_eq_square mult.commute fps_const_neg[symmetric] del: fps_const_neg) have eq: "inverse ?r $ 0 = 1" by (simp add: fps_inverse_def) from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + fps_X)" "- 1"] th'] eq show ?thesis by (simp add: fps_inverse_def) qed lemma fps_binomial_of_nat: "fps_binomial (of_nat n) = (1 + fps_X :: 'a :: field_char_0 fps) ^ n" proof (cases "n = 0") case [simp]: True have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = 0" by simp also have "… = fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)" by (simp add: fps_binomial_def) finally show ?thesis by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) simp_all next case False have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = fps_const (of_nat n) * (1 + fps_X) ^ (n - 1)" by (simp add: fps_deriv_power) also have "(1 + fps_X :: 'a fps) $ 0 ≠ 0" by simp hence "(1 + fps_X :: 'a fps) ≠ 0" by (intro notI) (simp only: , simp) with False have "(1 + fps_X :: 'a fps) ^ (n - 1) = (1 + fps_X) ^ n / (1 + fps_X)" by (cases n) (simp_all ) also have "fps_const (of_nat n :: 'a) * ((1 + fps_X) ^ n / (1 + fps_X)) = fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)" by (simp add: unit_div_mult_swap) finally show ?thesis by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) (simp_all add: fps_power_nth) qed lemma fps_binomial_0 [simp]: "fps_binomial 0 = 1" using fps_binomial_of_nat[of 0] by simp lemma fps_binomial_power: "fps_binomial a ^ n = fps_binomial (of_nat n * a)" by (induction n) (simp_all add: fps_binomial_add_mult ring_distribs) lemma fps_binomial_1: "fps_binomial 1 = 1 + fps_X" using fps_binomial_of_nat[of 1] by simp lemma fps_binomial_minus_of_nat: "fps_binomial (- of_nat n) = inverse ((1 + fps_X :: 'a :: field_char_0 fps) ^ n)" by (rule sym, rule fps_inverse_unique) (simp add: fps_binomial_of_nat [symmetric] fps_binomial_add_mult [symmetric]) lemma one_minus_const_fps_X_power: "c ≠ 0 ⟹ (1 - fps_const c * fps_X) ^ n = fps_compose (fps_binomial (of_nat n)) (-fps_const c * fps_X)" by (subst fps_binomial_of_nat) (simp add: fps_compose_power [symmetric] fps_compose_add_distrib fps_const_neg [symmetric] del: fps_const_neg) lemma one_minus_fps_X_const_neg_power: "inverse ((1 - fps_const c * fps_X) ^ n) = fps_compose (fps_binomial (-of_nat n)) (-fps_const c * fps_X)" proof (cases "c = 0") case False thus ?thesis by (subst fps_binomial_minus_of_nat) (simp add: fps_compose_power [symmetric] fps_inverse_compose fps_compose_add_distrib fps_const_neg [symmetric] del: fps_const_neg) qed simp lemma fps_X_plus_const_power: "c ≠ 0 ⟹ (fps_X + fps_const c) ^ n = fps_const (c^n) * fps_compose (fps_binomial (of_nat n)) (fps_const (inverse c) * fps_X)" by (subst fps_binomial_of_nat) (simp add: fps_compose_power [symmetric] fps_binomial_of_nat fps_compose_add_distrib fps_const_power [symmetric] power_mult_distrib [symmetric] algebra_simps inverse_mult_eq_1' del: fps_const_power) lemma fps_X_plus_const_neg_power: "c ≠ 0 ⟹ inverse ((fps_X + fps_const c) ^ n) = fps_const (inverse c^n) * fps_compose (fps_binomial (-of_nat n)) (fps_const (inverse c) * fps_X)" by (subst fps_binomial_minus_of_nat) (simp add: fps_compose_power [symmetric] fps_binomial_of_nat fps_compose_add_distrib fps_const_power [symmetric] power_mult_distrib [symmetric] fps_inverse_compose algebra_simps fps_const_inverse [symmetric] fps_inverse_mult [symmetric] fps_inverse_power [symmetric] inverse_mult_eq_1' del: fps_const_power) lemma one_minus_const_fps_X_neg_power': "n > 0 ⟹ inverse ((1 - fps_const (c :: 'a :: field_char_0) * fps_X) ^ n) = Abs_fps (λk. of_nat ((n + k - 1) choose k) * c^k)" apply (rule fps_ext) apply (subst one_minus_fps_X_const_neg_power, subst fps_const_neg, subst fps_compose_linear) apply (simp add: power_mult_distrib [symmetric] mult.assoc [symmetric] gbinomial_minus binomial_gbinomial of_nat_diff) done text ‹Vandermonde's Identity as a consequence.› lemma gbinomial_Vandermonde: "sum (λk. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n" proof - let ?ba = "fps_binomial a" let ?bb = "fps_binomial b" let ?bab = "fps_binomial (a + b)" from fps_binomial_add_mult[of a b] have "?bab $ n = (?ba * ?bb)$n" by simp then show ?thesis by (simp add: fps_mult_nth) qed lemma binomial_Vandermonde: "sum (λk. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n" using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n] by (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric] of_nat_sum[symmetric] of_nat_add[symmetric] of_nat_eq_iff) lemma binomial_Vandermonde_same: "sum (λk. (n choose k)⇧^{2}) {0..n} = (2 * n) choose n" using binomial_Vandermonde[of n n n, symmetric] unfolding mult_2 apply (simp add: power2_eq_square) apply (rule sum.cong) apply (auto intro: binomial_symmetric) done lemma Vandermonde_pochhammer_lemma: fixes a :: "'a::field_char_0" assumes b: "∀j∈{0 ..<n}. b ≠ of_nat j" shows "sum (λk. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) / (of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} = pochhammer (- (a + b)) n / pochhammer (- b) n" (is "?l = ?r") proof - let ?m1 = "λm. (- 1 :: 'a) ^ m" let ?f = "λm. of_nat (fact m)" let ?p = "λ(x::'a). pochhammer (- x)" from b have bn0: "?p b n ≠ 0" unfolding pochhammer_eq_0_iff by simp have th00: "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" (is ?gchoose) "pochhammer (1 + b - of_nat n) k ≠ 0" (is ?pochhammer) if kn: "k ∈ {0..n}" for k proof - from kn have "k ≤ n" by simp have nz: "pochhammer (1 + b - of_nat n) n ≠ 0" proof assume "pochhammer (1 + b - of_nat n) n = 0" then have c: "pochhammer (b - of_nat n + 1) n = 0" 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 show False using j by auto qed from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k ≠ 0" by (rule pochhammer_neq_0_mono) consider "k = 0 ∨ n = 0" | "k ≠ 0" "n ≠ 0" by blast then have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" proof cases case 1 then show ?thesis using kn by (cases "k = 0") (simp_all add: gbinomial_pochhammer) next case neq: 2 then obtain m where m: "n = Suc m" by (cases n) auto from neq(1) obtain h where h: "k = Suc h" by (cases k) auto show ?thesis proof (cases "k = n") case True then show ?thesis using pochhammer_minus'[where k=k and b=b] apply (simp add: pochhammer_same) using bn0 apply (simp add: field_simps power_add[symmetric]) done next case False with kn have kn': "k < n" by simp have m1nk: "?m1 n = prod (λi. - 1) {..m}" "?m1 k = prod (λi. - 1) {0..h}" by (simp_all add: prod_constant m h) have bnz0: "pochhammer (b - of_nat n + 1) k ≠ 0" using bn0 kn unfolding pochhammer_eq_0_iff apply auto apply (erule_tac x= "n - ka - 1" in allE) apply (auto simp add: algebra_simps of_nat_diff) done have eq1: "prod (λk. (1::'a) + of_nat m - of_nat k) {..h} = prod of_nat {Suc (m - h) .. Suc m}" using kn' h m by (intro prod.reindex_bij_witness[where i="λk. Suc m - k" and j="λk. Suc m - k"]) (auto simp: of_nat_diff) have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))" apply (simp add: pochhammer_minus field_simps) using ‹k ≤ n› apply (simp add: fact_split [of k n]) apply (simp add: pochhammer_prod) using prod.atLeastLessThan_shift_bounds [where ?'a = 'a, of "λi. 1 + of_nat i" 0 "n - k" k] apply (auto simp add: of_nat_diff field_simps) done have th20: "?m1 n * ?p b n = prod (λi. b - of_nat i) {0..m}" apply (simp add: pochhammer_minus field_simps m) apply (auto simp add: pochhammer_prod_rev of_nat_diff prod.atLeast_Suc_atMost_Suc_shift) done have th21:"pochhammer (b - of_nat n + 1) k = prod (λi. b - of_nat i) {n - k .. n - 1}" using kn apply (simp add: pochhammer_prod_rev m h prod.atLeast_Suc_atMost_Suc_shift) using prod.atLeastAtMost_shift_0 [of "m - h" m, where ?'a = 'a] apply (auto simp add: of_nat_diff field_simps) done have "?m1 n * ?p b n = prod (λi. b - of_nat i) {0.. n - k - 1} * pochhammer (b - of_nat n + 1) k" using kn' m h unfolding th20 th21 apply simp