Up to index of Isabelle/HOL/ex
theory Formal_Power_Series(* Title: Formal_Power_Series.thy Author: Amine Chaieb, University of Cambridge *) header{* A formalization of formal power series *} theory Formal_Power_Series imports Main Fact Parity begin subsection {* The type of formal power series*} typedef (open) '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] expand_fun_eq) lemma fps_ext: "(!!n. p $ n = q $ n) ==> p = q" by (simp add: expand_fps_eq) lemma fps_nth_Abs_fps [simp]: "Abs_fps f $ n = f n" by (simp add: Abs_fps_inverse) text{* Definition of the basic elements 0 and 1 and the basic operations of addition, negation and multiplication *} instantiation fps :: (zero) zero begin definition fps_zero_def: "0 = Abs_fps (λn. 0)" instance .. end lemma fps_zero_nth [simp]: "0 $ n = 0" unfolding fps_zero_def by simp instantiation fps :: ("{one,zero}") one begin definition fps_one_def: "1 = Abs_fps (λn. if n = 0 then 1 else 0)" instance .. end lemma fps_one_nth [simp]: "1 $ n = (if n = 0 then 1 else 0)" unfolding fps_one_def by simp instantiation fps :: (plus) plus begin definition fps_plus_def: "op + = (λf g. Abs_fps (λn. f $ n + g $ n))" instance .. end lemma fps_add_nth [simp]: "(f + g) $ n = f $ n + g $ n" unfolding fps_plus_def by simp instantiation fps :: (minus) minus begin definition fps_minus_def: "op - = (λf g. Abs_fps (λn. f $ n - g $ n))" instance .. end lemma fps_sub_nth [simp]: "(f - g) $ n = f $ n - g $ n" unfolding fps_minus_def by simp instantiation fps :: (uminus) uminus begin definition fps_uminus_def: "uminus = (λf. Abs_fps (λn. - (f $ n)))" instance .. end lemma fps_neg_nth [simp]: "(- f) $ n = - (f $ n)" unfolding fps_uminus_def by simp instantiation fps :: ("{comm_monoid_add, times}") times begin definition fps_times_def: "op * = (λf g. Abs_fps (λn. ∑i=0..n. f $ i * g $ (n - i)))" instance .. end lemma fps_mult_nth: "(f * g) $ n = (∑i=0..n. f$i * g$(n - i))" unfolding fps_times_def by simp declare atLeastAtMost_iff[presburger] declare Bex_def[presburger] declare Ball_def[presburger] lemma mult_delta_left: fixes x y :: "'a::mult_zero" shows "(if b then x else 0) * y = (if b then x * y else 0)" by simp lemma mult_delta_right: fixes x y :: "'a::mult_zero" shows "x * (if b then y else 0) = (if b then x * y else 0)" by simp lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)" by auto lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)" by auto subsection{* Formal power series form a commutative ring with unity, if the range of sequences they represent is a commutative ring with unity*} instance fps :: (semigroup_add) semigroup_add proof fix a b c :: "'a fps" show "a + b + c = a + (b + c)" by (simp add: fps_ext add_assoc) qed instance fps :: (ab_semigroup_add) ab_semigroup_add proof fix a b :: "'a fps" show "a + b = b + a" by (simp add: fps_ext add_commute) qed lemma fps_mult_assoc_lemma: fixes k :: nat and f :: "nat => nat => nat => 'a::comm_monoid_add" shows "(∑j=0..k. ∑i=0..j. f i (j - i) (n - j)) = (∑j=0..k. ∑i=0..k - j. f j i (n - j - i))" proof (induct k) case 0 show ?case by simp next case (Suc k) thus ?case by (simp add: Suc_diff_le setsum_addf add_assoc cong: strong_setsum_cong) qed 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) thus "((a * b) * c) $ n = (a * (b * c)) $ n" by (simp add: fps_mult_nth setsum_right_distrib setsum_left_distrib mult_assoc) qed qed lemma fps_mult_commute_lemma: fixes n :: nat and f :: "nat => nat => 'a::comm_monoid_add" shows "(∑i=0..n. f i (n - i)) = (∑i=0..n. f (n - i) i)" proof (rule setsum_reindex_cong) show "inj_on (λi. n - i) {0..n}" by (rule inj_onI) simp show "{0..n} = (λi. n - i) ` {0..n}" by (auto, rule_tac x="n - x" in image_eqI, simp_all) next fix i assume "i ∈ {0..n}" hence "n - (n - i) = i" by simp thus "f (n - i) i = f (n - i) (n - (n - i))" by simp qed instance fps :: (comm_semiring_0) ab_semigroup_mult proof fix a b :: "'a fps" show "a * b = b * a" proof (rule fps_ext) fix n :: nat have "(∑i=0..n. a$i * b$(n - i)) = (∑i=0..n. a$(n - i) * b$i)" by (rule fps_mult_commute_lemma) thus "(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) next fix a :: "'a fps" show "a + 0 = a " by (simp add: fps_ext) qed instance fps :: (comm_monoid_add) comm_monoid_add proof fix a :: "'a fps" show "0 + a = a " by (simp add: fps_ext) qed instance fps :: (semiring_1) monoid_mult proof fix a :: "'a fps" show "1 * a = a" by (simp add: fps_ext fps_mult_nth mult_delta_left setsum_delta) next fix a :: "'a fps" show "a * 1 = a" by (simp add: fps_ext fps_mult_nth mult_delta_right setsum_delta') qed instance fps :: (cancel_semigroup_add) cancel_semigroup_add proof fix a b c :: "'a fps" assume "a + b = a + c" then show "b = c" by (simp add: expand_fps_eq) next fix a b c :: "'a fps" assume "b + a = c + a" then show "b = c" by (simp add: expand_fps_eq) qed instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add proof fix a b c :: "'a fps" assume "a + b = a + c" then show "b = c" by (simp add: expand_fps_eq) qed instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add .. instance fps :: (group_add) group_add proof fix a :: "'a fps" show "- a + a = 0" by (simp add: fps_ext) next fix a b :: "'a fps" show "a - b = a + - b" by (simp add: fps_ext diff_minus) qed instance fps :: (ab_group_add) ab_group_add proof fix a :: "'a fps" show "- a + a = 0" by (simp add: fps_ext) next fix a b :: "'a fps" show "a - b = a + - b" by (simp add: fps_ext) qed instance fps :: (zero_neq_one) zero_neq_one by default (simp add: expand_fps_eq) instance fps :: (semiring_0) semiring proof fix a b c :: "'a fps" show "(a + b) * c = a * c + b * c" by (simp add: expand_fps_eq fps_mult_nth left_distrib setsum_addf) next fix a b c :: "'a fps" show "a * (b + c) = a * b + a * c" by (simp add: expand_fps_eq fps_mult_nth right_distrib setsum_addf) qed instance fps :: (semiring_0) semiring_0 proof fix a:: "'a fps" show "0 * a = 0" by (simp add: fps_ext fps_mult_nth) next fix a:: "'a fps" show "a * 0 = 0" by (simp add: fps_ext fps_mult_nth) qed instance fps :: (semiring_0_cancel) semiring_0_cancel .. subsection {* Selection of the nth power of the implicit variable in the infinite sum*} lemma fps_nonzero_nth: "f ≠ 0 <-> (∃ n. f $n ≠ 0)" by (simp add: expand_fps_eq) lemma fps_nonzero_nth_minimal: "f ≠ 0 <-> (∃n. f $ n ≠ 0 ∧ (∀m<n. f $ m = 0))" proof let ?n = "LEAST n. f $ n ≠ 0" assume "f ≠ 0" then have "∃n. f $ n ≠ 0" by (simp add: fps_nonzero_nth) then have "f $ ?n ≠ 0" by (rule LeastI_ex) moreover have "∀m<?n. f $ m = 0" by (auto dest: not_less_Least) ultimately have "f $ ?n ≠ 0 ∧ (∀m<?n. f $ m = 0)" .. then show "∃n. f $ n ≠ 0 ∧ (∀m<n. f $ m = 0)" .. next assume "∃n. f $ n ≠ 0 ∧ (∀m<n. f $ m = 0)" then show "f ≠ 0" by (auto simp add: expand_fps_eq) qed lemma fps_eq_iff: "f = g <-> (∀n. f $ n = g $n)" by (rule expand_fps_eq) lemma fps_setsum_nth: "(setsum f S) $ n = setsum (λk. (f k) $ n) S" proof (cases "finite S") assume "¬ finite S" then show ?thesis by simp next assume "finite S" then show ?thesis by (induct set: finite) auto 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_mult[simp]: "fps_const (c::'a::ring) * fps_const d = fps_const (c * d)" by (simp add: fps_eq_iff fps_mult_nth setsum_0') lemma fps_const_add_left: "fps_const (c::'a::monoid_add) + f = Abs_fps (λn. if n = 0 then c + f$0 else f$n)" by (simp add: fps_ext) lemma fps_const_add_right: "f + fps_const (c::'a::monoid_add) = Abs_fps (λn. if n = 0 then f$0 + c else f$n)" by (simp add: fps_ext) lemma fps_const_mult_left: "fps_const (c::'a::semiring_0) * f = Abs_fps (λn. c * f$n)" unfolding fps_eq_iff fps_mult_nth by (simp add: fps_const_def mult_delta_left setsum_delta) lemma fps_const_mult_right: "f * fps_const (c::'a::semiring_0) = Abs_fps (λn. f$n * c)" unfolding fps_eq_iff fps_mult_nth by (simp add: fps_const_def mult_delta_right setsum_delta') lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n" by (simp add: fps_mult_nth mult_delta_left setsum_delta) lemma fps_mult_right_const_nth [simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c" by (simp add: fps_mult_nth mult_delta_right setsum_delta') subsection {* Formal power series form an integral domain*} instance fps :: (ring) ring .. instance fps :: (ring_1) ring_1 by (intro_classes, auto simp add: diff_minus left_distrib) instance fps :: (comm_ring_1) comm_ring_1 by (intro_classes, auto simp add: diff_minus left_distrib) instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors proof fix a b :: "'a fps" assume a0: "a ≠ 0" and b0: "b ≠ 0" then obtain i j where i: "a$i≠0" "∀k<i. a$k=0" and j: "b$j ≠0" "∀k<j. b$k =0" unfolding fps_nonzero_nth_minimal by blast+ have "(a * b) $ (i+j) = (∑k=0..i+j. a$k * b$(i+j-k))" by (rule fps_mult_nth) also have "… = (a$i * b$(i+j-i)) + (∑k∈{0..i+j}-{i}. a$k * b$(i+j-k))" by (rule setsum_diff1') simp_all also have "(∑k∈{0..i+j}-{i}. a$k * b$(i+j-k)) = 0" proof (rule setsum_0' [rule_format]) fix k assume "k ∈ {0..i+j} - {i}" then have "k < i ∨ i+j-k < j" by auto then show "a$k * b$(i+j-k) = 0" using i j by auto qed also have "a$i * b$(i+j-i) + 0 = a$i * b$j" by simp also have "a$i * b$j ≠ 0" using i j by simp finally have "(a*b) $ (i+j) ≠ 0" . then show "a*b ≠ 0" unfolding fps_nonzero_nth by blast qed instance fps :: (idom) idom .. instantiation fps :: (comm_ring_1) number_ring begin definition number_of_fps_def: "(number_of k::'a fps) = of_int k" instance by (intro_classes, rule number_of_fps_def) end subsection{* Inverses of formal power series *} declare setsum_cong[fundef_cong] instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}") inverse begin fun natfun_inverse:: "'a fps => nat => 'a" where "natfun_inverse f 0 = inverse (f$0)" | "natfun_inverse f n = - inverse (f$0) * setsum (λi. f$i * natfun_inverse f (n - i)) {1..n}" definition fps_inverse_def: "inverse f = (if f$0 = 0 then 0 else Abs_fps (natfun_inverse f))" definition fps_divide_def: "divide = (λ(f::'a fps) g. f * inverse g)" instance .. end lemma fps_inverse_zero[simp]: "inverse (0 :: 'a::{comm_monoid_add,inverse, times, uminus} fps) = 0" by (simp add: fps_ext fps_inverse_def) lemma fps_inverse_one[simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1" apply (auto simp add: expand_fps_eq fps_inverse_def) by (case_tac n, auto) instance fps :: ("{comm_monoid_add,inverse, times, uminus}") division_by_zero by default (rule fps_inverse_zero) lemma inverse_mult_eq_1[intro]: assumes f0: "f$0 ≠ (0::'a::field)" shows "inverse f * f = 1" proof- have c: "inverse f * f = f * inverse f" by (simp add: mult_commute) from f0 have ifn: "!!n. inverse f $ n = natfun_inverse f n" by (simp add: fps_inverse_def) from f0 have th0: "(inverse f * f) $ 0 = 1" by (simp add: fps_mult_nth fps_inverse_def) {fix n::nat assume np: "n >0 " from np have eq: "{0..n} = {0} ∪ {1 .. n}" by auto have d: "{0} ∩ {1 .. n} = {}" by auto have f: "finite {0::nat}" "finite {1..n}" by auto from f0 np have th0: "- (inverse f$n) = (setsum (λi. f$i * natfun_inverse f (n - i)) {1..n}) / (f$0)" by (cases n, simp, simp add: divide_inverse fps_inverse_def) from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]] have th1: "setsum (λi. f$i * natfun_inverse f (n - i)) {1..n} = - (f$0) * (inverse f)$n" by (simp add: ring_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))" unfolding setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] by simp also have "… = 0" unfolding th1 ifn by simp finally have "(inverse f * f)$n = 0" unfolding c . } with th0 show ?thesis by (simp add: fps_eq_iff) qed lemma fps_inverse_0_iff[simp]: "(inverse f)$0 = (0::'a::division_ring) <-> f$0 = 0" by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero) lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) <-> f $0 = 0" proof- {assume "f$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)} moreover {assume h: "inverse f = 0" and c: "f $0 ≠ 0" from inverse_mult_eq_1[OF c] h have False by simp} ultimately show ?thesis by blast qed lemma fps_inverse_idempotent[intro]: assumes f0: "f$0 ≠ (0::'a::field)" shows "inverse (inverse f) = f" proof- from f0 have if0: "inverse f $ 0 ≠ 0" by simp from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0] have th0: "inverse f * f = inverse f * inverse (inverse f)" by (simp add: mult_ac) then show ?thesis using f0 unfolding mult_cancel_left by simp qed lemma fps_inverse_unique: assumes f0: "f$0 ≠ (0::'a::field)" and fg: "f*g = 1" shows "inverse f = g" proof- from inverse_mult_eq_1[OF f0] fg have th0: "inverse f * f = g * f" by (simp add: mult_ac) then show ?thesis using f0 unfolding mult_cancel_right by (auto simp add: expand_fps_eq) qed lemma fps_inverse_gp: "inverse (Abs_fps(λn. (1::'a::field))) = Abs_fps (λn. if n= 0 then 1 else if n=1 then - 1 else 0)" apply (rule fps_inverse_unique) apply simp apply (simp add: fps_eq_iff fps_mult_nth) proof(clarsimp) fix n::nat assume n: "n > 0" let ?f = "λi. if n = i then (1::'a) else if n - i = 1 then - 1 else 0" let ?g = "λi. if i = n then 1 else if i=n - 1 then - 1 else 0" let ?h = "λi. if i=n - 1 then - 1 else 0" have th1: "setsum ?f {0..n} = setsum ?g {0..n}" by (rule setsum_cong2) auto have th2: "setsum ?g {0..n - 1} = setsum ?h {0..n - 1}" using n apply - by (rule setsum_cong2) auto have eq: "{0 .. n} = {0.. n - 1} ∪ {n}" by auto from n have d: "{0.. n - 1} ∩ {n} = {}" by auto have f: "finite {0.. n - 1}" "finite {n}" by auto show "setsum ?f {0..n} = 0" unfolding th1 apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def) unfolding th2 by(simp add: setsum_delta) qed subsection{* Formal Derivatives, and the MacLaurin theorem around 0*} definition "fps_deriv f = Abs_fps (λn. of_nat (n + 1) * f $ (n + 1))" lemma fps_deriv_nth[simp]: "fps_deriv f $ n = of_nat (n +1) * f $ (n+1)" by (simp add: fps_deriv_def) lemma fps_deriv_linear[simp]: "fps_deriv (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = fps_const a * fps_deriv f + fps_const b * fps_deriv g" unfolding fps_eq_iff fps_add_nth fps_const_mult_left fps_deriv_nth by (simp add: ring_simps) lemma fps_deriv_mult[simp]: fixes f :: "('a :: comm_ring_1) fps" shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g" proof- let ?D = "fps_deriv" {fix n::nat let ?Zn = "{0 ..n}" let ?Zn1 = "{0 .. n + 1}" let ?f = "λi. i + 1" have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def) have eq: "{1.. n+1} = ?f ` {0..n}" by auto let ?g = "λi. of_nat (i+1) * g $ (i+1) * f $ (n - i) + of_nat (i+1)* f $ (i+1) * g $ (n - i)" let ?h = "λi. of_nat i * g $ i * f $ ((n+1) - i) + of_nat i* f $ i * g $ ((n + 1) - i)" {fix k assume k: "k ∈ {0..n}" have "?h (k + 1) = ?g k" using k by auto} note th0 = this have eq': "{0..n +1}- {1 .. n+1} = {0}" by auto have s0: "setsum (λi. of_nat i * f $ i * g $ (n + 1 - i)) ?Zn1 = setsum (λi. of_nat (n + 1 - i) * f $ (n + 1 - i) * g $ i) ?Zn1" apply (rule setsum_reindex_cong[where f="λi. n + 1 - i"]) apply (simp add: inj_on_def Ball_def) apply presburger apply (rule set_ext) apply (presburger add: image_iff) by simp have s1: "setsum (λi. f $ i * g $ (n + 1 - i)) ?Zn1 = setsum (λi. f $ (n + 1 - i) * g $ i) ?Zn1" apply (rule setsum_reindex_cong[where f="λi. n + 1 - i"]) apply (simp add: inj_on_def Ball_def) apply presburger apply (rule set_ext) apply (presburger add: image_iff) by simp have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n" by (simp only: mult_commute) also have "… = (∑i = 0..n. ?g i)" by (simp add: fps_mult_nth setsum_addf[symmetric]) also have "… = setsum ?h {1..n+1}" using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto also have "… = setsum ?h {0..n+1}" apply (rule setsum_mono_zero_left) apply simp apply (simp add: subset_eq) unfolding eq' by simp also have "… = (fps_deriv (f * g)) $ n" apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf) unfolding s0 s1 unfolding setsum_addf[symmetric] setsum_right_distrib apply (rule setsum_cong2) by (auto simp add: of_nat_diff ring_simps) finally have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" .} then show ?thesis unfolding fps_eq_iff by auto qed lemma fps_deriv_neg[simp]: "fps_deriv (- (f:: ('a:: comm_ring_1) fps)) = - (fps_deriv f)" by (simp add: fps_eq_iff fps_deriv_def) lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g" using fps_deriv_linear[of 1 f 1 g] by simp lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) - g) = fps_deriv f - fps_deriv g" unfolding diff_minus by simp lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0" by (simp add: fps_ext fps_deriv_def fps_const_def) lemma fps_deriv_mult_const_left[simp]: "fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f" by simp lemma fps_deriv_0[simp]: "fps_deriv 0 = 0" by (simp add: fps_deriv_def fps_eq_iff) lemma fps_deriv_1[simp]: "fps_deriv 1 = 0" by (simp add: fps_deriv_def fps_eq_iff ) lemma fps_deriv_mult_const_right[simp]: "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c" by simp lemma fps_deriv_setsum: "fps_deriv (setsum f S) = setsum (λi. fps_deriv (f i :: ('a::comm_ring_1) fps)) S" proof- {assume "¬ finite S" hence ?thesis by simp} moreover {assume fS: "finite S" have ?thesis by (induct rule: finite_induct[OF fS], simp_all)} ultimately show ?thesis by blast qed lemma fps_deriv_eq_0_iff[simp]: "fps_deriv f = 0 <-> (f = fps_const (f$0 :: 'a::{idom,semiring_char_0}))" proof- {assume "f= fps_const (f$0)" hence "fps_deriv f = fps_deriv (fps_const (f$0))" by simp hence "fps_deriv f = 0" by simp } moreover {assume z: "fps_deriv f = 0" hence "∀n. (fps_deriv f)$n = 0" by simp hence "∀n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def) hence "f = fps_const (f$0)" apply (clarsimp simp add: fps_eq_iff fps_const_def) apply (erule_tac x="n - 1" in allE) by simp} ultimately show ?thesis by blast qed lemma fps_deriv_eq_iff: fixes f:: "('a::{idom,semiring_char_0}) fps" shows "fps_deriv f = fps_deriv g <-> (f = fps_const(f$0 - g$0) + g)" proof- have "fps_deriv f = fps_deriv g <-> fps_deriv (f - g) = 0" by simp also have "… <-> f - g = fps_const ((f-g)$0)" unfolding fps_deriv_eq_0_iff .. finally show ?thesis by (simp add: ring_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)" apply auto unfolding fps_deriv_eq_iff by blast fun fps_nth_deriv :: "nat => ('a::semiring_1) fps => 'a fps" where "fps_nth_deriv 0 f = f" | "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)" lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)" by (induct n arbitrary: f, auto) lemma fps_nth_deriv_linear[simp]: "fps_nth_deriv n (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g" by (induct n arbitrary: f g, auto simp add: fps_nth_deriv_commute) lemma fps_nth_deriv_neg[simp]: "fps_nth_deriv n (- (f:: ('a:: comm_ring_1) fps)) = - (fps_nth_deriv n f)" by (induct n arbitrary: f, simp_all) lemma fps_nth_deriv_add[simp]: "fps_nth_deriv n ((f:: ('a::comm_ring_1) fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g" using fps_nth_deriv_linear[of n 1 f 1 g] by simp lemma fps_nth_deriv_sub[simp]: "fps_nth_deriv n ((f:: ('a::comm_ring_1) fps) - g) = fps_nth_deriv n f - fps_nth_deriv n g" unfolding diff_minus fps_nth_deriv_add by simp lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0" by (induct n, simp_all ) lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)" by (induct n, simp_all ) lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)" by (cases n, simp_all) lemma fps_nth_deriv_mult_const_left[simp]: "fps_nth_deriv n (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_nth_deriv n f" using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp lemma fps_nth_deriv_mult_const_right[simp]: "fps_nth_deriv n (f * fps_const (c::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c" using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute) lemma fps_nth_deriv_setsum: "fps_nth_deriv n (setsum f S) = setsum (λi. fps_nth_deriv n (f i :: ('a::comm_ring_1) fps)) S" proof- {assume "¬ finite S" hence ?thesis by simp} moreover {assume fS: "finite S" have ?thesis by (induct rule: finite_induct[OF fS], simp_all)} ultimately show ?thesis by blast 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: ring_simps of_nat_mult) subsection {* Powers*} instantiation fps :: (semiring_1) power begin fun fps_pow :: "nat => 'a fps => 'a fps" where "fps_pow 0 f = 1" | "fps_pow (Suc n) f = f * fps_pow n f" definition fps_power_def: "power (f::'a fps) n = fps_pow n f" instance .. end instantiation fps :: (comm_ring_1) recpower begin instance apply (intro_classes) by (simp_all add: fps_power_def) end lemma fps_power_zeroth_eq_one: "a$0 =1 ==> a^n $ 0 = (1::'a::semiring_1)" by (induct n, auto simp add: fps_power_def 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 thus ?case by (simp add: fps_power_def) next case (Suc n) note h = Suc.hyps[OF `a$0 = 1`] show ?case unfolding power_Suc fps_mult_nth using h `a$0 = 1` fps_power_zeroth_eq_one[OF `a$0=1`] by (simp add: ring_simps) qed lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) ==> a^n $ 0 = 1" by (induct n, auto simp add: fps_power_def 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_power_def fps_mult_nth) lemma startsby_power:"a $0 = (v::'a::{comm_ring_1, recpower}) ==> a^n $0 = v^n" by (induct n, auto simp add: fps_power_def fps_mult_nth power_Suc) lemma startsby_zero_power_iff[simp]: "a^n $0 = (0::'a::{idom, recpower}) <-> (n ≠ 0 ∧ a$0 = 0)" apply (rule iffI) apply (induct n, auto simp add: power_Suc fps_mult_nth) by (rule startsby_zero_power, simp_all) lemma startsby_zero_power_prefix: assumes a0: "a $0 = (0::'a::idom)" shows "∀n < k. a ^ k $ n = 0" using a0 proof(induct k rule: nat_less_induct) fix k assume H: "∀m<k. a $0 = 0 --> (∀n<m. a ^ m $ n = 0)" and a0: "a $0 = (0::'a)" let ?ths = "∀m<k. a ^ k $ m = 0" {assume "k = 0" then have ?ths by simp} moreover {fix l assume k: "k = Suc l" {fix m assume mk: "m < k" {assume "m=0" hence "a^k $ m = 0" using startsby_zero_power[of a k] k a0 by simp} moreover {assume m0: "m ≠ 0" have "a ^k $ m = (a^l * a) $m" by (simp add: k power_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 setsum_0') apply auto apply (case_tac "aa = m") using a0 apply simp apply (rule H[rule_format]) using a0 k mk by auto finally have "a^k $ m = 0" .} ultimately have "a^k $ m = 0" by blast} hence ?ths by blast} ultimately show ?ths by (cases k, auto) qed lemma startsby_zero_setsum_depends: assumes a0: "a $0 = (0::'a::idom)" and kn: "n ≥ k" shows "setsum (λi. (a ^ i)$k) {0 .. n} = setsum (λi. (a ^ i)$k) {0 .. k}" apply (rule setsum_mono_zero_right) using kn apply auto apply (rule startsby_zero_power_prefix[rule_format, OF a0]) by arith lemma startsby_zero_power_nth_same: assumes a0: "a$0 = (0::'a::{recpower, idom})" shows "a^n $ n = (a$1) ^ n" proof(induct n) case 0 thus ?case by (simp add: power_0) next case (Suc n) have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)" by (simp add: ring_simps power_Suc) also have "… = setsum (λi. a^n$i * a $ (Suc n - i)) {0.. Suc n}" by (simp add: fps_mult_nth) also have "… = setsum (λi. a^n$i * a $ (Suc n - i)) {n .. Suc n}" apply (rule setsum_mono_zero_right) apply simp apply clarsimp apply clarsimp apply (rule startsby_zero_power_prefix[rule_format, OF a0]) apply arith done also have "… = a^n $ n * a$1" using a0 by simp finally show ?case using Suc.hyps by (simp add: power_Suc) qed lemma fps_inverse_power: fixes a :: "('a::{field, recpower}) fps" shows "inverse (a^n) = inverse a ^ n" proof- {assume a0: "a$0 = 0" hence eq: "inverse a = 0" by (simp add: fps_inverse_def) {assume "n = 0" hence ?thesis by simp} moreover {assume n: "n > 0" from startsby_zero_power[OF a0 n] eq a0 n have ?thesis by (simp add: fps_inverse_def)} ultimately have ?thesis by blast} moreover {assume a0: "a$0 ≠ 0" have ?thesis apply (rule fps_inverse_unique) apply (simp add: a0) unfolding power_mult_distrib[symmetric] apply (rule ssubst[where t = "a * inverse a" and s= 1]) apply simp_all apply (subst mult_commute) by (rule inverse_mult_eq_1[OF a0])} ultimately show ?thesis by blast qed lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n - 1)" apply (induct n, auto simp add: power_Suc ring_simps fps_const_add[symmetric] simp del: fps_const_add) by (case_tac n, auto simp add: power_Suc ring_simps) lemma fps_inverse_deriv: fixes a:: "('a :: field) fps" assumes a0: "a$0 ≠ 0" shows "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2" proof- from inverse_mult_eq_1[OF a0] have "fps_deriv (inverse a * a) = 0" by simp hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0" by simp with inverse_mult_eq_1[OF a0] have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0" unfolding power2_eq_square apply (simp add: ring_simps) by (simp add: mult_assoc[symmetric]) hence "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: ring_simps) qed lemma fps_inverse_mult: fixes a::"('a :: field) fps" shows "inverse (a * b) = inverse a * inverse b" proof- {assume a0: "a$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth) from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all have ?thesis unfolding th by simp} moreover {assume b0: "b$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth) from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all have ?thesis unfolding th by simp} moreover {assume a0: "a$0 ≠ 0" and b0: "b$0 ≠ 0" from a0 b0 have ab0:"(a*b) $ 0 ≠ 0" by (simp add: fps_mult_nth) from inverse_mult_eq_1[OF ab0] have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b" by (simp add: ring_simps) then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp} ultimately show ?thesis by blast qed lemma fps_inverse_deriv': fixes a:: "('a :: field) fps" assumes a0: "a$0 ≠ 0" shows "fps_deriv (inverse a) = - fps_deriv a / a ^ 2" using fps_inverse_deriv[OF a0] unfolding power2_eq_square fps_divide_def fps_inverse_mult by simp lemma inverse_mult_eq_1': assumes f0: "f$0 ≠ (0::'a::field)" shows "f * inverse f= 1" by (metis mult_commute inverse_mult_eq_1 f0) lemma fps_divide_deriv: fixes a:: "('a :: field) fps" assumes a0: "b$0 ≠ 0" shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b ^ 2" using fps_inverse_deriv[OF a0] by (simp add: fps_divide_def ring_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0]) subsection{* The eXtractor series X*} lemma minus_one_power_iff: "(- (1::'a :: {recpower, comm_ring_1})) ^ n = (if even n then 1 else - 1)" by (induct n, auto) definition "X = Abs_fps (λn. if n = 1 then 1 else 0)" lemma fps_inverse_gp': "inverse (Abs_fps(λn. (1::'a::field))) = 1 - X" by (simp add: fps_inverse_gp fps_eq_iff X_def) lemma X_mult_nth[simp]: "(X * (f :: ('a::semiring_1) fps)) $n = (if n = 0 then 0 else f $ (n - 1))" proof- {assume n: "n ≠ 0" have fN: "finite {0 .. n}" by simp have "(X * f) $n = (∑i = 0..n. X $ i * f $ (n - i))" by (simp add: fps_mult_nth) also have "… = f $ (n - 1)" using n by (simp add: X_def mult_delta_left setsum_delta [OF fN]) finally have ?thesis using n by simp } moreover {assume n: "n=0" hence ?thesis by (simp add: fps_mult_nth X_def)} ultimately show ?thesis by blast qed lemma X_mult_right_nth[simp]: "((f :: ('a::comm_semiring_1) fps) * X) $n = (if n = 0 then 0 else f $ (n - 1))" by (metis X_mult_nth mult_commute) lemma X_power_iff: "X^k = Abs_fps (λn. if n = k then (1::'a::comm_ring_1) else 0)" proof(induct k) case 0 thus ?case by (simp add: X_def fps_power_def fps_eq_iff) next case (Suc k) {fix m have "(X^Suc k) $ m = (if m = 0 then (0::'a) else (X^k) $ (m - 1))" by (simp add: power_Suc del: One_nat_def) then have "(X^Suc k) $ m = (if m = Suc k then (1::'a) else 0)" using Suc.hyps by (auto cong del: if_weak_cong)} then show ?case by (simp add: fps_eq_iff) qed lemma X_power_mult_nth: "(X^k * (f :: ('a::comm_ring_1) fps)) $n = (if n < k then 0 else f $ (n - k))" apply (induct k arbitrary: n) apply (simp) unfolding power_Suc mult_assoc by (case_tac n, auto) lemma X_power_mult_right_nth: "((f :: ('a::comm_ring_1) fps) * X^k) $n = (if n < k then 0 else f $ (n - k))" by (metis X_power_mult_nth mult_commute) lemma fps_deriv_X[simp]: "fps_deriv X = 1" by (simp add: fps_deriv_def X_def fps_eq_iff) lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)" by (cases "n", simp_all) lemma X_nth[simp]: "X$n = (if n = 1 then 1 else 0)" by (simp add: X_def) lemma X_power_nth[simp]: "(X^k) $n = (if n = k then 1 else (0::'a::comm_ring_1))" by (simp add: X_power_iff) lemma fps_inverse_X_plus1: "inverse (1 + X) = Abs_fps (λn. (- (1::'a::{recpower, field})) ^ n)" (is "_ = ?r") proof- have eq: "(1 + X) * ?r = 1" unfolding minus_one_power_iff apply (auto simp add: ring_simps fps_eq_iff) by presburger+ show ?thesis by (auto simp add: eq intro: fps_inverse_unique) qed subsection{* Integration *} definition "fps_integral a a0 = Abs_fps (λn. if n = 0 then a0 else (a$(n - 1) / of_nat n))" lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a (a0 :: 'a :: {field, ring_char_0})) = a" by (simp add: fps_integral_def fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc) lemma fps_integral_linear: "fps_integral (fps_const (a::'a::{field, ring_char_0}) * f + fps_const b * g) (a*a0 + b*b0) = fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0" (is "?l = ?r") proof- have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral) moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def) ultimately show ?thesis unfolding fps_deriv_eq_iff by auto qed subsection {* Composition of FPSs *} definition fps_compose :: "('a::semiring_1) fps => 'a fps => 'a fps" (infixl "oo" 55) where fps_compose_def: "a oo b = Abs_fps (λn. setsum (λi. a$i * (b^i$n)) {0..n})" lemma fps_compose_nth: "(a oo b)$n = setsum (λi. a$i * (b^i$n)) {0..n}" by (simp add: fps_compose_def) lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)" by (simp add: fps_ext fps_compose_def mult_delta_right setsum_delta') lemma fps_const_compose[simp]: "fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)" by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta) lemma X_fps_compose_startby0[simp]: "a$0 = 0 ==> X oo a = (a :: ('a :: comm_ring_1) fps)" by (simp add: fps_eq_iff fps_compose_def mult_delta_left setsum_delta power_Suc not_le) subsection {* Rules from Herbert Wilf's Generatingfunctionology*} subsubsection {* Rule 1 *} (* {a_{n+k}}_0^infty Corresponds to (f - setsum (λi. a_i * x^i))/x^h, for h>0*) lemma fps_power_mult_eq_shift: "X^Suc k * Abs_fps (λn. a (n + Suc k)) = Abs_fps a - setsum (λi. fps_const (a i :: 'a:: field) * X^i) {0 .. k}" (is "?lhs = ?rhs") proof- {fix n:: nat have "?lhs $ n = (if n < Suc k then 0 else a n)" unfolding X_power_mult_nth by auto also have "… = ?rhs $ n" proof(induct k) case 0 thus ?case by (simp add: fps_setsum_nth power_Suc) next case (Suc k) note th = Suc.hyps[symmetric] have "(Abs_fps a - setsum (λi. fps_const (a i :: 'a:: field) * X^i) {0 .. Suc k})$n = (Abs_fps a - setsum (λi. fps_const (a i :: 'a:: field) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) $ n" by (simp add: ring_simps) also have "… = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)$n" using th unfolding fps_sub_nth by simp also have "… = (if n < Suc (Suc k) then 0 else a n)" unfolding X_power_mult_right_nth apply (auto simp add: not_less fps_const_def) apply (rule cong[of a a, OF refl]) by arith finally show ?case by simp qed finally have "?lhs $ n = ?rhs $ n" .} then show ?thesis by (simp add: fps_eq_iff) qed subsubsection{* Rule 2*} (* We can not reach the form of Wilf, but still near to it using rewrite rules*) (* If f reprents {a_n} and P is a polynomial, then P(xD) f represents {P(n) a_n}*) definition "XD = op * X o fps_deriv" lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)" by (simp add: XD_def ring_simps) lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a" by (simp add: XD_def ring_simps) lemma XD_linear[simp]: "XD (fps_const c * a + fps_const d * b) = fps_const c * XD a + fps_const d * XD (b :: ('a::comm_ring_1) fps)" by simp lemma XDN_linear: "(XD^n) (fps_const c * a + fps_const d * b) = fps_const c * (XD^n) a + fps_const d * (XD^n) (b :: ('a::comm_ring_1) fps)" by (induct n, simp_all) lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (λn. of_nat n* a$n)" by (simp add: fps_eq_iff) lemma fps_mult_XD_shift: "(XD ^k) (a:: ('a::{comm_ring_1, recpower, ring_char_0}) fps) = Abs_fps (λn. (of_nat n ^ k) * a$n)" by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def) subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*} subsubsection{* Rule 5 --- summation and "division" by (1 - X)*} lemma fps_divide_X_minus1_setsum_lemma: "a = ((1::('a::comm_ring_1) fps) - X) * Abs_fps (λn. setsum (λi. a $ i) {0..n})" proof- let ?X = "X::('a::comm_ring_1) fps" let ?sa = "Abs_fps (λn. setsum (λi. a $ i) {0..n})" have th0: "!!i. (1 - (X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)" by simp {fix n:: nat {assume "n=0" hence "a$n = ((1 - ?X) * ?sa) $ n" by (simp add: fps_mult_nth)} moreover {assume n0: "n ≠ 0" then have u: "{0} ∪ ({1} ∪ {2..n}) = {0..n}" "{1}∪{2..n} = {1..n}" "{0..n - 1}∪{n} = {0..n}" apply (simp_all add: expand_set_eq) by presburger+ have d: "{0} ∩ ({1} ∪ {2..n}) = {}" "{1} ∩ {2..n} = {}" "{0..n - 1}∩{n} ={}" using n0 by (simp_all add: expand_set_eq, presburger+) have f: "finite {0}" "finite {1}" "finite {2 .. n}" "finite {0 .. n - 1}" "finite {n}" by simp_all have "((1 - ?X) * ?sa) $ n = setsum (λi. (1 - ?X)$ i * ?sa $ (n - i)) {0 .. n}" by (simp add: fps_mult_nth) also have "… = a$n" unfolding th0 unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)] unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)] apply (simp) unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)] by simp finally have "a$n = ((1 - ?X) * ?sa) $ n" by simp} ultimately have "a$n = ((1 - ?X) * ?sa) $ n" by blast} then show ?thesis unfolding fps_eq_iff by blast qed lemma fps_divide_X_minus1_setsum: "a /((1::('a::field) fps) - X) = Abs_fps (λn. setsum (λi. a $ i) {0..n})" proof- let ?X = "1 - (X::('a::field) fps)" have th0: "?X $ 0 ≠ 0" by simp have "a /?X = ?X * Abs_fps (λn::nat. setsum (op $ a) {0..n}) * inverse ?X" using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0 by (simp add: fps_divide_def mult_assoc) also have "… = (inverse ?X * ?X) * Abs_fps (λn::nat. setsum (op $ a) {0..n}) " by (simp add: mult_ac) finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0]) qed subsubsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary finite product of FPS, also the relvant instance of powers of a FPS*} definition "natpermute n k = {l:: nat list. length l = k ∧ foldl op + 0 l = n}" lemma natlist_trivial_1: "natpermute n 1 = {[n]}" apply (auto simp add: natpermute_def) apply (case_tac x, auto) done lemma foldl_add_start0: "foldl op + x xs = x + foldl op + (0::nat) xs" apply (induct xs arbitrary: x) apply simp unfolding foldl.simps apply atomize apply (subgoal_tac "∀x::nat. foldl op + x xs = x + foldl op + (0::nat) xs") apply (erule_tac x="x + a" in allE) apply (erule_tac x="a" in allE) apply simp apply assumption done lemma foldl_add_append: "foldl op + (x::nat) (xs@ys) = foldl op + x xs + foldl op + 0 ys" apply (induct ys arbitrary: x xs) apply auto apply (subst (2) foldl_add_start0) apply simp apply (subst (2) foldl_add_start0) by simp lemma foldl_add_setsum: "foldl op + (x::nat) xs = x + setsum (nth xs) {0..<length xs}" proof(induct xs arbitrary: x) case Nil thus ?case by simp next case (Cons a as x) have eq: "setsum (op ! (a#as)) {1..<length (a#as)} = setsum (op ! as) {0..<length as}" apply (rule setsum_reindex_cong [where f=Suc]) by (simp_all add: inj_on_def) have f: "finite {0}" "finite {1 ..< length (a#as)}" by simp_all have d: "{0} ∩ {1..<length (a#as)} = {}" by simp have seq: "{0} ∪ {1..<length(a#as)} = {0 ..<length (a#as)}" by auto have "foldl op + x (a#as) = x + foldl op + a as " apply (subst foldl_add_start0) by simp also have "… = x + a + setsum (op ! as) {0..<length as}" unfolding Cons.hyps by simp also have "… = x + setsum (op ! (a#as)) {0..<length (a#as)}" unfolding eq[symmetric] unfolding setsum_Un_disjoint[OF f d, unfolded seq] by simp finally show ?case . qed lemma append_natpermute_less_eq: assumes h: "xs@ys ∈ natpermute n k" shows "foldl op + 0 xs ≤ n" and "foldl op + 0 ys ≤ n" proof- {from h have "foldl op + 0 (xs@ ys) = n" by (simp add: natpermute_def) hence "foldl op + 0 xs + foldl op + 0 ys = n" unfolding foldl_add_append .} note th = this {from th show "foldl op + 0 xs ≤ n" by simp} {from th show "foldl op + 0 ys ≤ n" by simp} qed lemma natpermute_split: assumes mn: "h ≤ k" shows "natpermute n k = (\<Union>m ∈{0..n}. {l1 @ l2 |l1 l2. l1 ∈ natpermute m h ∧ l2 ∈ natpermute (n - m) (k - h)})" (is "?L = ?R" is "?L = (\<Union>m ∈{0..n}. ?S m)") proof- {fix l assume l: "l ∈ ?R" from l obtain m xs ys where h: "m ∈ {0..n}" and xs: "xs ∈ natpermute m h" and ys: "ys ∈ natpermute (n - m) (k - h)" and leq: "l = xs@ys" by blast from xs have xs': "foldl op + 0 xs = m" by (simp add: natpermute_def) from ys have ys': "foldl op + 0 ys = n - m" by (simp add: natpermute_def) have "l ∈ ?L" using leq xs ys h apply simp apply (clarsimp simp add: natpermute_def simp del: foldl_append) apply (simp add: foldl_add_append[unfolded foldl_append]) unfolding xs' ys' using mn xs ys unfolding natpermute_def by simp} moreover {fix l assume l: "l ∈ natpermute n k" let ?xs = "take h l" let ?ys = "drop h l" let ?m = "foldl op + 0 ?xs" from l have ls: "foldl op + 0 (?xs @ ?ys) = n" by (simp add: natpermute_def) have xs: "?xs ∈ natpermute ?m h" using l mn by (simp add: natpermute_def) have ys: "?ys ∈ natpermute (n - ?m) (k - h)" using l mn ls[unfolded foldl_add_append] by (simp add: natpermute_def) from ls have m: "?m ∈ {0..n}" unfolding foldl_add_append by simp from xs ys ls have "l ∈ ?R" apply auto apply (rule bexI[where x = "?m"]) apply (rule exI[where x = "?xs"]) apply (rule exI[where x = "?ys"]) using ls l unfolding foldl_add_append by (auto simp add: natpermute_def)} ultimately show ?thesis by blast qed lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})" by (auto simp add: natpermute_def) lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})" apply (auto simp add: set_replicate_conv_if natpermute_def) apply (rule nth_equalityI) by simp_all lemma natpermute_finite: "finite (natpermute n k)" proof(induct k arbitrary: n) case 0 thus ?case apply (subst natpermute_split[of 0 0, simplified]) by (simp add: natpermute_0) 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 unfolding image_Collect[symmetric] unfolding Collect_def mem_def apply (rule finite_imageI) apply blast done qed lemma natpermute_contain_maximal: "{xs ∈ natpermute n (k+1). n ∈ set xs} = UNION {0 .. k} (λi. {(replicate (k+1) 0) [i:=n]})" (is "?A = ?B") proof- {fix xs assume H: "xs ∈ natpermute n (k+1)" and n: "n ∈ set xs" from n obtain i where i: "i ∈ {0.. k}" "xs!i = n" using H unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def) have eqs: "({0..k} - {i}) ∪ {i} = {0..k}" using i by auto have f: "finite({0..k} - {i})" "finite {i}" by auto have d: "({0..k} - {i}) ∩ {i} = {}" using i by auto from H have "n = setsum (nth xs) {0..k}" apply (simp add: natpermute_def) unfolding foldl_add_setsum by (auto simp add: atLeastLessThanSuc_atLeastAtMost) also have "… = n + setsum (nth xs) ({0..k} - {i})" unfolding setsum_Un_disjoint[OF f d, unfolded eqs] using i by simp finally have zxs: "∀ j∈ {0..k} - {i}. xs!j = 0" by auto from H have xsl: "length xs = k+1" by (simp add: natpermute_def) from i have i': "i < length (replicate (k+1) 0)" "i < k+1" unfolding length_replicate by arith+ 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 by (case_tac "ia=i", auto simp del: replicate.simps) then have "xs ∈ ?B" using i by blast} moreover {fix i assume i: "i ∈ {0..k}" let ?xs = "replicate (k+1) 0 [i:=n]" have nxs: "n ∈ set ?xs" apply (rule set_update_memI) using i by simp have xsl: "length ?xs = k+1" by (simp only: length_replicate length_list_update) have "foldl op + 0 ?xs = setsum (nth ?xs) {0..<k+1}" unfolding foldl_add_setsum add_0 length_replicate length_list_update .. also have "… = setsum (λj. if j = i then n else 0) {0..< k+1}" apply (rule setsum_cong2) by (simp del: replicate.simps) also have "… = n" using i by (simp add: setsum_delta) finally have "?xs ∈ natpermute n (k+1)" using xsl unfolding natpermute_def Collect_def mem_def by blast then have "?xs ∈ ?A" using nxs by blast} ultimately show ?thesis by auto qed (* The general form *) lemma fps_setprod_nth: fixes m :: nat and a :: "nat => ('a::comm_ring_1) fps" shows "(setprod a {0 .. m})$n = setsum (λv. setprod (λj. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))" (is "?P m n") proof(induct m arbitrary: n rule: nat_less_induct) fix m n assume H: "∀m' < m. ∀n. ?P m' n" {assume m0: "m = 0" hence "?P m n" apply simp unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] by simp} moreover {fix k assume k: "m = Suc k" have km: "k < m" using k by arith have u0: "{0 .. k} ∪ {m} = {0..m}" using k apply (simp add: expand_set_eq) by presburger have f0: "finite {0 .. k}" "finite {m}" by auto have d0: "{0 .. k} ∩ {m} = {}" using k by auto have "(setprod a {0 .. m}) $ n = (setprod a {0 .. k} * a m) $ n" unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0] by simp also have "… = (∑i = 0..n. (∑v∈natpermute i (k + 1). ∏j∈{0..k}. a j $ v ! j) * a m $ (n - i))" unfolding fps_mult_nth H[rule_format, OF km] .. also have "… = (∑v∈natpermute n (m + 1). ∏j∈{0..m}. a j $ v ! j)" apply (simp add: k) unfolding natpermute_split[of m "m + 1", simplified, of n, unfolded natlist_trivial_1[unfolded One_nat_def] k] apply (subst setsum_UN_disjoint) apply simp apply simp unfolding image_Collect[symmetric] apply clarsimp apply (rule finite_imageI) apply (rule natpermute_finite) apply (clarsimp simp add: expand_set_eq) apply auto apply (rule setsum_cong2) unfolding setsum_left_distrib apply (rule sym) apply (rule_tac f="λxs. xs @[n - x]" in setsum_reindex_cong) apply (simp add: inj_on_def) apply auto unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded k] apply (clarsimp simp add: natpermute_def nth_append) done finally have "?P m n" .} ultimately show "?P m n " by (cases m, auto) qed text{* The special form for powers *} lemma fps_power_nth_Suc: fixes m :: nat and a :: "('a::comm_ring_1) fps" shows "(a ^ Suc m)$n = setsum (λv. setprod (λj. a $ (v!j)) {0..m}) (natpermute n (m+1))" proof- have f: "finite {0 ..m}" by simp have th0: "a^Suc m = setprod (λi. a) {0..m}" unfolding setprod_constant[OF f, of a] by simp show ?thesis unfolding th0 fps_setprod_nth .. qed lemma fps_power_nth: fixes m :: nat and a :: "('a::comm_ring_1) fps" shows "(a ^m)$n = (if m=0 then 1$n else setsum (λv. setprod (λj. a $ (v!j)) {0..m - 1}) (natpermute n m))" by (cases m, simp_all add: fps_power_nth_Suc del: power_Suc) lemma fps_nth_power_0: fixes m :: nat and a :: "('a::{comm_ring_1, recpower}) fps" shows "(a ^m)$0 = (a$0) ^ m" proof- {assume "m=0" hence ?thesis by simp} moreover {fix n assume m: "m = Suc n" have c: "m = card {0..n}" using m by simp have "(a ^m)$0 = setprod (λi. a$0) {0..n}" by (simp add: m fps_power_nth del: replicate.simps power_Suc) also have "… = (a$0) ^ m" unfolding c by (rule setprod_constant, simp) finally have ?thesis .} ultimately show ?thesis by (cases m, auto) qed lemma fps_compose_inj_right: assumes a0: "a$0 = (0::'a::{recpower,idom})" and a1: "a$1 ≠ 0" shows "(b oo a = c oo a) <-> b = c" (is "?lhs <->?rhs") proof- {assume ?rhs then have "?lhs" by simp} moreover {assume h: ?lhs {fix n have "b$n = c$n" proof(induct n rule: nat_less_induct) fix n assume H: "∀m<n. b$m = c$m" {assume n0: "n=0" from h have "(b oo a)$n = (c oo a)$n" by simp hence "b$n = c$n" using n0 by (simp add: fps_compose_nth)} moreover {fix n1 assume n1: "n = Suc n1" have f: "finite {0 .. n1}" "finite {n}" by simp_all have eq: "{0 .. n1} ∪ {n} = {0 .. n}" using n1 by auto have d: "{0 .. n1} ∩ {n} = {}" using n1 by auto have seq: "(∑i = 0..n1. b $ i * a ^ i $ n) = (∑i = 0..n1. c $ i * a ^ i $ n)" apply (rule setsum_cong2) using H n1 by auto have th0: "(b oo a) $n = (∑i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n" unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq] seq using startsby_zero_power_nth_same[OF a0] by simp have th1: "(c oo a) $n = (∑i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n" unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq] using startsby_zero_power_nth_same[OF a0] by simp from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1 have "b$n = c$n" by auto} ultimately show "b$n = c$n" by (cases n, auto) qed} then have ?rhs by (simp add: fps_eq_iff)} ultimately show ?thesis by blast qed subsection {* Radicals *} declare setprod_cong[fundef_cong] function radical :: "(nat => 'a => 'a) => nat => ('a::{field, recpower}) fps => nat => 'a" where "radical r 0 a 0 = 1" | "radical r 0 a (Suc n) = 0" | "radical r (Suc k) a 0 = r (Suc k) (a$0)" | "radical r (Suc k) a (Suc n) = (a$ Suc n - setsum (λxs. setprod (λj. radical r (Suc k) a (xs ! j)) {0..k}) {xs. xs ∈ natpermute (Suc n) (Suc k) ∧ Suc n ∉ set xs}) / (of_nat (Suc k) * (radical r (Suc k) a 0)^k)" by pat_completeness auto termination radical proof let ?R = "measure (λ(r, k, a, n). n)" { show "wf ?R" by auto} {fix r k a n xs i assume xs: "xs ∈ {xs ∈ natpermute (Suc n) (Suc k). Suc n ∉ set xs}" and i: "i ∈ {0..k}" {assume c: "Suc n ≤ xs ! i" from xs i have "xs !i ≠ Suc n" by (auto simp add: in_set_conv_nth natpermute_def) with c have c': "Suc n < xs!i" by arith have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all have d: "{0 ..< i} ∩ ({i} ∪ {i+1 ..< Suc k}) = {}" "{i} ∩ {i+1..< Suc k} = {}" by auto have eqs: "{0..<Suc k} = {0 ..< i} ∪ ({i} ∪ {i+1 ..< Suc k})" using i by auto from xs have "Suc n = foldl op + 0 xs" by (simp add: natpermute_def) also have "… = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs by (simp add: natpermute_def) also have "… = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}" unfolding eqs setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)] unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)] by simp finally have False using c' by simp} then show "((r,Suc k,a,xs!i), r, Suc k, a, Suc n) ∈ ?R" apply auto by (metis not_less)} {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) by (case_tac n, auto) 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- {assume "k=0" hence ?thesis by simp } moreover {fix h assume h: "k = Suc h" have fh: "finite {0..h}" by simp 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 h by simp also have "… = (∏j∈{0..h}. r k (a$0))" apply (rule setprod_cong) apply simp using h apply (subgoal_tac "replicate k (0::nat) ! x = 0") by (auto intro: nth_replicate simp del: replicate.simps) also have "… = a$0" unfolding setprod_constant[OF fh] using r by (simp add: h) finally have ?thesis using h by simp} ultimately show ?thesis by (cases k, auto) qed lemma natpermute_max_card: assumes n0: "n≠0" shows "card {xs ∈ natpermute n (k+1). n ∈ set xs} = k+1" unfolding natpermute_contain_maximal proof- let ?A= "λi. {replicate (k + 1) 0[i := n]}" let ?K = "{0 ..k}" have fK: "finite ?K" by simp have fAK: "∀i∈?K. finite (?A i)" by auto have d: "∀i∈ ?K. ∀j∈ ?K. i ≠ j --> {replicate (k + 1) 0[i := n]} ∩ {replicate (k + 1) 0[j := n]} = {}" proof(clarify) fix i j assume i: "i ∈ ?K" and j: "j∈ ?K" and ij: "i≠j" {assume eq: "replicate (k+1) 0 [i:=n] = replicate (k+1) 0 [j:= n]" have "(replicate (k+1) 0 [i:=n] ! i) = n" using i by (simp del: replicate.simps) moreover have "(replicate (k+1) 0 [j:=n] ! i) = 0" using i ij by (simp del: replicate.simps) ultimately have False using eq n0 by (simp del: replicate.simps)} then show "{replicate (k + 1) 0[i := n]} ∩ {replicate (k + 1) 0[j := n]} = {}" by auto qed from card_UN_disjoint[OF fK fAK d] show "card (\<Union>i∈{0..k}. {replicate (k + 1) 0[i := n]}) = k+1" by simp qed lemma power_radical: fixes a:: "'a ::{field, ring_char_0, recpower} fps" assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" and a0: "a$0 ≠ 0" shows "(fps_radical r (Suc k) a) ^ (Suc k) = a" proof- let ?r = "fps_radical r (Suc k) a" from a0 r0 have r00: "r (Suc k) (a$0) ≠ 0" by auto {fix z have "?r ^ Suc k $ z = a$z" proof(induct z rule: nat_less_induct) fix n assume H: "∀m<n. ?r ^ Suc k $ m = a$m" {assume "n = 0" hence "?r ^ Suc k $ n = a $n" using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp} moreover {fix n1 assume n1: "n = Suc n1" have fK: "finite {0..k}" by simp have nz: "n ≠ 0" using n1 by arith let ?Pnk = "natpermute n (k + 1)" let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}" let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}" have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast have d: "?Pnkn ∩ ?Pnknn = {}" by blast have f: "finite ?Pnkn" "finite ?Pnknn" using finite_Un[of ?Pnkn ?Pnknn, unfolded eq] by (metis natpermute_finite)+ let ?f = "λv. ∏j∈{0..k}. ?r $ v ! j" have "setsum ?f ?Pnkn = setsum (λv. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn" proof(rule setsum_cong2) fix v assume v: "v ∈ {xs ∈ natpermute n (k + 1). n ∈ set xs}" let ?ths = "(∏j∈{0..k}. fps_radical r (Suc k) a $ v ! j) = fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k" from v obtain i where i: "i ∈ {0..k}" "v = replicate (k+1) 0 [i:= n]" unfolding natpermute_contain_maximal by auto have "(∏j∈{0..k}. fps_radical r (Suc k) a $ v ! j) = (∏j∈{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))" apply (rule setprod_cong, simp) using i r0 by (simp del: replicate.simps) also have "… = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k" unfolding setprod_gen_delta[OF fK] using i r0 by simp finally show ?ths . qed then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k" by (simp add: natpermute_max_card[OF nz, simplified]) also have "… = a$n - setsum ?f ?Pnknn" unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc ) finally have fn: "setsum ?f ?Pnkn = a$n - setsum ?f ?Pnknn" . have "(?r ^ Suc k)$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn" unfolding fps_power_nth_Suc setsum_Un_disjoint[OF f d, unfolded eq] .. also have "… = a$n" unfolding fn by simp finally have "?r ^ Suc k $ n = a $n" .} ultimately show "?r ^ Suc k $ n = a $n" by (cases n, auto) qed } then show ?thesis by (simp add: fps_eq_iff) qed lemma eq_divide_imp': assumes c0: "(c::'a::field) ~= 0" and eq: "a * c = b" shows "a = b / c" proof- from eq have "a * c * inverse c = b * inverse c" by simp hence "a * (inverse c * c) = b/c" by (simp only: field_simps divide_inverse) then show "a = b/c" unfolding field_inverse[OF c0] by simp qed lemma radical_unique: assumes r0: "(r (Suc k) (b$0)) ^ Suc k = b$0" and a0: "r (Suc k) (b$0 ::'a::{field, ring_char_0, recpower}) = a$0" and b0: "b$0 ≠ 0" shows "a^(Suc k) = b <-> a = fps_radical r (Suc k) b" proof- let ?r = "fps_radical r (Suc k) b" have r00: "r (Suc k) (b$0) ≠ 0" using b0 r0 by auto {assume H: "a = ?r" from H have "a^Suc k = b" using power_radical[of r k, OF r0 b0] by simp} moreover {assume H: "a^Suc k = b" (* Generally a$0 would need to be the k+1 st root of b$0 *) have ceq: "card {0..k} = Suc k" by simp have fk: "finite {0..k}" by simp from a0 have a0r0: "a$0 = ?r$0" by simp {fix n have "a $ n = ?r $ n" proof(induct n rule: nat_less_induct) fix n assume h: "∀m<n. a$m = ?r $m" {assume "n = 0" hence "a$n = ?r $n" using a0 by simp } moreover {fix n1 assume n1: "n = Suc n1" have fK: "finite {0..k}" by simp have nz: "n ≠ 0" using n1 by arith let ?Pnk = "natpermute n (Suc k)" let ?Pnkn = "{xs ∈ ?Pnk. n ∈ set xs}" let ?Pnknn = "{xs ∈ ?Pnk. n ∉ set xs}" have eq: "?Pnkn ∪ ?Pnknn = ?Pnk" by blast have d: "?Pnkn ∩ ?Pnknn = {}" by blast have f: "finite ?Pnkn" "finite ?Pnknn" using finite_Un[of ?Pnkn ?Pnknn, unfolded eq] by (metis natpermute_finite)+ let ?f = "λv. ∏j∈{0..k}. ?r $ v ! j" let ?g = "λv. ∏j∈{0..k}. a $ v ! j" have "setsum ?g ?Pnkn = setsum (λv. a $ n * (?r$0)^k) ?Pnkn" proof(rule setsum_cong2) fix v assume v: "v ∈ {xs ∈ natpermute n (Suc k). n ∈ set xs}" let ?ths = "(∏j∈{0..k}. a $ v ! j) = a $ n * (?r$0)^k" from v obtain i where i: "i ∈ {0..k}" "v = replicate (k+1) 0 [i:= n]" unfolding Suc_plus1 natpermute_contain_maximal by (auto simp del: replicate.simps) have "(∏j∈{0..k}. a $ v ! j) = (∏j∈{0..k}. if j = i then a $ n else r (Suc k) (b$0))" apply (rule setprod_cong, simp) using i a0 by (simp del: replicate.simps) also have "… = a $ n * (?r $ 0)^k" unfolding setprod_gen_delta[OF fK] using i by simp finally show ?ths . qed then have th0: "setsum ?g ?Pnkn = of_nat (k+1) * a $ n * (?r $ 0)^k" by (simp add: natpermute_max_card[OF nz, simplified]) have th1: "setsum ?g ?Pnknn = setsum ?f ?Pnknn" proof (rule setsum_cong2, rule setprod_cong, simp) fix xs i assume xs: "xs ∈ ?Pnknn" and i: "i ∈ {0..k}" {assume c: "n ≤ xs ! i" from xs i have "xs !i ≠ n" by (auto simp add: in_set_conv_nth natpermute_def) with c have c': "n < xs!i" by arith have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all have d: "{0 ..< i} ∩ ({i} ∪ {i+1 ..< Suc k}) = {}" "{i} ∩ {i+1..< Suc k} = {}" by auto have eqs: "{0..<Suc k} = {0 ..< i} ∪ ({i} ∪ {i+1 ..< Suc k})" using i by auto from xs have "n = foldl op + 0 xs" by (simp add: natpermute_def) also have "… = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs by (simp add: natpermute_def) also have "… = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}" unfolding eqs setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)] unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)] by simp finally have False using c' by simp} then have thn: "xs!i < n" by arith from h[rule_format, OF thn] show "a$(xs !i) = ?r$(xs!i)" . qed have th00: "!!(x::'a). of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x" by (simp add: field_simps del: of_nat_Suc) from H have "b$n = a^Suc k $ n" by (simp add: fps_eq_iff) also have "a ^ Suc k$n = setsum ?g ?Pnkn + setsum ?g ?Pnknn" unfolding fps_power_nth_Suc using setsum_Un_disjoint[OF f d, unfolded Suc_plus1[symmetric], unfolded eq, of ?g] by simp also have "… = of_nat (k+1) * a $ n * (?r $ 0)^k + setsum ?f ?Pnknn" unfolding th0 th1 .. finally have "of_nat (k+1) * a $ n * (?r $ 0)^k = b$n - setsum ?f ?Pnknn" by simp then have "a$n = (b$n - setsum ?f ?Pnknn) / (of_nat (k+1) * (?r $ 0)^k)" apply - apply (rule eq_divide_imp') using r00 apply (simp del: of_nat_Suc) by (simp add: mult_ac) then have "a$n = ?r $n" apply (simp del: of_nat_Suc) unfolding fps_radical_def n1 by (simp add: field_simps n1 th00 del: of_nat_Suc)} ultimately show "a$n = ?r $ n" by (cases n, auto) qed} then have "a = ?r" by (simp add: fps_eq_iff)} ultimately show ?thesis by blast qed lemma radical_power: assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0" and a0: "(a$0 ::'a::{field, ring_char_0, recpower}) ≠ 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, ring_char_0, recpower} 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 power_radical[of r, OF r0 a0] have "fps_deriv (?r ^ Suc k) = fps_deriv a" by simp hence "fps_deriv ?r * ?w = fps_deriv a" by (simp add: fps_deriv_power mult_ac del: power_Suc) hence "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a" by simp hence "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w" by (simp add: fps_divide_def) then show ?thesis unfolding th0 by simp qed lemma radical_mult_distrib: fixes a:: "'a ::{field, ring_char_0, recpower} fps" assumes ra0: "r (k) (a $ 0) ^ k = a $ 0" and rb0: "r (k) (b $ 0) ^ k = b $ 0" and r0': "r (k) ((a * b) $ 0) = r (k) (a $ 0) * r (k) (b $ 0)" and a0: "a$0 ≠ 0" and b0: "b$0 ≠ 0" shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)" proof- from r0' have r0: "(r (k) ((a*b)$0)) ^ k = (a*b)$0" by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib) {assume "k=0" hence ?thesis by simp} moreover {fix h assume k: "k = Suc h" let ?ra = "fps_radical r (Suc h) a" let ?rb = "fps_radical r (Suc h) b" have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0" using r0' k by (simp add: fps_mult_nth) have ab0: "(a*b) $ 0 ≠ 0" using a0 b0 by (simp add: fps_mult_nth) from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric] power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)} ultimately show ?thesis by (cases k, auto) qed lemma radical_inverse: fixes a:: "'a ::{field, ring_char_0, recpower} fps" assumes ra0: "r (k) (a $ 0) ^ k = a $ 0" and ria0: "r (k) (inverse (a $ 0)) = inverse (r (k) (a $ 0))" and r1: "(r (k) 1) = 1" and a0: "a$0 ≠ 0" shows "fps_radical r (k) (inverse a) = inverse (fps_radical r (k) a)" proof- {assume "k=0" then have ?thesis by simp} moreover {fix h assume k[simp]: "k = Suc h" let ?ra = "fps_radical r (Suc h) a" let ?ria = "fps_radical r (Suc h) (inverse a)" from ra0 a0 have th00: "r (Suc h) (a$0) ≠ 0" by auto have ria0': "r (Suc h) (inverse a $ 0) ^ Suc h = inverse a$0" using ria0 ra0 a0 by (simp add: fps_inverse_def nonzero_power_inverse[OF th00, symmetric] del: power_Suc) from inverse_mult_eq_1[OF a0] have th0: "a * inverse a = 1" by (simp add: mult_commute) from radical_unique[where a=1 and b=1 and r=r and k=h, simplified, OF r1[unfolded k]] have th01: "fps_radical r (Suc h) 1 = 1" . have th1: "r (Suc h) ((a * inverse a) $ 0) ^ Suc h = (a * inverse a) $ 0" "r (Suc h) ((a * inverse a) $ 0) = r (Suc h) (a $ 0) * r (Suc h) (inverse a $ 0)" using r1 unfolding th0 apply (simp_all add: ria0[symmetric]) apply (simp add: fps_inverse_def a0) unfolding ria0[unfolded k] using th00 by simp from nonzero_imp_inverse_nonzero[OF a0] a0 have th2: "inverse a $ 0 ≠ 0" by (simp add: fps_inverse_def) from radical_mult_distrib[of r "Suc h" a "inverse a", OF ra0[unfolded k] ria0' th1(2) a0 th2] have th3: "?ra * ?ria = 1" unfolding th0 th01 by simp from th00 have ra0: "?ra $ 0 ≠ 0" by simp from fps_inverse_unique[OF ra0 th3] have ?thesis by simp} ultimately show ?thesis by (cases k, auto) qed lemma fps_divide_inverse: "(a::('a::field) fps) / b = a * inverse b" by (simp add: fps_divide_def) lemma radical_divide: fixes a:: "'a ::{field, ring_char_0, recpower} fps" assumes ra0: "r k (a $ 0) ^ k = a $ 0" and rb0: "r k (b $ 0) ^ k = b $ 0" and r1: "r k 1 = 1" and rb0': "r k (inverse (b $ 0)) = inverse (r k (b $ 0))" and raib': "r k (a$0 / (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 raib' have raib: "r k (a$0 / (b$0)) = r k (a$0) * r k (inverse (b$0))" by (simp add: divide_inverse rb0'[symmetric]) {assume "k=0" hence ?thesis by (simp add: fps_divide_def)} moreover {assume k0: "k≠ 0" from b0 k0 rb0 have rbn0: "r k (b $0) ≠ 0" by (auto simp add: power_0_left) from rb0 rb0' have rib0: "(r k (inverse (b $ 0)))^k = inverse (b$0)" by (simp add: nonzero_power_inverse[OF rbn0, symmetric]) from rib0 have th0: "r k (inverse b $ 0) ^ k = inverse b $ 0" by (simp add:fps_inverse_def b0) from raib have th1: "r k ((a * inverse b) $ 0) = r k (a $ 0) * r k (inverse b $ 0)" by (simp add: divide_inverse fps_inverse_def b0 fps_mult_nth) from nonzero_imp_inverse_nonzero[OF b0] b0 have th2: "inverse b $ 0 ≠ 0" by (simp add: fps_inverse_def) from radical_mult_distrib[of r k a "inverse b", OF ra0 th0 th1 a0 th2] have th: "fps_radical r k (a/b) = fps_radical r k a * fps_radical r k (inverse b)" by (simp add: fps_divide_def) with radical_inverse[of r k b, OF rb0 rb0' r1 b0] have ?thesis by (simp add: fps_divide_def)} ultimately show ?thesis by blast qed subsection{* Derivative of composition *} lemma fps_compose_deriv: fixes a:: "('a::idom) fps" assumes b0: "b$0 = 0" shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * (fps_deriv b)" proof- {fix n have "(fps_deriv (a oo b))$n = setsum (λi. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}" by (simp add: fps_compose_def ring_simps setsum_right_distrib del: of_nat_Suc) also have "… = setsum (λi. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}" by (simp add: ring_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc) also have "… = setsum (λi. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}" unfolding fps_mult_left_const_nth by (simp add: ring_simps) also have "… = setsum (λi. of_nat i * a$i * (setsum (λj. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {0.. Suc n}" unfolding fps_mult_nth .. also have "… = setsum (λi. of_nat i * a$i * (setsum (λj. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {1.. Suc n}" apply (rule setsum_mono_zero_right) apply (auto simp add: mult_delta_left setsum_delta not_le) done also have "… = setsum (λi. of_nat (i + 1) * a$(i+1) * (setsum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" unfolding fps_deriv_nth apply (rule setsum_reindex_cong[where f="Suc"]) by (auto simp add: mult_assoc) finally have th0: "(fps_deriv (a oo b))$n = setsum (λi. of_nat (i + 1) * a$(i+1) * (setsum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" . have "(((fps_deriv a) oo b) * (fps_deriv b))$n = setsum (λi. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}" unfolding fps_mult_nth by (simp add: mult_ac) also have "… = setsum (λi. setsum (λj. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}" unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult_assoc apply (rule setsum_cong2) apply (rule setsum_mono_zero_left) apply (simp_all add: subset_eq) apply clarify apply (subgoal_tac "b^i$x = 0") apply simp apply (rule startsby_zero_power_prefix[OF b0, rule_format]) by simp also have "… = setsum (λi. of_nat (i + 1) * a$(i+1) * (setsum (λj. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" unfolding setsum_right_distrib apply (subst setsum_commute) by ((rule setsum_cong2)+) simp finally have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n" unfolding th0 by simp} then show ?thesis by (simp add: fps_eq_iff) qed lemma fps_mult_X_plus_1_nth: "((1+X)*a) $n = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))" proof- {assume "n = 0" hence ?thesis by (simp add: fps_mult_nth )} moreover {fix m assume m: "n = Suc m" have "((1+X)*a) $n = setsum (λi. (1+X)$i * a$(n-i)) {0..n}" by (simp add: fps_mult_nth) also have "… = setsum (λi. (1+X)$i * a$(n-i)) {0.. 1}" unfolding m apply (rule setsum_mono_zero_right) by (auto simp add: ) also have "… = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))" unfolding m by (simp add: ) finally have ?thesis .} ultimately show ?thesis by (cases n, auto) qed subsection{* Finite FPS (i.e. polynomials) and X *} lemma fps_poly_sum_X: assumes z: "∀i > n. a$i = (0::'a::comm_ring_1)" shows "a = setsum (λi. fps_const (a$i) * X^i) {0..n}" (is "a = ?r") proof- {fix i have "a$i = ?r$i" unfolding fps_setsum_nth fps_mult_left_const_nth X_power_nth by (simp add: mult_delta_right setsum_delta' z) } then show ?thesis unfolding fps_eq_iff by blast qed subsection{* Compositional inverses *} fun compinv :: "'a fps => nat => 'a::{recpower,field}" where "compinv a 0 = X$0" | "compinv a (Suc n) = (X$ Suc n - setsum (λi. (compinv a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n" definition "fps_inv a = Abs_fps (compinv a)" lemma fps_inv: assumes a0: "a$0 = 0" and a1: "a$1 ≠ 0" shows "fps_inv a oo a = X" proof- let ?i = "fps_inv a oo a" {fix n have "?i $n = X$n" proof(induct n rule: nat_less_induct) fix n assume h: "∀m<n. ?i$m = X$m" {assume "n=0" hence "?i $n = X$n" using a0 by (simp add: fps_compose_nth fps_inv_def)} moreover {fix n1 assume n1: "n = Suc n1" have "?i $ n = setsum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1" by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0] del: power_Suc) also have "… = setsum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + (X$ Suc n1 - setsum (λi. (fps_inv a $ i) * (a^i)$n) {0 .. n1})" using a0 a1 n1 by (simp add: fps_inv_def) also have "… = X$n" using n1 by simp finally have "?i $ n = X$n" .} ultimately show "?i $ n = X$n" by (cases n, auto) qed} then show ?thesis by (simp add: fps_eq_iff) qed fun gcompinv :: "'a fps => 'a fps => nat => 'a::{recpower,field}" where "gcompinv b a 0 = b$0" | "gcompinv b a (Suc n) = (b$ Suc n - setsum (λi. (gcompinv b a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n" definition "fps_ginv b a = Abs_fps (gcompinv b a)" lemma fps_ginv: assumes a0: "a$0 = 0" and a1: "a$1 ≠ 0" shows "fps_ginv b a oo a = b" proof- let ?i = "fps_ginv b a oo a" {fix n have "?i $n = b$n" proof(induct n rule: nat_less_induct) fix n assume h: "∀m<n. ?i$m = b$m" {assume "n=0" hence "?i $n = b$n" using a0 by (simp add: fps_compose_nth fps_ginv_def)} moreover {fix n1 assume n1: "n = Suc n1" have "?i $ n = setsum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1" by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0] del: power_Suc) also have "… = setsum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + (b$ Suc n1 - setsum (λi. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})" using a0 a1 n1 by (simp add: fps_ginv_def) also have "… = b$n" using n1 by simp finally have "?i $ n = b$n" .} ultimately show "?i $ n = b$n" by (cases n, auto) qed} then show ?thesis by (simp add: fps_eq_iff) qed lemma fps_inv_ginv: "fps_inv = fps_ginv X" apply (auto simp add: expand_fun_eq fps_eq_iff fps_inv_def fps_ginv_def) apply (induct_tac n rule: nat_less_induct, 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 fps_power_def mult_delta_left setsum_delta) lemma fps_compose_0[simp]: "0 oo a = 0" by (simp add: fps_eq_iff fps_compose_nth) lemma fps_pow_0: "fps_pow n 0 = (if n = 0 then 1 else 0)" by (induct n, simp_all) lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a$0)" by (auto simp add: fps_eq_iff fps_compose_nth fps_power_def fps_pow_0 setsum_0') lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)" by (simp add: fps_eq_iff fps_compose_nth ring_simps setsum_addf) lemma fps_compose_setsum_distrib: "(setsum f S) oo a = setsum (λi. f i oo a) S" proof- {assume "¬ finite S" hence ?thesis by simp} moreover {assume fS: "finite S" have ?thesis proof(rule finite_induct[OF fS]) show "setsum f {} oo a = (∑i∈{}. f i oo a)" by simp next fix x F assume fF: "finite F" and xF: "x ∉ F" and h: "setsum f F oo a = setsum (λi. f i oo a) F" show "setsum f (insert x F) oo a = setsum (λi. f i oo a) (insert x F)" using fF xF h by (simp add: fps_compose_add_distrib) qed} ultimately show ?thesis by blast qed lemma convolution_eq: "setsum (%i. a (i :: nat) * b (n - i)) {0 .. n} = setsum (%(i,j). a i * b j) {(i,j). i <= n ∧ j ≤ n ∧ i + j = n}" apply (rule setsum_reindex_cong[where f=fst]) apply (clarsimp simp add: inj_on_def) apply (auto simp add: expand_set_eq image_iff) apply (rule_tac x= "x" in exI) apply clarsimp apply (rule_tac x="n - x" in exI) apply arith done lemma product_composition_lemma: assumes c0: "c$0 = (0::'a::idom)" and d0: "d$0 = 0" shows "((a oo c) * (b oo d))$n = setsum (%(k,m). a$k * b$m * (c^k * d^m) $ n) {(k,m). k + m ≤ n}" (is "?l = ?r") proof- let ?S = "{(k::nat, m::nat). k + m ≤ n}" have s: "?S ⊆ {0..n} <*> {0..n}" by (auto simp add: subset_eq) have f: "finite {(k::nat, m::nat). k + m ≤ n}" apply (rule finite_subset[OF s]) by auto have "?r = setsum (%i. setsum (%(k,m). a$k * (c^k)$i * b$m * (d^m) $ (n - i)) {(k,m). k + m ≤ n}) {0..n}" apply (simp add: fps_mult_nth setsum_right_distrib) apply (subst setsum_commute) apply (rule setsum_cong2) by (auto simp add: ring_simps) also have "… = ?l" apply (simp add: fps_mult_nth fps_compose_nth setsum_product) apply (rule setsum_cong2) apply (simp add: setsum_cartesian_product mult_assoc) apply (rule setsum_mono_zero_right[OF f]) apply (simp add: subset_eq) apply presburger apply clarsimp apply (rule ccontr) apply (clarsimp simp add: not_le) apply (case_tac "x < aa") apply simp apply (frule_tac startsby_zero_power_prefix[rule_format, OF c0]) apply blast apply simp apply (frule_tac startsby_zero_power_prefix[rule_format, OF d0]) apply blast done finally show ?thesis by simp qed lemma product_composition_lemma': assumes c0: "c$0 = (0::'a::idom)" and d0: "d$0 = 0" shows "((a oo c) * (b oo d))$n = setsum (%k. setsum (%m. a$k * b$m * (c^k * d^m) $ n) {0..n}) {0..n}" (is "?l = ?r") unfolding product_composition_lemma[OF c0 d0] unfolding setsum_cartesian_product apply (rule setsum_mono_zero_left) apply simp apply (clarsimp simp add: subset_eq) apply clarsimp apply (rule ccontr) apply (subgoal_tac "(c^aa * d^ba) $ n = 0") apply simp unfolding fps_mult_nth apply (rule setsum_0') apply (clarsimp simp add: not_le) apply (case_tac "aaa < aa") apply (rule startsby_zero_power_prefix[OF c0, rule_format]) apply simp apply (subgoal_tac "n - aaa < ba") apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format]) apply simp apply arith done lemma setsum_pair_less_iff: "setsum (%((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m ≤ n} = setsum (%s. setsum (%i. a i * b (s - i) * c s) {0..s}) {0..n}" (is "?l = ?r") proof- let ?KM= "{(k,m). k + m ≤ n}" let ?f = "%s. UNION {(0::nat)..s} (%i. {(i,s - i)})" have th0: "?KM = UNION {0..n} ?f" apply (simp add: expand_set_eq) apply arith (* FIXME: VERY slow! *) done show "?l = ?r " unfolding th0 apply (subst setsum_UN_disjoint) apply auto apply (subst setsum_UN_disjoint) apply auto done qed lemma fps_compose_mult_distrib_lemma: assumes c0: "c$0 = (0::'a::idom)" shows "((a oo c) * (b oo c))$n = setsum (%s. setsum (%i. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}" (is "?l = ?r") unfolding product_composition_lemma[OF c0 c0] power_add[symmetric] unfolding setsum_pair_less_iff[where a = "%k. a$k" and b="%m. b$m" and c="%s. (c ^ s)$n" and n = n] .. lemma fps_compose_mult_distrib: assumes c0: "c$0 = (0::'a::idom)" shows "(a * b) oo c = (a oo c) * (b oo c)" (is "?l = ?r") apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma[OF c0]) by (simp add: fps_compose_nth fps_mult_nth setsum_left_distrib) lemma fps_compose_setprod_distrib: assumes c0: "c$0 = (0::'a::idom)" shows "(setprod a S) oo c = setprod (%k. a k oo c) S" (is "?l = ?r") apply (cases "finite S") apply simp_all apply (induct S rule: finite_induct) apply simp apply (simp add: fps_compose_mult_distrib[OF c0]) done lemma fps_compose_power: assumes c0: "c$0 = (0::'a::idom)" shows "(a oo c)^n = a^n oo c" (is "?l = ?r") proof- {assume "n=0" then have ?thesis by simp} moreover {fix m assume m: "n = Suc m" have th0: "a^n = setprod (%k. a) {0..m}" "(a oo c) ^ n = setprod (%k. a oo c) {0..m}" by (simp_all add: setprod_constant m) then have ?thesis by (simp add: fps_compose_setprod_distrib[OF c0])} ultimately show ?thesis by (cases n, auto) qed lemma fps_const_mult_apply_left: "fps_const c * (a oo b) = (fps_const c * a) oo b" by (simp add: fps_eq_iff fps_compose_nth setsum_right_distrib mult_assoc) lemma fps_const_mult_apply_right: "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b" by (auto simp add: fps_const_mult_apply_left mult_commute) lemma fps_compose_assoc: assumes c0: "c$0 = (0::'a::idom)" and b0: "b$0 = 0" shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r") proof- {fix n have "?l$n = (setsum (λi. (fps_const (a$i) * b^i) oo c) {0..n})$n" by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left setsum_right_distrib mult_assoc fps_setsum_nth) also have "… = ((setsum (λi. fps_const (a$i) * b^i) {0..n}) oo c)$n" by (simp add: fps_compose_setsum_distrib) also have "… = ?r$n" apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult_assoc) apply (rule setsum_cong2) apply (rule setsum_mono_zero_right) apply (auto simp add: not_le) by (erule startsby_zero_power_prefix[OF b0, rule_format]) finally have "?l$n = ?r$n" .} then show ?thesis by (simp add: fps_eq_iff) qed lemma fps_X_power_compose: assumes a0: "a$0=0" shows "X^k oo a = (a::('a::idom fps))^k" (is "?l = ?r") proof- {assume "k=0" hence ?thesis by simp} moreover {fix h assume h: "k = Suc h" {fix n {assume kn: "k>n" hence "?l $ n = ?r $n" using a0 startsby_zero_power_prefix[OF a0] h by (simp add: fps_compose_nth del: power_Suc)} moreover {assume kn: "k ≤ n" hence "?l$n = ?r$n" by (simp add: fps_compose_nth mult_delta_left setsum_delta)} moreover have "k >n ∨ k≤ n" by arith ultimately have "?l$n = ?r$n" by blast} then have ?thesis unfolding fps_eq_iff by blast} ultimately show ?thesis by (cases k, auto) qed lemma fps_inv_right: assumes a0: "a$0 = 0" and a1: "a$1 ≠ 0" shows "a oo fps_inv a = X" proof- let ?ia = "fps_inv a" let ?iaa = "a oo fps_inv a" have th0: "?ia $ 0 = 0" by (simp add: fps_inv_def) have th1: "?iaa $ 0 = 0" using a0 a1 by (simp add: fps_inv_def fps_compose_nth) have th2: "X$0 = 0" by simp from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo X" by simp then have "(a oo fps_inv a) oo a = X oo a" by (simp add: fps_compose_assoc[OF a0 th0] X_fps_compose_startby0[OF a0]) with fps_compose_inj_right[OF a0 a1] show ?thesis by simp qed lemma fps_inv_deriv: assumes a0:"a$0 = (0::'a::{recpower,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$0 ≠ 0" using a1 by (simp add: fps_compose_nth fps_deriv_nth) from fps_inv_right[OF a0 a1] have "?d * ?dia = 1" by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] ) hence "inverse ?d * ?d * ?dia = inverse ?d * 1" by simp with inverse_mult_eq_1[OF th0] show "?dia = inverse ?d" by simp qed subsection{* Elementary series *} subsubsection{* Exponential series *} definition "E x = Abs_fps (λn. x^n / of_nat (fact n))" lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::{field, recpower, ring_char_0}) * E a" (is "?l = ?r") proof- {fix n have "?l$n = ?r $ n" apply (auto simp add: E_def field_simps power_Suc[symmetric]simp del: fact_Suc of_nat_Suc power_Suc) by (simp add: of_nat_mult ring_simps)} then show ?thesis by (simp add: fps_eq_iff) qed lemma E_unique_ODE: "fps_deriv a = fps_const c * a <-> a = fps_const (a$0) * E (c :: 'a::{field, ring_char_0, recpower})" (is "?lhs <-> ?rhs") proof- {assume d: ?lhs from d have th: "!!n. a $ Suc n = c * a$n / of_nat (Suc n)" by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc) {fix n have "a$n = a$0 * c ^ n/ (of_nat (fact n))" apply (induct n) apply simp unfolding th using fact_gt_zero apply (simp add: field_simps del: of_nat_Suc fact.simps) apply (drule sym) by (simp add: ring_simps of_nat_mult power_Suc)} note th' = this have ?rhs by (auto simp add: fps_eq_iff fps_const_mult_left E_def intro : th')} moreover {assume h: ?rhs have ?lhs apply (subst h) apply simp apply (simp only: h[symmetric]) by simp} ultimately show ?thesis by blast qed lemma E_add_mult: "E (a + b) = E (a::'a::{ring_char_0, field, recpower}) * E b" (is "?l = ?r") proof- have "fps_deriv (?r) = fps_const (a+b) * ?r" by (simp add: fps_const_add[symmetric] ring_simps del: fps_const_add) then have "?r = ?l" apply (simp only: E_unique_ODE) by (simp add: fps_mult_nth E_def) then show ?thesis .. qed lemma E_nth[simp]: "E a $ n = a^n / of_nat (fact n)" by (simp add: E_def) lemma E0[simp]: "E (0::'a::{field, recpower}) = 1" by (simp add: fps_eq_iff power_0_left) lemma E_neg: "E (- a) = inverse (E (a::'a::{ring_char_0, field, recpower}))" proof- from E_add_mult[of a "- a"] have th0: "E a * E (- a) = 1" by (simp ) have th1: "E a $ 0 ≠ 0" by simp from fps_inverse_unique[OF th1 th0] show ?thesis by simp qed lemma E_nth_deriv[simp]: "fps_nth_deriv n (E (a::'a::{field, recpower, ring_char_0})) = (fps_const a)^n * (E a)" by (induct n, auto simp add: power_Suc) lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)" by (simp add: fps_eq_iff fps_compose_nth ring_simps setsum_negf[symmetric]) lemma fps_compose_sub_distrib: shows "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)" unfolding diff_minus fps_compose_uminus fps_compose_add_distrib .. lemma X_fps_compose:"X oo a = Abs_fps (λn. if n = 0 then (0::'a::comm_ring_1) else a$n)" by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta power_Suc) lemma X_compose_E[simp]: "X oo E (a::'a::{field, recpower}) = E a - 1" by (simp add: fps_eq_iff X_fps_compose) lemma LE_compose: assumes a: "a≠0" shows "fps_inv (E a - 1) oo (E a - 1) = X" and "(E a - 1) oo fps_inv (E a - 1) = X" proof- let ?b = "E a - 1" have b0: "?b $ 0 = 0" by simp have b1: "?b $ 1 ≠ 0" by (simp add: a) from fps_inv[OF b0 b1] show "fps_inv (E a - 1) oo (E a - 1) = X" . from fps_inv_right[OF b0 b1] show "(E a - 1) oo fps_inv (E a - 1) = X" . qed lemma fps_const_inverse: "inverse (fps_const (a::'a::{field, division_by_zero})) = fps_const (inverse a)" apply (auto simp add: fps_eq_iff fps_inverse_def) by (case_tac "n", auto) lemma inverse_one_plus_X: "inverse (1 + X) = Abs_fps (λn. (- 1 ::'a::{field, recpower})^n)" (is "inverse ?l = ?r") proof- have th: "?l * ?r = 1" apply (auto simp add: ring_simps fps_eq_iff X_mult_nth minus_one_power_iff) apply presburger+ done have th': "?l $ 0 ≠ 0" by (simp add: ) from fps_inverse_unique[OF th' th] show ?thesis . qed lemma E_power_mult: "(E (c::'a::{field,recpower,ring_char_0}))^n = E (of_nat n * c)" by (induct n, auto simp add: ring_simps E_add_mult power_Suc) subsubsection{* Logarithmic series *} definition "(L::'a::{field, ring_char_0,recpower} fps) = Abs_fps (λn. (- 1) ^ Suc n / of_nat n)" lemma fps_deriv_L: "fps_deriv L = inverse (1 + X)" unfolding inverse_one_plus_X by (simp add: L_def fps_eq_iff power_Suc del: of_nat_Suc) lemma L_nth: "L $ n = (- 1) ^ Suc n / of_nat n" by (simp add: L_def) lemma L_E_inv: assumes a: "a≠ (0::'a::{field,division_by_zero,ring_char_0,recpower})" shows "L = fps_const a * fps_inv (E a - 1)" (is "?l = ?r") proof- let ?b = "E a - 1" have b0: "?b $ 0 = 0" by simp have b1: "?b $ 1 ≠ 0" by (simp add: a) have "fps_deriv (E a - 1) oo fps_inv (E a - 1) = (fps_const a * (E a - 1) + fps_const a) oo fps_inv (E a - 1)" by (simp add: ring_simps) also have "… = fps_const a * (X + 1)" apply (simp add: fps_compose_add_distrib fps_const_mult_apply_left[symmetric] fps_inv_right[OF b0 b1]) by (simp add: ring_simps) finally have eq: "fps_deriv (E a - 1) oo fps_inv (E a - 1) = fps_const a * (X + 1)" . from fps_inv_deriv[OF b0 b1, unfolded eq] have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (X + 1)" by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult) hence "fps_deriv (fps_const a * fps_inv ?b) = inverse (X + 1)" using a by (simp add: fps_divide_def field_simps) hence "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_L add_commute) then show ?thesis unfolding fps_deriv_eq_iff by (simp add: L_nth fps_inv_def) qed subsubsection{* Formal trigonometric functions *} definition "fps_sin (c::'a::{field, recpower, ring_char_0}) = Abs_fps (λn. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))" definition "fps_cos (c::'a::{field, recpower, ring_char_0}) = Abs_fps (λn. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)" lemma fps_sin_deriv: "fps_deriv (fps_sin c) = fps_const c * fps_cos c" (is "?lhs = ?rhs") proof- {fix n::nat {assume en: "even n" have "?lhs$n = of_nat (n+1) * (fps_sin c $ (n+1))" by simp also have "… = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))" using en by (simp add: fps_sin_def) also have "… = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))" unfolding fact_Suc of_nat_mult by (simp add: field_simps del: of_nat_add of_nat_Suc) also have "… = (- 1)^(n div 2) *c^Suc n / of_nat (fact n)" by (simp add: field_simps del: of_nat_add of_nat_Suc) finally have "?lhs $n = ?rhs$n" using en by (simp add: fps_cos_def ring_simps power_Suc )} then have "?lhs $ n = ?rhs $ n" by (cases "even n", simp_all add: fps_deriv_def fps_sin_def fps_cos_def) } then show ?thesis by (auto simp add: fps_eq_iff) qed lemma fps_cos_deriv: "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)" (is "?lhs = ?rhs") proof- have th0: "!!n. - ((- 1::'a) ^ n) = (- 1)^Suc n" by (simp add: power_Suc) have th1: "!!n. odd n==> Suc ((n - 1) div 2) = Suc n div 2" by presburger (* FIXME: VERY slow! *) {fix n::nat {assume en: "odd n" from en have n0: "n ≠0 " by presburger have "?lhs$n = of_nat (n+1) * (fps_cos c $ (n+1))" by simp also have "… = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))" using en by (simp add: fps_cos_def) also have "… = (- 1)^((n + 1) div 2)*c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))" unfolding fact_Suc of_nat_mult by (simp add: field_simps del: of_nat_add of_nat_Suc) also have "… = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)" by (simp add: field_simps del: of_nat_add of_nat_Suc) also have "… = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)" unfolding th0 unfolding th1[OF en] by simp finally have "?lhs $n = ?rhs$n" using en by (simp add: fps_sin_def ring_simps power_Suc)} then have "?lhs $ n = ?rhs $ n" by (cases "even n", simp_all add: fps_deriv_def fps_sin_def fps_cos_def) } then show ?thesis by (auto simp add: fps_eq_iff) qed lemma fps_sin_cos_sum_of_squares: "fps_cos c ^ 2 + fps_sin c ^ 2 = 1" (is "?lhs = 1") proof- have "fps_deriv ?lhs = 0" apply (simp add: fps_deriv_power fps_sin_deriv fps_cos_deriv power_Suc) by (simp add: fps_power_def ring_simps fps_const_neg[symmetric] del: fps_const_neg) then have "?lhs = fps_const (?lhs $ 0)" unfolding fps_deriv_eq_0_iff . also have "… = 1" by (auto simp add: fps_eq_iff fps_power_def numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def) finally show ?thesis . qed definition "fps_tan c = fps_sin c / fps_cos c" lemma fps_tan_deriv: "fps_deriv(fps_tan c) = fps_const c/ (fps_cos c ^ 2)" proof- have th0: "fps_cos c $ 0 ≠ 0" by (simp add: fps_cos_def) show ?thesis using fps_sin_cos_sum_of_squares[of c] apply (simp add: fps_tan_def fps_divide_deriv[OF th0] fps_sin_deriv fps_cos_deriv add: fps_const_neg[symmetric] ring_simps power2_eq_square del: fps_const_neg) unfolding right_distrib[symmetric] by simp qed end