(*  Title:      HOL/Library/Poly_Deriv.thy

     Author:     Amine Chaieb

     Author:     Brian Huffman

 *)

 section\<open>Polynomials and Differentiation\<close>

 theory Poly_Deriv

 imports Deriv Polynomial

 begin

 subsection \<open>Derivatives of univariate polynomials\<close>

 function pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly"

 where

   [simp del]: "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"

   by (auto intro: pCons_cases)

 termination pderiv

   by (relation "measure degree") simp_all

 lemma pderiv_0 [simp]:

   "pderiv 0 = 0"

   using pderiv.simps [of 0 0] by simp

 lemma pderiv_pCons:

   "pderiv (pCons a p) = p + pCons 0 (pderiv p)"

   by (simp add: pderiv.simps)

 lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"

   by (induct p arbitrary: n) 

      (auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)

 primrec pderiv_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list"

 where

   "pderiv_coeffs [] = []"

 | "pderiv_coeffs (x # xs) = plus_coeffs xs (cCons 0 (pderiv_coeffs xs))"

 lemma coeffs_pderiv [code abstract]:

   "coeffs (pderiv p) = pderiv_coeffs (coeffs p)"

   by (rule sym, induct p) (simp_all add: pderiv_pCons coeffs_plus_eq_plus_coeffs cCons_def)

 lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"

   apply (rule iffI)

   apply (cases p, simp)

   apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)

   apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)

   done

 lemma degree_pderiv: "degree (pderiv p) = degree p - 1"

   apply (rule order_antisym [OF degree_le])

   apply (simp add: coeff_pderiv coeff_eq_0)

   apply (cases "degree p", simp)

   apply (rule le_degree)

   apply (simp add: coeff_pderiv del: of_nat_Suc)

   apply (metis degree_0 leading_coeff_0_iff nat.distinct(1))

   done

 lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"

 by (simp add: pderiv_pCons)

 lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"

 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)

 lemma pderiv_minus: "pderiv (- p) = - pderiv p"

 by (rule poly_eqI, simp add: coeff_pderiv)

 lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"

 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)

 lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"

 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)

 lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"

 by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)

 lemma pderiv_power_Suc:

   "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"

 apply (induct n)

 apply simp

 apply (subst power_Suc)

 apply (subst pderiv_mult)

 apply (erule ssubst)

 apply (simp only: of_nat_Suc smult_add_left smult_1_left)

 apply (simp add: algebra_simps)

 done

 lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"

 by (rule DERIV_cong, rule DERIV_pow, simp)

 declare DERIV_pow2 [simp] DERIV_pow [simp]

 lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"

 by (rule DERIV_cong, rule DERIV_add, auto)

 lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"

   by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)

 text\<open>Consequences of the derivative theorem above\<close>

 lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"

 apply (simp add: real_differentiable_def)

 apply (blast intro: poly_DERIV)

 done

 lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"

 by (rule poly_DERIV [THEN DERIV_isCont])

 lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]

       ==> \<exists>x. a < x & x < b & (poly p x = 0)"

 using IVT_objl [of "poly p" a 0 b]

 by (auto simp add: order_le_less)

 lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]

       ==> \<exists>x. a < x & x < b & (poly p x = 0)"

 by (insert poly_IVT_pos [where p = "- p" ]) simp

 lemma poly_MVT: "(a::real) < b ==>

      \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"

 using MVT [of a b "poly p"]

 apply auto

 apply (rule_tac x = z in exI)

 apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])

 done

 text\<open>Lemmas for Derivatives\<close>

 lemma order_unique_lemma:

   fixes p :: "'a::idom poly"

   assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p"

   shows "n = order a p"

 unfolding Polynomial.order_def

 apply (rule Least_equality [symmetric])

 apply (fact assms)

 apply (rule classical)

 apply (erule notE)

 unfolding not_less_eq_eq

 using assms(1) apply (rule power_le_dvd)

 apply assumption

 done

 lemma lemma_order_pderiv1:

   "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +

     smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"

 apply (simp only: pderiv_mult pderiv_power_Suc)

 apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)

 done

 lemma dvd_add_cancel1:

   fixes a b c :: "'a::comm_ring_1"

   shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"

   by (drule (1) Rings.dvd_diff, simp)

 lemma lemma_order_pderiv:

   assumes n: "0 < n" 

       and pd: "pderiv p \<noteq> 0" 

       and pe: "p = [:- a, 1:] ^ n * q" 

       and nd: "~ [:- a, 1:] dvd q"

     shows "n = Suc (order a (pderiv p))"

 using n 

 proof -

   have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0"

     using assms by auto

   obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0"

     using assms by (cases n) auto

   then have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l"

     by (metis dvd_add_cancel1 dvd_smult_iff dvd_triv_left of_nat_eq_0_iff old.nat.distinct(2))

   have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" 

   proof (rule order_unique_lemma)

     show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"

       apply (subst lemma_order_pderiv1)

       apply (rule dvd_add)

       apply (metis dvdI dvd_mult2 power_Suc2)

       apply (metis dvd_smult dvd_triv_right)

       done

   next

     show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"

      apply (subst lemma_order_pderiv1)

      by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one)

   qed

   then show ?thesis

     by (metis \<open>n = Suc n'\<close> pe)

 qed

 lemma order_decomp:

   assumes "p \<noteq> 0"

   shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q"

 proof -

   from assms have A: "[:- a, 1:] ^ order a p dvd p"

     and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order)

   from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" ..

   with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q"

     by simp

   then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q"

     by simp

   then have D: "\<not> [:- a, 1:] dvd q"

     using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q]

     by auto

   from C D show ?thesis by blast

 qed

 lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]

       ==> (order a p = Suc (order a (pderiv p)))"

 apply (case_tac "p = 0", simp)

 apply (drule_tac a = a and p = p in order_decomp)

 using neq0_conv

 apply (blast intro: lemma_order_pderiv)

 done

 lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"

 proof -

   def i \<equiv> "order a p"

   def j \<equiv> "order a q"

   def t \<equiv> "[:-a, 1:]"

   have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"

     unfolding t_def by (simp add: dvd_iff_poly_eq_0)

   assume "p * q \<noteq> 0"

   then show "order a (p * q) = i + j"

     apply clarsimp

     apply (drule order [where a=a and p=p, folded i_def t_def])

     apply (drule order [where a=a and p=q, folded j_def t_def])

     apply clarify

     apply (erule dvdE)+

     apply (rule order_unique_lemma [symmetric], fold t_def)

     apply (simp_all add: power_add t_dvd_iff)

     done

 qed

 text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close>

 lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"

 apply (cases "p = 0", auto)

 apply (drule order_2 [where a=a and p=p])

 apply (metis not_less_eq_eq power_le_dvd)

 apply (erule power_le_dvd [OF order_1])

 done

 lemma poly_squarefree_decomp_order:

   assumes "pderiv p \<noteq> 0"

   and p: "p = q * d"

   and p': "pderiv p = e * d"

   and d: "d = r * p + s * pderiv p"

   shows "order a q = (if order a p = 0 then 0 else 1)"

 proof (rule classical)

   assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"

   from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto

   with p have "order a p = order a q + order a d"

     by (simp add: order_mult)

   with 1 have "order a p \<noteq> 0" by (auto split: if_splits)

   have "order a (pderiv p) = order a e + order a d"

     using \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> by (simp add: order_mult)

   have "order a p = Suc (order a (pderiv p))"

     using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv)

   have "d \<noteq> 0" using \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> by simp

   have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"

     apply (simp add: d)

     apply (rule dvd_add)

     apply (rule dvd_mult)

     apply (simp add: order_divides \<open>p \<noteq> 0\<close>

            \<open>order a p = Suc (order a (pderiv p))\<close>)

     apply (rule dvd_mult)

     apply (simp add: order_divides)

     done

   then have "order a (pderiv p) \<le> order a d"

     using \<open>d \<noteq> 0\<close> by (simp add: order_divides)

   show ?thesis

     using \<open>order a p = order a q + order a d\<close>

     using \<open>order a (pderiv p) = order a e + order a d\<close>

     using \<open>order a p = Suc (order a (pderiv p))\<close>

     using \<open>order a (pderiv p) \<le> order a d\<close>

     by auto

 qed

 lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;

          p = q * d;

          pderiv p = e * d;

          d = r * p + s * pderiv p

       |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"

 by (blast intro: poly_squarefree_decomp_order)

 lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]

       ==> (order a (pderiv p) = n) = (order a p = Suc n)"

 by (auto dest: order_pderiv)

 definition

   rsquarefree :: "'a::idom poly => bool" where

   "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"

 lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"

 apply (simp add: pderiv_eq_0_iff)

 apply (case_tac p, auto split: if_splits)

 done

 lemma rsquarefree_roots:

   "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"

 apply (simp add: rsquarefree_def)

 apply (case_tac "p = 0", simp, simp)

 apply (case_tac "pderiv p = 0")

 apply simp

 apply (drule pderiv_iszero, clarsimp)

 apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree)

 apply (force simp add: order_root order_pderiv2)

 done

 lemma poly_squarefree_decomp:

   assumes "pderiv p \<noteq> 0"

     and "p = q * d"

     and "pderiv p = e * d"

     and "d = r * p + s * pderiv p"

   shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"

 proof -

   from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto

   with \<open>p = q * d\<close> have "q \<noteq> 0" by simp

   have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"

     using assms by (rule poly_squarefree_decomp_order2)

   with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis

     by (simp add: rsquarefree_def order_root)

 qed

 end