Theory Sturm_Tarski

(*  Title: Sturm-Tarski Theorem
    Author:     Wenda Li <wl302@cam.ac.uk / liwenda1990@hotmail.com>
*)

section "Sturm--Tarski Theorem"

theory Sturm_Tarski
  imports Complex_Main PolyMisc "HOL-Computational_Algebra.Field_as_Ring"
begin

subsection‹Misc›

lemma eventually_at_right:
  fixes x::"'a::{archimedean_field,linorder_topology}"
  shows "eventually P (at_right x) ⟷ (∃b>x. ∀y>x. y < b ⟶ P y)"
proof -
  obtain y where "y>x" using ex_less_of_int by auto  
  thus ?thesis using eventually_at_right[OF ‹y>x›] by auto
qed

lemma eventually_at_left:
  fixes x::"'a::{archimedean_field,linorder_topology}"
  shows "eventually P (at_left x) ⟷ (∃b<x. ∀y>b. y < x ⟶ P y)"
proof -
  obtain y where "y<x" 
    using linordered_field_no_lb by auto
  thus ?thesis using eventually_at_left[OF ‹y<x›] by auto
qed

lemma eventually_neg:
  assumes "F≠bot"  and eve:"eventually (λx. P x) F"
  shows "¬ eventually (λx. ¬ P x) F"
proof (rule ccontr)
  assume "¬ ¬ eventually (λx. ¬ P x) F"
  hence "eventually (λx. ¬ P x) F" by auto
  hence "eventually (λx. False) F" using eventually_conj[OF eve,of "(λx. ¬ P x)"] by auto
  thus False using ‹F≠bot› eventually_False by auto 
qed

lemma poly_tendsto[simp]:
    "(poly p ⤏ poly p x) (at (x::real))" 
    "(poly p ⤏ poly p x) (at_left (x::real))"
    "(poly p ⤏ poly p x) (at_right (x::real))"
  using isCont_def[where f="poly p"] by (auto simp add:filterlim_at_split)

lemma not_eq_pos_or_neg_iff_1:
  fixes p::"real poly" 
  shows "(∀z. lb<z∧z≤ub⟶poly p z≠0) ⟷ 
    (∀z. lb<z∧z≤ub⟶poly p z>0)∨(∀z. lb<z∧z≤ub⟶poly p z<0)" (is "?Q ⟷ ?P")
proof (rule,rule ccontr)
  assume "?Q" "¬?P" 
  then obtain z1 z2 where z1:"lb<z1" "z1≤ub" "poly p z1≤0" 
                      and z2:"lb<z2" "z2≤ub" "poly p z2≥0"
    by auto
  hence "∃z. lb<z∧z≤ub∧poly p z=0"
  proof (cases "poly p z1 = 0 ∨ poly p z2 =0 ∨ z1=z2")
    case True
    thus ?thesis using z1 z2 by auto
  next
    case False
    hence "poly p z1<0" and "poly p z2>0" and "z1≠z2" using z1(3) z2(3) by auto
    hence "(∃z>z1. z < z2 ∧ poly p z = 0) ∨ (∃z>z2. z < z1 ∧ poly p z = 0)"
      using poly_IVT_neg poly_IVT_pos by (subst (asm) linorder_class.neq_iff,auto) 
    thus ?thesis using z1(1,2) z2(1,2) by (metis less_eq_real_def order.strict_trans2)
  qed
  thus False using ‹?Q› by auto
next
  assume "?P"
  thus ?Q by auto
qed

lemma not_eq_pos_or_neg_iff_2:
  fixes p::"real poly" 
  shows "(∀z. lb≤z∧z<ub⟶poly p z≠0) 
    ⟷(∀z. lb≤z∧z<ub⟶poly p z>0)∨(∀z. lb≤z∧z<ub⟶poly p z<0)" (is "?Q⟷?P")
proof (rule,rule ccontr)
  assume "?Q" "¬?P"
  then obtain z1 z2 where z1:"lb≤z1" "z1<ub" "poly p z1≤0" 
                      and z2:"lb≤z2" "z2<ub" "poly p z2≥0"
    by auto
  hence "∃z. lb≤z∧z<ub∧poly p z=0"
  proof (cases "poly p z1 = 0 ∨ poly p z2 =0 ∨ z1=z2")
    case True
    thus ?thesis using z1 z2 by auto
  next
    case False
    hence "poly p z1<0" and "poly p z2>0" and "z1≠z2" using z1(3) z2(3) by auto
    hence "(∃z>z1. z < z2 ∧ poly p z = 0) ∨ (∃z>z2. z < z1 ∧ poly p z = 0)"
      using poly_IVT_neg poly_IVT_pos by (subst (asm) linorder_class.neq_iff,auto) 
    thus ?thesis using z1(1,2) z2(1,2) by (meson dual_order.strict_trans not_le)
  qed
  thus False using ‹?Q› by auto
next
  assume "?P"
  thus ?Q by auto
qed

lemma next_non_root_interval:
  fixes p::"real poly"
  assumes "p≠0"
  obtains ub where "ub>lb" and "(∀z. lb<z∧z≤ub⟶poly p z≠0)" 
proof (cases "(∃ r. poly p r=0 ∧ r>lb)")
  case False
  thus ?thesis by (intro that[of "lb+1"],auto)
next
  case True
  define lr where "lr≡Min {r . poly p r=0 ∧ r>lb}" 
  have "∀z. lb<z∧z<lr⟶poly p z≠0" and "lr>lb" 
    using True lr_def poly_roots_finite[OF ‹p≠0›] by auto
  thus ?thesis using that[of "(lb+lr)/2"] by auto 
qed

lemma last_non_root_interval:
  fixes p::"real poly"
  assumes "p≠0"
  obtains lb where "lb<ub" and "(∀z. lb≤z∧z<ub⟶poly p z≠0)"
proof (cases "(∃ r. poly p r=0 ∧ r<ub)")
  case False
  thus ?thesis by (intro that[of "ub - 1"]) auto
next
  case True
  define mr where  "mr≡Max {r . poly p r=0 ∧ r<ub}" 
  have "∀z. mr<z∧z<ub⟶poly p z≠0" and "mr<ub" 
    using True mr_def poly_roots_finite[OF ‹p≠0›] by auto
  thus ?thesis using that[of "(mr+ub)/2"] ‹mr<ub› by auto
qed

subsection‹Sign›

definition sign:: "'a::{zero,linorder} ⇒ int" where
  "sign x≡(if x>0 then 1 else if x=0 then 0 else -1)"

lemma sign_simps[simp]:
  "x>0 ⟹ sign x=1"
  "x=0 ⟹ sign x=0"
  "x<0 ⟹ sign x=-1"
unfolding sign_def by auto

lemma sign_cases [case_names neg zero pos]:
  "(sign x = -1 ⟹ P) ⟹ (sign x = 0 ⟹ P) ⟹ (sign x =1 ⟹ P) ⟹ P"  
unfolding Sturm_Tarski.sign_def by argo   

lemma sign_times:
  fixes x::"'a::linordered_ring_strict"
  shows "sign (x*y) = sign x * sign y"
  unfolding Sturm_Tarski.sign_def 
  by (auto simp add:zero_less_mult_iff)
   
lemma sign_power:
  fixes x::"'a::linordered_idom"
  shows "sign (x^n) = (if n=0 then 1 else if even n then ¦sign x¦ else sign x)"
  by (simp add: Sturm_Tarski.sign_def zero_less_power_eq)

(*
lemma sgn_sign_eq:
  fixes x::"'a::{linordered_idom}"
  shows "sgn x = of_int (sign x)" 
  unfolding sgn_if by auto
*)

lemma sgn_sign_eq:"sgn = sign"
  unfolding sign_def sgn_if by auto

lemma sign_sgn[simp]: "sign (sgn x) = sign (x::'b::linordered_idom)"
  by (simp add: sign_def)

lemma sign_uminus[simp]:"sign (- x) = - sign (x::'b::linordered_idom)"
  by (simp add: sign_def)


subsection‹Bound of polynomials›

definition sgn_pos_inf :: "('a ::linordered_idom) poly ⇒ 'a" where 
  "sgn_pos_inf p ≡ sgn (lead_coeff p)"
definition sgn_neg_inf :: "('a ::linordered_idom) poly ⇒ 'a" where 
  "sgn_neg_inf p ≡ if even (degree p) then sgn (lead_coeff p) else -sgn (lead_coeff p)"

lemma sgn_inf_sym:
  fixes p::"real poly"
  shows "sgn_pos_inf (pcompose p [:0,-1:]) = sgn_neg_inf p" (is "?L=?R")
proof -
  have "?L= sgn (lead_coeff p * (- 1) ^ degree p)" 
    unfolding sgn_pos_inf_def by (subst lead_coeff_comp,auto)
  thus ?thesis unfolding sgn_neg_inf_def 
    by (metis mult.right_neutral mult_minus1_right neg_one_even_power neg_one_odd_power sgn_minus)
qed

lemma poly_pinfty_gt_lc:
  fixes p:: "real poly"
  assumes  "lead_coeff p > 0" 
  shows "∃ n. ∀ x ≥ n. poly p x ≥ lead_coeff p" using assms
proof (induct p)
  case 0
  thus ?case by auto
next
  case (pCons a p)
  have "⟦a≠0;p=0⟧ ⟹ ?case" by auto
  moreover have "p≠0 ⟹ ?case"
  proof -
    assume "p≠0"
    then obtain n1 where gte_lcoeff:"∀x≥n1. lead_coeff p ≤ poly p x" using that pCons by auto
    have gt_0:"lead_coeff p >0" using pCons(3) ‹p≠0› by auto
    define n where "n≡max n1 (1+ ¦a¦/(lead_coeff p))"
    show ?thesis 
    proof (rule_tac x=n in exI,rule,rule) 
      fix x assume "n ≤ x"
      hence "lead_coeff p ≤ poly p x" 
        using gte_lcoeff unfolding n_def by auto
      hence " ¦a¦/(lead_coeff p) ≥ ¦a¦/(poly p x)" and "poly p x>0" using gt_0
        by (intro frac_le,auto)
      hence "x≥1+ ¦a¦/(poly p x)" using ‹n≤x›[unfolded n_def] by auto
      thus "lead_coeff (pCons a p) ≤ poly (pCons a p) x"
        using ‹lead_coeff p ≤ poly p x› ‹poly p x>0› ‹p≠0›
        by (auto simp add:field_simps)
      qed
    qed
  ultimately show ?case by fastforce
qed

lemma poly_sgn_eventually_at_top:
  fixes p::"real poly"
  shows "eventually (λx. sgn (poly p x) = sgn_pos_inf p) at_top"
proof (cases "p=0")
  case True
  thus ?thesis unfolding sgn_pos_inf_def by auto
next
  case False
  obtain ub where ub:"∀x≥ub. sgn (poly p x) = sgn_pos_inf p"
  proof (cases "lead_coeff p>0")
    case True
    thus ?thesis 
      using that poly_pinfty_gt_lc[of p] unfolding sgn_pos_inf_def by fastforce 
  next
    case False
    hence "lead_coeff (-p) > 0" and "lead_coeff p < 0" unfolding lead_coeff_minus
      using leading_coeff_neq_0[OF ‹p≠0›] 
      by (auto simp add:not_less_iff_gr_or_eq)
    then obtain n where "∀x≥n. lead_coeff p ≥ poly p x"
      using poly_pinfty_gt_lc[of "-p"] unfolding lead_coeff_minus by auto
    thus ?thesis using ‹lead_coeff p<0› that[of n] unfolding sgn_pos_inf_def by fastforce
  qed
  thus ?thesis unfolding eventually_at_top_linorder by auto
qed

lemma poly_sgn_eventually_at_bot:
  fixes p::"real poly"
  shows "eventually (λx. sgn (poly p x) = sgn_neg_inf p) at_bot"
using 
  poly_sgn_eventually_at_top[of "pcompose p [:0,-1:]",unfolded poly_pcompose sgn_inf_sym,simplified]
  eventually_filtermap[of _ uminus "at_bot::real filter",folded at_top_mirror]
by auto
   
lemma root_ub:
  fixes p:: "real poly"
  assumes "p≠0"
  obtains ub where "∀x. poly p x=0 ⟶ x<ub"
    and "∀x≥ub. sgn (poly p x) = sgn_pos_inf p"
proof -
  obtain ub1 where ub1:"∀x. poly p x=0 ⟶ x<ub1"
  proof (cases "∃ r. poly p r=0")
    case False
    thus ?thesis using that by auto
  next
    case True
    define max_r where "max_r≡Max {x . poly p x=0}"
    hence "∀x. poly p x=0 ⟶ x≤max_r"
      using  poly_roots_finite[OF ‹p≠0›] True by auto
    thus ?thesis using that[of "max_r+1"] 
      by (metis add.commute add_strict_increasing zero_less_one)
  qed
  obtain ub2 where ub2:"∀x≥ub2. sgn (poly p x) = sgn_pos_inf p"
    using poly_sgn_eventually_at_top[unfolded eventually_at_top_linorder] by auto
  define ub where "ub≡max ub1 ub2"
  have "∀x. poly p x=0 ⟶ x<ub" using ub1 ub_def 
    by (metis eq_iff less_eq_real_def less_linear max.bounded_iff)
  thus ?thesis using that[of ub] ub2 ub_def by auto
qed

lemma root_lb:
  fixes p:: "real poly"
  assumes "p≠0"
  obtains lb where "∀x. poly p x=0 ⟶ x>lb"
    and "∀x≤lb. sgn (poly p x) = sgn_neg_inf p"
proof -
  obtain lb1 where lb1:"∀x. poly p x=0 ⟶ x>lb1"
  proof (cases "∃ r. poly p r=0")
    case False
    thus ?thesis using that by auto
  next
    case True
    define min_r where "min_r≡Min {x . poly p x=0}"
    hence "∀x. poly p x=0 ⟶ x≥min_r"
      using  poly_roots_finite[OF ‹p≠0›] True by auto
    thus ?thesis using that[of "min_r - 1"] by (metis lt_ex order.strict_trans2 that) 
  qed
  obtain lb2 where lb2:"∀x≤lb2. sgn (poly p x) = sgn_neg_inf p"
    using poly_sgn_eventually_at_bot[unfolded eventually_at_bot_linorder] by auto
  define lb where  "lb≡min lb1 lb2"
  have "∀x. poly p x=0 ⟶ x>lb" using lb1 lb_def 
    by (metis (poly_guards_query) less_not_sym min_less_iff_conj neq_iff)
  thus ?thesis using that[of lb] lb2 lb_def by auto
qed

subsection ‹Variation and cross›

definition variation :: "real ⇒  real ⇒ int" where
  "variation x y=(if x*y≥0 then 0 else if x<y then 1 else -1)" 

definition cross :: "real poly ⇒ real ⇒ real ⇒ int" where
  "cross p a b=variation (poly p a) (poly p b)"

lemma variation_0[simp]: "variation 0 y=0" "variation x 0=0" 
  unfolding variation_def by auto

lemma variation_comm: "variation x y= - variation y x" unfolding variation_def
  by (auto simp add: mult.commute) 

lemma cross_0[simp]: "cross 0 a b=0" unfolding cross_def by auto

lemma variation_cases:
  "⟦x>0;y>0⟧⟹variation x y = 0" 
  "⟦x>0;y<0⟧⟹variation x y = -1"
  "⟦x<0;y>0⟧⟹variation x y = 1"
  "⟦x<0;y<0⟧⟹variation x y = 0"
proof -
  show "⟦x>0;y>0⟧⟹variation x y = 0"  unfolding variation_def by auto
  show "⟦x>0;y<0⟧⟹variation x y = -1" unfolding variation_def  
    using mult_pos_neg by fastforce
  show "⟦x<0;y>0⟧⟹variation x y = 1" unfolding variation_def 
    using mult_neg_pos by fastforce
  show "⟦x<0;y<0⟧⟹variation x y = 0" unfolding variation_def 
    using mult_neg_neg by fastforce
qed

lemma variation_congr:
  assumes "sgn x=sgn x'" "sgn y=sgn y'"
  shows "variation x y=variation x' y'" using assms
proof -
  have " 0 ≤ x * y =  (0≤ x' * y')" using assms by (metis Real_Vector_Spaces.sgn_mult zero_le_sgn_iff)
  moreover hence "¬ 0≤x * y ⟹ x < y = (x' < y')" using assms 
    by (metis less_eq_real_def mult_nonneg_nonneg mult_nonpos_nonpos not_le order.strict_trans2 
      zero_le_sgn_iff)
  ultimately show ?thesis unfolding variation_def by auto
qed

lemma variation_mult_pos: 
  assumes "c>0"  
  shows "variation (c*x) y =variation x y" and "variation x (c*y) =variation x y" 
proof -
  have "sgn (c*x) = sgn x" using ‹c>0› 
    by (simp add: Real_Vector_Spaces.sgn_mult)
  thus "variation (c*x) y =variation x y" using variation_congr by blast 
next
  have "sgn (c*y) = sgn y" using ‹c>0›
    by (simp add: Real_Vector_Spaces.sgn_mult)
  thus "variation x (c*y) =variation x y" using variation_congr by blast 
qed

lemma variation_mult_neg_1: 
  assumes "c<0"  
  shows "variation (c*x) y =variation x y + (if y=0 then 0 else sign x)" 
  apply (cases x  rule:linorder_cases[of 0] )
  apply (cases y  rule:linorder_cases[of 0], auto simp add: 
    variation_cases mult_neg_pos[OF ‹c<0›,of x]  mult_neg_neg[OF ‹c<0›,of x])+ 
done

lemma variation_mult_neg_2: 
  assumes "c<0"  
  shows "variation x (c*y) = variation x y + (if x=0 then 0 else - sign y)" 
unfolding variation_comm[of x "c*y", unfolded variation_mult_neg_1[OF ‹c<0›, of y x] ] 
  by (auto,subst variation_comm,simp)

lemma cross_no_root:
  assumes "a<b" and no_root:"∀x. a<x∧x<b ⟶ poly p x≠0"
  shows "cross p a b=0"
proof -
  have "⟦poly p a>0;poly p b<0⟧ ⟹ False" using poly_IVT_neg[OF ‹a<b›] no_root by auto
  moreover have "⟦poly p a<0;poly p b>0⟧ ⟹ False" using poly_IVT_pos[OF ‹a<b›] no_root by auto
  ultimately have "0 ≤ poly p a * poly p b" 
    by (metis less_eq_real_def linorder_neqE_linordered_idom mult_less_0_iff)
  thus ?thesis unfolding cross_def variation_def by simp
qed

subsection ‹Tarski query›

definition taq :: "'a::linordered_idom set ⇒ 'a poly ⇒ int" where
  "taq s q ≡ ∑x∈s. sign (poly q x)"

subsection ‹Sign at the right›

definition sign_r_pos :: "real poly ⇒ real ⇒ bool " 
  where
  "sign_r_pos p x≡ (eventually (λx. poly p x>0) (at_right x))"

lemma sign_r_pos_rec:
  fixes p:: "real poly"
  assumes "p≠0"
  shows "sign_r_pos p x= (if poly p x=0 then sign_r_pos (pderiv p) x else poly p x>0 )"
proof (cases "poly p x=0")
  case True
  have "sign_r_pos (pderiv p) x ⟹ sign_r_pos p x" 
  proof (rule ccontr)
    assume "sign_r_pos (pderiv p) x" "¬ sign_r_pos p x"
    obtain b where "b>x" and b:"∀z. x < z ∧ z < b ⟶ 0 < poly (pderiv p) z" 
      using ‹sign_r_pos (pderiv p) x› unfolding sign_r_pos_def eventually_at_right  by auto
    have "∀b>x. ∃z>x. z < b ∧ ¬ 0 < poly p z" using ‹¬ sign_r_pos p x› 
      unfolding sign_r_pos_def eventually_at_right by auto
    then obtain z where "z>x" and "z<b" and "poly p z≤0"
      using ‹b>x› b by auto
    hence "∃z'>x. z' < z ∧ poly p z = (z - x) * poly (pderiv p) z'"
      using poly_MVT[OF ‹z>x›] True by (metis diff_0_right)
    hence "∃z'>x. z' < z ∧ poly (pderiv p) z' ≤0"
      using ‹poly p z≤0›‹z>x› by (metis leD le_iff_diff_le_0 mult_le_0_iff)
    thus False using b by (metis ‹z < b› dual_order.strict_trans not_le)
  qed
  moreover have "sign_r_pos p x ⟹ sign_r_pos (pderiv p) x" 
  proof -
    assume "sign_r_pos p x"
    have "pderiv p≠0" using ‹poly p x=0› ‹p≠0› 
      by (metis monoid_add_class.add.right_neutral monom_0 monom_eq_0 mult_zero_right 
          pderiv_iszero poly_0 poly_pCons)
    obtain ub where "ub>x" and ub: "(∀z. x<z∧z<ub⟶poly (pderiv p) z>0) 
          ∨ (∀z. x<z∧z<ub⟶poly (pderiv p) z<0)"
      using next_non_root_interval[OF ‹pderiv p≠0›, of x,unfolded not_eq_pos_or_neg_iff_1] 
      by (metis order.strict_implies_order)
    have "∀z. x<z∧z<ub⟶poly (pderiv p) z<0 ⟹ False" 
    proof -
      assume assm:"∀z. x<z∧z<ub⟶poly (pderiv p) z<0"
      obtain ub' where "ub'>x" and ub':"∀z. x < z ∧ z < ub' ⟶ 0 < poly p z" 
        using ‹sign_r_pos p x› unfolding sign_r_pos_def eventually_at_right by auto
      obtain z' where "x<z'" and "z' < (x+(min ub' ub))/2"
          and z':"poly p ((x+min ub' ub)/2) = ((x+min ub' ub)/2 - x) * poly (pderiv p) z'"       
        using poly_MVT[of x "(x+min ub' ub)/2" p] ‹ub'>x› ‹ub>x› True by auto
      moreover have "0 < poly p ((x+min ub' ub)/2)"
        using ub'[THEN HOL.spec,of "(x+(min ub' ub))/2"] ‹z' < (x+min ub' ub)/2› ‹x<z'›
        by auto
      moreover have "(x+min ub' ub)/2 - x>0" using ‹ub'>x› ‹ub>x› by auto 
      ultimately have "poly (pderiv p) z'>0" by (metis zero_less_mult_pos)
      thus False using assm[THEN spec,of z'] ‹x<z'› ‹z' < (x+(min ub' ub))/2› by auto
    qed
    hence "∀z. x<z∧z<ub⟶poly (pderiv p) z>0" using ub by auto
    thus "sign_r_pos (pderiv p) x" unfolding sign_r_pos_def eventually_at_right 
      using ‹ub>x› by auto 
  qed
  ultimately show ?thesis using True by auto
next
  case False
  have "sign_r_pos p x ⟹ poly p x>0" 
  proof (rule ccontr)
    assume "sign_r_pos p x" "¬ 0 < poly p x"
    then obtain  ub where "ub>x" and ub: "∀z. x < z ∧ z < ub ⟶ 0 < poly p z" 
      unfolding sign_r_pos_def eventually_at_right by auto
    hence "poly p ((ub+x)/2) > 0" by auto
    moreover have "poly p x<0" using ‹¬ 0 < poly p x› False by auto
    ultimately have "∃z>x. z < (ub + x) / 2 ∧ poly p z = 0"  
      using poly_IVT_pos[of x "((ub + x) / 2)" p] ‹ub>x› by auto
    thus False using ub by auto
  qed
  moreover have "poly p x>0 ⟹ sign_r_pos p x" 
    unfolding sign_r_pos_def
    using order_tendstoD(1)[OF poly_tendsto(1),of 0 p x] eventually_at_split by auto 
  ultimately show ?thesis using False by auto
qed

lemma sign_r_pos_0[simp]:"¬ sign_r_pos 0 (x::real)" 
  using eventually_False[of "at_right x"] unfolding sign_r_pos_def by auto

lemma sign_r_pos_minus:
  fixes p:: "real poly"
  assumes "p≠0"
  shows "sign_r_pos p x = (¬ sign_r_pos (-p) x)"  
proof -
  have "sign_r_pos p x ∨ sign_r_pos (-p) x"
    unfolding sign_r_pos_def eventually_at_right 
    using next_non_root_interval[OF ‹p≠0›,unfolded not_eq_pos_or_neg_iff_1] 
    by (metis (erased, opaque_lifting) le_less minus_zero neg_less_iff_less poly_minus)
  moreover have "sign_r_pos p x ⟹ ¬ sign_r_pos (-p) x" unfolding sign_r_pos_def 
    using eventually_neg[OF trivial_limit_at_right_real, of "λx. poly p x > 0" x] poly_minus 
    by (metis (lifting) eventually_mono less_asym neg_less_0_iff_less)
  ultimately show ?thesis by auto
qed

lemma sign_r_pos_smult:
  fixes p :: "real poly"
  assumes "c≠0" "p≠0"
  shows "sign_r_pos (smult c p) x= (if c>0 then sign_r_pos p x else ¬ sign_r_pos p x)"
  (is "?L=?R")
proof (cases "c>0")
  assume "c>0"
  hence "∀x. (0 < poly (smult c p) x) = (0 < poly p x)" 
    by (subst poly_smult,metis mult_pos_pos zero_less_mult_pos) 
  thus ?thesis unfolding sign_r_pos_def using ‹c>0› by auto
next
  assume "¬(c>0)"
  hence "∀x. (0 < poly (smult c p) x) = (0 < poly (-p) x)"
    by (subst poly_smult, metis assms(1) linorder_neqE_linordered_idom mult_neg_neg mult_zero_right 
      neg_0_less_iff_less poly_minus zero_less_mult_pos2)
  hence "sign_r_pos (smult c p) x=sign_r_pos (-p) x"
    unfolding sign_r_pos_def using ‹¬ c>0› by auto
  thus ?thesis using sign_r_pos_minus[OF ‹p≠0›, of x] ‹¬ c>0› by auto
qed

lemma sign_r_pos_mult:
  fixes p q :: "real poly"
  assumes "p≠0" "q≠0"
  shows "sign_r_pos (p*q) x= (sign_r_pos p x ⟷ sign_r_pos q x)"
proof -
  obtain ub where "ub>x" 
      and ub:"(∀z. x < z ∧ z < ub ⟶ 0 < poly p z) ∨ (∀z. x < z ∧ z < ub ⟶ poly p z < 0)"
    using next_non_root_interval[OF ‹p≠0›,of x,unfolded not_eq_pos_or_neg_iff_1]  
    by (metis order.strict_implies_order)
  obtain ub' where "ub'>x" 
      and ub':"(∀z. x < z ∧ z < ub' ⟶ 0 < poly q z) ∨ (∀z. x < z ∧ z < ub' ⟶ poly q z < 0)"
    using next_non_root_interval[OF ‹q≠0›,unfolded not_eq_pos_or_neg_iff_1] 
    by (metis order.strict_implies_order)
  have "(∀z. x < z ∧ z < ub ⟶ 0 < poly p z) ⟹ (∀z. x < z ∧ z < ub' ⟶ 0 < poly q z) ⟹ ?thesis"
    proof -
      assume "(∀z. x < z ∧ z < ub ⟶ 0 < poly p z)" "(∀z. x < z ∧ z < ub' ⟶ 0 < poly q z)"
      hence "sign_r_pos p x" and "sign_r_pos q x" unfolding sign_r_pos_def eventually_at_right
        using ‹ub>x› ‹ub'>x› by auto
      moreover hence "eventually (λz. poly p z>0 ∧ poly q z>0) (at_right x)"
        unfolding sign_r_pos_def using eventually_conj_iff[of _ _ "at_right x"] by auto
      hence "sign_r_pos (p*q) x"
        unfolding sign_r_pos_def poly_mult 
        by (metis (lifting, mono_tags) eventually_mono mult_pos_pos)
      ultimately show ?thesis by auto
    qed
  moreover have "(∀z. x < z ∧ z < ub ⟶ 0 > poly p z) ⟹ (∀z. x < z ∧ z < ub' ⟶ 0 < poly q z) 
      ⟹ ?thesis"
  proof -
    assume "(∀z. x < z ∧ z < ub ⟶ 0 > poly p z)" "(∀z. x < z ∧ z < ub' ⟶ 0 < poly q z)"
    hence "sign_r_pos (-p) x" and "sign_r_pos q x" unfolding sign_r_pos_def eventually_at_right
      using ‹ub>x› ‹ub'>x› by auto
    moreover hence "eventually (λz. poly (-p) z>0 ∧ poly q z>0) (at_right x)"
      unfolding sign_r_pos_def using eventually_conj_iff[of _ _ "at_right x"] by auto
    hence "sign_r_pos (- p*q) x"
      unfolding sign_r_pos_def poly_mult 
      by (metis (lifting, mono_tags) eventually_mono mult_pos_pos)
    ultimately show ?thesis 
      using sign_r_pos_minus ‹p≠0› ‹q≠0› by (metis minus_mult_left no_zero_divisors)
  qed
  moreover have "(∀z. x < z ∧ z < ub ⟶ 0 < poly p z) ⟹ (∀z. x < z ∧ z < ub' ⟶ 0 > poly q z)
      ⟹ ?thesis"
  proof -
    assume "(∀z. x < z ∧ z < ub ⟶ 0 < poly p z)" "(∀z. x < z ∧ z < ub' ⟶ 0 > poly q z)"
    hence "sign_r_pos p x" and "sign_r_pos (-q) x" unfolding sign_r_pos_def eventually_at_right
      using ‹ub>x› ‹ub'>x› by auto
    moreover hence "eventually (λz. poly p z>0 ∧ poly (-q) z>0) (at_right x)"
      unfolding sign_r_pos_def using eventually_conj_iff[of _ _ "at_right x"] by auto
    hence "sign_r_pos ( p * (- q)) x"
      unfolding sign_r_pos_def poly_mult 
      by (metis (lifting, mono_tags) eventually_mono mult_pos_pos)
    ultimately show ?thesis 
      using sign_r_pos_minus ‹p≠0› ‹q≠0› 
      by (metis minus_mult_right no_zero_divisors)
  qed
  moreover have "(∀z. x < z ∧ z < ub ⟶ 0 > poly p z) ⟹ (∀z. x < z ∧ z < ub' ⟶ 0 > poly q z) 
      ⟹ ?thesis"
  proof -
    assume "(∀z. x < z ∧ z < ub ⟶ 0 > poly p z)" "(∀z. x < z ∧ z < ub' ⟶ 0 > poly q z)"
    hence "sign_r_pos (-p) x" and "sign_r_pos (-q) x" 
      unfolding sign_r_pos_def eventually_at_right using ‹ub>x› ‹ub'>x› by auto
    moreover hence "eventually (λz. poly (-p) z>0 ∧ poly (-q) z>0) (at_right x)"
      unfolding sign_r_pos_def using eventually_conj_iff[of _ _ "at_right x"] by auto
    hence "sign_r_pos (p * q) x"
      unfolding sign_r_pos_def poly_mult poly_minus
      apply (elim eventually_mono[of _ "at_right x"])
      by (auto intro:mult_neg_neg)
    ultimately show ?thesis 
      using sign_r_pos_minus ‹p≠0› ‹q≠0› by metis
  qed
  ultimately show ?thesis using ub ub' by auto
qed

lemma sign_r_pos_add:
  fixes p q :: "real poly"
  assumes "poly p x=0" "poly q x≠0"
  shows "sign_r_pos (p+q) x=sign_r_pos q x"
proof (cases "poly (p+q) x=0")
  case False
  hence "p+q≠0" by (metis poly_0)
  have "sign_r_pos (p+q) x = (poly q x > 0)"  
    using sign_r_pos_rec[OF ‹p+q≠0›] False poly_add ‹poly p x=0› by auto
  moreover have "sign_r_pos q x=(poly q x > 0)"
    using sign_r_pos_rec[of q x] ‹poly q x≠0› poly_0 by force
  ultimately show ?thesis by auto
next
  case True
  hence False using ‹poly p x=0› ‹poly q x≠0› poly_add by auto
  thus ?thesis by auto
qed

lemma sign_r_pos_mod:
  fixes p q :: "real poly"
  assumes "poly p x=0" "poly q x≠0"
  shows "sign_r_pos (q mod p) x=sign_r_pos q x"
proof -
  have "poly (q div p * p) x=0" using ‹poly p x=0› poly_mult by auto
  moreover hence "poly (q mod p) x ≠ 0" using ‹poly q x≠0› 
    by (simp add: assms(1) poly_mod)
  ultimately show ?thesis 
    by (metis div_mult_mod_eq sign_r_pos_add)
qed

lemma sign_r_pos_pderiv:
  fixes p:: "real poly"
  assumes "poly p x=0" "p≠0"
  shows "sign_r_pos (pderiv p * p) x"
proof -
  have "pderiv p ≠0" 
    by (metis assms(1) assms(2) monoid_add_class.add.right_neutral mult_zero_right pCons_0_0 
      pderiv_iszero poly_0 poly_pCons)
  have "?thesis = (sign_r_pos (pderiv p) x ⟷ sign_r_pos p x)"
    using sign_r_pos_mult[OF ‹pderiv p ≠ 0› ‹p≠0›] by auto
  also have "...=((sign_r_pos (pderiv p) x ⟷ sign_r_pos (pderiv p) x))"
    using sign_r_pos_rec[OF ‹p≠0›] ‹poly p x=0› by auto
  finally show ?thesis by auto
qed

lemma sign_r_pos_power:
  fixes p:: "real poly" and a::real
  shows "sign_r_pos ([:-a,1:]^n) a" 
proof (induct n)
  case 0
  thus ?case unfolding sign_r_pos_def eventually_at_right by (simp,metis gt_ex)
next
  case (Suc n)
  have "pderiv ([:-a,1:]^Suc n) = smult (Suc n) ([:-a,1:]^n)" 
  proof -
    have "pderiv [:- a, 1::real:] = 1" by (simp add: pderiv.simps)
    thus ?thesis unfolding pderiv_power_Suc by (metis mult_cancel_left1)
  qed                                                                                    
  moreover have " poly ([:- a, 1:] ^ Suc n) a=0" by (metis old.nat.distinct(2) poly_power_n_eq)
  hence "sign_r_pos ([:- a, 1:] ^ Suc n) a = sign_r_pos (smult (Suc n) ([:-a,1:]^n)) a"
    using sign_r_pos_rec by (metis (erased, opaque_lifting) calculation pderiv_0)
  hence "sign_r_pos ([:- a, 1:] ^ Suc n) a = sign_r_pos  ([:-a,1:]^n) a"
    using sign_r_pos_smult by auto
  ultimately show ?case using Suc.hyps by auto
qed

subsection‹Jump›

definition jump_poly :: "real poly ⇒ real poly ⇒real ⇒ int"
 where 
 " jump_poly q p x≡ (if p≠0 ∧ q≠0 ∧ odd((order x p) - (order x q) ) then 
                  if sign_r_pos (q*p) x then 1 else -1
                else 0 )"

lemma jump_poly_not_root:"poly p x≠0⟹ jump_poly q p x=0"
  unfolding  jump_poly_def by (metis even_zero order_root zero_diff)

lemma jump_poly0[simp]: 
    "jump_poly 0 p x = 0"
    "jump_poly q 0 x = 0"
  unfolding jump_poly_def by auto 

lemma jump_poly_smult_1:
  fixes p q::"real poly" and c::real
  shows "jump_poly (smult c q) p x= sign c * jump_poly q p x" (is "?L=?R")
proof (cases "c=0∨ q=0") 
  case True
  thus ?thesis unfolding jump_poly_def by auto
next
  case False
  hence "c≠0" and "q≠0" by auto
  thus ?thesis unfolding jump_poly_def
     using order_smult[OF ‹c≠0›] sign_r_pos_smult[OF ‹c≠0›, of "q*p" x] ‹q≠0› 
     by auto
qed

lemma jump_poly_mult:
  fixes p q p'::"real poly"
  assumes "p'≠0"
  shows "jump_poly (p'*q) (p'*p) x= jump_poly q p x"
proof (cases "q=0 ∨ p=0")
  case True
  thus ?thesis unfolding jump_poly_def by fastforce
next 
  case False
  then have "q≠0" "p≠0" by auto
  have "sign_r_pos (p' * q * (p' * p)) x=sign_r_pos (q * p) x"
  proof (unfold sign_r_pos_def,rule eventually_subst,unfold eventually_at_right)
    obtain b where "b>x" and b:"∀z. x < z ∧ z < b ⟶ poly (p' * p') z >0" 
    proof (cases "∃z. poly p' z=0 ∧ z>x")
      case True
      define lr where "lr≡Min {r . poly p' r=0 ∧ r>x}" 
      have "∀z. x<z∧z<lr⟶poly p' z≠0" and "lr>x" 
        using True lr_def poly_roots_finite[OF ‹p'≠0›] by auto
      hence "∀z. x < z ∧ z < lr ⟶ 0 < poly (p' * p') z" 
        by (metis not_real_square_gt_zero poly_mult)
      thus ?thesis using that[OF ‹lr>x›] by auto
    next
      case False
      have "∀z. x<z∧z<x+1⟶poly p' z≠0" and "x+1>x" 
        using False poly_roots_finite[OF ‹p'≠0›]  by auto
      hence "∀z. x < z ∧ z < x+1 ⟶ 0 < poly (p' * p') z" 
        by (metis not_real_square_gt_zero poly_mult)
      thus ?thesis using that[OF ‹x+1>x›] by auto
    qed
    show "∃b>x. ∀z>x. z < b ⟶ (0 < poly (p' * q * (p' * p)) z) = (0 < poly (q * p) z)"
    proof (rule_tac x="b" in exI, rule conjI[OF ‹b>x›],rule allI,rule impI,rule impI)
      fix z assume "x < z"  "z < b"
      hence "0<poly (p'*p') z" using b by auto
      have " (0 < poly (p' * q * (p' * p)) z)=(0<poly (p'*p') z * poly (q*p) z)" 
        by (simp add: mult.commute mult.left_commute)
      also have "...=(0<poly (q*p) z)"
        using ‹0<poly (p'*p') z› by (metis mult_pos_pos zero_less_mult_pos)
      finally show "(0 < poly (p' * q * (p' * p)) z) = (0 < poly (q * p) z)" .
    qed
  qed
  moreover have " odd (order x (p' * p) - order x (p' * q)) = odd (order x p - order x q)"
    using  False ‹p'≠0› ‹p≠0› mult_eq_0_iff order_mult
    by (metis add_diff_cancel_left)
  moreover have " p' * q ≠ 0 ⟷ q ≠ 0" 
    by (metis ‹p'≠0› mult_eq_0_iff)
  ultimately show "jump_poly (p' * q) (p' * p) x = jump_poly q p x" unfolding jump_poly_def by auto
qed

lemma jump_poly_1_mult:
  fixes p1 p2::"real poly"
  assumes "poly p1 x≠0 ∨ poly p2 x≠0" 
  shows "jump_poly 1 (p1*p2) x= sign (poly p2 x) * jump_poly 1 p1 x 
            + sign (poly p1 x) * jump_poly 1 p2 x" (is "?L=?R")
proof (cases "p1=0 ∨ p2 =0")
  case True
  then show ?thesis by auto
next
  case False
  then have "p1≠0" "p2≠0" "p1*p2≠0" by auto
  have ?thesis when "poly p1 x≠0"
  proof -
    have [simp]:"order x p1 = 0" using that order_root by blast
    define simpL where "simpL≡(if p2≠0 ∧ odd (order x p2) then if (poly p1 x>0) 
    ⟷  sign_r_pos p2 x then 1::int else -1 else 0)"
    have "?L=simpL"
      unfolding simpL_def jump_poly_def  
      using order_mult[OF ‹p1*p2≠0›]
        sign_r_pos_mult[OF ‹p1≠0› ‹p2≠0›] sign_r_pos_rec[OF ‹p1≠0›] ‹poly p1 x≠0›
      by auto
    moreover have "poly p1 x>0 ⟹ simpL =?R" 
      unfolding simpL_def jump_poly_def using jump_poly_not_root[OF ‹poly p1 x≠0›] 
      by auto
    moreover have "poly p1 x<0 ⟹ simpL =?R" 
      unfolding simpL_def jump_poly_def using jump_poly_not_root[OF ‹poly p1 x≠0›] 
      by auto
    ultimately show "?L=?R" using ‹poly p1 x≠0› by (metis linorder_neqE_linordered_idom)
  qed
  moreover have ?thesis when "poly p2 x≠0"
  proof -
    have [simp]:"order x p2 = 0" using that order_root by blast
    define simpL where "simpL≡(if p1≠0 ∧ odd (order x p1) then if (poly p2 x>0) 
    ⟷  sign_r_pos p1 x then 1::int else -1 else 0)"
    have "?L=simpL"
      unfolding simpL_def jump_poly_def  
      using order_mult[OF ‹p1*p2≠0›]
        sign_r_pos_mult[OF ‹p1≠0› ‹p2≠0›] sign_r_pos_rec[OF ‹p2≠0›] ‹poly p2 x≠0›
      by auto
    moreover have "poly p2 x>0 ⟹ simpL =?R" 
      unfolding simpL_def jump_poly_def using jump_poly_not_root[OF ‹poly p2 x≠0›] 
      by auto
    moreover have "poly p2 x<0 ⟹ simpL =?R" 
      unfolding simpL_def jump_poly_def using jump_poly_not_root[OF ‹poly p2 x≠0›] 
      by auto
    ultimately show "?L=?R" using ‹poly p2 x≠0› by (metis linorder_neqE_linordered_idom)
  qed  
  ultimately show ?thesis using assms by auto
qed  
  
lemma jump_poly_mod:
  fixes p q::"real poly" 
  shows "jump_poly q p x= jump_poly (q mod p) p x"
proof (cases "q=0 ∨ p=0")
  case True
  thus ?thesis by fastforce
next
  case False
  then have "p≠0" "q≠0" by auto
  define n where "n≡min (order x q) (order x p)"
  obtain q' where q':"q=[:-x,1:]^n * q'" 
    using n_def  power_le_dvd[OF order_1[of x q], of n] 
    by (metis dvdE min.cobounded2 min.commute)
  obtain p' where p':"p=[:-x,1:]^n * p'" 
    using n_def  power_le_dvd[OF order_1[of x p], of n] 
    by (metis dvdE min.cobounded2)
  have "q'≠0" and "p'≠0" using q' p' ‹p≠0› ‹q≠0› by auto
  have "order x q'=0 ∨ order x p'=0" 
  proof (rule ccontr)
    assume "¬ (order x q' = 0 ∨ order x p' = 0)"
    hence "order x q' > 0" and "order x p' > 0" by auto
    hence "order x q>n" and "order x p>n" unfolding q' p' 
      using order_mult[OF ‹q≠0›[unfolded q'],of x] order_mult[OF ‹p≠0›[unfolded p'],of x] 
          order_power_n_n[of x n]
      by auto
    thus False using n_def by auto
  qed
  have cond:"q' ≠ 0 ∧ odd (order x p' - order x q') 
    = (q' mod p' ≠0 ∧ odd(order x p' - order x (q' mod p')))" 
  proof (cases "order x p'=0")
    case True
    thus ?thesis by (metis ‹q' ≠ 0› even_zero zero_diff)
  next
    case False
    hence "order x q'=0" using ‹order x q'=0 ∨ order x p'=0› by auto
    hence "¬ [:-x,1:] dvd q'" 
      by (metis ‹q' ≠ 0› order_root poly_eq_0_iff_dvd)
    moreover have "[:-x,1:] dvd p'" using False 
      by (metis order_root poly_eq_0_iff_dvd)
    ultimately have "¬ [:-x,1:] dvd (q' mod p')"
      by (metis dvd_mod_iff)
    hence "order x (q' mod p') = 0" and "q' mod p' ≠0" 
       apply (metis order_root poly_eq_0_iff_dvd)
      by (metis ‹¬ [:- x, 1:] dvd q' mod p'› dvd_0_right)
    thus ?thesis using ‹order x q'=0› by auto
  qed
  moreover have "q' mod p'≠0 ⟹ poly p' x = 0  
      ⟹ sign_r_pos (q' * p') x= sign_r_pos (q' mod p' * p') x" 
  proof -
    assume "q' mod p'≠0" "poly p' x = 0"
    hence "poly q' x≠0" using ‹order x q'=0 ∨ order x p'=0› 
      by (metis ‹p' ≠ 0› ‹q' ≠ 0› order_root)
    hence "sign_r_pos q' x= sign_r_pos (q' mod p') x" 
      using sign_r_pos_mod[OF ‹poly p' x=0›] by auto 
    thus ?thesis 
      unfolding sign_r_pos_mult[OF ‹q'≠0› ‹p'≠0›] sign_r_pos_mult[OF ‹q' mod p'≠0› ‹p'≠0›]
      by auto
  qed
  moreover have "q' mod p' = 0 ∨ poly p' x ≠ 0 ⟹ jump_poly q' p' x = jump_poly (q' mod p') p' x" 
  proof -
    assume assm:"q' mod p' = 0 ∨ poly p' x ≠ 0"
    have "q' mod p' = 0 ⟹ ?thesis" unfolding jump_poly_def
      using cond by auto
    moreover have "poly p' x ≠ 0 
        ⟹ ¬ odd (order x p' - order x q') ∧ ¬ odd(order x p' - order x (q' mod p'))"
      by (metis even_zero order_root zero_diff)
    hence "poly p' x ≠ 0 ⟹ ?thesis" unfolding jump_poly_def by auto
    ultimately show ?thesis using assm by auto
  qed
  ultimately have " jump_poly q' p' x = jump_poly (q' mod p') p' x" unfolding jump_poly_def by force
  thus ?thesis using p' q' jump_poly_mult by auto
qed 
  
lemma jump_poly_coprime:
  fixes p q:: "real poly"
  assumes "poly p x=0" "coprime p q"
  shows "jump_poly q p x= jump_poly 1 (q*p) x"
proof (cases "p=0 ∨ q=0")
  case True
  then show ?thesis by auto
next
  case False
  then have "p≠0" "q≠0" by auto
  then have "poly p x≠0 ∨ poly q x≠0" using coprime_poly_0[OF ‹coprime p q›] by auto    
  then have "poly q x≠0" using ‹poly p x=0› by auto
  then have "order x q=0" using order_root by blast
  then have "order x p - order x q = order x (q * p)" 
    using ‹p≠0› ‹q≠0› order_mult [of q p x] by auto
  then show ?thesis unfolding jump_poly_def using ‹q≠0› by auto   
qed    
 
lemma jump_poly_sgn:
  fixes p q:: "real poly"
  assumes "p≠0" "poly p x=0"
  shows "jump_poly (pderiv p * q) p x = sign (poly q x)" 
proof (cases "q=0")
  case True
  thus ?thesis by auto 
next
  case False
  have "pderiv p≠0" using ‹p≠0› ‹poly p x=0›  
    by (metis mult_poly_0_left sign_r_pos_0 sign_r_pos_pderiv)
  have elim_p_order: "order x p - order x (pderiv p * q)=1 - order x q" 
  proof -
    have "order x p - order x (pderiv p * q) = order x p - order x (pderiv p) - order x q" 
      using order_mult ‹pderiv p≠0› False by (metis diff_diff_left mult_eq_0_iff) 
    moreover have "order x p - order x (pderiv p) = 1" 
      using order_pderiv[OF ‹pderiv p≠0›, of x] ‹poly p x=0› order_root[of p x] ‹p≠0› by auto
    ultimately show ?thesis by auto
  qed
  have elim_p_sign_r_pos:"sign_r_pos (pderiv p * q * p) x=sign_r_pos q x" 
  proof - 
    have "sign_r_pos (pderiv p * q * p) x = (sign_r_pos (pderiv p* p) x ⟷ sign_r_pos q x)"
      by (metis ‹q ≠ 0› ‹pderiv p ≠ 0› assms(1) no_zero_divisors sign_r_pos_mult)
    thus ?thesis using sign_r_pos_pderiv[OF ‹poly p x=0› ‹p≠0›] by auto
  qed 
  define simpleL where "simpleL≡if pderiv p * q ≠ 0 ∧ odd (1 - order x q) then 
      if sign_r_pos q x then 1::int else - 1 else 0"
  have " jump_poly (pderiv p * q) p x =simpleL" 
    unfolding simpleL_def jump_poly_def by (subst elim_p_order, subst elim_p_sign_r_pos,simp)
  moreover have "poly q x=0 ⟹ simpleL=sign (poly q x)"
  proof -
    assume "poly q x=0" 
    hence "1-order x q = 0" using ‹q≠0› by (metis less_one not_gr0 order_root zero_less_diff) 
    hence "simpleL=0" unfolding simpleL_def by auto
    moreover have "sign (poly q x)=0" using ‹poly q x=0› by auto
    ultimately show ?thesis by auto
  qed
  moreover have "poly q x≠0⟹ simpleL=sign (poly q x)"
  proof -
    assume "poly q x≠0"
    hence "odd (1 - order x q)" by (simp add: order_root)  
    moreover have "pderiv p * q ≠ 0" by (metis False ‹pderiv p ≠ 0› no_zero_divisors)
    moreover have "sign_r_pos q x = (poly q x > 0)" 
      using sign_r_pos_rec[OF False] ‹poly q x≠0› by auto 
    ultimately have "simpleL=(if poly q x>0 then 1 else - 1)" unfolding simpleL_def by auto
    thus ?thesis using ‹poly q x≠0› by auto
  qed
  ultimately show ?thesis by force
qed
 
subsection ‹Cauchy index›

definition cindex_poly:: "real ⇒ real ⇒ real poly ⇒ real poly ⇒ int" 
  where 
  "cindex_poly a b q p≡ (∑x∈{x. poly p x=0 ∧ a< x∧ x< b}. jump_poly q p x)"

lemma cindex_poly_0[simp]: "cindex_poly a b 0 p = 0" "cindex_poly a b q 0 = 0" 
  unfolding cindex_poly_def by auto

lemma cindex_poly_cross:
  fixes p::"real poly" and a b::real
  assumes  "a<b" "poly p a≠0" "poly p b≠0"
  shows "cindex_poly a b 1 p = cross p a b" 
    using ‹poly p a≠0› ‹poly p b≠0›
proof (cases "{x. poly p x=0 ∧ a< x∧ x< b}≠{}", induct "degree p" arbitrary:p rule:nat_less_induct) 
  case 1
  then have "p≠0" by force
  define roots where "roots≡{x.  poly p x=0 ∧ a< x∧ x< b}"
  have "finite roots" unfolding roots_def using poly_roots_finite[OF ‹p≠0›] by auto
  define max_r where "max_r≡Max roots"
  hence "poly p max_r=0" and "a<max_r" and "max_r<b" 
    using Max_in[OF ‹finite roots›] "1.prems"  unfolding roots_def by auto
  define max_rp where "max_rp≡[:-max_r,1:]^order max_r p"
  then obtain p' where p':"p=p'*max_rp" and not_dvd:"¬ [:-max_r,1:] dvd p'"  
    by (metis ‹p≠0› mult.commute order_decomp)
  hence "p'≠0" and "max_rp≠0" and "poly p' a≠0" and "poly p' b≠0" 
      and  "poly max_rp a≠0" and "poly max_rp b≠0" 
    using ‹p≠0› ‹poly p a≠0› ‹poly p b≠0›  by auto
  define max_r_sign where "max_r_sign≡if odd(order max_r p) then -1 else 1::int"
  define roots' where "roots'≡{x.  a< x∧ x< b ∧ poly p' x=0}"
  have "(∑x∈roots. jump_poly 1 p x)= (∑x∈roots'. jump_poly 1 p x)+ jump_poly 1 p max_r"
  proof -
    have "roots=roots' ∪ {x.  a< x∧ x< b ∧ poly max_rp x=0 }"
      unfolding roots_def roots'_def p' by auto
    moreover have "{x.  a < x ∧ x < b ∧  poly max_rp x = 0 }={max_r}"
      unfolding max_rp_def using ‹poly p max_r=0›  
      by (auto simp add: ‹a<max_r› ‹max_r<b›,metis "1.prems"(1) neq0_conv order_root)
    moreover hence "roots' ∩ {x. a< x∧ x< b ∧ poly max_rp x=0} ={}"
      unfolding roots'_def  using ‹¬ [:-max_r,1:] dvd p'› 
      by (metis (mono_tags) Int_insert_right_if0 inf_bot_right mem_Collect_eq poly_eq_0_iff_dvd)
    moreover have "finite roots'" 
      using  p' ‹p≠0› by (metis ‹finite roots› calculation(1) calculation(2) finite_Un)
    ultimately show ?thesis using sum.union_disjoint by auto
  qed
  moreover have "(∑x∈roots'. jump_poly 1 p x) = max_r_sign * cross p' a b"
  proof -
    have "(∑x∈roots'. jump_poly 1 p x) = (∑x∈roots'. max_r_sign * jump_poly 1 p' x)"
    proof (rule sum.cong,rule refl)
      fix x assume "x ∈ roots'" 
      hence "x≠max_r" using not_dvd unfolding roots'_def 
        by (metis (mono_tags, lifting) mem_Collect_eq poly_eq_0_iff_dvd )
      hence "poly max_rp x≠0" using poly_power_n_eq unfolding max_rp_def by auto
      hence "order x max_rp=0"  by (metis order_root)
      moreover have "jump_poly 1 max_rp x=0" 
        using ‹poly max_rp x≠0› by (metis jump_poly_not_root)
      moreover have "x∈roots"
        using ‹x ∈ roots'› unfolding roots_def roots'_def p' by auto
      hence "x<max_r" 
        using Max_ge[OF ‹finite roots›,of x] ‹x≠max_r› by (fold max_r_def,auto)
      hence "sign (poly max_rp x) = max_r_sign" 
        using ‹poly max_rp x ≠ 0› unfolding max_r_sign_def max_rp_def sign_def
        by (subst poly_power,simp add:linorder_class.not_less zero_less_power_eq)
      ultimately show "jump_poly 1 p x = max_r_sign * jump_poly 1 p' x" 
        using jump_poly_1_mult[of p' x max_rp]  unfolding p' 
        by (simp add: ‹poly max_rp x ≠ 0›)
      qed
      also have "... = max_r_sign * (∑x∈roots'. jump_poly 1 p' x)" 
          by (simp add: sum_distrib_left)
      also have "... = max_r_sign * cross p' a b"
      proof (cases "roots'={}")
        case True
        hence "cross p' a b=0" unfolding roots'_def using cross_no_root[OF ‹a<b›] by auto 
        thus ?thesis using True by simp
      next
        case False
        moreover have "degree max_rp≠0" 
          unfolding max_rp_def degree_linear_power 
          by (metis "1.prems"(1) ‹poly p max_r = 0› order_root)
        hence  "degree p' < degree p" unfolding p' degree_mult_eq[OF ‹p'≠0› ‹max_rp≠0›]
          by auto
        ultimately have "cindex_poly a b 1 p' = cross p' a b"
          unfolding roots'_def 
          using "1.hyps"[rule_format,of "degree p'" p'] ‹p'≠0› ‹poly p' a≠0› ‹poly p' b≠0› 
          by auto  
        moreover have "cindex_poly a b 1 p' = sum (jump_poly 1 p') roots'" 
          unfolding cindex_poly_def roots'_def
          apply simp
          by (metis (no_types, lifting) )
        ultimately show ?thesis by auto
      qed
      finally show ?thesis .
    qed
    moreover have "max_r_sign * cross p' a b + jump_poly 1 p max_r = cross p a b" (is "?L=?R")
    proof (cases "odd (order max_r p)")
      case True
      have "poly max_rp a < 0" 
        using poly_power_n_odd[OF True,of max_r a] ‹poly max_rp a≠0› ‹max_r>a›  
        unfolding max_rp_def by linarith
      moreover have "poly max_rp b>0 "
        using poly_power_n_odd[OF True,of max_r b] ‹max_r<b›  
        unfolding max_rp_def by linarith
      ultimately have "?R=cross p' a b + sign (poly p' a)"
        unfolding p' cross_def poly_mult
        using variation_mult_neg_1[of "poly max_rp a", simplified mult.commute]
          variation_mult_pos(2)[of "poly max_rp b", simplified mult.commute] ‹poly p' b≠0›
        by auto
      moreover have "?L=- cross p' a b + sign (poly p' b)" 
      proof -
        have " sign_r_pos p' max_r = (poly p' max_r >0)" 
          using sign_r_pos_rec[OF ‹p'≠0›] not_dvd by (metis poly_eq_0_iff_dvd)
        moreover have "(poly p' max_r>0) = (poly p' b>0)"
        proof (rule ccontr)             
          assume "(0 < poly p' max_r) ≠ (0 < poly p' b)"
          hence "poly p' max_r * poly p' b <0"
            using ‹poly p' b≠0› not_dvd[folded poly_eq_0_iff_dvd] 
            by (metis (poly_guards_query) linorder_neqE_linordered_idom mult_less_0_iff)
          then obtain r where "r>max_r" and "r<b" and "poly p' r=0"
            using poly_IVT[OF ‹max_r<b›] by auto
          hence "r∈roots" unfolding roots_def p' using ‹max_r>a› by auto
          thus False using ‹r>max_r› Max_ge[OF ‹finite roots›,of r] unfolding max_r_def by auto
        qed
        moreover have "sign_r_pos max_rp max_r"
          using sign_r_pos_power unfolding max_rp_def by auto
        ultimately show ?thesis 
          using True ‹poly p' b≠0› ‹max_rp≠0› ‹p'≠0› sign_r_pos_mult[OF ‹p'≠0› ‹max_rp≠0›]
          unfolding max_r_sign_def  p' jump_poly_def 
          by simp  
      qed  
      moreover have "variation (poly p' a) (poly p' b) + sign (poly p' a) 
          = - variation (poly p' a) (poly p' b) + sign (poly p' b)" unfolding cross_def
        by (cases "poly p' b" rule:linorder_cases[of 0], (cases "poly p' a" rule:linorder_cases[of 0], 
          auto simp add:variation_cases ‹poly p' a ≠ 0› ‹poly p' b ≠ 0›)+)
      ultimately show ?thesis unfolding cross_def by auto
    next
      case False
      hence "poly max_rp a > 0" and "poly max_rp b > 0"
        unfolding max_rp_def poly_power 
        using ‹poly max_rp a≠0› ‹poly max_rp b ≠ 0›  "1.prems"(1-2) ‹poly p max_r = 0›
        apply (unfold zero_less_power_eq)
        by auto
      moreover have "poly max_rp b > 0"
        unfolding max_rp_def poly_power
        using ‹poly max_rp b ≠ 0› False max_rp_def poly_power 
          zero_le_even_power[of "order max_r p" "b - max_r"]
        by (auto simp add: le_less)
      ultimately have "?R=cross p' a b"
        apply (simp only: p' mult.commute cross_def) using variation_mult_pos
        by auto
      thus ?thesis unfolding max_r_sign_def jump_poly_def using False by auto
    qed
    ultimately have "sum (jump_poly 1 p) roots = cross p a b " by auto
    then show ?case unfolding roots_def cindex_poly_def by simp
next
  case False 
  hence "cross p a b=0" using cross_no_root[OF ‹a<b›] by auto
  thus ?thesis using False unfolding cindex_poly_def by (metis sum.empty)
qed 

lemma cindex_poly_mult:
  fixes p q p'::"real poly"
  assumes "p'≠ 0"  
  shows "cindex_poly a b (p' * q ) (p' * p) = cindex_poly a b q p"
proof (cases "p=0")
  case True
  then show ?thesis by auto
next
  case False
  show ?thesis unfolding cindex_poly_def
    apply (rule sum.mono_neutral_cong_right)
    subgoal using ‹p≠0› ‹p'≠0› by (simp add: poly_roots_finite) 
    subgoal by auto
    subgoal using jump_poly_mult jump_poly_not_root assms by fastforce
    subgoal for x using jump_poly_mult[OF ‹p'≠0›] by auto
    done
qed    

lemma cindex_poly_smult_1: 
  fixes p q::"real poly" and c::real
  shows "cindex_poly a b (smult c q) p =  (sign c) * cindex_poly a b q p"
unfolding cindex_poly_def
using sum_distrib_left[THEN sym, of "sign c" "λx. jump_poly q p x"
    "{x. poly p x = (0::real) ∧ a < x ∧ x < b}"] jump_poly_smult_1
  by auto

lemma cindex_poly_mod:
  fixes p q::"real poly" 
  shows "cindex_poly a b q p =  cindex_poly a b (q mod p) p"
unfolding cindex_poly_def using jump_poly_mod by auto

lemma cindex_poly_inverse_add:
  fixes p q::"real poly" 
  assumes "coprime p q"
  shows "cindex_poly a b q p + cindex_poly a b p q=cindex_poly a b 1 (q*p)"
    (is "?L=?R")
proof (cases "p=0 ∨ q=0")
  case True
  then show ?thesis by auto
next
  case False
  then have "p≠0" "q≠0" by auto
  define A where "A≡{x. poly p x = 0 ∧ a < x ∧ x < b}"
  define B where "B≡{x. poly q x = 0 ∧ a < x ∧ x < b}"
  have "?L = sum (λx. jump_poly 1 (q*p) x) A + sum (λx. jump_poly 1 (q*p) x) B" 
  proof -
    have "cindex_poly a b q p = sum (λx. jump_poly 1 (q*p) x) A" unfolding A_def cindex_poly_def
      using jump_poly_coprime[OF _ ‹coprime p q›] by auto 
    moreover have "coprime q p" using ‹coprime p q›
      by (simp add: ac_simps)
    hence "cindex_poly a b p q = sum (λx. jump_poly 1 (q*p) x) B" unfolding B_def cindex_poly_def
      using jump_poly_coprime [of q _ p] by (auto simp add: ac_simps)
    ultimately show ?thesis by auto
  qed
  moreover have "A ∪ B= {x. poly (q*p) x=0 ∧ a<x ∧ x<b }" unfolding poly_mult A_def B_def by auto
  moreover have "A ∩ B={}" 
  proof (rule ccontr)
    assume "A ∩ B≠{}"
    then obtain x where "x∈A" and "x∈B" by auto
    hence "poly p x=0" and "poly q x=0" unfolding A_def B_def by auto
    hence "gcd p q≠1" by (metis poly_1 poly_eq_0_iff_dvd gcd_greatest zero_neq_one)
    thus False using ‹coprime p q› by auto
  qed
  moreover have "finite A" and "finite B" 
    unfolding A_def B_def using poly_roots_finite ‹p≠0› ‹q≠0› by fast+
  ultimately have "cindex_poly a b q p + cindex_poly a b p q 
      = sum (jump_poly 1 (q * p)) {x. poly (q*p) x=0 ∧ a<x ∧ x<b}"
    using sum.union_disjoint by metis 
  then show ?thesis unfolding cindex_poly_def by auto 
qed  

lemma cindex_poly_inverse_add_cross:
  fixes p q::"real poly"
  assumes "a < b" "poly (p * q) a