Theory Polynomials

(*  Title:       Executable multivariate polynomials
    Author:      Christian Sternagel <christian.sternagel@uibk.ac.at>
                 René Thiemann       <rene.thiemann@uibk.ac.at>
    Maintainer:  Christian Sternagel and René Thiemann
    License:     LGPL
*)

(*
Copyright 2010 Christian Sternagel, René Thiemann

This file is part of IsaFoR/CeTA.

IsaFoR/CeTA is free software: you can redistribute it and/or modify it under the
terms of the GNU Lesser General Public License as published by the Free Software
Foundation, either version 3 of the License, or (at your option) any later
version.

IsaFoR/CeTA is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A
PARTICULAR PURPOSE.  See the GNU Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public License along
with IsaFoR/CeTA. If not, see <http://www.gnu.org/licenses/>.
*)

section ‹Polynomials›

(* TODO: attempt to turn polynomials into type *)

theory Polynomials
imports 
  "Abstract-Rewriting.SN_Orders"
  Matrix.Utility 
begin

subsection ‹
Polynomials represented as trees
›
datatype (vars_tpoly: 'v, nums_tpoly: 'a)tpoly = PVar 'v | PNum 'a | PSum "('v,'a)tpoly list" | PMult "('v,'a)tpoly list"

type_synonym ('v,'a)assign = "'v ⇒ 'a"

primrec eval_tpoly :: "('v,'a::{monoid_add,monoid_mult})assign ⇒ ('v,'a)tpoly ⇒ 'a"
where "eval_tpoly α (PVar x) = α x"
   |  "eval_tpoly α (PNum a) = a"
   |  "eval_tpoly α (PSum ps) = sum_list (map (eval_tpoly α) ps)"
   |  "eval_tpoly α (PMult ps) = prod_list (map (eval_tpoly α) ps)"

subsection ‹Polynomials represented in normal form as lists of monomials›
text ‹
  The internal representation of polynomials is a sum of products of monomials with coefficients
  where all coefficients are non-zero, and all monomials are different
›
 
text ‹Definition of type ‹monom››


type_synonym 'v monom_list = "('v × nat)list" 
text ‹
\begin{itemize}
\item $[(x,n),(y,m)]$ represent $x^n \cdot y^m$
\item invariants: all powers are $\geq 1$ and each variable occurs at most once \\
   hence: $[(x,1),(y,2),(x,2)]$ will not occur, but $[(x,3),(y,2)]$;
          $[(x,1),(y,0)]$ will not occur, but $[(x,1)]$
\end{itemize}
›

context linorder
begin
definition monom_inv :: "'a monom_list ⇒ bool" where 
  "monom_inv m ≡ (∀ (x,n) ∈ set m. 1 ≤ n) ∧ distinct (map fst m) ∧ sorted (map fst m)"

fun eval_monom_list :: "('a,'b :: comm_semiring_1)assign ⇒ ('a monom_list) ⇒ 'b" where 
  "eval_monom_list α [] = 1"
| "eval_monom_list α ((x,p) # m) = eval_monom_list α m * (α x)^p"

lemma eval_monom_list[simp]: "eval_monom_list α (m @ n) = eval_monom_list α m * eval_monom_list α n"
  by (induct m, auto simp: field_simps)

definition sum_var_list :: "'a monom_list ⇒ 'a ⇒ nat" where
  "sum_var_list m x ≡ sum_list (map (λ (y,c). if x = y then c else 0) m)"

lemma sum_var_list_not: "x ∉ fst ` set m ⟹ sum_var_list m x = 0"
  unfolding sum_var_list_def by (induct m, auto)

text ‹
show that equality of monomials is equivalent to statement that 
all variables occur with the same (accumulated) power;
afterwards properties like transitivity, etc. are easy to prove›

lemma monom_inv_Cons: assumes "monom_inv ((x,p) # m)" 
  and "y ≤ x" shows "y ∉ fst ` set m" 
proof -
  define M where "M = map fst m" 
  from assms[unfolded monom_inv_def] 
  have "distinct (x # map fst m)" "sorted (x # map fst m)"  by auto
  with assms(2) have "y ∉ set (map fst m)" unfolding M_def[symmetric]
    by (induct M, auto)
  thus ?thesis by auto
qed

lemma eq_monom_sum_var_list: assumes "monom_inv m" and "monom_inv n"
  shows "(m = n) = (∀ x. sum_var_list m x = sum_var_list n x)" (is "?l = ?r")
using assms
proof (induct m arbitrary: n)
  case Nil
  show ?case 
  proof (cases n)
    case (Cons yp nn)
    obtain y p where yp: "yp = (y,p)" by (cases yp, auto)
    with Cons Nil(2)[unfolded monom_inv_def] have p: "0 < p" by auto
    show ?thesis by (simp add: Cons, rule exI[of _ y], simp add: sum_var_list_def yp p)
  qed simp
next
  case (Cons xp m)
  obtain x p where xp: "xp = (x,p)" by (cases xp, auto)
  with Cons(2) have p: "0 < p" and x: "x ∉ fst ` set m" and m: "monom_inv m" unfolding monom_inv_def 
    by (auto)
  show ?case 
  proof (cases n)
    case Nil
    thus ?thesis by (auto simp: xp sum_var_list_def p intro!: exI[of _ x])
  next
    case n: (Cons yq n')
    from Cons(3)[unfolded n] have n': "monom_inv n'" by (auto simp: monom_inv_def)
    show ?thesis
    proof (cases "yq = xp")
      case True
      show ?thesis unfolding n True using Cons(1)[OF m n'] by (auto simp: xp sum_var_list_def)
    next
      case False
      obtain y q where yq: "yq = (y,q)" by force
      from Cons(3)[unfolded n yq monom_inv_def] have q: "q > 0" by auto
      define z where "z = min x y" 
      have zm: "z ∉ fst ` set m" using Cons(2) unfolding xp z_def
        by (rule monom_inv_Cons, simp)
      have zn': "z ∉ fst ` set n'" using Cons(3) unfolding n yq z_def
        by (rule monom_inv_Cons, simp)
      have smz: "sum_var_list (xp # m) z = sum_var_list [(x,p)] z" 
        using sum_var_list_not[OF zm] by (simp add: sum_var_list_def xp)
      also have "… ≠ sum_var_list [(y,q)] z" using False unfolding xp yq 
        by (auto simp: sum_var_list_def z_def p q min_def)
      also have "sum_var_list [(y,q)] z = sum_var_list n z" 
        using sum_var_list_not[OF zn'] by (simp add: sum_var_list_def n yq)
      finally show ?thesis using False unfolding n by auto
    qed
  qed
qed

text ‹
  equality of monomials is also a complete for several carriers, e.g. the naturals, integers, where $x^p = x^q$ implies $p = q$.
  note that it is not complete for carriers like the Booleans where e.g. $x^{Suc(m)} = x^{Suc(n)}$ for all $n,m$.
›
(*
lemma eq_monom_inv: 
  fixes m :: "'v :: linorder monom_list"
  assumes exp_inject: "⋀ p q :: nat. ∃ base :: 'a :: poly_carrier. base^p = base^q ⟹ p = q" 
  and m: "monom_inv m" and n: "monom_inv n" 
  shows "(m = n) = (∀ α :: ('v,'a :: poly_carrier)assign. eval_monom_list α m = eval_monom_list α n)"
proof(intro iffI allI, rule eq_monom_list)
  assume "∀ α :: ('v,'a :: poly_carrier)assign. eval_monom_list α m = eval_monom_list α n"
  with m n show "m = n"
  proof (induct m arbitrary: n)
    case Nil
    show ?case 
    proof (cases n)
      case (Cons yq nn)
      with Nil obtain y q where yq: "yq = (y,q)" and "1 ≤ q" by (cases yq, auto simp: monom_inv_def)
      then obtain qq where q: "q = Suc (qq)" by (cases q, auto)
      from Nil(3) have "1 = eval_monom_list (λ x. 0 :: 'a) n" (is "?one = _") by simp
      also have "… = 0" by (simp add: Cons yq q)
      finally show ?thesis by simp
    qed simp
  next
    case (Cons xp m) note mCons = this
    show ?case
    proof (cases xp)
      case (Pair x p)
      let ?ass = "(λ v y. if x = y then v else 1 :: 'a)"
      {
        fix v :: 'a and m :: "'v monom_list"
        assume "x ∉ fst ` (set m)"
        hence "eval_monom_list (?ass v) m = 1"
        proof (induct m)
          case (Cons yp m)
          thus ?case 
            by (cases yp, cases "fst yp = x", auto)
        qed simp
      } note ass = this
      from Cons(2)[unfolded Pair] obtain pp where p: "p = Suc pp" and xm: "x ∉ fst ` (set m)" unfolding monom_inv_def by (cases p, auto)
      from ass[OF xm] have "⋀ v. eval_monom_list (?ass v) (xp # m) = v * v^pp" by (simp add: Pair p)
      with Cons(4) have eval: "⋀ v. eval_monom_list (?ass v) n = v * v^pp" by auto
      show ?thesis 
      proof (cases "List.extract (λ yq. fst yq = x) n")
        case None
        with ass[of n] have "⋀ v. eval_monom_list (?ass v) n = 1" by (auto simp: extract_None_iff)
        from this[of 0] eval[of 0] show ?thesis by simp
      next
        case (Some res)
        obtain n1 yq n2 where "res = (n1,yq,n2)" by (cases res, auto)
        then obtain y q where "res = (n1,(y,q),n2)" by (cases yq, auto)        
        from extract_SomeE[OF Some[simplified this]] mCons(2)  Some Pair this have n: "n = n1 @ (x,q) # n2" and res: "res = (n1,(x,q),n2)" by auto
        from mCons(3)[unfolded n] have xn: "x ∉ fst ` (set (n1 @ n2))" unfolding monom_inv_def by auto
        have "⋀ v. eval_monom_list (?ass v) n = v^q * eval_monom_list  (?ass v) (n1 @ n2)" unfolding n by (auto simp: field_simps)
        from eval[unfolded this ass[OF xn]] have id: "⋀ v :: 'a. v^p = v^q" using p by auto
        from exp_inject[of p q] id have pq: "p = q" by auto
        have rec: "((xp # m) =m n) = (m =m (n1 @ n2))" by (simp add: Pair Some res pq)
        have ass: "∀α :: ('v,'a)assign. eval_monom_list  α m = eval_monom_list α (n1 @ n2)"
        proof
          fix α :: "('v,'a)assign"
          show "eval_monom_list α m = eval_monom_list α (n1 @ n2)"
          proof (rule ccontr)
            assume neq: "¬ ?thesis"
            let ?ass =  "λ y :: 'v. if x = y then 1 :: 'a else α y"
            {
              fix m :: "'v monom_list"
              assume "x ∉ fst ` set m"
              hence "eval_monom_list α m = eval_monom_list ?ass m"
                by (induct m, auto)
            } note ass = this
            have "eval_monom_list α (n1 @ n2) = eval_monom_list ?ass (n1 @ n2)" using ass[OF xn] .
            also have "… = eval_monom_list ?ass n" unfolding n by auto
            also have "… = eval_monom_list ?ass ((xp # m))" using mCons(4) by auto
            also have "… = eval_monom_list ?ass m" unfolding Pair by simp
            also have "… = eval_monom_list α m" using ass[OF xm] by simp 
            also have "… ≠ eval_monom_list α (n1 @ n2)" by (rule neq)
            finally show False by simp
          qed
        qed
        from mCons(2) mCons(3) have "monom_inv m" and "monom_inv (n1 @ n2)" unfolding monom_inv_def n by auto
        from mCons(1)[OF this ass] rec show ?thesis by simp
      qed
    qed    
  qed
qed simp  *)

abbreviation (input) monom_list_vars :: "'a monom_list ⇒ 'a set"
  where "monom_list_vars m ≡ fst ` set m"

fun monom_mult_list :: "'a monom_list ⇒ 'a monom_list ⇒ 'a monom_list" where 
  "monom_mult_list [] n = n"
| "monom_mult_list ((x,p) # m) n = (case n of
     Nil ⇒ (x,p) # m 
   | (y,q) # n' ⇒ if x = y then (x,p + q) # monom_mult_list m n' else
       if x < y then (x,p) # monom_mult_list m n else (y,q) # monom_mult_list ((x,p) # m) n')"

lemma monom_list_mult_list_vars: "monom_list_vars (monom_mult_list m1 m2) = monom_list_vars m1 ∪ monom_list_vars m2"
  by (induct m1 m2 rule: monom_mult_list.induct, auto split: list.splits)


lemma monom_mult_list_inv: "monom_inv m1 ⟹ monom_inv m2 ⟹ monom_inv (monom_mult_list m1 m2)"
proof (induct m1 m2 rule: monom_mult_list.induct)
  case (2 x p m n')
  note IH = 2(1-3)
  note xpm = 2(4)
  note n' = 2(5)
  show ?case
  proof (cases n')
    case Nil
    with xpm show ?thesis by auto
  next
    case (Cons yq n)
    then obtain y q where id: "n' = ((y,q) # n)" by (cases yq, auto)
    from xpm have m: "monom_inv m" and p: "p > 0" and x: "x ∉ fst ` set m" 
      and xm: "⋀ z. z ∈ fst ` set m ⟹ x ≤ z" 
      unfolding monom_inv_def by (auto)
    from n'[unfolded id] have n: "monom_inv n" and q: "q > 0" and y: "y ∉ fst ` set n" 
      and yn: "⋀ z. z ∈ fst ` set n ⟹ y ≤ z" 
      unfolding monom_inv_def by (auto)
    show ?thesis
    proof (cases "x = y")
      case True
      hence res: "monom_mult_list ((x, p) # m) n' = (x, p + q) # monom_mult_list m n" 
        by (simp add: id)
      from IH(1)[OF id refl True m n] have inv: "monom_inv (monom_mult_list m n)" by simp
      show ?thesis unfolding res using inv p x y True xm yn
        by (fastforce simp add: monom_inv_def monom_list_mult_list_vars)
    next
      case False
      show ?thesis
      proof (cases "x < y")
        case True
        hence res: "monom_mult_list ((x, p) # m) n' = (x,p) # monom_mult_list m n'" 
          by (auto simp add: id)
        from IH(2)[OF id refl False True m n'] have inv: "monom_inv (monom_mult_list m n')" .
        show ?thesis unfolding res using inv p x y True xm yn unfolding id
          by (fastforce simp add: monom_inv_def monom_list_mult_list_vars)
      next
        case gt: False
        with False have lt: "y < x" by auto        
        hence res: "monom_mult_list ((x, p) # m) n' = (y,q) # monom_mult_list ((x, p) # m) n" 
          using False by (auto simp add: id)
        from lt have zm: "z ≤ x ⟹ (z,b) ∉ set m" for z b using xm[of z] x by force
        from zm[of y] lt have ym: "(y,b) ∉ set m" for b by auto
        from yn have yn': "(a, b) ∈ set n ⟹ y ≤ a" for a b by force
        from IH(3)[OF id refl False gt xpm n] have inv: "monom_inv (monom_mult_list ((x, p) # m) n)" .
        define xpm where "xpm = ((x,p) # m)" 
        have xpm': "fst ` set xpm = insert x (fst ` set m)" unfolding xpm_def by auto
        show ?thesis unfolding res using inv p q x y False gt ym lt xm yn' zm xpm' unfolding id xpm_def[symmetric]
          by (auto simp add: monom_inv_def monom_list_mult_list_vars)
      qed
    qed
  qed
qed auto

lemma monom_inv_ConsD: "monom_inv (x # xs) ⟹ monom_inv xs" 
  by (auto simp: monom_inv_def)

lemma sum_var_list_monom_mult_list:  "sum_var_list (monom_mult_list m n) x = sum_var_list m x + sum_var_list n x"
proof (induct m n rule: monom_mult_list.induct)
  case (2 x p m n)
  thus ?case by (cases n; cases "hd n", auto split: if_splits simp: sum_var_list_def)
qed (auto simp: sum_var_list_def)

lemma monom_mult_list_inj: assumes m: "monom_inv m" and m1: "monom_inv m1" and m2: "monom_inv m2"
  and eq: "monom_mult_list m m1 = monom_mult_list m m2"
  shows "m1 = m2"
proof -
  from eq sum_var_list_monom_mult_list[of m] show ?thesis
    by (auto simp: eq_monom_sum_var_list[OF m1 m2] eq_monom_sum_var_list[OF monom_mult_list_inv[OF m m1] monom_mult_list_inv[OF m m2]])
qed

lemma monom_mult_list[simp]: "eval_monom_list α (monom_mult_list m n) = eval_monom_list α m * eval_monom_list α n"
  by (induct m n rule: monom_mult_list.induct, auto split: list.splits prod.splits simp: field_simps power_add)
end

declare monom_mult_list.simps[simp del]

typedef (overloaded) 'v monom = "Collect (monom_inv :: 'v :: linorder monom_list ⇒ bool)"
  by (rule exI[of _ Nil], auto simp: monom_inv_def)

setup_lifting type_definition_monom

lift_definition eval_monom :: "('v :: linorder,'a :: comm_semiring_1)assign ⇒ 'v monom ⇒ 'a"
  is eval_monom_list .

lift_definition sum_var :: "'v :: linorder monom ⇒ 'v ⇒ nat" is sum_var_list .

instantiation monom :: (linorder) comm_monoid_mult
begin

lift_definition times_monom :: "'a monom ⇒ 'a monom ⇒ 'a monom" is monom_mult_list
  using monom_mult_list_inv by auto

lift_definition one_monom :: "'a monom" is Nil
  by (auto simp: monom_inv_def)

instance
proof
  fix a b c :: "'a monom" 
  show "a * b * c = a * (b * c)" 
    by (transfer, auto simp: eq_monom_sum_var_list monom_mult_list_inv sum_var_list_monom_mult_list)
  show "a * b = b * a"
    by (transfer, auto simp: eq_monom_sum_var_list monom_mult_list_inv sum_var_list_monom_mult_list)
  show "1 * a = a" 
    by (transfer, auto simp: eq_monom_sum_var_list monom_mult_list_inv sum_var_list_monom_mult_list monom_mult_list.simps)
qed
end

lemma eq_monom_sum_var: "m = n ⟷ (∀ x. sum_var m x = sum_var n x)"
  by (transfer, auto simp: eq_monom_sum_var_list)

lemma eval_monom_mult[simp]: "eval_monom α (m * n) = eval_monom α m * eval_monom α n"
  by (transfer, rule monom_mult_list)

lemma sum_var_monom_mult:  "sum_var (m * n) x = sum_var m x + sum_var n x"
  by (transfer, rule sum_var_list_monom_mult_list)

lemma monom_mult_inj: fixes m1 :: "_ monom"
  shows "m * m1 = m * m2 ⟹ m1 = m2"
  by (transfer, rule monom_mult_list_inj, auto) 


lemma one_monom_inv_sum_var_inv[simp]: "sum_var 1 x = 0" 
  by (transfer, auto simp: sum_var_list_def)

lemma eval_monom_1[simp]: "eval_monom  α 1 = 1" 
  by (transfer, auto)

lift_definition var_monom :: "'v :: linorder ⇒ 'v monom" is "λ x. [(x,1)]" 
  by (auto simp: monom_inv_def)

lemma var_monom_1[simp]: "var_monom x ≠ 1" 
  by (transfer, auto)

lemma eval_var_monom[simp]: "eval_monom α (var_monom x) = α x" 
  by (transfer, auto)

lemma sum_var_monom_var: "sum_var (var_monom x) y = (if x = y then 1 else 0)"
  by (transfer, auto simp: sum_var_list_def)

instantiation monom :: ("{equal,linorder}")equal
begin

lift_definition equal_monom :: "'a monom ⇒ 'a monom ⇒ bool" is "(=)" .

instance by (standard, transfer, auto)
end

text ‹
Polynomials are represented with as sum of monomials multiplied by some coefficient 
›
type_synonym ('v,'a)poly = "('v monom × 'a)list"

text ‹
The polynomials we construct satisfy the following invariants:
\begin{itemize}
\item all coefficients are non-zero
\item the monomial list is distinct 
\end{itemize}
›

definition poly_inv :: "('v,'a :: zero)poly ⇒ bool"
  where "poly_inv p ≡ (∀ c ∈ snd ` set p. c ≠ 0) ∧ distinct (map fst p)"

abbreviation eval_monomc where "eval_monomc α mc ≡ eval_monom α (fst mc) * (snd mc)"

primrec eval_poly :: "('v :: linorder, 'a :: comm_semiring_1)assign ⇒ ('v,'a)poly ⇒ 'a" where 
  "eval_poly α [] = 0"
| "eval_poly α (mc # p) = eval_monomc α mc + eval_poly α p"


definition poly_const :: "'a :: zero ⇒ ('v :: linorder,'a)poly" where
  "poly_const a = (if a = 0 then [] else [(1,a)])" 

lemma poly_const[simp]: "eval_poly α (poly_const a) = a" 
  unfolding poly_const_def by auto

lemma poly_const_inv: "poly_inv (poly_const a)" 
  unfolding poly_const_def poly_inv_def by auto

fun poly_add :: "('v,'a)poly ⇒ ('v,'a :: semiring_0)poly ⇒ ('v,'a)poly" where
  "poly_add [] q = q"
| "poly_add ((m,c) # p) q = (case List.extract (λ mc. fst mc = m) q of
    None ⇒ (m,c) # poly_add p q
  | Some (q1,(_,d),q2) ⇒ if (c+d = 0) then poly_add p (q1 @ q2) else (m,c+d) # poly_add p (q1 @ q2))"
 
lemma eval_poly_append[simp]: "eval_poly α (mc1 @ mc2) = eval_poly α mc1 + eval_poly α mc2"
  by (induct mc1, auto simp: field_simps)

abbreviation poly_monoms :: "('v,'a)poly ⇒ 'v monom set"
  where "poly_monoms p ≡ fst ` set p"

lemma poly_add_monoms: "poly_monoms (poly_add p1 p2) ⊆ poly_monoms p1 ∪ poly_monoms p2"
proof (induct p1 arbitrary: p2)
  case (Cons mc p)
  obtain m c where mc: "mc = (m,c)" by (cases mc, auto)
  hence m: "m ∈ poly_monoms (mc # p1)" by auto
  show ?case
  proof (cases "List.extract (λ nd. fst nd = m) p2")
    case None
    with Cons m show ?thesis by (auto simp: mc)
  next
    case (Some res)
    obtain q1 md q2 where res: "res = (q1,md,q2)" by (cases res, auto)
    from extract_SomeE[OF Some[simplified res]] res obtain d where q: "p2 = q1 @ (m,d) # q2" and res: "res = (q1,(m,d),q2)" by (cases md, auto)
    show ?thesis
      by (simp add: mc Some res, rule subset_trans[OF Cons[of "q1 @ q2"]], auto simp: q)
  qed
qed simp
  

lemma poly_add_inv: "poly_inv p ⟹ poly_inv q ⟹ poly_inv (poly_add p q)"
proof (induct p arbitrary: q)
  case (Cons mc p)
  obtain m c where mc: "mc = (m,c)" by (cases mc, auto)
  with Cons(2) have p: "poly_inv p" and c: "c ≠ 0" and mp: "∀ mm ∈ fst ` set p. (¬ mm = m)" unfolding poly_inv_def by auto
  show ?case
  proof (cases "List.extract (λ mc. fst mc = m) q")
    case None
    hence mq: "∀ mm ∈ fst ` set q. ¬ mm = m" by (auto simp: extract_None_iff)
    { 
      fix mm 
      assume "mm ∈ fst ` set (poly_add p q)" 
      then obtain dd where "(mm,dd) ∈ set (poly_add p q)" by auto
      with poly_add_monoms have "mm ∈ poly_monoms p ∨ mm ∈ poly_monoms q" by force
      hence "¬ mm = m" using mp mq by auto
    } note main = this
    show ?thesis using Cons(1)[OF p Cons(3)] unfolding poly_inv_def using main by (auto simp add: None mc c)
  next
    case (Some res)
    obtain q1 md q2 where res: "res = (q1,md,q2)" by (cases res, auto)
    from extract_SomeE[OF Some[simplified res]] res obtain d where q: "q = q1 @ (m,d) # q2" and res: "res = (q1,(m,d),q2)" by (cases md, auto)
    from q Cons(3) have q1q2: "poly_inv (q1 @ q2)" unfolding poly_inv_def by auto
    from Cons(1)[OF p q1q2]  have main1: "poly_inv (poly_add p (q1 @ q2))" .
    {
      fix mm
      assume "mm ∈ fst ` set (poly_add p (q1 @ q2))"
      then obtain dd where "(mm,dd) ∈ set (poly_add p (q1 @ q2))" by auto
      with poly_add_monoms have "mm ∈ poly_monoms p ∨ mm ∈ poly_monoms (q1 @ q2)" by force
      hence "mm ≠ m"
      proof
        assume "mm ∈ poly_monoms p"
        thus ?thesis using mp  by auto
      next
        assume member: "mm ∈ poly_monoms (q1 @ q2)"
        from member have "mm ∈ poly_monoms q1 ∨ mm ∈ poly_monoms q2" by auto
        thus "mm ≠ m"
        proof
          assume "mm ∈ poly_monoms q2"
          with Cons(3)[simplified q]
          show ?thesis unfolding poly_inv_def by auto
        next
          assume "mm ∈ poly_monoms q1"
          with Cons(3)[simplified q]
          show ?thesis unfolding poly_inv_def by auto
        qed
      qed
    } note main2 = this
    show ?thesis using  main1[unfolded poly_inv_def] main2
      by (auto simp: poly_inv_def mc Some res)
  qed
qed simp

lemma poly_add[simp]: "eval_poly α (poly_add p q) = eval_poly α p + eval_poly α q"
proof (induct p arbitrary: q)
  case (Cons mc p)
  obtain m c where mc: "mc = (m,c)" by (cases mc, auto)
  show ?case
  proof (cases "List.extract (λ mc. fst mc = m) q")
    case None
    show ?thesis by (simp add: Cons[of q] mc None field_simps)
  next
    case (Some res)
    obtain q1 md q2 where res: "res = (q1,md,q2)" by (cases res, auto)
    from extract_SomeE[OF Some[simplified res]] res obtain d where q: "q = q1 @ (m,d) # q2" and res: "res = (q1,(m,d),q2)" by (cases md, auto)
    {
      fix x
      assume c: "c + d = 0"
      have "c * x + d * x = (c + d) * x" by (auto simp: field_simps)
      also have "… = 0 * x" by (simp only: c)
      finally have "c * x + d * x = 0" by simp
    } note id = this
    show ?thesis 
      by (simp add: Cons[of "q1 @ q2"] mc Some res, simp only: q, simp add: field_simps, auto simp: field_simps id)
  qed
qed simp

declare poly_add.simps[simp del]

fun monom_mult_poly :: "('v :: linorder monom × 'a) ⇒ ('v,'a :: semiring_0)poly ⇒ ('v,'a)poly" where 
  "monom_mult_poly _ [] = []"
| "monom_mult_poly (m,c) ((m',d) # p) = (if c * d = 0 then monom_mult_poly (m,c) p else (m * m', c * d) # monom_mult_poly (m,c) p)"

lemma monom_mult_poly_inv: "poly_inv p ⟹ poly_inv (monom_mult_poly (m,c) p)"
proof (induct p)
  case Nil thus ?case by (simp add: poly_inv_def)
next
  case (Cons md p)
  obtain m' d where md: "md = (m',d)" by (cases md, auto)
  with Cons(2) have p: "poly_inv p" unfolding poly_inv_def by auto
  from Cons(1)[OF p] have prod: "poly_inv (monom_mult_poly (m,c) p)" .
  {
    fix mm 
    assume "mm ∈ fst ` set (monom_mult_poly (m,c) p)" 
       and two: "mm = m * m'"
    then obtain dd where one: "(mm,dd) ∈ set (monom_mult_poly (m,c) p)" by auto
    have "poly_monoms (monom_mult_poly (m,c) p) ⊆ (*) m ` poly_monoms p" 
    proof (induct p, simp)
      case (Cons md p)
      thus ?case
        by (cases md, auto)
    qed
    with one have "mm ∈ (*) m ` poly_monoms p" by force
    then obtain mmm where mmm: "mmm ∈ poly_monoms p" and mm: "mm = m * mmm" by blast
    from Cons(2)[simplified md] mmm have not1: "¬ mmm = m'" unfolding poly_inv_def by auto
    from mm two have "m * mmm = m * m'" by simp
    from monom_mult_inj[OF this] not1 
    have False by simp
  } 
  thus ?case 
    by (simp add: md prod, intro impI, auto simp: poly_inv_def prod[simplified poly_inv_def])
qed

lemma monom_mult_poly[simp]: "eval_poly α (monom_mult_poly mc p) = eval_monomc α mc * eval_poly α p"
proof (cases mc)
  case (Pair m c)
  show ?thesis
  proof (simp add: Pair, induct p)
    case (Cons nd q)
    obtain n d where nd: "nd = (n,d)" by (cases nd, auto)
    show ?case
    proof (cases "c * d = 0")
      case False
      thus ?thesis by (simp add: nd Cons field_simps)
    next
      case True
      let ?l = "c * (d * (eval_monom α m * eval_monom α n))"
      have "?l = (c * d) * (eval_monom α m * eval_monom α n)" 
        by (simp only: field_simps)
      also have "… = 0" by (simp only: True, simp add: field_simps)
      finally have l: "?l = 0" .
      show ?thesis 
        by (simp add: nd Cons True, simp add: field_simps l) 
    qed
  qed simp
qed

declare monom_mult_poly.simps[simp del]

definition poly_minus :: "('v :: linorder,'a :: ring_1)poly ⇒ ('v,'a)poly ⇒ ('v,'a)poly" where
  "poly_minus f g = poly_add f (monom_mult_poly (1,-1) g)" 
  
lemma poly_minus[simp]: "eval_poly α (poly_minus f g) = eval_poly α f - eval_poly α g" 
  unfolding poly_minus_def by simp

lemma poly_minus_inv: "poly_inv f ⟹ poly_inv g ⟹ poly_inv (poly_minus f g)" 
  unfolding poly_minus_def by (intro poly_add_inv monom_mult_poly_inv)

fun poly_mult :: "('v :: linorder, 'a :: semiring_0)poly ⇒ ('v,'a)poly ⇒ ('v,'a)poly" where 
  "poly_mult [] q = []"
| "poly_mult (mc # p) q = poly_add (monom_mult_poly mc q) (poly_mult p q)"

lemma poly_mult_inv: assumes p: "poly_inv p" and q: "poly_inv q"
  shows "poly_inv (poly_mult p q)"
using p
proof (induct p)
  case Nil thus ?case by (simp add: poly_inv_def)
next
  case (Cons mc p)
  obtain m c where mc: "mc = (m,c)" by (cases mc, auto)
  with Cons(2) have p: "poly_inv p" unfolding poly_inv_def by auto
  show ?case
    by (simp add: mc, rule poly_add_inv[OF monom_mult_poly_inv[OF q] Cons(1)[OF p]])
qed

lemma poly_mult[simp]: "eval_poly α (poly_mult p q) = eval_poly α p * eval_poly α q"
  by (induct p, auto simp: field_simps)

declare poly_mult.simps[simp del]

definition zero_poly :: "('v,'a)poly"
where "zero_poly ≡ []"

lemma zero_poly_inv: "poly_inv zero_poly" unfolding zero_poly_def poly_inv_def by auto

definition one_poly :: "('v :: linorder,'a :: semiring_1)poly" where 
  "one_poly ≡ [(1,1)]"

lemma one_poly_inv: "poly_inv one_poly" unfolding one_poly_def poly_inv_def monom_inv_def by auto

lemma poly_one[simp]: "eval_poly α one_poly = 1" 
  unfolding one_poly_def by simp

lemma poly_zero_add: "poly_add zero_poly p = p" unfolding zero_poly_def using poly_add.simps by auto

lemma poly_zero_mult: "poly_mult zero_poly p = zero_poly" unfolding zero_poly_def using poly_mult.simps by auto

text ‹equality of polynomials›
definition eq_poly :: "('v :: linorder, 'a :: comm_semiring_1)poly ⇒ ('v,'a)poly ⇒ bool" (infix ‹=p› 51)
where "p =p q ≡ ∀ α. eval_poly α p = eval_poly α q"

lemma poly_one_mult: "poly_mult one_poly p =p p" 
  unfolding eq_poly_def one_poly_def by simp

lemma eq_poly_refl[simp]: "p =p p" unfolding eq_poly_def by auto

lemma eq_poly_trans[trans]: "⟦p1 =p p2; p2 =p p3⟧ ⟹ p1 =p p3"
unfolding eq_poly_def by auto

lemma poly_add_comm: "poly_add p q =p poly_add q p" unfolding eq_poly_def by (auto simp: field_simps)

lemma poly_add_assoc: "poly_add p1 (poly_add p2 p3) =p poly_add (poly_add p1 p2) p3" unfolding eq_poly_def by (auto simp: field_simps)

lemma poly_mult_comm: "poly_mult p q =p poly_mult q p" unfolding eq_poly_def by (auto simp: field_simps)

lemma poly_mult_assoc: "poly_mult p1 (poly_mult p2 p3) =p poly_mult (poly_mult p1 p2) p3" unfolding eq_poly_def by (auto simp: field_simps)

lemma poly_distrib: "poly_mult p (poly_add q1 q2) =p poly_add (poly_mult p q1) (poly_mult p q2)" unfolding eq_poly_def by (auto simp: field_simps)


subsection ‹Computing normal forms of polynomials›
fun
  poly_of :: "('v :: linorder,'a :: comm_semiring_1)tpoly ⇒ ('v,'a)poly"
where "poly_of (PNum i) = (if i = 0 then [] else [(1,i)])"
    | "poly_of (PVar x) = [(var_monom x,1)]"
    | "poly_of (PSum []) = zero_poly" 
    | "poly_of (PSum (p # ps)) = (poly_add (poly_of p) (poly_of (PSum ps)))"
    | "poly_of (PMult []) = one_poly" 
    | "poly_of (PMult (p # ps)) = (poly_mult (poly_of p) (poly_of (PMult ps)))"

text ‹
  evaluation is preserved by poly\_of
›
lemma poly_of: "eval_poly α (poly_of p) = eval_tpoly α p"
by (induct p rule: poly_of.induct, (simp add: zero_poly_def one_poly_def)+)

text ‹
  poly\_of only generates polynomials that satisfy the invariant
›
lemma poly_of_inv: "poly_inv (poly_of p)"
by (induct p rule: poly_of.induct, 
    simp add: poly_inv_def monom_inv_def,
    simp add: poly_inv_def monom_inv_def,
    simp add: zero_poly_inv,
    simp add: poly_add_inv,
    simp add: one_poly_inv,
    simp add: poly_mult_inv)


subsection ‹Powers and substitutions of polynomials›
fun poly_power :: "('v :: linorder, 'a :: comm_semiring_1)poly ⇒ nat ⇒ ('v,'a)poly" where 
  "poly_power _ 0 = one_poly"
| "poly_power p (Suc n) = poly_mult p (poly_power p n)"

lemma poly_power[simp]: "eval_poly α (poly_power p n) = (eval_poly α p) ^ n"
  by (induct n, auto simp: one_poly_def)

lemma poly_power_inv: assumes p: "poly_inv p" 
  shows "poly_inv (poly_power p n)"
  by (induct n, simp add: one_poly_inv, simp add: poly_mult_inv[OF p])

declare poly_power.simps[simp del]

fun monom_list_subst :: "('v ⇒ ('w :: linorder,'a :: comm_semiring_1)poly) ⇒ 'v monom_list ⇒ ('w,'a)poly" where 
  "monom_list_subst σ [] = one_poly"
| "monom_list_subst σ ((x,p) # m) = poly_mult (poly_power (σ x) p) (monom_list_subst σ m)"

lift_definition monom_list :: "'v :: linorder monom ⇒ 'v monom_list" is "λ x. x" .

definition monom_subst :: "('v :: linorder ⇒ ('w :: linorder,'a :: comm_semiring_1)poly) ⇒ 'v monom ⇒ ('w,'a)poly" where 
  "monom_subst σ m = monom_list_subst σ (monom_list m)"  

lemma monom_list_subst_inv: assumes sub: "⋀ x. poly_inv (σ x)" 
  shows "poly_inv (monom_list_subst σ m)"
proof (induct m)
  case Nil thus ?case by (simp add: one_poly_inv)
next
  case (Cons xp m)
  obtain x p where xp: "xp = (x,p)" by (cases xp, auto)
  show ?case by (simp add: xp, rule poly_mult_inv[OF poly_power_inv[OF sub] Cons])
qed

lemma monom_subst_inv: assumes sub: "⋀ x. poly_inv (σ x)" 
  shows "poly_inv (monom_subst σ m)"
  unfolding monom_subst_def by (rule monom_list_subst_inv[OF sub])

lemma monom_subst[simp]: "eval_poly α (monom_subst σ m) = eval_monom (λ v. eval_poly α (σ v)) m"
  unfolding monom_subst_def
proof (transfer fixing: α σ, clarsimp)
  fix m 
  show "monom_inv m ⟹ eval_poly α (monom_list_subst σ m) = eval_monom_list (λv. eval_poly α (σ v)) m"
    by (induct m, simp add: one_poly_def, auto simp: field_simps monom_inv_ConsD)
qed

fun poly_subst :: "('v :: linorder ⇒ ('w :: linorder,'a :: comm_semiring_1)poly) ⇒ ('v,'a)poly ⇒ ('w,'a)poly" where 
  "poly_subst σ [] = zero_poly"
| "poly_subst σ ((m,c) # p) = poly_add (poly_mult [(1,c)] (monom_subst σ m)) (poly_subst σ p)"

lemma poly_subst_inv: assumes sub: "⋀ x. poly_inv (σ x)" and p: "poly_inv p"
  shows "poly_inv (poly_subst σ p)"
using p
proof (induct p)
  case Nil thus ?case by (simp add: zero_poly_inv)
next
  case (Cons mc p)
  obtain m c where mc: "mc = (m,c)" by (cases mc, auto)
  with Cons(2) have c: "c ≠ 0" and p: "poly_inv p" unfolding poly_inv_def by auto
  from c have c: "poly_inv [(1,c)]" unfolding poly_inv_def monom_inv_def by auto
  show ?case 
    by (simp add: mc, rule poly_add_inv[OF poly_mult_inv[OF c monom_subst_inv[OF sub]] Cons(1)[OF p]])
qed

lemma poly_subst: "eval_poly α (poly_subst σ p) = eval_poly (λ v. eval_poly α (σ v)) p"
  by (induct p, simp add: zero_poly_def, auto simp: field_simps)

lemma eval_poly_subst: 
  assumes eq: "⋀ w. f w = eval_poly g (q w)"
  shows "eval_poly f p = eval_poly g (poly_subst q p)" 
proof (induct p)
  case Nil thus ?case by (simp add: zero_poly_def)
next
  case (Cons mc p)
  obtain m c where mc: "mc = (m,c)" by (cases mc, auto)
  have id: "eval_monom f m =  eval_monom (λv. eval_poly g (q v)) m"
  proof (transfer fixing: f g q, clarsimp)
    fix m
    show "eval_monom_list f m = eval_monom_list (λv. eval_poly g (q v)) m"
    proof (induct m)
      case (Cons wp m)
      obtain w p where wp: "wp = (w,p)" by (cases wp, auto)
      show ?case
        by (simp add: wp Cons eq)
    qed simp
  qed
  show ?case
    by (simp add: mc Cons id, simp add: field_simps)
qed

lift_definition monom_vars_list :: "'v :: linorder monom ⇒ 'v list" is "map fst" .

lemma monom_vars_list_subst: assumes "⋀ w. w ∈ set (monom_vars_list m) ⟹ f w = g w" 
  shows "monom_subst f m = monom_subst g m" 
  unfolding monom_subst_def using assms
proof (transfer fixing: f g)
  fix m :: "'a monom_list" 
  assume eq: "⋀w. w ∈ set (map fst m) ⟹ f w = g w" 
  thus "monom_list_subst f m = monom_list_subst g m" 
  proof (induct m)
    case (Cons wn m)
    hence rec: "monom_list_subst f m = monom_list_subst g m" and eq: "f (fst wn) = g (fst wn)" by auto
    show ?case
    proof (cases wn)
      case (Pair w n)
      with eq rec show ?thesis by auto
    qed
  qed simp
qed

lemma eval_monom_vars_list: assumes "⋀ x. x ∈ set (monom_vars_list xs) ⟹ α x = β x"
  shows "eval_monom α xs = eval_monom β xs" using assms
proof (transfer fixing: α β)
  fix xs :: "'a monom_list" 
  assume eq: "⋀w. w ∈ set (map fst xs) ⟹ α w = β w" 
  thus "eval_monom_list α xs = eval_monom_list β xs" 
  proof (induct xs)
    case (Cons xi xs)
    hence IH: "eval_monom_list α xs = eval_monom_list β xs" by auto
    obtain x i where xi: "xi = (x,i)" by force
    from Cons(2) xi have "α x = β x" by auto
    with IH show ?case unfolding xi by auto
  qed simp
qed

definition monom_vars where "monom_vars m = set (monom_vars_list m)" 

lemma monom_vars_list_1[simp]: "monom_vars_list 1 = []" 
  by transfer auto

lemma monom_vars_list_var_monom[simp]: "monom_vars_list (var_monom x) = [x]" 
  by transfer auto

lemma monom_vars_eval_monom: 
  "(⋀ x. x ∈ monom_vars m ⟹ f x = g x) ⟹ eval_monom f m = eval_monom g m"
  by (rule eval_monom_vars_list, auto simp: monom_vars_def)


(* the list of variables occurring in p *)
definition poly_vars_list :: "('v :: linorder,'a)poly ⇒ 'v list" where 
  "poly_vars_list p = remdups (concat (map (monom_vars_list o fst) p))"

definition poly_vars :: "('v :: linorder,'a)poly ⇒ 'v set" where 
  "poly_vars p = set (concat (map (monom_vars_list o fst) p))"

lemma poly_vars_list[simp]: "set (poly_vars_list p) = poly_vars p" 
  unfolding poly_vars_list_def poly_vars_def by auto

lemma poly_vars: assumes eq: "⋀ w. w ∈ poly_vars p ⟹ f w = g w"
  shows "poly_subst f p = poly_subst g p" 
using eq
proof (induct p)
  case (Cons mc p)
  hence rec: "poly_subst f p = poly_subst g p" unfolding poly_vars_def by auto
  show ?case
  proof (cases mc)
    case (Pair m c)
    with Cons(2) have "⋀ w. w ∈ set (monom_vars_list m) ⟹ f w = g w" unfolding poly_vars_def by auto
    hence "monom_subst f m = monom_subst g m"
      by (rule monom_vars_list_subst)
    with rec Pair show ?thesis by auto
  qed
qed simp

lemma poly_var: assumes pv: "v ∉ poly_vars p" and diff: "⋀ w. v ≠ w ⟹ f w = g w"
  shows "poly_subst f p = poly_subst g p"
proof (rule poly_vars)
  fix w
  assume "w ∈ poly_vars p"
  thus "f w = g w" using pv diff by (cases "v = w", auto)
qed


lemma eval_poly_vars: assumes "⋀ x. x ∈ poly_vars p ⟹ α x = β x"
  shows "eval_poly α p = eval_poly β p"
using assms
proof (induct p)
  case Nil thus ?case by simp
next
  case (Cons m p)
  from Cons(2) have "⋀ x. x ∈ poly_vars p ⟹ α x = β x" unfolding poly_vars_def by auto
  from Cons(1)[OF this] have IH: "eval_poly α p = eval_poly β p" .
  obtain xs c where m: "m = (xs,c)" by force
  from Cons(2) have "⋀ x. x ∈ set (monom_vars_list xs) ⟹ α x = β x" unfolding poly_vars_def m by auto
  hence "eval_monom α xs = eval_monom β xs"
    by (rule eval_monom_vars_list)
  thus ?case unfolding eval_poly.simps IH m by auto
qed

           
declare poly_subst.simps[simp del]



subsection ‹
  Polynomial orders
›

definition pos_assign :: "('v,'a :: ordered_semiring_0)assign ⇒ bool"
where "pos_assign α = (∀ x. α x ≥ 0)"

definition poly_ge :: "('v :: linorder,'a :: poly_carrier)poly ⇒ ('v,'a)poly ⇒ bool" (infix ‹≥p› 51)
where "p ≥p q = (∀ α. pos_assign α ⟶ eval_poly α p ≥ eval_poly α q)"

lemma poly_ge_refl[simp]: "p ≥p p"
unfolding poly_ge_def using ge_refl by auto

lemma poly_ge_trans[trans]: "⟦p1 ≥p p2; p2 ≥p p3⟧ ⟹ p1 ≥p p3"
unfolding poly_ge_def using ge_trans by blast


lemma pos_assign_monom_list: fixes α :: "('v :: linorder, 'a :: poly_carrier)assign"
  assumes pos: "pos_assign α"
  shows "eval_monom_list α m ≥ 0"
proof (induct m)
  case Nil thus ?case by (simp add: one_ge_zero)
next
  case (Cons xp m)
  show ?case
  proof (cases xp)
    case (Pair x p)
    from pos[unfolded pos_assign_def] have ge: "α x ≥ 0" by simp
    have ge: "α x ^ p ≥ 0"
    proof (induct p)
      case 0 thus ?case by (simp add: one_ge_zero)
    next
      case (Suc p)
      from ge_trans[OF times_left_mono[OF ge Suc] times_right_mono[OF ge_refl ge]]
      show ?case by (simp add: field_simps)
    qed
    from ge_trans[OF times_right_mono[OF Cons ge] times_left_mono[OF ge_refl Cons]]
    show ?thesis
      by (simp add: Pair)
  qed
qed

lemma pos_assign_monom: fixes α :: "('v :: linorder, 'a :: poly_carrier)assign"
  assumes pos: "pos_assign α"
  shows "eval_monom α m ≥ 0"
  by (transfer fixing: α, rule pos_assign_monom_list[OF pos])


lemma pos_assign_poly:   assumes pos: "pos_assign α"
  and p: "p ≥p zero_poly"
  shows "eval_poly α p ≥ 0"
proof -
  from p[unfolded poly_ge_def zero_poly_def] pos 
  show ?thesis by auto
qed


lemma poly_add_ge_mono: assumes "p1 ≥p p2" shows "poly_add p1 q ≥p poly_add p2 q"
using assms unfolding poly_ge_def by (auto simp: field_simps plus_left_mono)

lemma poly_mult_ge_mono: assumes "p1 ≥p p2" and "q ≥p zero_poly"
  shows "poly_mult p1 q ≥p poly_mult p2 q"
using assms unfolding poly_ge_def zero_poly_def by (auto simp: times_left_mono)

context poly_order_carrier
begin

definition poly_gt :: "('v :: linorder,'a)poly ⇒ ('v,'a)poly ⇒ bool" (infix ‹>p› 51)
where "p >p q = (∀ α. pos_assign α ⟶ eval_poly α p ≻ eval_poly α q)"

lemma poly_gt_imp_poly_ge: "p >p q ⟹ p ≥p q" unfolding poly_ge_def poly_gt_def using gt_imp_ge by blast

abbreviation poly_GT :: "('v :: linorder,'a)poly rel"
where "poly_GT ≡ {(p,q) | p q. p >p q ∧ q ≥p zero_poly}"

lemma poly_compat: "⟦p1 ≥p p2; p2 >p p3⟧ ⟹ p1 >p p3"
unfolding poly_ge_def poly_gt_def using compat by blast

lemma poly_compat2: "⟦p1 >p p2; p2 ≥p p3⟧ ⟹ p1 >p p3"
unfolding poly_ge_def poly_gt_def using compat2 by blast

lemma poly_gt_trans[trans]: "⟦p1 >p p2; p2 >p p3⟧ ⟹ p1 >p p3"
unfolding poly_gt_def using gt_trans by blast

lemma poly_GT_SN: "SN poly_GT"
proof
  fix f :: "nat ⇒ ('c :: linorder,'a)poly"
  assume f: "∀ i. (f i, f (Suc i)) ∈ poly_GT"
  have pos: "pos_assign ((λ x. 0) :: ('v,'a)assign)" (is "pos_assign ?ass") unfolding pos_assign_def using ge_refl by auto
  obtain g where g: "⋀ i. g i = eval_poly ?ass (f i)" by auto
  from f pos have "∀ i. g (Suc i) ≥ 0 ∧ g i ≻ g (Suc i)" unfolding poly_gt_def g using pos_assign_poly by auto
  with SN show False unfolding SN_defs by blast 
qed
end

text ‹monotonicity of polynomials›

lemma eval_monom_list_mono: assumes fg: "⋀ x. (f :: ('v :: linorder,'a :: poly_carrier)assign) x ≥ g x" 
  and g: "⋀ x. g x ≥ 0"
  shows "eval_monom_list f m ≥ eval_monom_list g m" "eval_monom_list g m ≥ 0"
proof (atomize(full), induct m)
  case Nil show ?case using one_ge_zero by (auto simp: ge_refl)
next
  case (Cons xd m)
  hence IH1: " eval_monom_list f m ≥ eval_monom_list g m" and IH2: "eval_monom_list g m ≥ 0" by auto
  obtain x d where xd: "xd = (x,d)" by force
  from pow_mono[OF fg g, of x d] have fgd: "f x ^ d ≥ g x ^ d" and gd: "g x ^ d ≥ 0" by auto
  show ?case unfolding xd eval_monom_list.simps
  proof (rule conjI, rule ge_trans[OF times_left_mono[OF pow_ge_zero IH1] times_right_mono[OF IH2 fgd]])
    show "f x ≥ 0" by (rule ge_trans[OF fg g])
    show "eval_monom_list g m * g x ^ d ≥ 0"
      by (rule mult_ge_zero[OF IH2 gd])
  qed
qed

lemma eval_monom_mono: assumes fg: "⋀ x. (f :: ('v :: linorder,'a :: poly_carrier)assign) x ≥ g x" 
  and g: "⋀ x. g x ≥ 0"
shows "eval_monom f m ≥ eval_monom g m" "eval_monom g m ≥ 0"
  by (atomize(full), transfer fixing: f g, insert eval_monom_list_mono[of g f, OF fg g], auto)


definition poly_weak_mono_all :: "('v :: linorder,'a :: poly_carrier)poly ⇒ bool" where 
  "poly_weak_mono_all p ≡ ∀ (α :: ('v,'a)assign) β. (∀ x. α x ≥ β x) 
    ⟶ pos_assign β ⟶ eval_poly α p ≥ eval_poly β p"

lemma poly_weak_mono_all_E: assumes p: "poly_weak_mono_all p" and 
  ge: "⋀ x. f x ≥p g x ∧ g x ≥p zero_poly"
  shows "poly_subst f p ≥p poly_subst g p"
  unfolding poly_ge_def poly_subst
proof (intro allI impI, rule p[unfolded poly_weak_mono_all_def, rule_format])
  fix α :: "('c,'b)assign" and x
  show "pos_assign α ⟹ eval_poly α (f x) ≥ eval_poly α (g x)" using ge[of x] unfolding poly_ge_def by auto
next
  fix α :: "('c,'b)assign"
  assume alpha: "pos_assign α"
  show "pos_assign (λv. eval_poly α (g v))" 
    unfolding pos_assign_def
  proof
    fix x
    show "eval_poly α (g x) ≥ 0"
    using ge[of x] unfolding poly_ge_def zero_poly_def using alpha by auto
  qed
qed

definition poly_weak_mono :: "('v :: linorder,'a :: poly_carrier)poly ⇒ 'v ⇒ bool" where 
  "poly_weak_mono p v ≡ ∀ (α :: ('v,'a)assign) β. (∀ x. v ≠ x ⟶ α x = β x) ⟶ pos_assign β ⟶ α v ≥ β v ⟶ eval_poly α p ≥ eval_poly β p"

lemma poly_weak_mono_E: assumes p: "poly_weak_mono p v"
  and fgw: "⋀ w. v ≠ w ⟹ f w = g w"
  and g: "⋀ w. g w ≥p zero_poly" 
  and fgv: "f v ≥p g v"
  shows "poly_subst f p ≥p poly_subst g p"
  unfolding poly_ge_def poly_subst
proof (intro allI impI, rule p[unfolded poly_weak_mono_def, rule_format])
  fix α :: "('c,'b)assign"
  show "pos_assign α ⟹ eval_poly α (f v) ≥ eval_poly α (g v)" using fgv unfolding poly_ge_def by auto
next
  fix α :: "('c,'b)assign"
  assume alpha: "pos_assign α"
  show "pos_assign (λv. eval_poly α (g v))" 
    unfolding pos_assign_def
  proof
    fix x
    show "eval_poly α (g x) ≥ 0"
    using g[of x] unfolding poly_ge_def zero_poly_def using alpha by auto
  qed
next
  fix α :: "('c,'b)assign" and x
  assume v: "v ≠ x"
  show "pos_assign α ⟹ eval_poly α (f x) = eval_poly α (g x)" using fgw[OF v] unfolding poly_ge_def by auto
qed

definition poly_weak_anti_mono :: "('v :: linorder,'a :: poly_carrier)poly ⇒ 'v ⇒ bool" where 
  "poly_weak_anti_mono p v ≡ ∀ (α :: ('v,'a)assign) β. (∀ x. v ≠ x ⟶ α x = β x) ⟶ pos_assign β ⟶ α v ≥ β v ⟶ eval_poly β p ≥ eval_poly α p"

lemma poly_weak_anti_mono_E: assumes p: "poly_weak_anti_mono p v"
  and fgw: "⋀ w. v ≠ w ⟹ f w = g w"
  and g: "⋀ w. g w ≥p zero_poly" 
  and fgv: "f v ≥p g v"
  shows "poly_subst g p ≥p poly_subst f p"
  unfolding poly_ge_def poly_subst
proof (intro allI impI, rule p[unfolded poly_weak_anti_mono_def, rule_format])
  fix α :: "('c,'b)assign"
  show "pos_assign α ⟹ eval_poly α (f v) ≥ eval_poly α (g v)" using fgv unfolding poly_ge_def by auto
next
  fix α :: "('c,'b)assign"
  assume alpha: "pos_assign α"
  show "pos_assign (λv. eval_poly α (g v))" 
    unfolding pos_assign_def
  proof
    fix x
    show "eval_poly α (g x) ≥ 0"
    using g[of x] unfolding poly_ge_def zero_poly_def using alpha by auto
  qed
next
  fix α :: "('c,'b)assign" and x
  assume v: "v ≠ x"
  show "pos_assign α ⟹ eval_poly α (f x) = eval_poly α (g x)" using fgw[OF v] unfolding poly_ge_def by auto
qed

lemma poly_weak_mono: fixes p :: "('v :: linorder,'a :: poly_carrier)poly"
  assumes mono: "⋀ v. v ∈ poly_vars p ⟹ poly_weak_mono p v"
  shows "poly_weak_mono_all p"
unfolding poly_weak_mono_all_def
proof (intro allI impI)
  fix α β :: "('v,'a)assign"
  assume all: "∀ x. α x ≥ β x"
  assume pos: "pos_assign β"
  let ?ab = "λ vs v. if (v ∈ set vs) then α v else β v"
  {
    fix vs :: "'v list"
    assume "set vs ⊆ poly_vars p"
    hence "eval_poly (?ab vs) p ≥ eval_poly β p"
    proof (induct vs)
      case Nil show ?case by (simp add: ge_refl)
    next
      case (Cons v vs)
      hence subset: "set vs ⊆ poly_vars p"  and v: "v ∈ poly_vars p" by auto
      show ?case
      proof (rule ge_trans[OF mono[OF v, unfolded poly_weak_mono_def, rule_format] Cons(1)[OF subset]])
        show "pos_assign (?ab vs)" unfolding pos_assign_def
        proof
          fix x
          from pos[unfolded pos_assign_def] have beta: "β x ≥ 0" by simp
          from ge_trans[OF all[rule_format] this] have alpha: "α x ≥ 0" .
          from alpha beta show "?ab vs x ≥ 0" by auto
        qed
        show "(?ab (v # vs) v) ≥ (?ab vs v)" using all ge_refl by auto
      next
        fix x
        assume "v ≠ x"
        thus "(?ab (v # vs) x) = (?ab vs x)" by simp
      qed
    qed
  }
  from this[of "poly_vars_list p", unfolded poly_vars_list]
  have "eval_poly (λv. if v ∈ poly_vars p then α v else β v) p ≥ eval_poly β p" by auto
  also have "eval_poly (λv. if v ∈ poly_vars p then α v else β v) p = eval_poly α p"
    by (rule eval_poly_vars, auto)
  finally
  show "eval_poly α p ≥ eval_poly β p" .
qed  

lemma poly_weak_mono_all: fixes p :: "('v :: linorder,'a :: poly_carrier)poly" 
  assumes p: "poly_weak_mono_all p"
  shows "poly_weak_mono p v"
unfolding poly_weak_mono_def
proof (intro allI impI)
  fix α β :: "('v,'a)assign"
  assume all: "∀x. v ≠ x ⟶ α x = β x"
  assume pos: "pos_assign β"
  assume v: "α v ≥ β v"
  show "eval_poly α p ≥ eval_poly β p" 
  proof (rule p[unfolded poly_weak_mono_all_def, rule_format, OF _ pos])
    fix x 
    show "α x ≥ β x"
    using v all ge_refl[of "β x"] by auto
  qed
qed

lemma poly_weak_mono_all_pos: 
  fixes p :: "('v :: linorder,'a :: poly_carrier)poly"
  assumes pos_at_zero: "eval_poly (λ w. 0) p ≥ 0"
  and mono: "poly_weak_mono_all p"
  shows "p ≥p zero_poly"
unfolding poly_ge_def zero_poly_def
proof (intro allI impI, simp)
  fix  α :: "('v,'a)assign"
  assume pos: "pos_assign α"
  show "eval_poly α p ≥ 0"
  proof -
    let ?id = "λ w. poly_of (PVar w)"
    let ?z = "λ w. zero_poly"
    have "poly_subst ?id p ≥p poly_subst ?z p" 
      by (rule poly_weak_mono_all_E[OF mono],  
        simp, simp add: poly_ge_def zero_poly_def pos_assign_def) 
    hence "eval_poly α (poly_subst ?id p) ≥ eval_poly α (poly_subst ?z p)" (is "_ ≥ ?res")
      unfolding poly_ge_def using pos by simp
    also have "?res = eval_poly (λ w. 0) p" by (simp add: poly_subst zero_poly_def)
    also have "… ≥ 0" by (rule pos_at_zero)
    finally show ?thesis by  (simp add: poly_subst)
  qed
qed

context poly_order_carrier
begin

definition poly_strict_mono :: "('v :: linorder,'a)poly ⇒ 'v ⇒ bool" where 
  "poly_strict_mono p v ≡ ∀ (α :: ('v,'a)assign) β. (∀ x. (v ≠ x ⟶ α x = β x)) ⟶ pos_assign β ⟶ α v ≻ β v ⟶ eval_poly α p ≻ eval_poly β p"

lemma poly_strict_mono_E: assumes p: "poly_strict_mono p v"
  and fgw: "⋀ w. v ≠ w ⟹ f w = g w"
  and g: "⋀ w. g w ≥p zero_poly" 
  and fgv: "f v >p g v"
  shows "poly_subst f p >p poly_subst g p"
  unfolding poly_gt_def poly_subst
proof (intro allI impI, rule p[unfolded poly_strict_mono_def, rule_format])
  fix α :: "('c,'a)assign"
  show "pos_assign α ⟹ eval_poly α (f v) ≻ eval_poly α (g v)" using fgv unfolding poly_gt_def by auto
next
  fix α :: "('c,'a)assign"
  assume alpha: "pos_assign α"
  show "pos_assign (λv. eval_poly α (g v))" 
    unfolding pos_assign_def
  proof
    fix x
    show "eval_poly α (g x) ≥ 0"
    using g[of x] unfolding poly_ge_def zero_poly_def using alpha by auto
  qed
next
  fix α :: "('c,'a)assign" and x
  assume v: "v ≠ x"
  show "pos_assign α ⟹ eval_poly α (f x) = eval_poly α (g x)" using fgw[OF v] unfolding poly_ge_def by auto
qed

lemma poly_add_gt_mono: assumes "p1 >p p2" shows "poly_add p1 q >p poly_add p2 q"
using assms unfolding poly_gt_def by (auto simp: field_simps plus_gt_left_mono)

lemma poly_mult_gt_mono: 
  fixes q :: "('v :: linorder,'a)poly"
  assumes gt: "p1 >p p2" and mono: "q ≥p one_poly"
  shows "poly_mult p1 q >p poly_mult p2 q"
proof (unfold poly_gt_def, intro impI allI)
  fix α :: "('v,'a)assign"
  assume p: "pos_assign α"
  with gt have gt: "eval_poly α p1 ≻ eval_poly α p2" unfolding poly_gt_def by simp
  from mono p have one: "eval_poly α q ≥ 1" unfolding poly_ge_def one_poly_def by auto
  show "eval_poly α (poly_mult p1 q) ≻ eval_poly α (poly_mult p2 q)"
    using times_gt_mono[OF gt one] by simp
qed
end

subsection ‹Degree of polynomials›

definition monom_list_degree :: "'v monom_list ⇒ nat" where 
  "monom_list_degree xps ≡ sum_list (map snd xps)"

lift_definition monom_degree :: "'v :: linorder monom ⇒ nat" is monom_list_degree .

definition poly_degree :: "(_,'a) poly ⇒ nat" where
  "poly_degree p ≡ max_list (map (λ (m,c). monom_degree m) p)"

definition poly_coeff_sum :: "('v,'a :: ordered_ab_semigroup) poly ⇒ 'a" where
  "poly_coeff_sum p ≡ sum_list (map (λ mc. max 0 (snd mc)) p)"

lemma monom_list_degree: "eval_monom_list (λ _. x) m = x ^ monom_list_degree m"
  unfolding monom_list_degree_def
proof (induct m)
  case Nil show ?case by simp
next 
  case (Cons mc m)
  thus ?case by (cases mc, auto simp: power_add field_simps)
qed

lemma monom_list_var_monom[simp]: "monom_list (var_monom x) = [(x,1)]" 
  by (transfer, simp)

lemma monom_list_1[simp]: "monom_list 1 = []" 
  by (transfer, simp)

lemma monom_degree: "eval_monom (λ _. x) m = x ^ monom_degree m"
  by (transfer, rule monom_list_degree)

lemma poly_coeff_sum: "poly_coeff_sum p ≥ 0"
  unfolding poly_coeff_sum_def
proof (induct p)
  case Nil show ?case by (simp add: ge_refl)
next
  case (Cons mc p)
  have "(∑mc←mc # p. max 0 (snd mc)) = max 0 (snd mc) + (∑mc←p. max 0 (snd mc))" by auto
  also have "… ≥ 0 + 0"
    by (rule ge_trans[OF plus_left_mono plus_right_mono[OF Cons]], auto)
  finally show ?case by simp
qed

lemma poly_degree: assumes x: "x ≥ (1 :: 'a :: poly_carrier)" 
  shows "poly_coeff_sum p * (x ^ poly_degree p) ≥ eval_poly (λ _. x) p"
proof (induct p)
  case Nil show ?case by (simp add: ge_refl poly_degree_def poly_coeff_sum_def)
next
  case (Cons mc p)  
  obtain m c where mc: "mc = (m,c)" by force
  from ge_trans[OF x one_ge_zero] have x0: "x ≥ 0" .
  have id1: "eval_poly (λ_. x) (mc # p) = x ^ monom_degree m  * c + eval_poly (λ_. x) p" unfolding mc by (simp add: monom_degree)
  have id2: "poly_coeff_sum (mc # p) * x ^ poly_degree (mc # p) = 
    x ^ max (monom_degree m) (poly_degree p) * (max 0 c) + poly_coeff_sum p * x ^ max (monom_degree m) (poly_degree p)"
    unfolding poly_coeff_sum_def poly_degree_def by (simp add: mc field_simps)
  show "poly_coeff_sum (mc # p) * x ^ poly_degree (mc # p) ≥ eval_poly (λ_. x) (mc # p)"
    unfolding id1 id2
  proof (rule ge_trans[OF plus_left_mono plus_right_mono])
    show "x ^ max (monom_degree m) (poly_degree p) * max 0 c ≥ x ^ monom_degree m * c"
      by (rule ge_trans[OF times_left_mono[OF _ pow_mono_exp] times_right_mono[OF pow_ge_zero]], insert x x0, auto)
    show "poly_coeff_sum p * x ^ max (monom_degree m) (poly_degree p) ≥ eval_poly (λ_. x) p"
      by (rule ge_trans[OF times_right_mono[OF poly_coeff_sum pow_mono_exp[OF x]] Cons], auto)
  qed
qed

lemma poly_degree_bound: assumes x: "x ≥ (1 :: 'a :: poly_carrier)" 
  and c: "c ≥ poly_coeff_sum p"
  and d: "d ≥ poly_degree p"
  shows "c * (x ^ d) ≥ eval_poly (λ _. x) p"
  by (rule ge_trans[OF ge_trans[OF 
    times_left_mono[OF pow_ge_zero[OF ge_trans[OF x one_ge_zero]] c]   
    times_right_mono[OF poly_coeff_sum pow_mono_exp[OF x d]]] poly_degree[OF x]])


subsection ‹Executable and sufficient criteria to compare polynomials and ensure monotonicity› 

text ‹poly\_split extracts the coefficient for a given monomial and returns additionally the remaining polynomial›
definition poly_split :: "('v monom) ⇒ ('v,'a :: zero)poly ⇒ 'a × ('v,'a)poly" 
  where "poly_split m p ≡ case List.extract (λ (n,_). m = n) p of None ⇒ (0,p) | Some (p1,(_,c),p2) ⇒ (c, p1 @ p2)"

lemma poly_split: assumes "poly_split m p = (c,q)"
  shows "p =p (m,c) # q"
proof (cases "List.extract (λ (n,_). m = n) p")
  case None
  with assms have "(c,q) = (0,p)" unfolding poly_split_def by auto
  thus ?thesis unfolding eq_poly_def by auto
next
  case (Some res)
  obtain p1 mc p2 where "res = (p1,mc,p2)" by (cases res, auto)
  with extract_SomeE[OF Some[simplified this]] obtain a where p: "p = p1 @ (m,a) # p2" and res: "res = (p1,(m,a),p2)" by (cases mc, auto)
  from Some res assms have c: "c = a" and q: "q = p1 @ p2" unfolding poly_split_def by auto
  show ?thesis unfolding eq_poly_def by (simp add: p c q field_simps)
qed 

lemma poly_split_eval: assumes "poly_split m p = (c,q)" 
  shows "eval_poly α p = (eval_monom α m * c) + eval_poly α q"
using poly_split[OF assms] unfolding eq_poly_def by auto

(* we assume that the polynomial invariant is present, otherwise this check might fail, e.g., on 0 =p 0 + 0 *)
fun check_poly_eq :: "('v,'a :: semiring_0)poly ⇒ ('v,'a)poly ⇒ bool" where 
  "check_poly_eq [] q = (q = [])"
| "check_poly_eq ((m,c) # p) q = (case List.extract (λ nd. fst nd = m) q of
       None ⇒ False
     | Some (q1,(_,d),q2) ⇒ c = d ∧ check_poly_eq p (q1 @ q2))"

lemma check_poly_eq: fixes p :: "('v :: linorder,'a :: poly_carrier)poly"
  assumes chk: "check_poly_eq p q"
  shows "p =p q" unfolding eq_poly_def
proof
  fix α
  from chk show "eval_poly α p = eval_poly α q"
  proof (induct p arbitrary: q)
    case Nil
    thus ?case by auto
  next
    case (Cons mc p)
    obtain m c where mc: "mc = (m,c)" by (cases mc, auto)
    show ?case
    proof (cases "List.extract (λ mc. fst mc = m) q")
      case None
      with Cons(2) show ?thesis unfolding mc by simp
    next
      case (Some res)
      obtain q1 md q2 where "res = (q1,md,q2)" by (cases res, auto)
      with extract_SomeE[OF Some[simplified this]] obtain d where q: "q = q1 @ (m,d) # q2" and res: "res = (q1,(m,d),q2)" 
        by (cases md, auto)
      from Cons(2) Some mc res have rec: "check_poly_eq p (q1 @ q2)" and c: "c = d" by auto
      from Cons(1)[OF rec] have p: "eval_poly α p = eval_poly α (q1 @ q2)" .
      show ?thesis unfolding mc eval_poly.simps c p q by (simp add: ac_simps)
    qed
  qed
qed

declare check_poly_eq.simps[simp del]


fun check_poly_ge :: "('v,'a :: ordered_semiring_0)poly ⇒ ('v,'a)poly ⇒ bool" where 
  "check_poly_ge [] q = list_all (λ (_,d). 0 ≥ d) q"
| "check_poly_ge ((m,c) # p) q = (case List.extract (λ nd. fst nd = m) q of
     None ⇒ c ≥ 0 ∧ check_poly_ge p q
   | Some (q1,(_,d),q2) ⇒ c ≥ d ∧ check_poly_ge p (q1 @ q2))"

lemma check_poly_ge: fixes p :: "('v :: linorder,'a :: poly_carrier)poly"
  shows "check_poly_ge p q ⟹ p ≥p q"
proof (induct p arbitrary: q)
  case Nil
  hence "∀ (n,d) ∈ set q. 0 ≥ d" using list_all_iff[of _ q] by auto
  hence "[] ≥p q" 
  proof (induct q)
    case Nil thus ?case by (simp)
  next
    case (Cons nd q)
    hence rec: "[] ≥p q" by simp
    show ?case
    proof (cases nd)
      case (Pair n d)
      with Cons have ge: "0 ≥ d" by auto
      show ?thesis 
      proof (simp only: Pair, unfold poly_ge_def, intro allI impI)
        fix α :: "('v,'a)assign"
        assume pos: "pos_assign α"
        have ge: "0 ≥ eval_monom α n * d"
          using times_right_mono[OF pos_assign_monom[OF pos, of n] ge] by simp
        from rec[unfolded poly_ge_def] pos have ge2: "0 ≥ eval_poly α q" by auto
        show "eval_poly α [] ≥ eval_poly α ((n,d) # q)" using ge_trans[OF plus_left_mono[OF ge] plus_right_mono[OF ge2]]
          by simp
      qed
    qed
  qed
  thus ?case by simp
next
  case (Cons mc p)
  obtain m c where mc: "mc = (m,c)" by (cases mc, auto)
  show ?case
  proof (cases "List.extract (λ mc. fst mc = m) q")
    case None
    with Cons(2) have rec: "check_poly_ge p q" and c: "c ≥ 0" using mc by auto
    from Cons(1)[OF rec] have rec: "p ≥p q" .
    show ?thesis 
    proof (simp only: mc, unfold poly_ge_def, intro allI impI)
      fix α :: "('v,'a)assign"
      assume pos: "pos_assign α"
      have ge: "eval_monom α m * c ≥ 0"
        using times_right_mono[OF pos_assign_monom[OF pos, of m] c] by simp
      from rec have pq: "eval_poly α p ≥ eval_poly α q" unfolding poly_ge_def using pos by auto
      show "eval_poly α ((m,c) # p) ≥ eval_poly α q"
        using ge_trans[OF plus_left_mono[OF ge] plus_right_mono[OF pq]] by simp
    qed
  next
    case (Some res)
    obtain q1 md q2 where "res = (q1,md,q2)" by (cases res, auto)
    with extract_SomeE[OF Some[simplified this]] obtain d where q: "q = q1 @ (m,d) # q2" and res: "res = (q1,(m,d),q2)" 
      by (cases md, auto)
    from Cons(2) Some mc res have rec: "check_poly_ge p (q1 @ q2)" and c: "c ≥ d" by auto
    from Cons(1)[OF rec] have p: "p ≥p q1 @ q2" .
    show ?thesis
    proof (simp only: mc, unfold poly_ge_def, intro allI impI)
      fix α :: "('v,'a)assign"
      assume pos: "pos_assign α"
      have ge: "eval_monom α m * c ≥ eval_monom α m * d"
        using times_right_mono[OF pos_assign_monom[OF pos, of m] c] 
          by simp
      from p have ge2: "eval_poly α p ≥ eval_poly α (q1 @ q2)" unfolding poly_ge_def using pos by auto
      show "eval_poly α ((m,c) # p) ≥ eval_poly α q" using ge_trans[OF plus_left_mono[OF ge] plus_right_mono[OF ge2]]
        by (simp add: q field_simps)
    qed
  qed
qed

declare check_poly_ge.simps[simp del]

definition check_poly_weak_mono_all :: "('v,'a :: ordered_semiring_0)poly ⇒ bool"
where "check_poly_weak_mono_all p ≡ list_all (λ (m,c). c ≥ 0) p"

lemma check_poly_weak_mono_all: fixes p :: "('v :: linorder,'a :: poly_carrier)poly"
  assumes "check_poly_weak_mono_all p" shows  "poly_weak_mono_all p"
unfolding poly_weak_mono_all_def
proof (intro allI impI)
  fix f g :: "('v,'a)assign"
  assume fg: "∀ x. f x ≥ g x"
  and pos: "pos_assign g"
  hence fg: "⋀ x. f x ≥ g x" by auto
  from pos[unfolded pos_assign_def] have g: "⋀ x. g x ≥ 0" ..
  from assms have "⋀ m c. (m,c) ∈ set p ⟹ c ≥ 0" unfolding check_poly_weak_mono_all_def by (auto simp: list_all_iff)
  thus "eval_poly f p ≥ eval_poly g p"
  proof (induct p)
    case Nil thus ?case by (simp add: ge_refl)
  next
    case (Cons mc p)
    hence IH: "eval_poly f p ≥ eval_poly g p" by auto
    show ?case 
    proof (cases mc)
      case (Pair m c)
      with Cons have c: "c ≥ 0" by auto
      show ?thesis unfolding Pair eval_poly.simps fst_conv snd_conv
      proof (rule ge_trans[OF plus_left_mono[OF times_left_mono[OF c]] plus_right_mono[OF IH]])
        show "eval_monom f m ≥ eval_monom g m"
          by (rule eval_monom_mono(1)[OF fg g])
      qed
    qed
  qed
qed

lemma check_poly_weak_mono_all_pos: 
  assumes "check_poly_weak_mono_all p" shows  "p ≥p zero_poly"
unfolding zero_poly_def
proof (rule check_poly_ge)
  from assms have "⋀ m c. (m,c) ∈ set p ⟹ c ≥ 0" unfolding check_poly_weak_mono_all_def by (auto simp: list_all_iff)
  thus "check_poly_ge p []"
    by (induct p, simp add: check_poly_ge.simps,  clarify, auto simp: check_poly_ge.simps extract_Nil_code)
qed


text ‹better check for weak monotonicity for discrete carriers: 
   $p$ is monotone in $v$ if $p(\ldots v+1 \ldots) \geq p(\ldots v \ldots)$›
definition check_poly_weak_mono_discrete :: "('v :: linorder,'a :: poly_carrier)poly ⇒ 'v ⇒ bool"
  where "check_poly_weak_mono_discrete p v ≡ check_poly_ge (poly_subst (λ w. poly_of (if w = v then PSum [PNum 1, PVar v] else PVar w)) p) p"

definition check_poly_weak_mono_and_pos :: "bool ⇒ ('v :: linorder,'a :: poly_carrier)poly ⇒ bool"
  where "check_poly_weak_mono_and_pos discrete p ≡ 
            if discrete then list_all (λ v. check_poly_weak_mono_discrete p v) (poly_vars_list p) ∧ eval_poly (λ w. 0) p ≥  0
                        else check_poly_weak_mono_all p"

definition check_poly_weak_anti_mono_discrete :: "('v :: linorder,'a :: poly_carrier)poly ⇒ 'v ⇒ bool"
  where "check_poly_weak_anti_mono_discrete p v ≡ check_poly_ge p (poly_subst (λ w. poly_of (if w = v then PSum [PNum 1, PVar v] else PVar w)) p)"

context poly_order_carrier
begin

lemma check_poly_weak_mono_discrete: 
  fixes v :: "'v :: linorder" and p :: "('v,'a)poly"
  assumes discrete and check: "check_poly_weak_mono_discrete p v"
  shows "poly_weak_mono p v"
unfolding poly_weak_mono_def 
proof (intro allI impI)
  fix f g :: "('v,'a)assign"
  assume fgw: "∀ w. (v ≠ w ⟶ f w = g w)"
  and gass: "pos_assign g"
  and v: "f v ≥ g v"
  from fgw have w: "⋀ w. v ≠ w ⟹ f w = g w" by auto
  from assms check_poly_ge have ge: "poly_ge (poly_subst (λ w. poly_of (if w = v then PSum [PNum 1, PVar v] else PVar w)) p) p" (is "poly_ge ?p1 p") unfolding check_poly_weak_mono_discrete_def by blast
  from discrete[OF ‹discrete› v] obtain k' where id: "f v = (((+) 1)^^k') (g v)" by auto
  show "eval_poly f p ≥ eval_poly g p"
  proof (cases k')
    case 0
    {
      fix x
      have "f x = g x" using id 0 w by (cases "x = v", auto)
    }
    hence "f = g" ..
    thus ?thesis using ge_refl by simp
  next
    case (Suc k)
    with id have "f v = (((+) 1)^^(Suc k))  (g v)" by simp 
    with w gass show "eval_poly f p ≥ eval_poly g p"
    proof (induct k arbitrary: f g rule: less_induct)
      case (less k)
      show ?case 
      proof (cases k)
        case 0
        with less have id0: "f v = 1 + g v" by simp
        have id1: "eval_poly f p = eval_poly g ?p1"
        proof (rule eval_poly_subst)
          fix w
          show "f w = eval_poly g (poly_of (if w = v then PSum [PNum 1, PVar v] else PVar w))"
          proof (cases "w = v")
            case True
            show ?thesis by (simp add: True id0 zero_poly_def)
          next
            case False
            with less have "f w = g w" by simp
            thus ?thesis by (simp add: False)
          qed
        qed
        have "eval_poly g ?p1 ≥ eval_poly g p" using ge less unfolding poly_ge_def by simp
        with id1 show ?thesis by simp
      next
        case (Suc kk)        
        obtain g' where g': "g' = (λ w. if (w = v) then 1 + g w else g w)" by auto
        have "(1 :: 'a) + g v ≥ 1 + 0" 
          by (rule plus_right_mono, simp add: less(3)[unfolded pos_assign_def])
        also have "1 + (0 :: 'a) = 1" by simp
        also have "… ≥ 0" by (rule one_ge_zero)
        finally have g'pos: "pos_assign g'" using less(3) unfolding pos_assign_def 
          by (simp add: g')
        {
          fix w
          assume "v ≠ w"
          hence "f w = g' w"
            unfolding g' by (simp add: less)
        } note w = this
        have eq: "f v = ((+) (1 :: 'a) ^^ Suc kk) ((g' v))"
          by (simp add: less(4) g' Suc, rule arg_cong[where f = "(+) 1"], induct kk, auto)
        from Suc have kk: "kk < k" by simp
        from less(1)[OF kk w g'pos] eq
        have rec1: "eval_poly f p ≥ eval_poly g' p" by simp
        { 
          fix w
          assume "v ≠ w"
          hence "g' w = g w"
            unfolding g' by simp
        } note w = this
        from Suc have z: "0 < k" by simp
        from less(1)[OF z w less(3)] g'
        have rec2: "eval_poly g' p ≥ eval_poly g p" by simp
        show ?thesis by (rule ge_trans[OF rec1 rec2])
      qed
    qed
  qed
qed

lemma check_poly_weak_anti_mono_discrete: 
  fixes v :: "'v :: linorder" and p :: "('v,'a)poly"
  assumes discrete and check: "check_poly_weak_anti_mono_discrete p v"
  shows "poly_weak_anti_mono p v"
unfolding poly_weak_anti_mono_def 
proof (intro allI impI)
  fix f g :: "('v,'a)assign"
  assume fgw: "∀ w. (v ≠ w ⟶ f w = g w)"
  and gass: "pos_assign g"
  and v: "f v ≥ g v"
  from fgw have w: "⋀ w. v ≠ w ⟹ f w = g w" by auto
  from assms check_poly_ge have ge: "poly_ge p (poly_subst (λ w. poly_of (if w = v then PSum [PNum 1, PVar v] else PVar w)) p)" (is "poly_ge p ?p1") unfolding check_poly_weak_anti_mono_discrete_def by blast
  from discrete[OF ‹discrete› v] obtain k' where id: "f v = (((+) 1)^^k') (g v)" by auto
  show "eval_poly g p ≥ eval_poly f p"
  proof (cases k')
    case 0
    {
      fix x
      have "f x = g x" using id 0 w by (cases "x = v", auto)
    }
    hence "f = g" ..
    thus ?thesis using ge_refl by simp
  next
    case (Suc k)
    with id have "f v = (((+) 1)^^(Suc k))  (g v)" by simp 
    with w gass show "eval_poly g p ≥ eval_poly f p"
    proof (induct k arbitrary: f g rule: less_induct)
      case (less k)
      show ?case 
      proof (cases k)
        case 0
        with less have id0: "f v = 1 + g v" by simp
        have id1: "eval_poly f p = eval_poly g ?p1"
        proof (rule eval_poly_subst)
          fix w
          show "f w = eval_poly g (poly_of (if w = v then PSum [PNum 1, PVar v] else PVar w))"
          proof (cases "w = v")
            case True
            show ?thesis by (simp add: True id0 zero_poly_def)
          next
            case False
            with less have "f w = g w" by simp
            thus ?thesis by (simp add: False)
          qed
        qed
        have "eval_poly g p ≥ eval_poly g ?p1" using ge less unfolding poly_ge_def by simp
        with id1 show ?thesis by simp
      next
        case (Suc kk)        
        obtain g' where g': "g' = (λ w. if (w = v) then 1 + g w else g w)" by auto
        have "(1 :: 'a) + g v ≥ 1 + 0" 
          by (rule plus_right_mono, simp add: less(3)[unfolded pos_assign_def])
        also have "(1 :: 'a) + 0 = 1" by simp
        also have "… ≥ 0" by (rule one_ge_zero)
        finally have g'pos: "pos_assign g'" using less(3) unfolding pos_assign_def 
          by (simp add: g')
        {
          fix w
          assume "v ≠ w"
          hence "f w = g' w"
            unfolding g' by (simp add: less)
        } note w = this
        have eq: "f v = ((+) (1 :: 'a) ^^ Suc kk) ((g' v))"
          by (simp add: less(4) g' Suc, rule arg_cong[where f = "(+) 1"], induct kk, auto)
        from Suc have kk: "kk < k" by simp
        from less(1)[OF kk w g'pos] eq
        have rec1: "eval_poly g' p ≥ eval_poly f p" by simp
        { 
          fix w
          assume "v ≠ w"
          hence "g' w = g w"
            unfolding g' by simp
        } note w = this
        from Suc have z: "0 < k" by simp
        from less(1)[OF z w less(3)] g'
        have rec2: "eval_poly g p ≥ eval_poly g' p" by simp
        show ?thesis by (rule ge_trans[OF rec2 rec1])
      qed
    qed
  qed
qed

lemma check_poly_weak_mono_and_pos: 
  fixes p :: "('v :: linorder,'a)poly"
  assumes "check_poly_weak_mono_and_pos discrete p"
  shows "poly_weak_mono_all p ∧ (p ≥p zero_poly)"
proof (cases discrete)
  case False
  with assms have c: "check_poly_weak_mono_all p" unfolding check_poly_weak_mono_and_pos_def
    by auto
  from check_poly_weak_mono_all[OF c] check_poly_weak_mono_all_pos[OF c] show ?thesis by auto
next
  case True
  with assms have c: "list_all (λ v. check_poly_weak_mono_discrete p v) (poly_vars_list p)" and g: "eval_poly (λ w. 0) p ≥ 0"
    unfolding check_poly_weak_mono_and_pos_def by auto
  have m: "poly_weak_mono_all p"
  proof (rule poly_weak_mono)
    fix v :: 'v
    assume v: "v ∈ poly_vars p"
    show "poly_weak_mono p v"
      by (rule check_poly_weak_mono_discrete[OF True], insert c[unfolded list_all_iff] v, auto)  
  qed
  have m': "poly_weak_mono_all  p"
  proof (rule poly_weak_mono)
    fix v :: 'v
    assume v: "v ∈ poly_vars p"
    show "poly_weak_mono p v"
      by (rule check_poly_weak_mono_discrete[OF True], insert c[unfolded list_all_iff] v, auto) 
  qed
  from poly_weak_mono_all_pos[OF g m'] m show ?thesis by auto
qed

end

definition check_poly_weak_mono :: "('v :: linorder,'a :: ordered_semiring_0)poly ⇒ 'v ⇒ bool"
  where "check_poly_weak_mono p v ≡ list_all (λ (m,c). c ≥ 0 ∨ v ∉ monom_vars m) p"

lemma check_poly_weak_mono: fixes p :: "('v :: linorder,'a :: poly_carrier)poly"
  assumes "check_poly_weak_mono p v" shows  "poly_weak_mono p v"
unfolding poly_weak_mono_def
proof (intro allI impI)
  fix f g :: "('v,'a)assign"
  assume "∀ x. v ≠ x ⟶ f x = g x"
  and pos: "pos_assign g" 
  and ge: "f v ≥ g v"
  hence fg: "⋀ x. v ≠ x ⟹ f x = g x" by auto
  from pos[unfolded pos_assign_def] have g: "⋀ x. g x ≥ 0" ..
  from assms have "⋀ m c. (m,c) ∈ set p ⟹ c ≥ 0 ∨ v ∉ monom_vars m" unfolding check_poly_weak_mono_def by (auto simp: list_all_iff)
  thus "eval_poly f p ≥ eval_poly g p"
  proof (induct p)
    case (Cons mc p)
    hence IH: "eval_poly f p ≥ eval_poly g p" by auto
    obtain m c where mc: "mc = (m,c)" by force
    with Cons have c: "c ≥ 0 ∨ v ∉ monom_vars m" by auto
    show ?case unfolding mc eval_poly.simps fst_conv snd_conv
    proof (rule ge_trans[OF plus_left_mono plus_right_mono[OF IH]])
      from c show "eval_monom f m * c ≥ eval_monom g m * c"
      proof
        assume c: "c ≥ 0"
        show ?thesis
        proof (rule times_left_mono[OF c], rule eval_monom_mono(1)[OF _ g])
          fix x
          show "f x ≥ g x" using ge fg[of x] by (cases "x = v", auto simp: ge_refl)
        qed
      next
        assume v: "v ∉ monom_vars m"
        have "eval_monom f m = eval_monom g m"
          by (rule monom_vars_eval_monom, insert fg v, fast)
        thus ?thesis by (simp add: ge_refl)        
      qed
    qed
  qed (simp add: ge_refl)
qed

definition check_poly_weak_mono_smart :: "bool ⇒ ('v :: linorder,'a :: poly_carrier)poly ⇒ 'v ⇒ bool"
  where "check_poly_weak_mono_smart discrete ≡ if discrete then check_poly_weak_mono_discrete else check_poly_weak_mono"

lemma (in poly_order_carrier) check_poly_weak_mono_smart: fixes p :: "('v :: linorder,'a :: poly_carrier)poly"
  shows "check_poly_weak_mono_smart discrete p v ⟹ poly_weak_mono p v"
  unfolding check_poly_weak_mono_smart_def
  using check_poly_weak_mono check_poly_weak_mono_discrete by (cases discrete, auto)

definition check_poly_weak_anti_mono :: "('v :: linorder,'a :: ordered_semiring_0)poly ⇒ 'v ⇒ bool"
  where "check_poly_weak_anti_mono p v ≡ list_all (λ (m,c). 0 ≥ c ∨ v ∉ monom_vars m) p"

lemma check_poly_weak_anti_mono: fixes p :: "('v :: linorder,'a :: poly_carrier)poly"
  assumes "check_poly_weak_anti_mono p v" shows  "poly_weak_anti_mono p v"
unfolding poly_weak_anti_mono_def
proof (intro allI impI)
  fix f g :: "('v,'a)assign"
  assume "∀ x. v ≠ x ⟶ f x = g x"
  and pos: "pos_assign g" 
  and ge: "f v ≥ g v"
  hence fg: "⋀ x. v ≠ x ⟹ f x = g x" by auto
  from pos[unfolded pos_assign_def] have g: "⋀ x. g x ≥ 0" ..
  from assms have "⋀ m c. (m,c) ∈ set p ⟹ 0 ≥ c ∨ v ∉ monom_vars m" unfolding check_poly_weak_anti_mono_def by (auto simp: list_all_iff)
  thus "eval_poly g p ≥ eval_poly f p"
  proof (induct p)
    case Nil thus ?case by (simp add: ge_refl)
  next
    case (Cons mc p)
    hence IH: "eval_poly g p ≥ eval_poly f p" by auto
    obtain m c where mc: "mc = (m,c)" by force
    with Cons have c: "0 ≥ c ∨ v ∉ monom_vars m" by auto
    show ?case unfolding mc eval_poly.simps fst_conv snd_conv
    proof (rule ge_trans[OF plus_left_mono plus_right_mono[OF IH]])
      from c show "eval_monom g m * c ≥ eval_monom f m * c"
      proof
        assume c: "0 ≥ c"
        show ?thesis
        proof (rule times_left_anti_mono[OF eval_monom_mono(1)[OF _ g] c])
          fix x
          show "f x ≥ g x" using ge fg[of x] by (cases "x = v", auto simp: ge_refl)
        qed
      next
        assume v: "v ∉ monom_vars m"
        have "eval_monom f m = eval_monom g m"
          by (rule monom_vars_eval_monom, insert fg v, fast)
        thus ?thesis by (simp add: ge_refl)        
      qed
    qed
  qed
qed

definition check_poly_weak_anti_mono_smart :: "bool ⇒ ('v :: linorder,'a :: poly_carrier)poly ⇒ 'v ⇒ bool"
  where "check_poly_weak_anti_mono_smart discrete ≡ if discrete then check_poly_weak_anti_mono_discrete else check_poly_weak_anti_mono"

lemma (in poly_order_carrier) check_poly_weak_anti_mono_smart: fixes p :: "('v :: linorder,'a :: poly_carrier)poly"
  shows "check_poly_weak_anti_mono_smart discrete p v ⟹ poly_weak_anti_mono p v"
  unfolding check_poly_weak_anti_mono_smart_def
  using check_poly_weak_anti_mono[of p v] check_poly_weak_anti_mono_discrete[of p v] 
  by (cases discrete, auto)

definition check_poly_gt :: "('a ⇒ 'a ⇒ bool) ⇒ ('v :: linorder,'a :: ordered_semiring_0)poly ⇒ ('v,'a)poly ⇒ bool"
where "check_poly_gt gt p q ≡ let (a1,p1) = poly_split 1 p; (b1,q1) = poly_split 1 q in gt a1 b1 ∧ check_poly_ge p1 q1"

fun univariate_power_list :: "'v ⇒ 'v monom_list ⇒ nat option" where
  "univariate_power_list x [(y,n)] = (if x = y then Some n else None)" 
| "univariate_power_list _ _ = None" 

lemma univariate_power_list: assumes "monom_inv m" "univariate_power_list x m = Some n" 
  shows "sum_var_list m = (λ y. if x = y then n else 0)" 
   "eval_monom_list α m = ((α x)^n)" 
   "n ≥ 1" 
proof -
  have m: "m = [(x,n)]" using assms
    by (induct x m rule: univariate_power_list.induct, auto split: if_splits)
  show "eval_monom_list α m = ((α x)^n)" "sum_var_list m = (λ y. if x = y then n else 0)"
    "n ≥ 1" using assms(1)
    unfolding m monom_inv_def by (auto simp: sum_var_list_def)
qed

lift_definition univariate_power :: "'v :: linorder ⇒ 'v monom ⇒ nat option" 
  is univariate_power_list .

lemma univariate_power: assumes "univariate_power x m = Some n" 
  shows "sum_var m = (λ y. if x = y then n else 0)" 
   "eval_monom α m = ((α x)^n)"
   "n ≥ 1" 
  by (atomize(full), insert assms, transfer, auto dest: univariate_power_list)

lemma univariate_power_var_monom: "univariate_power y (var_monom x) = (if x = y then Some 1 else None)"
  by (transfer, auto)

definition check_monom_strict_mono :: "bool ⇒ 'v :: linorder monom ⇒ 'v ⇒ bool" where
  "check_monom_strict_mono pm m v ≡ case univariate_power v m of
     Some p ⇒ pm ∨ p = 1
   | None ⇒ False"

definition check_poly_strict_mono :: "bool ⇒ ('v :: linorder, 'a :: poly_carrier)poly ⇒ 'v ⇒ bool"
  where "check_poly_strict_mono pm p v ≡ list_ex (λ (m,c). (c ≥ 1) ∧ check_monom_strict_mono pm m v) p"

definition check_poly_strict_mono_discrete :: "('a :: poly_carrier ⇒ 'a ⇒ bool) ⇒ ('v :: linorder,'a)poly ⇒ 'v ⇒ bool"
  where "check_poly_strict_mono_discrete gt p v ≡ check_poly_gt gt (poly_subst (λ w. poly_of (if w = v then PSum [PNum 1, PVar v] else PVar w)) p) p "

definition check_poly_strict_mono_smart :: "bool ⇒ bool ⇒ ('a :: poly_carrier ⇒ 'a ⇒ bool) ⇒ ('v :: linorder,'a)poly ⇒ 'v ⇒ bool"
  where "check_poly_strict_mono_smart discrete pm gt p v ≡ 
            if discrete then check_poly_strict_mono_discrete gt p v else check_poly_strict_mono pm p v"

context poly_order_carrier
begin
lemma check_monom_strict_mono: fixes α β :: "('v :: linorder,'a)assign" and v :: 'v and m :: "'v monom"
  assumes check: "check_monom_strict_mono power_mono m v"
  and gt: "α v ≻ β v"
  and ge: "β v ≥ 0"
shows "eval_monom α m ≻ eval_monom β m"
proof -
  from check[unfolded check_monom_strict_mono_def] obtain n where
    uni: "univariate_power v m = Some n" and 1: "¬ power_mono ⟹ n = 1" 
    by (auto split: option.splits)
  from univariate_power[OF uni] 
  have n1: "n ≥ 1" and eval: "eval_monom a m = a v ^ n" for a :: "('v,'a)assign"
    by auto
  show ?thesis
  proof (cases power_mono)
    case False
    with gt 1[OF this] show ?thesis unfolding eval by auto
  next
    case True
    from power_mono[OF True gt ge n1] show ?thesis unfolding eval .
  qed
qed

lemma check_poly_strict_mono: 
  assumes check1: "check_poly_strict_mono power_mono p v"
  and check2: "check_poly_weak_mono_all p"
  shows "poly_strict_mono p v"
unfolding poly_strict_mono_def
proof (intro allI impI)
  fix f g :: "('b,'a)assign"
  assume fgw: "∀ w. (v ≠ w ⟶ f w = g w)"
  and pos: "pos_assign g"
  and fgv: "f v ≻ g v"
  from pos[unfolded pos_assign_def] have g: "⋀ x. g x ≥ 0" ..
  {
    fix w
    have "f w ≥ g w"
    proof (cases "v = w")
      case False
      with fgw ge_refl show ?thesis by auto
    next
      case True
      from fgv[unfolded True] show ?thesis by (rule gt_imp_ge)
    qed
  } note fgw2 = this
  let ?e = "eval_poly"
  show "?e f p ≻ ?e g p"
    using check1[unfolded check_poly_strict_mono_def, simplified list_ex_iff]
      check2[unfolded check_poly_weak_mono_all_def, simplified list_all_iff, THEN bspec]
  proof (induct p)
    case Nil thus ?case by simp
  next
    case (Cons mc p)
    obtain m c where mc: "mc = (m,c)" by (cases mc, auto)
    show ?case 
    proof (cases "c ≥ 1 ∧ check_monom_strict_mono power_mono m v")
      case True
      hence c: "c ≥ 1" and m: "check_monom_strict_mono power_mono m v" by blast+
      from times_gt_mono[OF check_monom_strict_mono[OF m, of f g, OF fgv g] c]
      have gt: "eval_monom f m * c ≻ eval_monom g m * c" .
      from Cons(3) have "check_poly_weak_mono_all p" unfolding check_poly_weak_mono_all_def list_all_iff by auto
      from check_poly_weak_mono_all[OF this, unfolded poly_weak_mono_all_def, rule_format, OF fgw2 pos]
      have ge: "?e f p ≥ ?e g p" .
      from compat2[OF plus_gt_left_mono[OF gt] plus_right_mono[OF ge]]
      show ?thesis unfolding mc by simp
    next
      case False
      with Cons(2) mc have "∃ mc ∈ set p. (λ (m,c). c ≥ 1 ∧ check_monom_strict_mono power_mono m v) mc" by auto
      from Cons(1)[OF this] Cons(3) have rec: "?e f p ≻ ?e g p" by simp
      from Cons(3) mc have c: "c ≥ 0" by auto
      from times_left_mono[OF c eval_monom_mono(1)[OF fgw2 g]] 
      have ge: "eval_monom f m * c ≥ eval_monom g m * c" .
      from compat2[OF plus_gt_left_mono[OF rec] plus_right_mono[OF ge]]
      show ?thesis by (simp add: mc field_simps)
    qed
  qed 
qed     
      

lemma check_poly_gt: 
  fixes p :: "('v :: linorder,'a)poly"
  assumes "check_poly_gt gt p q" shows "p >p q"
proof -
  obtain a1 p1 where p: "poly_split 1 p = (a1,p1)" by force
  obtain b1 q1 where q: "poly_split 1 q = (b1,q1)" by force
  from p q assms have gt: "a1 ≻ b1" and ge: "p1 ≥p q1" unfolding check_poly_gt_def using check_poly_ge[of p1 q1]  by auto
  show ?thesis
  proof (unfold poly_gt_def, intro impI allI)
    fix α :: "('v,'a)assign"
    assume "pos_assign α"
    with ge have ge: "eval_poly α p1 ≥ eval_poly α q1" unfolding poly_ge_def by simp
    from plus_gt_left_mono[OF gt] compat[OF plus_left_mono[OF ge]] have gt: "a1 + eval_poly α p1 ≻ b1 + eval_poly α q1" by (force simp: field_simps)
    show "eval_poly α p ≻ eval_poly α q"
      by (simp add: poly_split[OF p, unfolded eq_poly_def] poly_split[OF q, unfolded eq_poly_def] gt)
  qed
qed

lemma check_poly_strict_mono_discrete: 
  fixes v :: "'v :: linorder" and p :: "('v,'a)poly"
  assumes discrete and check: "check_poly_strict_mono_discrete gt p v"
  shows "poly_strict_mono p v"
unfolding poly_strict_mono_def 
proof (intro allI impI)
  fix f g :: "('v,'a)assign"
  assume fgw: "∀ w. (v ≠ w ⟶ f w = g w)"
  and gass: "pos_assign g"
  and v: "f v ≻ g v"
  from gass have g: "⋀ x. g x ≥ 0" unfolding pos_assign_def ..
  from fgw have w: "⋀ w. v ≠ w ⟹ f w = g w" by auto
  from assms check_poly_gt have gt: "poly_gt (poly_subst (λ w. poly_of (if w = v then PSum [PNum 1, PVar v] else PVar w)) p) p" (is "poly_gt ?p1 p") unfolding check_poly_strict_mono_discrete_def by blast
  from discrete[OF ‹discrete› gt_imp_ge[OF v]] obtain k' where id: "f v = (((+) 1)^^k') (g v)" by auto
  {
    assume "k' = 0"
    from v[unfolded id this] have "g v ≻ g v" by simp
    hence False using SN g[of v] unfolding SN_defs by auto
  }
  with id obtain k where id: "f v = (((+) 1)^^(Suc k)) (g v)" by (cases k', auto)
  with w gass
  show "eval_poly f p ≻ eval_poly g p"
  proof (induct k arbitrary: f g rule: less_induct)
    case (less k)
    show ?case
    proof (cases k)
      case 0
      with less(4) have id0: "f v = 1 + g v" by simp
      have id1: "eval_poly f p = eval_poly g ?p1"
      proof (rule eval_poly_subst)
        fix w
        show "f w = eval_poly g (poly_of (if w = v then PSum [PNum 1, PVar v] else PVar w))"
        proof (cases "w = v")
          case True
          show ?thesis by (simp add: True id0 zero_poly_def)
        next
          case False
          with less have "f w = g w" by simp
          thus ?thesis by (simp add: False)
        qed
      qed
      have "eval_poly g ?p1 ≻ eval_poly g p" using gt less unfolding poly_gt_def by simp
      with id1 show ?thesis by simp
    next
      case (Suc kk)        
      obtain g' where g': "g' = (λ w. if (w = v) then 1 + g w else g w)" by auto
      have "(1 :: 'a) + g v ≥ 1 + 0" 
        by (rule plus_right_mono, simp add: less(3)[unfolded pos_assign_def])
      also have "(1 :: 'a) + 0 = 1" by simp
      also have "… ≥ 0" by (rule one_ge_zero)
      finally have g'pos: "pos_assign g'" using less(3) unfolding pos_assign_def 
        by (simp add: g')
      {
        fix w
        assume "v ≠ w"
        hence "f w = g' w"
          unfolding g' by (simp add: less)
      } note w = this
      have eq: "f v = ((+) (1 :: 'a) ^^ Suc kk) ((g' v))"
        by (simp add: less(4) g' Suc, rule arg_cong[where f = "(+) 1"], induct kk, auto)
      from Suc have kk: "kk < k" by simp
      from less(1)[OF kk w g'pos] eq
      have rec1: "eval_poly f p ≻ eval_poly g' p" by simp
      { 
        fix w
        assume "v ≠ w"
        hence "g' w = g w"
          unfolding g' by simp
      } note w = this
      from Suc have z: "0 < k" by simp
      from less(1)[OF z w less(3)] g'
      have rec2: "eval_poly g' p ≻ eval_poly g p" by simp
      show ?thesis by (rule gt_trans[OF rec1 rec2])
    qed
  qed
qed

lemma check_poly_strict_mono_smart: 
  assumes check1: "check_poly_strict_mono_smart discrete power_mono gt p v"
  and check2: "check_poly_weak_mono_and_pos discrete p"
  shows "poly_strict_mono p v"
proof (cases discrete)
  case True
  with check1[unfolded check_poly_strict_mono_smart_def]
    check_poly_strict_mono_discrete[OF True]
  show ?thesis by auto
next
  case False
  from check_poly_strict_mono[OF check1[unfolded check_poly_strict_mono_smart_def, simplified False, simplified]]
    check2[unfolded check_poly_weak_mono_and_pos_def, simplified False, simplified]
  show ?thesis by auto
qed

end

end