Theory Reflected_Multivariate_Polynomial

theory Reflected_Multivariate_Polynomial
imports Rat_Pair Polynomial_List
```(*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
Author:     Amine Chaieb
*)

section ‹Implementation and verification of multivariate polynomials›

theory Reflected_Multivariate_Polynomial
imports Complex_Main Rat_Pair Polynomial_List
begin

subsection ‹Datatype of polynomial expressions›

datatype poly = C Num | Bound nat | Add poly poly | Sub poly poly
| Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly

abbreviation poly_0 :: "poly" ("0⇩p") where "0⇩p ≡ C (0⇩N)"
abbreviation poly_p :: "int ⇒ poly" ("'((_)')⇩p") where "(i)⇩p ≡ C (i)⇩N"

subsection‹Boundedness, substitution and all that›

primrec polysize:: "poly ⇒ nat"
where
"polysize (C c) = 1"
| "polysize (Bound n) = 1"
| "polysize (Neg p) = 1 + polysize p"
| "polysize (Add p q) = 1 + polysize p + polysize q"
| "polysize (Sub p q) = 1 + polysize p + polysize q"
| "polysize (Mul p q) = 1 + polysize p + polysize q"
| "polysize (Pw p n) = 1 + polysize p"
| "polysize (CN c n p) = 4 + polysize c + polysize p"

primrec polybound0:: "poly ⇒ bool" ― ‹a poly is INDEPENDENT of Bound 0›
where
"polybound0 (C c) ⟷ True"
| "polybound0 (Bound n) ⟷ n > 0"
| "polybound0 (Neg a) ⟷ polybound0 a"
| "polybound0 (Add a b) ⟷ polybound0 a ∧ polybound0 b"
| "polybound0 (Sub a b) ⟷ polybound0 a ∧ polybound0 b"
| "polybound0 (Mul a b) ⟷ polybound0 a ∧ polybound0 b"
| "polybound0 (Pw p n) ⟷ polybound0 p"
| "polybound0 (CN c n p) ⟷ n ≠ 0 ∧ polybound0 c ∧ polybound0 p"

primrec polysubst0:: "poly ⇒ poly ⇒ poly" ― ‹substitute a poly into a poly for Bound 0›
where
"polysubst0 t (C c) = C c"
| "polysubst0 t (Bound n) = (if n = 0 then t else Bound n)"
| "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
| "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
| "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
| "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
| "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
| "polysubst0 t (CN c n p) =
(if n = 0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
else CN (polysubst0 t c) n (polysubst0 t p))"

fun decrpoly:: "poly ⇒ poly"
where
"decrpoly (Bound n) = Bound (n - 1)"
| "decrpoly (Neg a) = Neg (decrpoly a)"
| "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
| "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
| "decrpoly (Pw p n) = Pw (decrpoly p) n"
| "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
| "decrpoly a = a"

subsection ‹Degrees and heads and coefficients›

fun degree :: "poly ⇒ nat"
where
"degree (CN c 0 p) = 1 + degree p"
| "degree p = 0"

fun head :: "poly ⇒ poly"
where

text ‹More general notions of degree and head.›

fun degreen :: "poly ⇒ nat ⇒ nat"
where
"degreen (CN c n p) = (λm. if n = m then 1 + degreen p n else 0)"
| "degreen p = (λm. 0)"

fun headn :: "poly ⇒ nat ⇒ poly"
where
"headn (CN c n p) = (λm. if n ≤ m then headn p m else CN c n p)"
| "headn p = (λm. p)"

fun coefficients :: "poly ⇒ poly list"
where
"coefficients (CN c 0 p) = c # coefficients p"
| "coefficients p = [p]"

fun isconstant :: "poly ⇒ bool"
where
"isconstant (CN c 0 p) = False"
| "isconstant p = True"

fun behead :: "poly ⇒ poly"
where
"behead (CN c 0 p) = (let p' = behead p in if p' = 0⇩p then c else CN c 0 p')"

fun headconst :: "poly ⇒ Num"
where
| "headconst (C n) = n"

subsection ‹Operations for normalization›

declare if_cong[fundef_cong del]
declare let_cong[fundef_cong del]

fun polyadd :: "poly ⇒ poly ⇒ poly"  (infixl "+⇩p" 60)
where
"polyadd (C c) (C c') = C (c +⇩N c')"
| "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
| "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
| "polyadd (CN c n p) (CN c' n' p') =
(if n < n' then CN (polyadd c (CN c' n' p')) n p
else if n' < n then CN (polyadd (CN c n p) c') n' p'
else
let
in if pp' = 0⇩p then cc' else CN cc' n pp')"

fun polyneg :: "poly ⇒ poly" ("~⇩p")
where
"polyneg (C c) = C (~⇩N c)"
| "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
| "polyneg a = Neg a"

definition polysub :: "poly ⇒ poly ⇒ poly"  (infixl "-⇩p" 60)
where "p -⇩p q = polyadd p (polyneg q)"

fun polymul :: "poly ⇒ poly ⇒ poly"  (infixl "*⇩p" 60)
where
"polymul (C c) (C c') = C (c *⇩N c')"
| "polymul (C c) (CN c' n' p') =
(if c = 0⇩N then 0⇩p else CN (polymul (C c) c') n' (polymul (C c) p'))"
| "polymul (CN c n p) (C c') =
(if c' = 0⇩N  then 0⇩p else CN (polymul c (C c')) n (polymul p (C c')))"
| "polymul (CN c n p) (CN c' n' p') =
(if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
else if n' < n then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
else polyadd (polymul (CN c n p) c') (CN 0⇩p n' (polymul (CN c n p) p')))"
| "polymul a b = Mul a b"

declare if_cong[fundef_cong]
declare let_cong[fundef_cong]

fun polypow :: "nat ⇒ poly ⇒ poly"
where
"polypow 0 = (λp. (1)⇩p)"
| "polypow n =
(λp.
let
q = polypow (n div 2) p;
d = polymul q q
in if even n then d else polymul p d)"

abbreviation poly_pow :: "poly ⇒ nat ⇒ poly"  (infixl "^⇩p" 60)
where "a ^⇩p k ≡ polypow k a"

function polynate :: "poly ⇒ poly"
where
"polynate (Bound n) = CN 0⇩p n (1)⇩p"
| "polynate (Add p q) = polynate p +⇩p polynate q"
| "polynate (Sub p q) = polynate p -⇩p polynate q"
| "polynate (Mul p q) = polynate p *⇩p polynate q"
| "polynate (Neg p) = ~⇩p (polynate p)"
| "polynate (Pw p n) = polynate p ^⇩p n"
| "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
| "polynate (C c) = C (normNum c)"
by pat_completeness auto
termination by (relation "measure polysize") auto

fun poly_cmul :: "Num ⇒ poly ⇒ poly"
where
"poly_cmul y (C x) = C (y *⇩N x)"
| "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
| "poly_cmul y p = C y *⇩p p"

definition monic :: "poly ⇒ poly × bool"
where "monic p =
in if h = 0⇩N then (p, False) else (C (Ninv h) *⇩p p, 0>⇩N h))"

subsection ‹Pseudo-division›

definition shift1 :: "poly ⇒ poly"
where "shift1 p = CN 0⇩p 0 p"

abbreviation funpow :: "nat ⇒ ('a ⇒ 'a) ⇒ 'a ⇒ 'a"
where "funpow ≡ compow"

partial_function (tailrec) polydivide_aux :: "poly ⇒ nat ⇒ poly ⇒ nat ⇒ poly ⇒ nat × poly"
where
"polydivide_aux a n p k s =
(if s = 0⇩p then (k, s)
else
let
m = degree s
in
if m < n then (k,s)
else
let p' = funpow (m - n) shift1 p
in
if a = b then polydivide_aux a n p k (s -⇩p p')
else polydivide_aux a n p (Suc k) ((a *⇩p s) -⇩p (b *⇩p p')))"

definition polydivide :: "poly ⇒ poly ⇒ nat × poly"
where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s"

fun poly_deriv_aux :: "nat ⇒ poly ⇒ poly"
where
"poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)⇩N) c) 0 (poly_deriv_aux (n + 1) p)"
| "poly_deriv_aux n p = poly_cmul ((int n)⇩N) p"

fun poly_deriv :: "poly ⇒ poly"
where
"poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
| "poly_deriv p = 0⇩p"

subsection ‹Semantics of the polynomial representation›

primrec Ipoly :: "'a list ⇒ poly ⇒ 'a::{field_char_0,power}"
where
"Ipoly bs (C c) = INum c"
| "Ipoly bs (Bound n) = bs!n"
| "Ipoly bs (Neg a) = - Ipoly bs a"
| "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
| "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
| "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
| "Ipoly bs (Pw t n) = Ipoly bs t ^ n"
| "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p"

abbreviation Ipoly_syntax :: "poly ⇒ 'a list ⇒'a::{field_char_0,power}"  ("⦇_⦈⇩p⇗_⇖")
where "⦇p⦈⇩p⇗bs⇖ ≡ Ipoly bs p"

lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i"

lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"

lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat

subsection ‹Normal form and normalization›

fun isnpolyh:: "poly ⇒ nat ⇒ bool"
where
"isnpolyh (C c) = (λk. isnormNum c)"
| "isnpolyh (CN c n p) = (λk. n ≥ k ∧ isnpolyh c (Suc n) ∧ isnpolyh p n ∧ p ≠ 0⇩p)"
| "isnpolyh p = (λk. False)"

lemma isnpolyh_mono: "n' ≤ n ⟹ isnpolyh p n ⟹ isnpolyh p n'"
by (induct p rule: isnpolyh.induct) auto

definition isnpoly :: "poly ⇒ bool"
where "isnpoly p = isnpolyh p 0"

lemma polyadd_normh: "isnpolyh p n0 ⟹ isnpolyh q n1 ⟹ isnpolyh (polyadd p q) (min n0 n1)"
proof (induct p q arbitrary: n0 n1 rule: polyadd.induct)
case (2 ab c' n' p' n0 n1)
from 2 have  th1: "isnpolyh (C ab) (Suc n')"
by simp
from 2(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' ≥ n1"
by simp_all
with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
by simp
with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +⇩p c') (Suc n')"
by simp
from nplen1 have n01len1: "min n0 n1 ≤ n'"
by simp
then show ?case using 2 th3
by simp
next
case (3 c' n' p' ab n1 n0)
from 3 have  th1: "isnpolyh (C ab) (Suc n')"
by simp
from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' ≥ n1"
by simp_all
with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
by simp
with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +⇩p C ab) (Suc n')"
by simp
from nplen1 have n01len1: "min n0 n1 ≤ n'"
by simp
then show ?case using 3 th3
by simp
next
case (4 c n p c' n' p' n0 n1)
then have nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n"
by simp_all
from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'"
by simp_all
from 4 have ngen0: "n ≥ n0"
by simp
from 4 have n'gen1: "n' ≥ n1"
by simp
consider (eq) "n = n'" | (lt) "n < n'" | (gt) "n > n'"
by arith
then show ?case
proof cases
case eq
with "4.hyps"(3)[OF nc nc']
have ncc':"isnpolyh (c +⇩p c') (Suc n)"
by auto
then have ncc'n01: "isnpolyh (c +⇩p c') (min n0 n1)"
using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1
by auto
from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +⇩p p') n"
by simp
have minle: "min n0 n1 ≤ n'"
using ngen0 n'gen1 eq by simp
from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' show ?thesis
next
case lt
have "min n0 n1 ≤ n0"
by simp
with 4 lt have th1:"min n0 n1 ≤ n"
by auto
from 4 have th21: "isnpolyh c (Suc n)"
by simp
from 4 have th22: "isnpolyh (CN c' n' p') n'"
by simp
from lt have th23: "min (Suc n) n' = Suc n"
by arith
from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
using th23 by simp
with 4 lt th1 show ?thesis
by simp
next
case gt
then have gt': "n' < n ∧ ¬ n < n'"
by simp
have "min n0 n1 ≤ n1"
by simp
with 4 gt have th1: "min n0 n1 ≤ n'"
by auto
from 4 have th21: "isnpolyh c' (Suc n')"
by simp_all
from 4 have th22: "isnpolyh (CN c n p) n"
by simp
from gt have th23: "min n (Suc n') = Suc n'"
by arith
from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
using th23 by simp
with 4 gt th1 show ?thesis
by simp
qed
qed auto

lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
by (induct p q rule: polyadd.induct)
(auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left_NO_MATCH)

lemma polyadd_norm: "isnpoly p ⟹ isnpoly q ⟹ isnpoly (polyadd p q)"
using polyadd_normh[of p 0 q 0] isnpoly_def by simp

text ‹The degree of addition and other general lemmas needed for the normal form of polymul.›

assumes "isnpolyh p n0"
and "isnpolyh q n1"
and "degreen p m ≠ degreen q m"
and "m ≤ min n0 n1"
shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
using assms
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
case (4 c n p c' n' p' m n0 n1)
show ?case
proof (cases "n = n'")
case True
with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
show ?thesis by (auto simp: Let_def)
next
case False
with 4 show ?thesis by auto
qed
qed auto

lemma headnz[simp]: "isnpolyh p n ⟹ p ≠ 0⇩p ⟹ headn p m ≠ 0⇩p"
by (induct p arbitrary: n rule: headn.induct) auto

lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) ⟹ degree p = 0"
by (induct p arbitrary: n rule: degree.induct) auto

lemma degreen_0[simp]: "isnpolyh p n ⟹ m < n ⟹ degreen p m = 0"
by (induct p arbitrary: n rule: degreen.induct) auto

lemma degree_isnpolyh_Suc': "n > 0 ⟹ isnpolyh p n ⟹ degree p = 0"
by (induct p arbitrary: n rule: degree.induct) auto

lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 ⟹ degree c = 0"
using degree_isnpolyh_Suc by auto

lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 ⟹ degreen c n = 0"
using degreen_0 by auto

assumes np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
and m: "m ≤ max n0 n1"
shows "degreen (p +⇩p q) m ≤ max (degreen p m) (degreen q m)"
using np nq m
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
case (2 c c' n' p' n0 n1)
then show ?case
by (cases n') simp_all
next
case (3 c n p c' n0 n1)
then show ?case
by (cases n) auto
next
case (4 c n p c' n' p' n0 n1 m)
show ?case
proof (cases "n = n'")
case True
with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
show ?thesis by (auto simp: Let_def)
next
case False
then show ?thesis by simp
qed
qed auto

assumes "isnpolyh p n0"
and "isnpolyh q n1"
and "polyadd p q = C c"
shows "degreen p m = degreen q m"
using assms
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
case (4 c n p c' n' p' m n0 n1 x)
consider "n = n'" | "n > n' ∨ n < n'" by arith
then show ?case
proof cases
case 1
with 4 show ?thesis
by (cases "p +⇩p p' = 0⇩p") (auto simp add: Let_def)
next
case 2
with 4 show ?thesis by auto
qed
qed simp_all

lemma polymul_properties:
assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
and m: "m ≤ min n0 n1"
shows "isnpolyh (p *⇩p q) (min n0 n1)"
and "p *⇩p q = 0⇩p ⟷ p = 0⇩p ∨ q = 0⇩p"
and "degreen (p *⇩p q) m = (if p = 0⇩p ∨ q = 0⇩p then 0 else degreen p m + degreen q m)"
using np nq m
proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
case (2 c c' n' p')
{
case (1 n0 n1)
with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
show ?case by (auto simp add: min_def)
next
case (2 n0 n1)
then show ?case by auto
next
case (3 n0 n1)
then show ?case using "2.hyps" by auto
}
next
case (3 c n p c')
{
case (1 n0 n1)
with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
show ?case by (auto simp add: min_def)
next
case (2 n0 n1)
then show ?case by auto
next
case (3 n0 n1)
then show ?case  using "3.hyps" by auto
}
next
case (4 c n p c' n' p')
let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
{
case (1 n0 n1)
then have cnp: "isnpolyh ?cnp n"
and cnp': "isnpolyh ?cnp' n'"
and np: "isnpolyh p n"
and nc: "isnpolyh c (Suc n)"
and np': "isnpolyh p' n'"
and nc': "isnpolyh c' (Suc n')"
and nn0: "n ≥ n0"
and nn1: "n' ≥ n1"
by simp_all
consider "n < n'" | "n' < n" | "n' = n" by arith
then show ?case
proof cases
case 1
with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
show ?thesis by (simp add: min_def)
next
case 2
with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
show ?thesis by (cases "Suc n' = n") (simp_all add: min_def)
next
case 3
with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
show ?thesis
qed
next
fix n0 n1 m
assume np: "isnpolyh ?cnp n0"
assume np':"isnpolyh ?cnp' n1"
assume m: "m ≤ min n0 n1"
let ?d = "degreen (?cnp *⇩p ?cnp') m"
let ?d1 = "degreen ?cnp m"
let ?d2 = "degreen ?cnp' m"
let ?eq = "?d = (if ?cnp = 0⇩p ∨ ?cnp' = 0⇩p then 0  else ?d1 + ?d2)"
consider "n' < n ∨ n < n'" | "n' = n" by linarith
then show ?eq
proof cases
case 1
with "4.hyps"(3,6,18) np np' m show ?thesis by auto
next
case 2
have nn': "n' = n" by fact
then have nn: "¬ n' < n ∧ ¬ n < n'" by arith
from "4.hyps"(16,18)[of n n' n]
"4.hyps"(13,14)[of n "Suc n'" n]
np np' nn'
have norm:
"isnpolyh ?cnp n"
"isnpolyh c' (Suc n)"
"isnpolyh (?cnp *⇩p c') n"
"isnpolyh p' n"
"isnpolyh (?cnp *⇩p p') n"
"isnpolyh (CN 0⇩p n (CN c n p *⇩p p')) n"
"?cnp *⇩p c' = 0⇩p ⟷ c' = 0⇩p"
"?cnp *⇩p p' ≠ 0⇩p"
show ?thesis
proof (cases "m = n")
case mn: True
from "4.hyps"(17,18)[OF norm(1,4), of n]
"4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
have degs:
"degreen (?cnp *⇩p c') n = (if c' = 0⇩p then 0 else ?d1 + degreen c' n)"
"degreen (?cnp *⇩p p') n = ?d1  + degreen p' n"
from degs norm have th1: "degreen (?cnp *⇩p c') n < degreen (CN 0⇩p n (?cnp *⇩p p')) n"
by simp
then have neq: "degreen (?cnp *⇩p c') n ≠ degreen (CN 0⇩p n (?cnp *⇩p p')) n"
by simp
have nmin: "n ≤ min n n"
from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
have deg: "degreen (CN c n p *⇩p c' +⇩p CN 0⇩p n (CN c n p *⇩p p')) n =
degreen (CN 0⇩p n (CN c n p *⇩p p')) n"
by simp
from "4.hyps"(16-18)[OF norm(1,4), of n]
"4.hyps"(13-15)[OF norm(1,2), of n]
mn norm m nn' deg
show ?thesis by simp
next
case mn: False
then have mn': "m < n"
using m np by auto
from nn' m np have max1: "m ≤ max n n"
by simp
then have min1: "m ≤ min n n"
by simp
then have min2: "m ≤ min n (Suc n)"
by simp
from "4.hyps"(16-18)[OF norm(1,4) min1]
"4.hyps"(13-15)[OF norm(1,2) min2]
have "degreen (?cnp *⇩p c' +⇩p CN 0⇩p n (?cnp *⇩p p')) m ≤
max (degreen (?cnp *⇩p c') m) (degreen (CN 0⇩p n (?cnp *⇩p p')) m)"
using mn nn' np np' by simp
with "4.hyps"(16-18)[OF norm(1,4) min1]
"4.hyps"(13-15)[OF norm(1,2) min2]
degreen_0[OF norm(3) mn']
nn' mn np np'
show ?thesis by clarsimp
qed
qed
}
{
case (2 n0 n1)
then have np: "isnpolyh ?cnp n0"
and np': "isnpolyh ?cnp' n1"
and m: "m ≤ min n0 n1"
by simp_all
then have mn: "m ≤ n" by simp
let ?c0p = "CN 0⇩p n (?cnp *⇩p p')"
have False if C: "?cnp *⇩p c' +⇩p ?c0p = 0⇩p" "n' = n"
proof -
from C have nn: "¬ n' < n ∧ ¬ n < n'"
by simp
from "4.hyps"(16-18) [of n n n]
"4.hyps"(13-15)[of n "Suc n" n]
np np' C(2) mn
have norm:
"isnpolyh ?cnp n"
"isnpolyh c' (Suc n)"
"isnpolyh (?cnp *⇩p c') n"
"isnpolyh p' n"
"isnpolyh (?cnp *⇩p p') n"
"isnpolyh (CN 0⇩p n (CN c n p *⇩p p')) n"
"?cnp *⇩p c' = 0⇩p ⟷ c' = 0⇩p"
"?cnp *⇩p p' ≠ 0⇩p"
"degreen (?cnp *⇩p c') n = (if c' = 0⇩p then 0 else degreen ?cnp n + degreen c' n)"
"degreen (?cnp *⇩p p') n = degreen ?cnp n + degreen p' n"
from norm have cn: "isnpolyh (CN 0⇩p n (CN c n p *⇩p p')) n"
by simp
have degneq: "degreen (?cnp *⇩p c') n < degreen (CN 0⇩p n (?cnp *⇩p p')) n"
using norm by simp
from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
show ?thesis by simp
qed
then show ?case using "4.hyps" by clarsimp
}
qed auto

lemma polymul[simp]: "Ipoly bs (p *⇩p q) = Ipoly bs p * Ipoly bs q"
by (induct p q rule: polymul.induct) (auto simp add: field_simps)

lemma polymul_normh:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpolyh p n0 ⟹ isnpolyh q n1 ⟹ isnpolyh (p *⇩p q) (min n0 n1)"
using polymul_properties(1) by blast

lemma polymul_eq0_iff:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpolyh p n0 ⟹ isnpolyh q n1 ⟹ p *⇩p q = 0⇩p ⟷ p = 0⇩p ∨ q = 0⇩p"
using polymul_properties(2) by blast

lemma polymul_degreen:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpolyh p n0 ⟹ isnpolyh q n1 ⟹ m ≤ min n0 n1 ⟹
degreen (p *⇩p q) m = (if p = 0⇩p ∨ q = 0⇩p then 0 else degreen p m + degreen q m)"
by (fact polymul_properties(3))

lemma polymul_norm:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpoly p ⟹ isnpoly q ⟹ isnpoly (polymul p q)"
using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp

lemma headconst_zero: "isnpolyh p n0 ⟹ headconst p = 0⇩N ⟷ p = 0⇩p"
by (induct p arbitrary: n0 rule: headconst.induct) auto

by (induct p arbitrary: n0) auto

lemma monic_eqI:
assumes np: "isnpolyh p n0"
shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
(Ipoly bs p ::'a::{field_char_0, power})"
unfolding monic_def Let_def
assume pz: "p ≠ 0⇩p"
{
assume hz: "INum ?h = (0::'a)"
from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0⇩N"
by simp_all
from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0⇩N"
by simp
with headconst_zero[OF np] have "p = 0⇩p"
by blast
with pz have False
by blast
}
then show "INum (headconst p) = (0::'a) ⟶ ⦇p⦈⇩p⇗bs⇖ = 0"
by blast
qed

text ‹polyneg is a negation and preserves normal forms›

lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
by (induct p rule: polyneg.induct) auto

lemma polyneg0: "isnpolyh p n ⟹ (~⇩p p) = 0⇩p ⟷ p = 0⇩p"
by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)

lemma polyneg_polyneg: "isnpolyh p n0 ⟹ ~⇩p (~⇩p p) = p"
by (induct p arbitrary: n0 rule: polyneg.induct) auto

lemma polyneg_normh: "isnpolyh p n ⟹ isnpolyh (polyneg p) n"
by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)

lemma polyneg_norm: "isnpoly p ⟹ isnpoly (polyneg p)"
using isnpoly_def polyneg_normh by simp

text ‹polysub is a substraction and preserves normal forms›

lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"

lemma polysub_normh: "isnpolyh p n0 ⟹ isnpolyh q n1 ⟹ isnpolyh (polysub p q) (min n0 n1)"

lemma polysub_norm: "isnpoly p ⟹ isnpoly q ⟹ isnpoly (polysub p q)"

lemma polysub_same_0[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpolyh p n0 ⟹ polysub p p = 0⇩p"
unfolding polysub_def split_def fst_conv snd_conv
by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])

lemma polysub_0:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpolyh p n0 ⟹ isnpolyh q n1 ⟹ p -⇩p q = 0⇩p ⟷ p = q"
unfolding polysub_def split_def fst_conv snd_conv
by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
(auto simp: Nsub0[simplified Nsub_def] Let_def)

text ‹polypow is a power function and preserves normal forms›

lemma polypow[simp]: "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::field_char_0) ^ n"
proof (induct n rule: polypow.induct)
case 1
then show ?case by simp
next
case (2 n)
let ?q = "polypow ((Suc n) div 2) p"
let ?d = "polymul ?q ?q"
consider "odd (Suc n)" | "even (Suc n)" by auto
then show ?case
proof cases
case odd: 1
have *: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
by arith
from odd have "Ipoly bs (p ^⇩p Suc n) = Ipoly bs (polymul p ?d)"
also have "… = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
using "2.hyps" by simp
also have "… = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
by (simp only: power_add power_one_right) simp
also have "… = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
by (simp only: *)
finally show ?thesis
unfolding numeral_2_eq_2 [symmetric]
using odd_two_times_div_two_nat [OF odd] by simp
next
case even: 2
from even have "Ipoly bs (p ^⇩p Suc n) = Ipoly bs ?d"
also have "… = (Ipoly bs p) ^ (2 * (Suc n div 2))"
using "2.hyps" by (simp only: mult_2 power_add) simp
finally show ?thesis
using even_two_times_div_two [OF even] by simp
qed
qed

lemma polypow_normh:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpolyh p n ⟹ isnpolyh (polypow k p) n"
proof (induct k arbitrary: n rule: polypow.induct)
case 1
then show ?case by auto
next
case (2 k n)
let ?q = "polypow (Suc k div 2) p"
let ?d = "polymul ?q ?q"
from 2 have *: "isnpolyh ?q n" and **: "isnpolyh p n"
by blast+
from polymul_normh[OF * *] have dn: "isnpolyh ?d n"
by simp
from polymul_normh[OF ** dn] have on: "isnpolyh (polymul p ?d) n"
by simp
from dn on show ?case by (simp, unfold Let_def) auto
qed

lemma polypow_norm:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpoly p ⟹ isnpoly (polypow k p)"

text ‹Finally the whole normalization›

lemma polynate [simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::field_char_0)"
by (induct p rule:polynate.induct) auto

lemma polynate_norm[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpoly (polynate p)"
by (induct p rule: polynate.induct)

text ‹shift1›

lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"

lemma shift1_isnpoly:
assumes "isnpoly p"
and "p ≠ 0⇩p"
shows "isnpoly (shift1 p) "
using assms by (simp add: shift1_def isnpoly_def)

lemma shift1_nz[simp]:"shift1 p ≠ 0⇩p"

lemma funpow_shift1_isnpoly: "isnpoly p ⟹ p ≠ 0⇩p ⟹ isnpoly (funpow n shift1 p)"
by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)

lemma funpow_isnpolyh:
assumes "⋀p. isnpolyh p n ⟹ isnpolyh (f p) n"
and "isnpolyh p n"
shows "isnpolyh (funpow k f p) n"
using assms by (induct k arbitrary: p) auto

lemma funpow_shift1:
"(Ipoly bs (funpow n shift1 p) :: 'a :: field_char_0) =
Ipoly bs (Mul (Pw (Bound 0) n) p)"
by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)

lemma shift1_isnpolyh: "isnpolyh p n0 ⟹ p ≠ 0⇩p ⟹ isnpolyh (shift1 p) 0"
using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)

lemma funpow_shift1_1:
"(Ipoly bs (funpow n shift1 p) :: 'a :: field_char_0) =
Ipoly bs (funpow n shift1 (1)⇩p *⇩p p)"

lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)

assumes "isnpolyh p n"
(Ipoly bs p :: 'a :: field_char_0)"
using assms
proof (induct p arbitrary: n rule: behead.induct)
case (1 c p n)
then have pn: "isnpolyh p n" by simp
from 1(1)[OF pn]
have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
then show ?case using "1.hyps"
done

assumes "isnpolyh p n"
using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)

subsection ‹Miscellaneous lemmas about indexes, decrementation, substitution  etc ...›

lemma isnpolyh_polybound0: "isnpolyh p (Suc n) ⟹ polybound0 p"
proof (induct p arbitrary: n rule: poly.induct, auto, goal_cases)
case prems: (1 c n p n')
then have "n = Suc (n - 1)"
by simp
then have "isnpolyh p (Suc (n - 1))"
using ‹isnpolyh p n› by simp
with prems(2) show ?case
by simp
qed

lemma isconstant_polybound0: "isnpolyh p n0 ⟹ isconstant p ⟷ polybound0 p"
by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)

lemma decrpoly_zero[simp]: "decrpoly p = 0⇩p ⟷ p = 0⇩p"
by (induct p) auto

lemma decrpoly_normh: "isnpolyh p n0 ⟹ polybound0 p ⟹ isnpolyh (decrpoly p) (n0 - 1)"
apply (induct p arbitrary: n0)
apply auto
apply atomize
apply (rename_tac nat a b, erule_tac x = "Suc nat" in allE)
apply auto
done

by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)

lemma polybound0_I:
assumes "polybound0 a"
shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
using assms by (induct a rule: poly.induct) auto

lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
by (induct t) simp_all

lemma polysubst0_I':
assumes "polybound0 a"
shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"])

lemma decrpoly:
assumes "polybound0 t"
shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
using assms by (induct t rule: decrpoly.induct) simp_all

lemma polysubst0_polybound0:
assumes "polybound0 t"
shows "polybound0 (polysubst0 t a)"
using assms by (induct a rule: poly.induct) auto

lemma degree0_polybound0: "isnpolyh p n ⟹ degree p = 0 ⟹ polybound0 p"
by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)

primrec maxindex :: "poly ⇒ nat"
where
"maxindex (Bound n) = n + 1"
| "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
| "maxindex (Add p q) = max (maxindex p) (maxindex q)"
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
| "maxindex (Neg p) = maxindex p"
| "maxindex (Pw p n) = maxindex p"
| "maxindex (C x) = 0"

definition wf_bs :: "'a list ⇒ poly ⇒ bool"
where "wf_bs bs p ⟷ length bs ≥ maxindex p"

lemma wf_bs_coefficients: "wf_bs bs p ⟹ ∀c ∈ set (coefficients p). wf_bs bs c"
proof (induct p rule: coefficients.induct)
case (1 c p)
show ?case
proof
fix x
assume "x ∈ set (coefficients (CN c 0 p))"
then consider "x = c" | "x ∈ set (coefficients p)"
by auto
then show "wf_bs bs x"
proof cases
case prems: 1
then show ?thesis
using "1.prems" by (simp add: wf_bs_def)
next
case prems: 2
from "1.prems" have "wf_bs bs p"
with "1.hyps" prems show ?thesis
by blast
qed
qed
qed simp_all

lemma maxindex_coefficients: "∀c ∈ set (coefficients p). maxindex c ≤ maxindex p"
by (induct p rule: coefficients.induct) auto

lemma wf_bs_I: "wf_bs bs p ⟹ Ipoly (bs @ bs') p = Ipoly bs p"
by (induct p) (auto simp add: nth_append wf_bs_def)

lemma take_maxindex_wf:
assumes wf: "wf_bs bs p"
shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
proof -
let ?ip = "maxindex p"
let ?tbs = "take ?ip bs"
from wf have "length ?tbs = ?ip"
unfolding wf_bs_def by simp
then have wf': "wf_bs ?tbs p"
unfolding wf_bs_def by  simp
have eq: "bs = ?tbs @ drop ?ip bs"
by simp
from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
using eq by simp
qed

lemma decr_maxindex: "polybound0 p ⟹ maxindex (decrpoly p) = maxindex p - 1"
by (induct p) auto

lemma wf_bs_insensitive: "length bs = length bs' ⟹ wf_bs bs p = wf_bs bs' p"

lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"

lemma wf_bs_coefficients': "∀c ∈ set (coefficients p). wf_bs bs c ⟹ wf_bs (x # bs) p"
by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)

lemma coefficients_Nil[simp]: "coefficients p ≠ []"
by (induct p rule: coefficients.induct) simp_all

by (induct p rule: coefficients.induct) auto

lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) ⟹ wf_bs (x # bs) p"
unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto

lemma length_le_list_ex: "length xs ≤ n ⟹ ∃ys. length (xs @ ys) = n"
by (rule exI[where x="replicate (n - length xs) z" for z]) simp

lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) ⟹ isconstant p"
apply (cases p)
apply auto
apply (rename_tac nat a, case_tac "nat")
apply simp_all
done

lemma wf_bs_polyadd: "wf_bs bs p ∧ wf_bs bs q ⟶ wf_bs bs (p +⇩p q)"

lemma wf_bs_polyul: "wf_bs bs p ⟹ wf_bs bs q ⟹ wf_bs bs (p *⇩p q)"
apply (induct p q arbitrary: bs rule: polymul.induct)
apply clarsimp
apply auto
done

lemma wf_bs_polyneg: "wf_bs bs p ⟹ wf_bs bs (~⇩p p)"
by (induct p rule: polyneg.induct) (auto simp: wf_bs_def)

lemma wf_bs_polysub: "wf_bs bs p ⟹ wf_bs bs q ⟹ wf_bs bs (p -⇩p q)"
unfolding polysub_def split_def fst_conv snd_conv

subsection ‹Canonicity of polynomial representation, see lemma isnpolyh_unique›

definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
definition "polypoly' bs p = map (Ipoly bs ∘ decrpoly) (coefficients p)"
definition "poly_nate bs p = map (Ipoly bs ∘ decrpoly) (coefficients (polynate p))"

lemma coefficients_normh: "isnpolyh p n0 ⟹ ∀q ∈ set (coefficients p). isnpolyh q n0"
proof (induct p arbitrary: n0 rule: coefficients.induct)
case (1 c p n0)
have cp: "isnpolyh (CN c 0 p) n0"
by fact
then have norm: "isnpolyh c 0" "isnpolyh p 0" "p ≠ 0⇩p" "n0 = 0"
by (auto simp add: isnpolyh_mono[where n'=0])
from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
by simp
qed auto

lemma coefficients_isconst: "isnpolyh p n ⟹ ∀q ∈ set (coefficients p). isconstant q"
by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)

lemma polypoly_polypoly':
assumes np: "isnpolyh p n0"
shows "polypoly (x # bs) p = polypoly' bs p"
proof -
let ?cf = "set (coefficients p)"
from coefficients_normh[OF np] have cn_norm: "∀ q∈ ?cf. isnpolyh q n0" .
have "polybound0 q" if "q ∈ ?cf" for q
proof -
from that cn_norm have *: "isnpolyh q n0"
by blast
from coefficients_isconst[OF np] that have "isconstant q"
by blast
with isconstant_polybound0[OF *] show ?thesis
by blast
qed
then have "∀q ∈ ?cf. polybound0 q" ..
then have "∀q ∈ ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
by auto
then show ?thesis
unfolding polypoly_def polypoly'_def by simp
qed

lemma polypoly_poly:
assumes "isnpolyh p n0"
shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
using assms
by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)

lemma polypoly'_poly:
assumes "isnpolyh p n0"
shows "⦇p⦈⇩p⇗x # bs⇖ = poly (polypoly' bs p) x"
using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .

lemma polypoly_poly_polybound0:
assumes "isnpolyh p n0"
and "polybound0 p"
shows "polypoly bs p = [Ipoly bs p]"
using assms
unfolding polypoly_def
apply (cases p)
apply auto
apply (rename_tac nat a, case_tac nat)
apply auto
done

by (induct p rule: head.induct) auto

lemma headn_nz[simp]: "isnpolyh p n0 ⟹ headn p m = 0⇩p ⟷ p = 0⇩p"
by (cases p) auto

by (induct p rule: head.induct) simp_all

lemma head_nz[simp]: "isnpolyh p n0 ⟹ head p = 0⇩p ⟷ p = 0⇩p"

lemma isnpolyh_zero_iff:
assumes nq: "isnpolyh p n0"
and eq :"∀bs. wf_bs bs p ⟶ ⦇p⦈⇩p⇗bs⇖ = (0::'a::{field_char_0, power})"
shows "p = 0⇩p"
using nq eq
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
case less
note np = ‹isnpolyh p n0› and zp = ‹∀bs. wf_bs bs p ⟶ ⦇p⦈⇩p⇗bs⇖ = (0::'a)›
show "p = 0⇩p"
proof (cases "maxindex p = 0")
case True
with np obtain c where "p = C c" by (cases p) auto
with zp np show ?thesis by (simp add: wf_bs_def)
next
case nz: False
let ?hd = "decrpoly ?h"
let ?ihd = "maxindex ?hd"
have h: "isnpolyh ?h n0" "polybound0 ?h"
by simp_all
then have nhd: "isnpolyh ?hd (n0 - 1)"
using decrpoly_normh by blast

from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
have mihn: "maxindex ?h ≤ maxindex p"
by auto
with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
by auto

have "⦇?hd⦈⇩p⇗bs⇖ = 0" if bs: "wf_bs bs ?hd" for bs :: "'a list"
proof -
let ?ts = "take ?ihd bs"
let ?rs = "drop ?ihd bs"
from bs have ts: "wf_bs ?ts ?hd"
have bs_ts_eq: "?ts @ ?rs = bs"
by simp
from wf_bs_decrpoly[OF ts] have tsh: " ∀x. wf_bs (x # ?ts) ?h"
by simp
from ihd_lt_n have "∀x. length (x # ?ts) ≤ maxindex p"
by simp
with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
by blast
then have "∀x. wf_bs ((x # ?ts) @ xs) p"
with zp have "∀x. Ipoly ((x # ?ts) @ xs) p = 0"
by blast
then have "∀x. Ipoly (x # (?ts @ xs)) p = 0"
by simp
with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
have "∀x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
by simp
then have "poly (polypoly' (?ts @ xs) p) = poly []"
by auto
then have "∀c ∈ set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
using poly_zero[where ?'a='a] by (simp add: polypoly'_def)
have *: "Ipoly (?ts @ xs) ?hd = 0"
by simp
from bs have wf'': "wf_bs ?ts ?hd"
with * wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
by simp
with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq show ?thesis
by simp
qed
then have hdz: "∀bs. wf_bs bs ?hd ⟶ ⦇?hd⦈⇩p⇗bs⇖ = (0::'a)"
by blast
from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0⇩p"
by blast
then have "?h = 0⇩p" by simp
with head_nz[OF np] show ?thesis by simp
qed
qed

lemma isnpolyh_unique:
assumes np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
shows "(∀bs. ⦇p⦈⇩p⇗bs⇖ = (⦇q⦈⇩p⇗bs⇖ :: 'a::{field_char_0,power})) ⟷ p = q"
proof auto
assume "∀bs. (⦇p⦈⇩p⇗bs⇖ ::'a) = ⦇q⦈⇩p⇗bs⇖"
then have "∀bs.⦇p -⇩p q⦈⇩p⇗bs⇖= (0::'a)"
by simp
then have "∀bs. wf_bs bs (p -⇩p q) ⟶ ⦇p -⇩p q⦈⇩p⇗bs⇖ = (0::'a)"
using wf_bs_polysub[where p=p and q=q] by auto
with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
by blast
qed

text ‹Consequences of unicity on the algorithms for polynomial normalization.›

assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
shows "p +⇩p q = q +⇩p p"
by simp

lemma zero_normh: "isnpolyh 0⇩p n"
by simp

lemma one_normh: "isnpolyh (1)⇩p n"
by simp

assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
shows "p +⇩p 0⇩p = p"
and "0⇩p +⇩p p = p"
using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all

lemma polymul_1[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
shows "p *⇩p (1)⇩p = p"
and "(1)⇩p *⇩p p = p"
using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all

lemma polymul_0[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
shows "p *⇩p 0⇩p = 0⇩p"
and "0⇩p *⇩p p = 0⇩p"
using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all

lemma polymul_commute:
assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
shows "p *⇩p q = q *⇩p p"
using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
where ?'a = "'a::{field_char_0, power}"]
by simp

declare polyneg_polyneg [simp]

lemma isnpolyh_polynate_id [simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
shows "polynate p = p"
using isnpolyh_unique[where ?'a= "'a::field_char_0",
OF polynate_norm[of p, unfolded isnpoly_def] np]
polynate[where ?'a = "'a::field_char_0"]
by simp

lemma polynate_idempotent[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "polynate (polynate p) = polynate p"
using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .

lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
unfolding poly_nate_def polypoly'_def ..

lemma poly_nate_poly:
"poly (poly_nate bs p) = (λx:: 'a ::field_char_0. ⦇p⦈⇩p⇗x # bs⇖)"
using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
unfolding poly_nate_polypoly' by auto

subsection ‹Heads, degrees and all that›

lemma degree_eq_degreen0: "degree p = degreen p 0"
by (induct p rule: degree.induct) simp_all

lemma degree_polyneg:
assumes "isnpolyh p n"
shows "degree (polyneg p) = degree p"
apply (induct p rule: polyneg.induct)
using assms
apply simp_all
apply (case_tac na)
apply auto
done

assumes np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
shows "degree (p +⇩p q) ≤ max (degree p) (degree q)"
using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp

lemma degree_polysub:
assumes np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
shows "degree (p -⇩p q) ≤ max (degree p) (degree q)"
proof-
from nq have nq': "isnpolyh (~⇩p q) n1"
using polyneg_normh by simp
from degree_polyadd[OF np nq'] show ?thesis
by (simp add: polysub_def degree_polyneg[OF nq])
qed

assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
and d: "degree p = degree q"
shows "degree (p -⇩p q) < degree p ∨ (p -⇩p q = 0⇩p)"
unfolding polysub_def split_def fst_conv snd_conv
using np nq h d
proof (induct p q rule: polyadd.induct)
case (1 c c')
then show ?case
by (simp add: Nsub_def Nsub0[simplified Nsub_def])
next
case (2 c c' n' p')
from 2 have "degree (C c) = degree (CN c' n' p')"
by simp
then have nz: "n' > 0"
by (cases n') auto
then have "head (CN c' n' p') = CN c' n' p'"
by (cases n') auto
with 2 show ?case
by simp
next
case (3 c n p c')
then have "degree (C c') = degree (CN c n p)"
by simp
then have nz: "n > 0"
by (cases n) auto
then have "head (CN c n p) = CN c n p"
by (cases n) auto
with 3 show ?case by simp
next
case (4 c n p c' n' p')
then have H:
"isnpolyh (CN c n p) n0"
"isnpolyh (CN c' n' p') n1"
"degree (CN c n p) = degree (CN c' n' p')"
by simp_all
then have degc: "degree c = 0" and degc': "degree c' = 0"
by simp_all
then have degnc: "degree (~⇩p c) = 0" and degnc': "degree (~⇩p c') = 0"
using H(1-2) degree_polyneg by auto
from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
by simp_all
from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
have degcmc': "degree (c +⇩p  ~⇩pc') = 0"
by simp
from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
by auto
consider "n = n'" | "n < n'" | "n > n'"
by arith
then show ?case
proof cases
case nn': 1
consider "n = 0" | "n > 0" by arith
then show ?thesis
proof cases
case 1
with 4 nn' show ?thesis
by (auto simp add: Let_def degcmc')
next
case 2
with nn' H(3) have "c = c'" and "p = p'"
by (cases n; auto)+
with nn' 4 show ?thesis
using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
using polysub_same_0[OF c'nh, simplified polysub_def]
qed
next
case nn': 2
then have n'p: "n' > 0"
by simp
then have headcnp':"head (CN c' n' p') = CN c' n' p'"
by (cases n') simp_all
with 4 nn' have degcnp': "degree (CN c' n' p') = 0"
and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
by (cases n', simp_all)
then have "n > 0"
by (cases n) simp_all
then have headcnp: "head (CN c n p) = CN c n p"
by (cases n) auto
by auto
next
case nn': 3
then have np: "n > 0" by simp
then have headcnp:"head (CN c n p) = CN c n p"
by (cases n) simp_all
from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
by simp
from np have degcnp: "degree (CN c n p) = 0"
by (cases n) simp_all
with degcnpeq have "n' > 0"
by (cases n') simp_all
then have headcnp': "head (CN c' n' p') = CN c' n' p'"
by (cases n') auto
qed
qed auto

lemma funpow_shift1_head: "isnpolyh p n0 ⟹ p ≠ 0⇩p ⟹ head (funpow k shift1 p) = head p"
proof (induct k arbitrary: n0 p)
case 0
then show ?case
by auto
next
case (Suc k n0 p)
then have "isnpolyh (shift1 p) 0"
with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
then show ?case
qed

lemma shift1_degree: "degree (shift1 p) = 1 + degree p"

lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
by (induct k arbitrary: p) (auto simp add: shift1_degree)

lemma funpow_shift1_nz: "p ≠ 0⇩p ⟹ funpow n shift1 p ≠ 0⇩p"
by (induct n arbitrary: p) simp_all

by (induct p arbitrary: n rule: degree.induct) auto
lemma headn_0[simp]: "isnpolyh p n ⟹ m < n ⟹ headn p m = p"
by (induct p arbitrary: n rule: degreen.induct) auto
lemma head_isnpolyh_Suc': "n > 0 ⟹ isnpolyh p n ⟹ head p = p"
by (induct p arbitrary: n rule: degree.induct) auto
by (induct p rule: head.induct) auto

"isnpolyh p n0 ⟹ isnpolyh q n1 ⟹ polyadd p q = C c ⟹ degree p = degree q"

assumes np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
and deg: "degree p ≠ degree q"
shows "head (p +⇩p q) = (if degree p < degree q then head q else head p)"
using np nq deg
apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
apply simp_all
apply (case_tac n', simp, simp)
apply (case_tac n, simp, simp)
apply (case_tac n, case_tac n', simp add: Let_def)
apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
done

assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpolyh p n0 ⟹ isnpolyh q n1 ⟹ p ≠ 0⇩p ⟹ q ≠ 0⇩p ⟹ head (p *⇩p q) = head p *⇩p head q"
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
case (2 c c' n' p' n0 n1)
then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
then show ?case
using 2 by (cases n') auto
next
case (3 c n p c' n0 n1)
then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
then show ?case
using 3 by (cases n) auto
next
case (4 c n p c' n' p' n0 n1)
then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
"isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
by simp_all
consider "n < n'" | "n' < n" | "n' = n" by arith
then show ?case
proof cases
case nn': 1
then show ?thesis
using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
apply simp
apply (cases n)
apply simp
apply (cases n')
apply simp_all
done
next
case nn': 2
then show ?thesis
using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
apply simp
apply (cases n')
apply simp
apply (cases n)
apply auto
done
next
case nn': 3
from nn' polymul_normh[OF norm(5,4)]
have ncnpc': "isnpolyh (CN c n p *⇩p c') n" by (simp add: min_def)
from nn' polymul_normh[OF norm(5,3)] norm
have ncnpp': "isnpolyh (CN c n p *⇩p p') n" by simp
from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
have ncnpp0': "isnpolyh (CN 0⇩p n (CN c n p *⇩p p')) n" by simp
have nth: "isnpolyh ((CN c n p *⇩p c') +⇩p (CN 0⇩p n (CN c n p *⇩p p'))) n"
consider "n > 0" | "n = 0" by auto
then show ?thesis
proof cases
case np: 1
show ?thesis by simp
next
case nz: 2
from polymul_degreen[OF norm(5,4), where m="0"]
polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
norm(5,6) degree_npolyhCN[OF norm(6)]
have dth: "degree (CN c 0 p *⇩p c') < degree (CN 0⇩p 0 (CN c 0 p *⇩p p'))"
by simp
then have dth': "degree (CN c 0 p *⇩p c') ≠ degree (CN 0⇩p 0 (CN c 0 p *⇩p p'))"
by simp
show ?thesis
using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
by simp
qed
qed
qed simp_all

lemma degree_coefficients: "degree p = length (coefficients p) - 1"
by (induct p rule: degree.induct) auto

by (induct p rule: head.induct) auto

lemma degree_CN: "isnpolyh p n ⟹ degree (CN c n p) ≤ 1 + degree p"
by (cases n) simp_all

lemma degree_CN': "isnpolyh p n ⟹ degree (CN c n p) ≥  degree p"
by (cases n) simp_all

"isnpolyh p n0 ⟹ isnpolyh q n1 ⟹ degree p ≠ degree q ⟹
degree (polyadd p q) = max (degree p) (degree q)"

lemma degreen_polyneg: "isnpolyh p n0 ⟹ degreen (~⇩p p) m = degreen p m"
by (induct p arbitrary: n0 rule: polyneg.induct) auto

lemma degree_polymul:
assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
and nq: "isnpolyh q n1"
shows "degree (p *⇩p q) ≤ degree p + degree q"
using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp

lemma polyneg_degree: "isnpolyh p n ⟹ degree (polyneg p) = degree p"
by (induct p arbitrary: n rule: degree.induct) auto

by (induct p arbitrary: n rule: degree.induct) auto

subsection ‹Correctness of polynomial pseudo division›

lemma polydivide_aux_properties:
assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
and ns: "isnpolyh s n1"
and ap: "head p = a"
and ndp: "degree p = n"
and pnz: "p ≠ 0⇩p"
shows "polydivide_aux a n p k s = (k', r) ⟶ k' ≥ k ∧ (degree r = 0 ∨ degree r < degree p) ∧
(∃nr. isnpolyh r nr) ∧ (∃q n1. isnpolyh q n1 ∧ (polypow (k' - k) a) *⇩p s = p *⇩p q +⇩p r)"
using ns
proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
case less
let ?qths = "∃q n1. isnpolyh q n1 ∧ (a ^⇩p (k' - k) *⇩p s = p *⇩p q +⇩p r)"
let ?ths = "polydivide_aux a n p k s = (k', r) ⟶  k ≤ k' ∧
(degree r = 0 ∨ degree r < degree p) ∧ (∃nr. isnpolyh r nr) ∧ ?qths"
let ?p' = "funpow (degree s - n) shift1 p"
let ?xdn = "funpow (degree s - n) shift1 (1)⇩p"
let ?akk' = "a ^⇩p (k' - k)"
note ns = ‹isnpolyh s n1›
from np have np0: "isnpolyh p 0"
using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
have np': "isnpolyh ?p' 0"
using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
by simp
using funpow_shift1_head[OF np pnz] by simp
from funpow_shift1_isnpoly[where p="(1)⇩p"] have nxdn: "isnpolyh ?xdn 0"
from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
have nakk':"isnpolyh ?akk' 0" by blast
show ?ths
proof (cases "s = 0⇩p")
case True
with np show ?thesis
apply (clarsimp simp: polydivide_aux.simps)
apply (rule exI[where x="0⇩p"])
apply simp
done
next
case sz: False
show ?thesis
proof (cases "degree s < n")
case True
then show ?thesis
using ns ndp np polydivide_aux.simps
apply auto
apply (rule exI[where x="0⇩p"])
apply simp
done
next
case dn': False
then have dn: "degree s ≥ n"
by arith
have degsp': "degree s = degree ?p'"
using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
by simp
show ?thesis
proof (cases "?b = a")
case ba: True
have nr: "isnpolyh (s -⇩p ?p') 0"
using polysub_normh[OF ns np'] by simp
consider "degree (s -⇩p ?p') < degree s" | "s -⇩p ?p' = 0⇩p" by auto
then show ?thesis
proof cases
case deglt: 1
from polydivide_aux.simps sz dn' ba
have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -⇩p ?p')"
have "k ≤ k' ∧ (degree r = 0 ∨ degree r < degree p) ∧ (∃nr. isnpolyh r nr) ∧ ?qths"
if h1: "polydivide_aux a n p k s = (k', r)"
proof -
from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
have kk': "k ≤ k'"
and nr: "∃nr. isnpolyh r nr"
and dr: "degree r = 0 ∨ degree r < degree p"
and q1: "∃q nq. isnpolyh q nq ∧ a ^⇩p k' - k *⇩p (s -⇩p ?p') = p *⇩p q +⇩p r"
by auto
from q1 obtain q n1 where nq: "isnpolyh q n1"
and asp: "a^⇩p (k' - k) *⇩p (s -⇩p ?p') = p *⇩p q +⇩p r"
by blast
from nr obtain nr where nr': "isnpolyh r nr"
by blast
from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^⇩p (k' - k) *⇩p s) 0"
by simp
from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
have nq': "isnpolyh (?akk' *⇩p ?xdn +⇩p q) 0" by simp
polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
have nqr': "isnpolyh (p *⇩p (?akk' *⇩p ?xdn +⇩p q) +⇩p r) 0"
by simp
from asp have "∀bs :: 'a::field_char_0 list.
Ipoly bs (a^⇩p (k' - k) *⇩p (s -⇩p ?p')) = Ipoly bs (p *⇩p q +⇩p r)"
by simp
then have "∀bs :: 'a::field_char_0 list.
Ipoly bs (a^⇩p (k' - k)*⇩p s) =
Ipoly bs (a^⇩p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
then have "∀bs :: 'a::field_char_0 list.
Ipoly bs (a ^⇩p (k' - k) *⇩p s) =
Ipoly bs (a^⇩p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)⇩p *⇩p p) +
Ipoly bs p * Ipoly bs q + Ipoly bs r"
by (auto simp only: funpow_shift1_1)
then have "∀bs:: 'a::field_char_0 list.
Ipoly bs (a ^⇩p (k' - k) *⇩p s) =
Ipoly bs p * (Ipoly bs (a^⇩p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)⇩p) +
Ipoly bs q) + Ipoly bs r"
then have "∀bs:: 'a::field_char_0 list.
Ipoly bs (a ^⇩p (k' - k) *⇩p s) =
Ipoly bs (p *⇩p ((a^⇩p (k' - k)) *⇩p (funpow (degree s - n) shift1 (1)⇩p) +⇩p q) +⇩p r)"
by simp
with isnpolyh_unique[OF nakks' nqr']
have "a ^⇩p (k' - k) *⇩p s =
p *⇩p ((a^⇩p (k' - k)) *⇩p (funpow (degree s - n) shift1 (1)⇩p) +⇩p q) +⇩p r"
by blast
with nq' have ?qths
apply (rule_tac x="(a^⇩p (k' - k)) *⇩p (funpow (degree s - n) shift1 (1)⇩p) +⇩p q" in exI)
apply (rule_tac x="0" in exI)
apply simp
done
with kk' nr dr show ?thesis
by blast
qed
then show ?thesis by blast
next
case spz: 2
from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0⇩p", symmetric, where ?'a = "'a::field_char_0"]
have "∀bs:: 'a::field_char_0 list. Ipoly bs s = Ipoly bs ?p'"
by simp
with np nxdn have "∀bs:: 'a::field_char_0 list. Ipoly bs s = Ipoly bs (?xdn *⇩p p)"
by (simp only: funpow_shift1_1) simp
then have sp': "s = ?xdn *⇩p p"
using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
by blast
have ?thesis if h1: "polydivide_aux a n p k s = (k', r)"
proof -
from sz dn' ba
have "polydivide_aux a n p k s = polydivide_aux a n p k (s -⇩p ?p')"
also have "… = (k,0⇩p)"
using spz by (simp add: polydivide_aux.simps)
finally have "(k', r) = (k, 0⇩p)"
with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
polyadd_0(2)[OF polymul_normh[OF np nxdn]] show ?thesis
apply auto
apply (rule exI[where x="?xdn"])
apply (auto simp add: polymul_commute[of p])
done
qed
then show ?thesis by blast
qed
next
case ba: False
from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
have nth: "isnpolyh ((a *⇩p s) -⇩p (?b *⇩p ?p')) 0"
have nzths: "a *⇩p s ≠ 0⇩p" "?b *⇩p ?p' ≠ 0⇩p"
using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
funpow_shift1_nz[OF pnz]
by simp_all
funpow_shift1_nz[OF pnz, where n="degree s - n"]
from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
funpow_shift1_nz[OF pnz, where n="degree s - n"]
ndp dn
have degth: "degree (a *⇩p s) = degree (?b *⇩p ?p')"

consider "degree ((a *⇩p s) -⇩p (?b *⇩p ?p')) < degree s" | "a *⇩p s -⇩p (?b *⇩p ?p') = 0⇩p"
polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth]
polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
then show ?thesis
proof cases
case dth: 1
from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
have nasbp': "isnpolyh ((a *⇩p s) -⇩p (?b *⇩p ?p')) 0"
by simp
have ?thesis if h1: "polydivide_aux a n p k s = (k', r)"
proof -
from h1 polydivide_aux.simps sz dn' ba
have eq:"polydivide_aux a n p (Suc k) ((a *⇩p s) -⇩p (?b *⇩p ?p')) = (k',r)"
with less(1)[OF dth nasbp', of "Suc k" k' r]
obtain q nq nr where kk': "Suc k ≤ k'"
and nr: "isnpolyh r nr"
and nq: "isnpolyh q nq"
and dr: "degree r = 0 ∨ degree r < degree p"
and qr: "a ^⇩p (k' - Suc k) *⇩p ((a *⇩p s) -⇩p (?b *⇩p ?p')) = p *⇩p q +⇩p r"
by auto
from kk' have kk'': "Suc (k' - Suc k) = k' - k"
by arith
have "Ipoly bs (a ^⇩p (k' - k) *⇩p s) =
Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
for bs :: "'a::field_char_0 list"
proof -
from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
have "Ipoly bs (a ^⇩p (k' - Suc k) *⇩p ((a *⇩p s) -⇩p (?b *⇩p ?p'))) = Ipoly bs (p *⇩p q +⇩p r)"
by simp
then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
then show ?thesis
qed
then have ieq: "∀bs :: 'a::field_char_0 list.
Ipoly bs (a ^⇩p (k' - k) *⇩p s) =
Ipoly bs (p *⇩p (q +⇩p (a ^⇩p (k' - Suc k) *⇩p ?b *⇩p ?xdn)) +⇩p r)"
by auto
let ?q = "q +⇩p (a ^⇩p (k' - Suc k) *⇩p ?b *⇩p ?xdn)"
from polyadd_normh[OF nq polymul_normh[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k"] head_isnpolyh[OF ns], simplified ap] nxdn]]
have nqw: "isnpolyh ?q 0"
by simp
from ieq isnpolyh_unique[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - k"] ns, simplified ap] polyadd_normh[OF polymul_normh[OF np nqw] nr]]
have asth: "(a ^⇩p (k' - k) *⇩p s) = p *⇩p (q +⇩p (a ^⇩p (k' - Suc k) *⇩p ?b *⇩p ?xdn)) +⇩p r"
by blast
from dr kk' nr h1 asth nqw show ?thesis
apply simp
apply (rule conjI)
apply (rule exI[where x="nr"], simp)
apply (rule exI[where x="(q +⇩p (a ^⇩p (k' - Suc k) *⇩p ?b *⇩p ?xdn))"], simp)
apply (rule exI[where x="0"], simp)
done
qed
then show ?thesis by blast
next
case spz: 2
have hth: "∀bs :: 'a::field_char_0 list.
Ipoly bs (a *⇩p s) = Ipoly bs (p *⇩p (?b *⇩p ?xdn))"
proof
fix bs :: "'a::field_char_0 list"
from isnpolyh_unique[OF nth, where ?'a="'a" and q="0⇩p",simplified,symmetric] spz
have "Ipoly bs (a*⇩p s) = Ipoly bs ?b * Ipoly bs ?p'"
by simp
then have "Ipoly bs (a*⇩p s) = Ipoly bs (?b *⇩p ?xdn) * Ipoly bs p"
by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
then show "Ipoly bs (a*⇩p s) = Ipoly bs (p *⇩p (?b *⇩p ?xdn))"
by simp
qed
from hth have asq: "a *⇩p s = p *⇩p (?b *⇩p ?xdn)"
using isnpolyh_unique[where ?'a = "'a::field_char_0", OF polymul_normh[OF head_isnpolyh[OF np] ns]
polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
simplified ap]
by simp
have ?ths if h1: "polydivide_aux a n p k s = (k', r)"
proof -
from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
have "(k', r) = (Suc k, 0⇩p)"
with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
show ?thesis
apply (rule exI[where x="?b *⇩p ?xdn"])
apply simp
apply (rule exI[where x="0"], simp)
done
qed
then show ?thesis by blast
qed
qed
qed
qed
qed

lemma polydivide_properties:
assumes "SORT_CONSTRAINT('a::field_char_0)"
and np: "isnpolyh p n0"
and ns: "isnpolyh s n1"
and pnz: "p ≠ 0⇩p"
shows "∃k r. polydivide s p = (k, r) ∧
(∃nr. isnpolyh r nr) ∧ (degree r = 0 ∨ degree r < degree p) ∧
(∃q n1. isnpolyh q n1 ∧ polypow k (head p) *⇩p s = p *⇩p q +⇩p r)"
proof -
have trv: "head p = head p" "degree p = degree p"
by simp_all
from polydivide_def[where s="s" and p="p"] have ex: "∃ k r. polydivide s p = (k,r)"
by auto
then obtain k r where kr: "polydivide s p = (k,r)"
by blast
from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
have "(degree r = 0 ∨ degree r < degree p) ∧
(∃nr. isnpolyh r nr) ∧ (∃q n1. isnpolyh q n1 ∧ head p ^⇩p k - 0 *⇩p s = p *⇩p q +⇩p r)"
by blast
with kr show ?thesis
apply -
apply (rule exI[where x="k"])
apply (rule exI[where x="r"])
apply simp
done
qed

subsection ‹More about polypoly and pnormal etc›

definition "isnonconstant p ⟷ ¬ isconstant p"

lemma isnonconstant_pnormal_iff:
assumes "isnonconstant p"
shows "pnormal (polypoly bs p) ⟷ Ipoly bs (head p) ≠ 0"
proof
let ?p = "polypoly bs p"
assume *: "pnormal ?p"
have "coefficients p ≠ []"
using assms by (cases p) auto
from coefficients_head[of p] last_map[OF this, of "Ipoly bs"] pnormal_last_nonzero[OF *]
show "Ipoly bs (head p) ≠ 0"
next
assume *: "⦇head p⦈⇩p⇗bs⇖ ≠ 0"
let ?p = "polypoly bs p"
have csz: "coefficients p ≠ []"
using assms by (cases p) auto
then have pz: "?p ≠ []"
then have lg: "length ?p > 0"
by simp
from * coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
have lz: "last ?p ≠ 0"
from pnormal_last_length[OF lg lz] show "pnormal ?p" .
qed

lemma isnonconstant_coefficients_length: "isnonconstant p ⟹ length (coefficients p) > 1"
unfolding isnonconstant_def
apply (cases p)
apply simp_all
apply (rename_tac nat a, case_tac nat)
apply auto
done

lemma isnonconstant_nonconstant:
assumes "isnonconstant p"
shows "nonconstant (polypoly bs p) ⟷ Ipoly bs (head p) ≠ 0"
proof
let ?p = "polypoly bs p"
assume "nonconstant ?p"
with isnonconstant_pnormal_iff[OF assms, of bs] show "⦇head p⦈⇩p⇗bs⇖ ≠ 0"
unfolding nonconstant_def by blast
next
let ?p = "polypoly bs p"
with isnonconstant_pnormal_iff[OF assms, of bs] have pn: "pnormal ?p"
by blast
have False if H: "?p = [x]" for x
proof -
from H have "length (coefficients p) = 1"
by (auto simp: polypoly_def)
with isnonconstant_coefficients_length[OF assms]
show ?thesis by arith
qed
then show "nonconstant ?p"
using pn unfolding nonconstant_def by blast
qed

lemma pnormal_length: "p ≠ [] ⟹ pnormal p ⟷ length (pnormalize p) = length p"
apply (induct p)
apply (case_tac "p = []")
apply simp_all
done

lemma degree_degree:
assumes "isnonconstant p"
shows "degree p = Polynomial_List.degree (polypoly bs p) ⟷ ⦇head p⦈⇩p⇗bs⇖ ≠ 0"
(is "?lhs ⟷ ?rhs")
proof
let ?p = "polypoly bs p"
{
assume ?lhs
from isnonconstant_coefficients_length[OF assms] have "?p ≠ []"
by (auto simp: polypoly_def)
from ‹?lhs› degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
have "length (pnormalize ?p) = length ?p"
with pnormal_length[OF ‹?p ≠ []›] have "pnormal ?p"
by blast
with isnonconstant_pnormal_iff[OF assms] show ?rhs
by blast
next
assume ?rhs
with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
by blast
with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms] show ?lhs
by (auto simp: polypoly_def pnormal_def Polynomial_List.degree_def)
}
qed

section ‹Swaps -- division by a certain variable›

primrec swap :: "nat ⇒ nat ⇒ poly ⇒ poly"
where
"swap n m (C x) = C x"
| "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
| "swap n m (Neg t) = Neg (swap n m t)"
| "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
| "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
| "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
| "swap n m (Pw t k) = Pw (swap n m t) k"
| "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"

lemma swap:
assumes "n < length bs"
and "m < length bs"
shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
proof (induct t)
case (Bound k)
then show ?case
using assms by simp
next
case (CN c k p)
then show ?case
using assms by simp
qed simp_all

lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
by (induct t) simp_all

lemma swap_commute: "swap n m p = swap m n p"
by (induct p) simp_all

lemma swap_same_id[simp]: "swap n n t = t"
by (induct t) simp_all

definition "swapnorm n m t = polynate (swap n m t)"

lemma swapnorm:
assumes nbs: "n < length bs"
and mbs: "m < length bs"
shows "((Ipoly bs (swapnorm n m t) :: 'a::field_char_0)) =
Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
using swap[OF assms] swapnorm_def by simp

lemma swapnorm_isnpoly [simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "isnpoly (swapnorm n m p)"
unfolding swapnorm_def by simp

definition "polydivideby n s p =
(let
ss = swapnorm 0 n s;
sp = swapnorm 0 n p;
(k, r) = polydivide ss sp
in (k, swapnorm 0 n h, swapnorm 0 n r))"

lemma swap_nz [simp]: "swap n m p = 0⇩p ⟷ p = 0⇩p"
by (induct p) simp_all

fun isweaknpoly :: "poly ⇒ bool"
where
"isweaknpoly (C c) = True"
| "isweaknpoly (CN c n p) ⟷ isweaknpoly c ∧ isweaknpoly p"
| "isweaknpoly p = False"

lemma isnpolyh_isweaknpoly: "isnpolyh p n0 ⟹ isweaknpoly p"
by (induct p arbitrary: n0) auto

lemma swap_isweanpoly: "isweaknpoly p ⟹ isweaknpoly (swap n m p)"
by (induct p) auto

end
```