--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy Sun Oct 25 08:57:36 2009 +0100
@@ -0,0 +1,1743 @@
+(* Title: HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
+ Author: Amine Chaieb
+*)
+
+header {* Implementation and verification of mutivariate polynomials Library *}
+
+
+theory Reflected_Multivariate_Polynomial
+imports Parity Abstract_Rat Efficient_Nat List Polynomial_List
+begin
+
+ (* Impelementation *)
+
+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
+
+ML{* @{term "Add"}*}
+syntax "_poly0" :: "poly" ("0\<^sub>p")
+translations "0\<^sub>p" \<rightleftharpoons> "C (0\<^sub>N)"
+syntax "_poly" :: "int \<Rightarrow> poly" ("_\<^sub>p")
+translations "i\<^sub>p" \<rightleftharpoons> "C (i\<^sub>N)"
+
+subsection{* Boundedness, substitution and all that *}
+consts polysize:: "poly \<Rightarrow> nat"
+primrec
+ "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"
+
+consts
+ polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *)
+ polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *)
+primrec
+ "polybound0 (C c) = True"
+ "polybound0 (Bound n) = (n>0)"
+ "polybound0 (Neg a) = polybound0 a"
+ "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
+ "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)"
+ "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
+ "polybound0 (Pw p n) = (polybound0 p)"
+ "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
+primrec
+ "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))"
+
+consts
+ decrpoly:: "poly \<Rightarrow> poly"
+recdef decrpoly "measure polysize"
+ "decrpoly (Bound n) = Bound (n - 1)"
+ "decrpoly (Neg a) = Neg (decrpoly a)"
+ "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
+ "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 *}
+
+consts degree:: "poly \<Rightarrow> nat"
+recdef degree "measure size"
+ "degree (CN c 0 p) = 1 + degree p"
+ "degree p = 0"
+consts head:: "poly \<Rightarrow> poly"
+
+recdef head "measure size"
+ "head (CN c 0 p) = head p"
+ "head p = p"
+ (* More general notions of degree and head *)
+consts degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
+recdef degreen "measure size"
+ "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
+ "degreen p = (\<lambda>m. 0)"
+
+consts headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
+recdef headn "measure size"
+ "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
+ "headn p = (\<lambda>m. p)"
+
+consts coefficients:: "poly \<Rightarrow> poly list"
+recdef coefficients "measure size"
+ "coefficients (CN c 0 p) = c#(coefficients p)"
+ "coefficients p = [p]"
+
+consts isconstant:: "poly \<Rightarrow> bool"
+recdef isconstant "measure size"
+ "isconstant (CN c 0 p) = False"
+ "isconstant p = True"
+
+consts behead:: "poly \<Rightarrow> poly"
+recdef behead "measure size"
+ "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
+ "behead p = 0\<^sub>p"
+
+consts headconst:: "poly \<Rightarrow> Num"
+recdef headconst "measure size"
+ "headconst (CN c n p) = headconst p"
+ "headconst (C n) = n"
+
+subsection{* Operations for normalization *}
+consts
+ polyadd :: "poly\<times>poly \<Rightarrow> poly"
+ polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
+ polysub :: "poly\<times>poly \<Rightarrow> poly"
+ polymul :: "poly\<times>poly \<Rightarrow> poly"
+ polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
+syntax "_polyadd" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
+translations "a +\<^sub>p b" \<rightleftharpoons> "polyadd (a,b)"
+syntax "_polymul" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
+translations "a *\<^sub>p b" \<rightleftharpoons> "polymul (a,b)"
+syntax "_polysub" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
+translations "a -\<^sub>p b" \<rightleftharpoons> "polysub (a,b)"
+syntax "_polypow" :: "nat \<Rightarrow> poly \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
+translations "a ^\<^sub>p k" \<rightleftharpoons> "polypow k a"
+
+recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)"
+ "polyadd (C c, C c') = C (c+\<^sub>Nc')"
+ "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"
+stupid: "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 cc' = polyadd (c,c') ;
+ pp' = polyadd (p,p')
+ in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
+ "polyadd (a, b) = Add a b"
+(hints recdef_simp add: Let_def measure_def split_def inv_image_def)
+
+(*
+declare stupid [simp del, code del]
+
+lemma [simp,code]: "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 cc' = polyadd (c,c') ;
+ pp' = polyadd (p,p')
+ in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
+ by (simp add: Let_def stupid)
+*)
+
+recdef polyneg "measure size"
+ "polyneg (C c) = C (~\<^sub>N c)"
+ "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
+ "polyneg a = Neg a"
+
+defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)"
+
+recdef polymul "measure (\<lambda>(a,b). size a + size b)"
+ "polymul(C c, C c') = C (c*\<^sub>Nc')"
+ "polymul(C c, CN c' n' p') =
+ (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul(C c,c')) n' (polymul(C c, p')))"
+ "polymul(CN c n p, C c') =
+ (if c' = 0\<^sub>N then 0\<^sub>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\<^sub>p n' (polymul(CN c n p, p'))))"
+ "polymul (a,b) = Mul a b"
+recdef polypow "measure id"
+ "polypow 0 = (\<lambda>p. 1\<^sub>p)"
+ "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in
+ if even n then d else polymul(p,d))"
+
+consts polynate :: "poly \<Rightarrow> poly"
+recdef polynate "measure polysize"
+ "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
+ "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
+ "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
+ "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
+ "polynate (Neg p) = (~\<^sub>p (polynate p))"
+ "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
+ "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
+ "polynate (C c) = C (normNum c)"
+
+fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
+ "poly_cmul y (C x) = C (y *\<^sub>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 *\<^sub>p p"
+
+constdefs monic:: "poly \<Rightarrow> (poly \<times> bool)"
+ "monic p \<equiv> (let h = headconst p in if h = 0\<^sub>N then (p,False) else ((C (Ninv h)) *\<^sub>p p, 0>\<^sub>N h))"
+
+subsection{* Pseudo-division *}
+
+constdefs shift1:: "poly \<Rightarrow> poly"
+ "shift1 p \<equiv> CN 0\<^sub>p 0 p"
+consts funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
+
+primrec
+ "funpow 0 f x = x"
+ "funpow (Suc n) f x = funpow n f (f x)"
+function (tailrec) polydivide_aux :: "(poly \<times> nat \<times> poly \<times> nat \<times> poly) \<Rightarrow> (nat \<times> poly)"
+ where
+ "polydivide_aux (a,n,p,k,s) =
+ (if s = 0\<^sub>p then (k,s)
+ else (let b = head s; 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 -\<^sub>p p')
+ else polydivide_aux (a,n,p,Suc k, (a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
+ by pat_completeness auto
+
+
+constdefs polydivide:: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)"
+ "polydivide s p \<equiv> polydivide_aux (head p,degree p,p,0, s)"
+
+fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
+ "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
+| "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
+
+fun poly_deriv :: "poly \<Rightarrow> poly" where
+ "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
+| "poly_deriv p = 0\<^sub>p"
+
+ (* Verification *)
+lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
+using Nat.gr0_conv_Suc
+by clarsimp
+
+subsection{* Semantics of the polynomial representation *}
+
+consts Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{ring_char_0,power,division_by_zero,field}"
+primrec
+ "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)"
+syntax "_Ipoly" :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{ring_char_0,power,division_by_zero,field}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
+translations "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>" \<rightleftharpoons> "Ipoly bs p"
+
+lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
+ by (simp add: INum_def)
+lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
+ by (simp add: INum_def)
+
+lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
+
+subsection {* Normal form and normalization *}
+
+consts isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
+recdef isnpolyh "measure size"
+ "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
+ "isnpolyh (CN c n p) = (\<lambda>k. n\<ge> k \<and> (isnpolyh c (Suc n)) \<and> (isnpolyh p n) \<and> (p \<noteq> 0\<^sub>p))"
+ "isnpolyh p = (\<lambda>k. False)"
+
+lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
+by (induct p rule: isnpolyh.induct, auto)
+
+constdefs isnpoly:: "poly \<Rightarrow> bool"
+ "isnpoly p \<equiv> isnpolyh p 0"
+
+text{* polyadd preserves normal forms *}
+
+lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
+ \<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
+proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
+ case (2 a b c' n' p' n0 n1)
+ from prems have th1: "isnpolyh (C (a,b)) (Suc n')" by simp
+ from prems(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" by simp_all
+ with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
+ with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
+ from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
+ thus ?case using prems th3 by simp
+next
+ case (3 c' n' p' a b n1 n0)
+ from prems have th1: "isnpolyh (C (a,b)) (Suc n')" by simp
+ from prems(2) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" by simp_all
+ with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
+ with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
+ from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
+ thus ?case using prems th3 by simp
+next
+ case (4 c n p c' n' p' n0 n1)
+ hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
+ from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
+ from prems have ngen0: "n \<ge> n0" by simp
+ from prems have n'gen1: "n' \<ge> n1" by simp
+ have "n < n' \<or> n' < n \<or> n = n'" by auto
+ moreover {assume eq: "n = n'" hence eq': "\<not> n' < n \<and> \<not> n < n'" by simp
+ with prems(2)[rule_format, OF eq' nc nc']
+ have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
+ hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
+ using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
+ from eq prems(1)[rule_format, OF eq' np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
+ have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
+ from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
+ moreover {assume lt: "n < n'"
+ have "min n0 n1 \<le> n0" by simp
+ with prems have th1:"min n0 n1 \<le> n" by auto
+ from prems have th21: "isnpolyh c (Suc n)" by simp
+ from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
+ from lt have th23: "min (Suc n) n' = Suc n" by arith
+ from prems(4)[rule_format, OF lt th21 th22]
+ have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
+ with prems th1 have ?case by simp }
+ moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
+ have "min n0 n1 \<le> n1" by simp
+ with prems have th1:"min n0 n1 \<le> n'" by auto
+ from prems have th21: "isnpolyh c' (Suc n')" by simp_all
+ from prems have th22: "isnpolyh (CN c n p) n" by simp
+ from gt have th23: "min n (Suc n') = Suc n'" by arith
+ from prems(3)[rule_format, OF gt' th22 th21]
+ have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
+ with prems th1 have ?case by simp}
+ ultimately show ?case by blast
+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 ring_simps right_distrib[symmetric] simp del: right_distrib)
+
+lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> 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*}
+
+lemma polyadd_different_degreen:
+ "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
+ degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)"
+proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
+ case (4 c n p c' n' p' m n0 n1)
+ thus ?case
+ apply (cases "n' < n", simp_all add: Let_def)
+ apply (cases "n = n'", simp_all)
+ apply (cases "n' = m", simp_all add: Let_def)
+ by (erule allE[where x="m"], erule allE[where x="Suc m"],
+ erule allE[where x="m"], erule allE[where x="Suc m"],
+ clarsimp,erule allE[where x="m"],erule allE[where x="Suc m"], simp)
+qed simp_all
+
+lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
+ by (induct p arbitrary: n rule: headn.induct, auto)
+lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
+ by (induct p arbitrary: n rule: degree.induct, auto)
+lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
+ by (induct p arbitrary: n rule: degreen.induct, auto)
+
+lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
+ by (induct p arbitrary: n rule: degree.induct, auto)
+
+lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
+ using degree_isnpolyh_Suc by auto
+lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
+ using degreen_0 by auto
+
+
+lemma degreen_polyadd:
+ assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
+ shows "degreen (p +\<^sub>p q) m \<le> 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) thus ?case by (cases n', simp_all)
+next
+ case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
+next
+ case (4 c n p c' n' p' n0 n1 m)
+ thus ?case
+ apply (cases "n < n'", simp_all add: Let_def)
+ apply (cases "n' < n", simp_all)
+ apply (erule allE[where x="n"],erule allE[where x="Suc n"],clarify)
+ apply (erule allE[where x="n'"],erule allE[where x="Suc n'"],clarify)
+ by (erule allE[where x="m"],erule allE[where x="m"], auto)
+qed auto
+
+
+lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk>
+ \<Longrightarrow> degreen p m = degreen q m"
+proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
+ case (4 c n p c' n' p' m n0 n1 x)
+ hence z: "CN c n p +\<^sub>p CN c' n' p' = C x" by simp
+ {assume nn': "n' < n" hence ?case using prems by simp}
+ moreover
+ {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
+ moreover {assume "n < n'" with prems have ?case by simp }
+ moreover {assume eq: "n = n'" hence ?case using prems
+ by (cases "p +\<^sub>p p' = 0\<^sub>p", auto simp add: Let_def) }
+ ultimately have ?case by blast}
+ ultimately show ?case by blast
+qed simp_all
+
+lemma polymul_properties:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
+ shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
+ and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
+ and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>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 a b c' n' p')
+ let ?c = "(a,b)"
+ { case (1 n0 n1)
+ hence n: "isnpolyh (C ?c) n'" "isnpolyh c' (Suc n')" "isnpolyh p' n'" "isnormNum ?c"
+ "isnpolyh (CN c' n' p') n1"
+ by simp_all
+ {assume "?c = 0\<^sub>N" hence ?case by auto}
+ moreover {assume cnz: "?c \<noteq> 0\<^sub>N"
+ from "2.hyps"(1)[rule_format,where xb="n'", OF cnz n(1) n(3)]
+ "2.hyps"(2)[rule_format, where x="Suc n'"
+ and xa="Suc n'" and xb = "n'", OF cnz ] cnz n have ?case
+ by (auto simp add: min_def)}
+ ultimately show ?case by blast
+ next
+ case (2 n0 n1) thus ?case by auto
+ next
+ case (3 n0 n1) thus ?case using "2.hyps" by auto }
+next
+ case (3 c n p a b){
+ let ?c' = "(a,b)"
+ case (1 n0 n1)
+ hence n: "isnpolyh (C ?c') n" "isnpolyh c (Suc n)" "isnpolyh p n" "isnormNum ?c'"
+ "isnpolyh (CN c n p) n0"
+ by simp_all
+ {assume "?c' = 0\<^sub>N" hence ?case by auto}
+ moreover {assume cnz: "?c' \<noteq> 0\<^sub>N"
+ from "3.hyps"(1)[rule_format,where xb="n", OF cnz n(3) n(1)]
+ "3.hyps"(2)[rule_format, where x="Suc n"
+ and xa="Suc n" and xb = "n", OF cnz ] cnz n have ?case
+ by (auto simp add: min_def)}
+ ultimately show ?case by blast
+ next
+ case (2 n0 n1) thus ?case apply auto done
+ next
+ case (3 n0 n1) thus ?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'"
+ {fix n0 n1
+ assume "isnpolyh ?cnp n0" and "isnpolyh ?cnp' n1"
+ hence 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 \<ge> n0" and nn1:"n' \<ge> n1"
+ by simp_all
+ have "n < n' \<or> n' < n \<or> n' = n" by auto
+ moreover
+ {assume nn': "n < n'"
+ with "4.hyps"(5)[rule_format, OF nn' np cnp', where xb ="n"]
+ "4.hyps"(6)[rule_format, OF nn' nc cnp', where xb="n"] nn' nn0 nn1 cnp
+ have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
+ by (simp add: min_def) }
+ moreover
+
+ {assume nn': "n > n'" hence stupid: "n' < n \<and> \<not> n < n'" by arith
+ with "4.hyps"(3)[rule_format, OF stupid cnp np', where xb="n'"]
+ "4.hyps"(4)[rule_format, OF stupid cnp nc', where xb="Suc n'"]
+ nn' nn0 nn1 cnp'
+ have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
+ by (cases "Suc n' = n", simp_all add: min_def)}
+ moreover
+ {assume nn': "n' = n" hence stupid: "\<not> n' < n \<and> \<not> n < n'" by arith
+ from "4.hyps"(1)[rule_format, OF stupid cnp np', where xb="n"]
+ "4.hyps"(2)[rule_format, OF stupid cnp nc', where xb="n"] nn' cnp cnp' nn1
+
+ have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
+ by simp (rule polyadd_normh,simp_all add: min_def isnpolyh_mono[OF nn0]) }
+ ultimately show "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)" by blast }
+ note th = this
+ {fix n0 n1 m
+ assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
+ and m: "m \<le> min n0 n1"
+ let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
+ let ?d1 = "degreen ?cnp m"
+ let ?d2 = "degreen ?cnp' m"
+ let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0 else ?d1 + ?d2)"
+ have "n'<n \<or> n < n' \<or> n' = n" by auto
+ moreover
+ {assume "n' < n \<or> n < n'"
+ with "4.hyps" np np' m
+ have ?eq apply (cases "n' < n", simp_all)
+ apply (erule allE[where x="n"],erule allE[where x="n"],auto)
+ done }
+ moreover
+ {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
+ from "4.hyps"(1)[rule_format, OF nn, where x="n" and xa ="n'" and xb="n"]
+ "4.hyps"(2)[rule_format, OF nn, where x="n" and xa ="Suc n'" and xb="n"]
+ np np' nn'
+ have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
+ "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
+ "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
+ "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
+ {assume mn: "m = n"
+ from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
+ "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"] norm nn' mn
+ have degs: "degreen (?cnp *\<^sub>p c') n =
+ (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
+ "degreen (?cnp *\<^sub>p p') n = ?d1 + degreen p' n" by (simp_all add: min_def)
+ from degs norm
+ have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
+ hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
+ by simp
+ have nmin: "n \<le> min n n" by (simp add: min_def)
+ from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
+ have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n = degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
+ from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
+ "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"]
+ mn norm m nn' deg
+ have ?eq by simp}
+ moreover
+ {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
+ from nn' m np have max1: "m \<le> max n n" by simp
+ hence min1: "m \<le> min n n" by simp
+ hence min2: "m \<le> min n (Suc n)" by simp
+ {assume "c' = 0\<^sub>p"
+ from `c' = 0\<^sub>p` have ?eq
+ using "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
+ "4.hyps"(2)[rule_format, OF nn norm(1,2) min2] mn nn'
+ apply simp
+ done}
+ moreover
+ {assume cnz: "c' \<noteq> 0\<^sub>p"
+ from "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
+ "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
+ degreen_polyadd[OF norm(3,6) max1]
+
+ have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
+ \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
+ using mn nn' cnz np np' by simp
+ with "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
+ "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
+ degreen_0[OF norm(3) mn'] have ?eq using nn' mn cnz np np' by clarsimp}
+ ultimately have ?eq by blast }
+ ultimately have ?eq by blast}
+ ultimately show ?eq by blast}
+ note degth = this
+ { case (2 n0 n1)
+ hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
+ and m: "m \<le> min n0 n1" by simp_all
+ hence mn: "m \<le> n" by simp
+ let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
+ {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
+ hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
+ from "4.hyps"(1) [rule_format, OF nn, where x="n" and xa = "n" and xb="n"]
+ "4.hyps"(2) [rule_format, OF nn, where x="n" and xa = "Suc n" and xb="n"]
+ np np' C(2) mn
+ have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
+ "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
+ "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
+ "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
+ "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
+ "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
+ by (simp_all add: min_def)
+
+ from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
+ have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
+ using norm by simp
+ from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
+ have "False" by simp }
+ thus ?case using "4.hyps" by clarsimp}
+qed auto
+
+lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
+by(induct p q rule: polymul.induct, auto simp add: ring_simps)
+
+lemma polymul_normh:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
+ using polymul_properties(1) by blast
+lemma polymul_eq0_iff:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p) "
+ using polymul_properties(2) by blast
+lemma polymul_degreen:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 else degreen p m + degreen q m)"
+ using polymul_properties(3) by blast
+lemma polymul_norm:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
+ using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
+
+lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
+ by (induct p arbitrary: n0 rule: headconst.induct, auto)
+
+lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
+ 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::{ring_char_0,power,division_by_zero,field})"
+ unfolding monic_def Let_def
+proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
+ let ?h = "headconst p"
+ assume pz: "p \<noteq> 0\<^sub>p"
+ {assume hz: "INum ?h = (0::'a)"
+ from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
+ from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
+ with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
+ thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
+qed
+
+
+
+
+text{* polyneg is a negation and preserves normal form *}
+lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
+by (induct p rule: polyneg.induct, auto)
+
+lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
+ by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
+lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
+ by (induct p arbitrary: n0 rule: polyneg.induct, auto)
+lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
+by (induct p rule: polyneg.induct, auto simp add: polyneg0)
+
+lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
+ using isnpoly_def polyneg_normh by simp
+
+
+text{* polysub is a substraction and preserves normalform *}
+lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
+by (simp add: polysub_def polyneg polyadd)
+lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
+by (simp add: polysub_def polyneg_normh polyadd_normh)
+
+lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
+ using polyadd_norm polyneg_norm by (simp add: polysub_def)
+lemma polysub_same_0[simp]: assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>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::{ring_char_0,division_by_zero,field})"
+ shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
+ unfolding polysub_def split_def fst_conv snd_conv
+ apply (induct p q arbitrary: n0 n1 rule:polyadd.induct, simp_all add: Nsub0[simplified Nsub_def])
+ apply (clarsimp simp add: Let_def)
+ apply (case_tac "n < n'", simp_all)
+ apply (case_tac "n' < n", simp_all)
+ apply (erule impE)+
+ apply (rule_tac x="Suc n" in exI, simp)
+ apply (rule_tac x="n" in exI, simp)
+ apply (erule impE)+
+ apply (rule_tac x="n" in exI, simp)
+ apply (rule_tac x="Suc n" in exI, simp)
+ apply (erule impE)+
+ apply (rule_tac x="Suc n" in exI, simp)
+ apply (rule_tac x="n" in exI, simp)
+ apply (erule impE)+
+ apply (rule_tac x="Suc n" in exI, simp)
+ apply clarsimp
+ done
+
+text{* polypow is a power function and preserves normal forms *}
+lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{ring_char_0,division_by_zero,field})) ^ n"
+proof(induct n rule: polypow.induct)
+ case 1 thus ?case by simp
+next
+ case (2 n)
+ let ?q = "polypow ((Suc n) div 2) p"
+ let ?d = "polymul(?q,?q)"
+ have "odd (Suc n) \<or> even (Suc n)" by simp
+ moreover
+ {assume odd: "odd (Suc n)"
+ have th: "(Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat))))) = Suc n div 2 + Suc n div 2 + 1" by arith
+ from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
+ also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
+ using "2.hyps" by simp
+ also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
+ apply (simp only: power_add power_one_right) by simp
+ also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
+ by (simp only: th)
+ finally have ?case
+ using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp }
+ moreover
+ {assume even: "even (Suc n)"
+ have th: "(Suc (Suc (0\<Colon>nat))) * (Suc n div Suc (Suc (0\<Colon>nat))) = Suc n div 2 + Suc n div 2" by arith
+ from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
+ also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
+ using "2.hyps" apply (simp only: power_add) by simp
+ finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
+ ultimately show ?case by blast
+qed
+
+lemma polypow_normh:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
+proof (induct k arbitrary: n rule: polypow.induct)
+ case (2 k n)
+ let ?q = "polypow (Suc k div 2) p"
+ let ?d = "polymul (?q,?q)"
+ from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
+ from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
+ from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
+ from dn on show ?case by (simp add: Let_def)
+qed auto
+
+lemma polypow_norm:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
+ by (simp add: polypow_normh isnpoly_def)
+
+text{* Finally the whole normalization*}
+
+lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{ring_char_0,division_by_zero,field})"
+by (induct p rule:polynate.induct, auto)
+
+lemma polynate_norm[simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ shows "isnpoly (polynate p)"
+ by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
+
+text{* shift1 *}
+
+
+lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
+by (simp add: shift1_def polymul)
+
+lemma shift1_isnpoly:
+ assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
+ using pn pnz by (simp add: shift1_def isnpoly_def )
+
+lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
+ by (simp add: shift1_def)
+lemma funpow_shift1_isnpoly:
+ "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
+ by (induct n arbitrary: p, auto simp add: shift1_isnpoly)
+
+lemma funpow_isnpolyh:
+ assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
+ shows "isnpolyh (funpow k f p) n"
+ using f np by (induct k arbitrary: p, auto)
+
+lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {ring_char_0,division_by_zero,field}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
+ by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
+
+lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> 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 :: {ring_char_0,division_by_zero,field}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"
+ by (simp add: funpow_shift1)
+
+lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
+by (induct p arbitrary: n0 rule: poly_cmul.induct, auto simp add: ring_simps)
+
+lemma behead:
+ assumes np: "isnpolyh p n"
+ shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {ring_char_0,division_by_zero,field})"
+ using np
+proof (induct p arbitrary: n rule: behead.induct)
+ case (1 c p n) hence pn: "isnpolyh p n" by simp
+ from prems(2)[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" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
+ by (simp_all add: th[symmetric] ring_simps power_Suc)
+qed (auto simp add: Let_def)
+
+lemma behead_isnpolyh:
+ assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
+ using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
+
+subsection{* Miscilanious lemmas about indexes, decrementation, substitution etc ... *}
+lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
+proof(induct p arbitrary: n rule: polybound0.induct, auto)
+ case (goal1 c n p n')
+ hence "n = Suc (n - 1)" by simp
+ hence "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 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
+by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
+
+lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
+
+lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
+ apply (induct p arbitrary: n0, auto)
+ apply (atomize)
+ apply (erule_tac x = "Suc nat" in allE)
+ apply auto
+ done
+
+lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
+ by (induct p arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
+
+lemma polybound0_I:
+ assumes nb: "polybound0 a"
+ shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
+using nb
+by (induct a rule: polybound0.induct) auto
+lemma polysubst0_I:
+ shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
+ by (induct t) simp_all
+
+lemma polysubst0_I':
+ assumes nb: "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 nb, where b="b" and b'="b'"])
+
+lemma decrpoly: assumes nb: "polybound0 t"
+ shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
+ using nb by (induct t rule: decrpoly.induct, simp_all)
+
+lemma polysubst0_polybound0: assumes nb: "polybound0 t"
+ shows "polybound0 (polysubst0 t a)"
+using nb by (induct a rule: polysubst0.induct, auto)
+
+lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
+ by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
+
+fun maxindex :: "poly \<Rightarrow> 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 \<Rightarrow> poly \<Rightarrow> bool" where
+ "wf_bs bs p = (length bs \<ge> maxindex p)"
+
+lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
+proof(induct p rule: coefficients.induct)
+ case (1 c p)
+ show ?case
+ proof
+ fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
+ hence "x = c \<or> x \<in> set (coefficients p)" by simp
+ moreover
+ {assume "x = c" hence "wf_bs bs x" using "1.prems" unfolding wf_bs_def by simp}
+ moreover
+ {assume H: "x \<in> set (coefficients p)"
+ from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
+ with "1.hyps" H have "wf_bs bs x" by blast }
+ ultimately show "wf_bs bs x" by blast
+ qed
+qed simp_all
+
+lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
+by (induct p rule: coefficients.induct, auto)
+
+lemma length_exists: "\<exists>xs. length xs = n" by (rule exI[where x="replicate n x"], simp)
+
+lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
+ unfolding wf_bs_def by (induct p, auto simp add: nth_append)
+
+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
+ hence 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 \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
+ by (induct p, auto)
+
+lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
+ unfolding wf_bs_def by simp
+
+lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
+ unfolding wf_bs_def by simp
+
+
+
+lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
+by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
+lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
+ by (induct p rule: coefficients.induct, simp_all)
+
+
+lemma coefficients_head: "last (coefficients p) = head p"
+ by (induct p rule: coefficients.induct, auto)
+
+lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
+ unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
+
+lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
+ apply (rule exI[where x="replicate (n - length xs) z"])
+ by simp
+lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
+by (cases p, auto) (case_tac "nat", simp_all)
+
+lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
+ unfolding wf_bs_def
+ apply (induct p q rule: polyadd.induct)
+ apply (auto simp add: Let_def)
+ done
+
+lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
+
+ unfolding wf_bs_def
+ apply (induct p q arbitrary: bs rule: polymul.induct)
+ apply (simp_all add: wf_bs_polyadd)
+ apply clarsimp
+ apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
+ apply auto
+ done
+
+lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
+ unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
+
+lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
+ unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
+
+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 o decrpoly)) (coefficients p)"
+definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
+
+lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> 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
+ hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>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 \<Longrightarrow> \<forall>q\<in>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: "\<forall> q\<in> ?cf. isnpolyh q n0" .
+ {fix q assume q: "q \<in> ?cf"
+ from q cn_norm have th: "isnpolyh q n0" by blast
+ from coefficients_isconst[OF np] q have "isconstant q" by blast
+ with isconstant_polybound0[OF th] have "polybound0 q" by blast}
+ hence "\<forall>q \<in> ?cf. polybound0 q" ..
+ hence "\<forall>q \<in> ?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
+
+ thus ?thesis unfolding polypoly_def polypoly'_def by simp
+qed
+
+lemma polypoly_poly:
+ assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
+ using np
+by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
+
+lemma polypoly'_poly:
+ assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
+ using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
+
+
+lemma polypoly_poly_polybound0:
+ assumes np: "isnpolyh p n0" and nb: "polybound0 p"
+ shows "polypoly bs p = [Ipoly bs p]"
+ using np nb unfolding polypoly_def
+ by (cases p, auto, case_tac nat, auto)
+
+lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
+ by (induct p rule: head.induct, auto)
+
+lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
+ by (cases p,auto)
+
+lemma head_eq_headn0: "head p = headn p 0"
+ by (induct p rule: head.induct, simp_all)
+
+lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
+ by (simp add: head_eq_headn0)
+
+lemma isnpolyh_zero_iff:
+ assumes nq: "isnpolyh p n0" and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{ring_char_0,power,division_by_zero,field})"
+ shows "p = 0\<^sub>p"
+using nq eq
+proof (induct n\<equiv>"maxindex p" arbitrary: p n0 rule: nat_less_induct)
+ fix n p n0
+ assume H: "\<forall>m<n. \<forall>p n0. isnpolyh p n0 \<longrightarrow>
+ (\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)) \<longrightarrow> m = maxindex p \<longrightarrow> p = 0\<^sub>p"
+ and np: "isnpolyh p n0" and zp: "\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" and n: "n = maxindex p"
+ {assume nz: "n = 0"
+ then obtain c where "p = C c" using n np by (cases p, auto)
+ with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
+ moreover
+ {assume nz: "n \<noteq> 0"
+ let ?h = "head p"
+ let ?hd = "decrpoly ?h"
+ let ?ihd = "maxindex ?hd"
+ from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
+ by simp_all
+ hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
+
+ from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
+ have mihn: "maxindex ?h \<le> n" unfolding n by auto
+ with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < n" by auto
+ {fix bs:: "'a list" assume bs: "wf_bs bs ?hd"
+ let ?ts = "take ?ihd bs"
+ let ?rs = "drop ?ihd bs"
+ have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
+ have bs_ts_eq: "?ts@ ?rs = bs" by simp
+ from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
+ from ihd_lt_n have "ALL x. length (x#?ts) \<le> n" by simp
+ with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = n" by blast
+ hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" using n unfolding wf_bs_def by simp
+ with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
+ hence "\<forall> 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 "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x" by simp
+ hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext)
+ hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
+ thm poly_zero
+ using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
+ with coefficients_head[of p, symmetric]
+ have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
+ from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
+ with th0 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 have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
+ then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
+
+ from H[rule_format, OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
+ hence "?h = 0\<^sub>p" by simp
+ with head_nz[OF np] have "p = 0\<^sub>p" by simp}
+ ultimately show "p = 0\<^sub>p" by blast
+qed
+
+lemma isnpolyh_unique:
+ assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
+ shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{ring_char_0,power,division_by_zero,field})) \<longleftrightarrow> p = q"
+proof(auto)
+ assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
+ hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
+ hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (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{* consequenses of unicity on the algorithms for polynomial normalization *}
+
+lemma polyadd_commute: assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
+ using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
+
+lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
+lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
+lemma polyadd_0[simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>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::{ring_char_0,division_by_zero,field})"
+ and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>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::{ring_char_0,division_by_zero,field})"
+ and np: "isnpolyh p n0" shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" and "0\<^sub>p *\<^sub>p p = 0\<^sub>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::{ring_char_0,division_by_zero,field})"
+ and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
+ shows "p *\<^sub>p q = q *\<^sub>p p"
+using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a\<Colon>{ring_char_0,power,division_by_zero,field}"] by simp
+
+declare polyneg_polyneg[simp]
+
+lemma isnpolyh_polynate_id[simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ and np:"isnpolyh p n0" shows "polynate p = p"
+ using isnpolyh_unique[where ?'a= "'a::{ring_char_0,division_by_zero,field}", OF polynate_norm[of p, unfolded isnpoly_def] np] polynate[where ?'a = "'a::{ring_char_0,division_by_zero,field}"] by simp
+
+lemma polynate_idempotent[simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ 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: shows "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{ring_char_0,division_by_zero,field}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
+ using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
+ unfolding poly_nate_polypoly' by (auto intro: ext)
+
+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 n: "isnpolyh p n"
+ shows "degree (polyneg p) = degree p"
+ using n
+ by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
+
+lemma degree_polyadd:
+ assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
+ shows "degree (p +\<^sub>p q) \<le> 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 -\<^sub>p q) \<le> max (degree p) (degree q)"
+proof-
+ from nq have nq': "isnpolyh (~\<^sub>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
+
+lemma degree_polysub_samehead:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
+ and d: "degree p = degree q"
+ shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
+unfolding polysub_def split_def fst_conv snd_conv
+using np nq h d
+proof(induct p q rule:polyadd.induct)
+ case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
+next
+ case (2 a b c' n' p')
+ let ?c = "(a,b)"
+ from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
+ hence nz:"n' > 0" by (cases n', auto)
+ hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
+ with prems show ?case by simp
+next
+ case (3 c n p a' b')
+ let ?c' = "(a',b')"
+ from prems have "degree (C ?c') = degree (CN c n p)" by simp
+ hence nz:"n > 0" by (cases n, auto)
+ hence "head (CN c n p) = CN c n p" by (cases n, auto)
+ with prems show ?case by simp
+next
+ case (4 c n p c' n' p')
+ hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
+ "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
+ hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
+ hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>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+
+ from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p ~\<^sub>pc') = 0" by simp
+ from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
+ have "n = n' \<or> n < n' \<or> n > n'" by arith
+ moreover
+ {assume nn': "n = n'"
+ have "n = 0 \<or> n >0" by arith
+ moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
+ moreover {assume nz: "n > 0"
+ with nn' H(3) have cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
+ hence ?case using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] polysub_same_0[OF c'nh, simplified polysub_def split_def fst_conv snd_conv] using nn' prems by (simp add: Let_def)}
+ ultimately have ?case by blast}
+ moreover
+ {assume nn': "n < n'" hence n'p: "n' > 0" by simp
+ hence headcnp':"head (CN c' n' p') = CN c' n' p'" by (cases n', simp_all)
+ have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" using prems by (cases n', simp_all)
+ hence "n > 0" by (cases n, simp_all)
+ hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
+ from H(3) headcnp headcnp' nn' have ?case by auto}
+ moreover
+ {assume nn': "n > n'" hence np: "n > 0" by simp
+ hence headcnp:"head (CN c n p) = CN c n p" by (cases n, simp_all)
+ from prems 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)
+ hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
+ from H(3) headcnp headcnp' nn' have ?case by auto}
+ ultimately show ?case by blast
+qed auto
+
+lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
+by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
+
+lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
+proof(induct k arbitrary: n0 p)
+ case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
+ with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
+ and "head (shift1 p) = head p" by (simp_all add: shift1_head)
+ thus ?case by simp
+qed auto
+
+lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
+ by (simp add: shift1_def)
+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 \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
+ by (induct n arbitrary: p, simp_all add: funpow_def)
+
+lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
+ by (induct p arbitrary: n rule: degree.induct, auto)
+lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
+ by (induct p arbitrary: n rule: degreen.induct, auto)
+lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
+ by (induct p arbitrary: n rule: degree.induct, auto)
+lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
+ by (induct p rule: head.induct, auto)
+
+lemma polyadd_eq_const_degree:
+ "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q"
+ using polyadd_eq_const_degreen degree_eq_degreen0 by simp
+
+lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
+ and deg: "degree p \<noteq> degree q"
+ shows "head (p +\<^sub>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,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 (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
+apply (clarsimp simp add: polyadd_eq_const_degree)
+apply clarsimp
+apply (erule_tac impE,blast)
+apply (erule_tac impE,blast)
+apply clarsimp
+apply simp
+apply (case_tac n', simp_all)
+done
+
+lemma polymul_head_polyeq:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ shows "\<lbrakk>isnpolyh p n0; isnpolyh q n1 ; p \<noteq> 0\<^sub>p ; q \<noteq> 0\<^sub>p \<rbrakk> \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
+proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
+ case (2 a b c' n' p' n0 n1)
+ hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)" by (simp_all add: head_isnpolyh)
+ thus ?case using prems by (cases n', auto)
+next
+ case (3 c n p a' b' n0 n1)
+ hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')" by (simp_all add: head_isnpolyh)
+ thus ?case using prems by (cases n, auto)
+next
+ case (4 c n p c' n' p' n0 n1)
+ hence 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
+ have "n < n' \<or> n' < n \<or> n = n'" by arith
+ moreover
+ {assume nn': "n < n'" hence ?case
+ thm prems
+ using norm
+ prems(6)[rule_format, OF nn' norm(1,6)]
+ prems(7)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
+ moreover {assume nn': "n'< n"
+ hence stupid: "n' < n \<and> \<not> n < n'" by simp
+ hence ?case using norm prems(4) [rule_format, OF stupid norm(5,3)]
+ prems(5)[rule_format, OF stupid norm(5,4)]
+ by (simp,cases n',simp,cases n,auto)}
+ moreover {assume nn': "n' = n"
+ hence stupid: "\<not> n' < n \<and> \<not> n < n'" by simp
+ from nn' polymul_normh[OF norm(5,4)]
+ have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
+ from nn' polymul_normh[OF norm(5,3)] norm
+ have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
+ from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
+ have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
+ from polyadd_normh[OF ncnpc' ncnpp0']
+ have nth: "isnpolyh ((CN c n p *\<^sub>p c') +\<^sub>p (CN 0\<^sub>p n (CN c n p *\<^sub>p p'))) n"
+ by (simp add: min_def)
+ {assume np: "n > 0"
+ with nn' head_isnpolyh_Suc'[OF np nth]
+ head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
+ have ?case by simp}
+ moreover
+ {moreover assume nz: "n = 0"
+ 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 *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
+ hence dth':"degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
+ from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
+ have ?case using norm prems(2)[rule_format, OF stupid norm(5,3)]
+ prems(3)[rule_format, OF stupid norm(5,4)] nn' nz by simp }
+ ultimately have ?case by (cases n) auto}
+ ultimately show ?case by blast
+qed simp_all
+
+lemma degree_coefficients: "degree p = length (coefficients p) - 1"
+ by(induct p rule: degree.induct, auto)
+
+lemma degree_head[simp]: "degree (head p) = 0"
+ by (induct p rule: head.induct, auto)
+
+lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p"
+ by (cases n, simp_all)
+lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge> degree p"
+ by (cases n, simp_all)
+
+lemma polyadd_different_degree: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow> degree (polyadd(p,q)) = max (degree p) (degree q)"
+ using polyadd_different_degreen degree_eq_degreen0 by simp
+
+lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
+ by (induct p arbitrary: n0 rule: polyneg.induct, auto)
+
+lemma degree_polymul:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
+ shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
+ using polymul_degreen[OF np nq, where m="0"] degree_eq_degreen0 by simp
+
+lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
+ by (induct p arbitrary: n rule: degree.induct, auto)
+
+lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
+ by (induct p arbitrary: n rule: degree.induct, auto)
+
+subsection {* Correctness of polynomial pseudo division *}
+
+lemma polydivide_aux_real_domintros:
+ assumes call1: "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a = head s\<rbrakk>
+ \<Longrightarrow> polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)"
+ and call2 : "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a \<noteq> head s\<rbrakk>
+ \<Longrightarrow> polydivide_aux_dom(a, n, p,Suc k, a *\<^sub>p s -\<^sub>p (head s *\<^sub>p funpow (degree s - n) shift1 p))"
+ shows "polydivide_aux_dom (a, n, p, k, s)"
+proof (rule accpI, erule polydivide_aux_rel.cases)
+ fix y aa ka na pa sa x xa xb
+ assume eqs: "y = (aa, na, pa, ka, sa -\<^sub>p xb)" "(a, n, p, k, s) = (aa, na, pa, ka, sa)"
+ and \<Gamma>1': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na"
+ "xb = funpow (xa - na) shift1 pa" "aa = x"
+
+ hence \<Gamma>1: "s \<noteq> 0\<^sub>p" "a = head s" "xa = degree s" "\<not> degree s < n" "\<not> xa < na"
+ "xb = funpow (xa - na) shift1 pa" "aa = x" by auto
+
+ with call1 have "polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)"
+ by auto
+ with eqs \<Gamma>1 show "polydivide_aux_dom y" by auto
+next
+ fix y aa ka na pa sa x xa xb
+ assume eqs: "y = (aa, na, pa, Suc ka, aa *\<^sub>p sa -\<^sub>p (x *\<^sub>p xb))"
+ "(a, n, p, k, s) =(aa, na, pa, ka, sa)"
+ and \<Gamma>2': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na"
+ "xb = funpow (xa - na) shift1 pa" "aa \<noteq> x"
+ hence \<Gamma>2: "s \<noteq> 0\<^sub>p" "a \<noteq> head s" "xa = degree s" "\<not> degree s < n"
+ "xb = funpow (xa - na) shift1 pa" "aa \<noteq> x" by auto
+ with call2 have "polydivide_aux_dom (a, n, p, Suc k, a *\<^sub>p s -\<^sub>p (head s *\<^sub>p funpow (degree s - n) shift1 p))" by auto
+ with eqs \<Gamma>2' show "polydivide_aux_dom y" by auto
+qed
+
+lemma polydivide_aux_properties:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
+ and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
+ shows "polydivide_aux_dom (a,n,p,k,s) \<and>
+ (polydivide_aux (a,n,p,k,s) = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
+ \<and> (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
+ using ns
+proof(induct d\<equiv>"degree s" arbitrary: s k k' r n1 rule: nat_less_induct)
+ fix d s k k' r n1
+ let ?D = "polydivide_aux_dom"
+ let ?dths = "?D (a, n, p, k, s)"
+ let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
+ let ?qrths = "polydivide_aux (a, n, p, k, s) = (k', r) \<longrightarrow> k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
+ \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
+ let ?ths = "?dths \<and> ?qrths"
+ let ?b = "head s"
+ let ?p' = "funpow (d - n) shift1 p"
+ let ?xdn = "funpow (d - n) shift1 1\<^sub>p"
+ let ?akk' = "a ^\<^sub>p (k' - k)"
+ assume H: "\<forall>m<d. \<forall>x xa xb xc xd.
+ isnpolyh x xd \<longrightarrow>
+ m = degree x \<longrightarrow> ?D (a, n, p, xa, x) \<and>
+ (polydivide_aux (a, n, p, xa, x) = (xb, xc) \<longrightarrow>
+ xa \<le> xb \<and> (degree xc = 0 \<or> degree xc < degree p) \<and>
+ (\<exists> nr. isnpolyh xc nr) \<and>
+ (\<exists>q n1. isnpolyh q n1 \<and> a ^\<^sub>p xb - xa *\<^sub>p x = p *\<^sub>p q +\<^sub>p xc))"
+ and ns: "isnpolyh s n1" and ds: "d = degree s"
+ 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="d -n"] isnpoly_def by simp
+ have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
+ from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
+ from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
+ have nakk':"isnpolyh ?akk' 0" by blast
+ {assume sz: "s = 0\<^sub>p"
+ hence dom: ?dths apply - apply (rule polydivide_aux_real_domintros) apply simp_all done
+ from polydivide_aux.psimps[OF dom] sz
+ have ?qrths using np apply clarsimp by (rule exI[where x="0\<^sub>p"], simp)
+ hence ?ths using dom by blast}
+ moreover
+ {assume sz: "s \<noteq> 0\<^sub>p"
+ {assume dn: "d < n"
+ with sz ds have dom:"?dths" by - (rule polydivide_aux_real_domintros,simp_all)
+ from polydivide_aux.psimps[OF dom] sz dn ds
+ have "?qrths" using ns ndp np by auto (rule exI[where x="0\<^sub>p"],simp)
+ with dom have ?ths by blast}
+ moreover
+ {assume dn': "\<not> d < n" hence dn: "d \<ge> n" by arith
+ have degsp': "degree s = degree ?p'"
+ using ds dn ndp funpow_shift1_degree[where k = "d - n" and p="p"] by simp
+ {assume ba: "?b = a"
+ hence headsp': "head s = head ?p'" using ap headp' by simp
+ have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
+ from ds degree_polysub_samehead[OF ns np' headsp' degsp']
+ have "degree (s -\<^sub>p ?p') < d \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
+ moreover
+ {assume deglt:"degree (s -\<^sub>p ?p') < d"
+ from H[rule_format, OF deglt nr,simplified]
+ have domsp: "?D (a, n, p, k, s -\<^sub>p ?p')" by blast
+ have dom: ?dths apply (rule polydivide_aux_real_domintros)
+ using ba ds dn' domsp by simp_all
+ from polydivide_aux.psimps[OF dom] sz dn' ba ds
+ have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
+ by (simp add: Let_def)
+ {assume h1: "polydivide_aux (a, n, p, k, s) = (k', r)"
+ from H[rule_format, OF deglt nr, where xa = "k" and xb="k'" and xc="r", simplified]
+ trans[OF eq[symmetric] h1]
+ have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
+ and q1:"\<exists>q nq. isnpolyh q nq \<and> (a ^\<^sub>p k' - k *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r)" by auto
+ from q1 obtain q n1 where nq: "isnpolyh q n1"
+ and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>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 ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
+ from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
+ have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
+ from polyadd_normh[OF polymul_normh[OF np
+ polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
+ have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
+ from asp have "\<forall> (bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) =
+ Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
+ hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
+ Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
+ by (simp add: ring_simps)
+ hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
+ Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (d - n) shift1 1\<^sub>p *\<^sub>p p)
+ + Ipoly bs p * Ipoly bs q + Ipoly bs r"
+ by (auto simp only: funpow_shift1_1)
+ hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
+ Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (d - n) shift1 1\<^sub>p)
+ + Ipoly bs q) + Ipoly bs r" by (simp add: ring_simps)
+ hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
+ Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (d - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r)" by simp
+ with isnpolyh_unique[OF nakks' nqr']
+ have "a ^\<^sub>p (k' - k) *\<^sub>p s =
+ p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (d - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r" by blast
+ hence ?qths using nq'
+ apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (d - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
+ apply (rule_tac x="0" in exI) by simp
+ with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
+ by blast } hence ?qrths by blast
+ with dom have ?ths by blast}
+ moreover
+ {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
+ hence domsp: "?D (a, n, p, k, s -\<^sub>p ?p')"
+ apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
+ have dom: ?dths apply (rule polydivide_aux_real_domintros)
+ using ba ds dn' domsp by simp_all
+ from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{ring_char_0,division_by_zero,field}"]
+ have " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs ?p'" by simp
+ hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)" using np nxdn apply simp
+ by (simp only: funpow_shift1_1) simp
+ hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
+ {assume h1: "polydivide_aux (a,n,p,k,s) = (k',r)"
+ from polydivide_aux.psimps[OF dom] sz dn' ba ds
+ have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
+ by (simp add: Let_def)
+ also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.psimps[OF domsp] spz by simp
+ finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
+ with sp' ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
+ polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?qrths
+ apply auto
+ apply (rule exI[where x="?xdn"])
+ apply auto
+ apply (rule polymul_commute)
+ apply simp_all
+ done}
+ with dom have ?ths by blast}
+ ultimately have ?ths by blast }
+ moreover
+ {assume ba: "?b \<noteq> a"
+ from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
+ polymul_normh[OF head_isnpolyh[OF ns] np']]
+ have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
+ have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
+ using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
+ polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
+ funpow_shift1_nz[OF pnz] by simp_all
+ from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
+ polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="d - n"]
+ have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
+ using head_head[OF ns] funpow_shift1_head[OF np pnz]
+ polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
+ by (simp add: ap)
+ from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
+ head_nz[OF np] pnz sz ap[symmetric]
+ funpow_shift1_nz[OF pnz, where n="d - n"]
+ polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
+ ndp ds[symmetric] dn
+ have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
+ by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
+ {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < d"
+ have th: "?D (a, n, p, Suc k, (a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))"
+ using H[rule_format, OF dth nth, simplified] by blast
+ have dom: ?dths
+ using ba ds dn' th apply simp apply (rule polydivide_aux_real_domintros)
+ using ba ds dn' by simp_all
+ from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
+ ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
+ {assume h1:"polydivide_aux (a,n,p,k,s) = (k', r)"
+ from h1 polydivide_aux.psimps[OF dom] sz dn' ba ds
+ have eq:"polydivide_aux (a,n,p,Suc k,(a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
+ by (simp add: Let_def)
+ with H[rule_format, OF dth nasbp', simplified, where xa="Suc k" and xb="k'" and xc="r"]
+ obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
+ and dr: "degree r = 0 \<or> degree r < degree p"
+ and qr: "a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = p *\<^sub>p q +\<^sub>p r" by auto
+ from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
+ {fix bs:: "'a::{ring_char_0,division_by_zero,field} list"
+
+ 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 ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
+ hence "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"
+ by (simp add: ring_simps power_Suc)
+ hence "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="d - n" and p="p"])
+ hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
+ by (simp add: ring_simps)}
+ hence ieq:"\<forall>(bs :: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
+ Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)" by auto
+ let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>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 ^\<^sub>p (k' - k) *\<^sub>p s) = p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r" by blast
+ from dr kk' nr h1 asth nqw have ?qrths apply simp
+ apply (rule conjI)
+ apply (rule exI[where x="nr"], simp)
+ apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
+ apply (rule exI[where x="0"], simp)
+ done}
+ hence ?qrths by blast
+ with dom have ?ths by blast}
+ moreover
+ {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
+ hence domsp: "?D (a, n, p, Suc k, a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p'))"
+ apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
+ have dom: ?dths using sz ba dn' ds domsp
+ by - (rule polydivide_aux_real_domintros, simp_all)
+ {fix bs :: "'a::{ring_char_0,division_by_zero,field} list"
+ from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
+ have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
+ hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
+ by (simp add: funpow_shift1_1[where n="d - n" and p="p"])
+ hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
+ }
+ hence hth: "\<forall> (bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
+ from hth
+ have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
+ using isnpolyh_unique[where ?'a = "'a::{ring_char_0,division_by_zero,field}", 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
+ {assume h1: "polydivide_aux (a,n,p,k,s) = (k', r)"
+ from h1 sz ds ba dn' spz polydivide_aux.psimps[OF dom] polydivide_aux.psimps[OF domsp]
+ have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
+ 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
+ have ?qrths apply (clarsimp simp add: Let_def)
+ apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
+ apply (rule exI[where x="0"], simp)
+ done}
+ hence ?qrths by blast
+ with dom have ?ths by blast}
+ ultimately have ?ths using degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
+ head_nz[OF np] pnz sz ap[symmetric] ds[symmetric]
+ by (simp add: degree_eq_degreen0[symmetric]) blast }
+ ultimately have ?ths by blast
+ }
+ ultimately have ?ths by blast}
+ ultimately show ?ths by blast
+qed
+
+lemma polydivide_properties:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
+ shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
+ \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
+proof-
+ have trv: "head p = head p" "degree p = degree p" by simp_all
+ from polydivide_aux_properties[OF np ns trv pnz, where k="0"]
+ have d: "polydivide_aux_dom (head p, degree p, p, 0, s)" by blast
+ from polydivide_def[where s="s" and p="p"] polydivide_aux.psimps[OF d]
+ have ex: "\<exists> 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 meta_eq_to_obj_eq[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 \<or> degree r < degree p) \<and>
+ (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> head p ^\<^sub>p k - 0 *\<^sub>p s = p *\<^sub>p q +\<^sub>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 = (\<not> isconstant p)"
+
+lemma last_map: "xs \<noteq> [] ==> last (map f xs) = f (last xs)" by (induct xs, auto)
+
+lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p"
+ shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
+proof
+ let ?p = "polypoly bs p"
+ assume H: "pnormal ?p"
+ have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
+
+ from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
+ pnormal_last_nonzero[OF H]
+ show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
+next
+ assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
+ let ?p = "polypoly bs p"
+ have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
+ hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
+ hence lg: "length ?p > 0" by simp
+ from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
+ have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
+ from pnormal_last_length[OF lg lz] show "pnormal ?p" .
+qed
+
+lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
+ unfolding isnonconstant_def
+ apply (cases p, simp_all)
+ apply (case_tac nat, auto)
+ done
+lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
+ shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
+proof
+ let ?p = "polypoly bs p"
+ assume nc: "nonconstant ?p"
+ from isnonconstant_pnormal_iff[OF inc, of bs] nc
+ show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
+next
+ let ?p = "polypoly bs p"
+ assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
+ from isnonconstant_pnormal_iff[OF inc, of bs] h
+ have pn: "pnormal ?p" by blast
+ {fix x assume H: "?p = [x]"
+ from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
+ with isnonconstant_coefficients_length[OF inc] have False by arith}
+ thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
+qed
+
+lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
+ unfolding pnormal_def
+ apply (induct p rule: pnormalize.induct, simp_all)
+ apply (case_tac "p=[]", simp_all)
+ done
+
+lemma degree_degree: assumes inc: "isnonconstant p"
+ shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
+proof
+ let ?p = "polypoly bs p"
+ assume H: "degree p = Polynomial_List.degree ?p"
+ from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
+ unfolding polypoly_def by auto
+ from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
+ have lg:"length (pnormalize ?p) = length ?p"
+ unfolding Polynomial_List.degree_def polypoly_def by simp
+ hence "pnormal ?p" using pnormal_length[OF pz] by blast
+ with isnonconstant_pnormal_iff[OF inc]
+ show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
+next
+ let ?p = "polypoly bs p"
+ assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
+ with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
+ with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
+ show "degree p = Polynomial_List.degree ?p"
+ unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
+qed
+
+section{* Swaps ; Division by a certain variable *}
+consts swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
+primrec
+ "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 nbs: "n < length bs" and mbs: "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) thus ?case using nbs mbs by simp
+next
+ case (CN c k p) thus ?case using nbs mbs 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\<Colon>{ring_char_0,division_by_zero,field})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
+ using swap[OF prems] swapnorm_def by simp
+
+lemma swapnorm_isnpoly[simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
+ 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 ; h = head sp; (k,r) = polydivide ss sp
+ in (k,swapnorm 0 n h,swapnorm 0 n r))"
+
+lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
+
+consts isweaknpoly :: "poly \<Rightarrow> bool"
+recdef isweaknpoly "measure size"
+ "isweaknpoly (C c) = True"
+ "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
+ "isweaknpoly p = False"
+
+lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
+ by (induct p arbitrary: n0, auto)
+
+lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
+ by (induct p, auto)
+
+end
\ No newline at end of file