(* Title: HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
Author: Amine Chaieb
*)
header {* Implementation and verification of multivariate polynomials *}
theory Reflected_Multivariate_Polynomial
imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
begin
(* Implementation *)
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\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
abbreviation poly_p :: "int \<Rightarrow> poly" ("_\<^sub>p") where "i\<^sub>p \<equiv> C (i\<^sub>N)"
subsection{* Boundedness, substitution and all that *}
primrec polysize:: "poly \<Rightarrow> 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 \<Rightarrow> 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 \<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:: "poly \<Rightarrow> poly \<Rightarrow> 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 \<Rightarrow> poly"
where
"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 *}
fun degree:: "poly \<Rightarrow> nat"
where
"degree (CN c 0 p) = 1 + degree p"
| "degree p = 0"
fun head:: "poly \<Rightarrow> poly"
where
"head (CN c 0 p) = head p"
| "head p = p"
(* More general notions of degree and head *)
fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
where
"degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
|"degreen p = (\<lambda>m. 0)"
fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
where
"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)"
fun coefficients:: "poly \<Rightarrow> poly list"
where
"coefficients (CN c 0 p) = c#(coefficients p)"
| "coefficients p = [p]"
fun isconstant:: "poly \<Rightarrow> bool"
where
"isconstant (CN c 0 p) = False"
| "isconstant p = True"
fun behead:: "poly \<Rightarrow> poly"
where
"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"
fun headconst:: "poly \<Rightarrow> Num"
where
"headconst (CN c n p) = headconst p"
| "headconst (C n) = n"
subsection{* Operations for normalization *}
consts
polysub :: "poly\<times>poly \<Rightarrow> poly"
polymul :: "poly\<times>poly \<Rightarrow> poly"
abbreviation poly_mul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
where "a *\<^sub>p b \<equiv> polymul (a,b)"
abbreviation poly_sub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
where "a -\<^sub>p b \<equiv> polysub (a,b)"
declare if_cong[fundef_cong del]
declare let_cong[fundef_cong del]
fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
where
"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"
| "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"
fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
where
"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"
(hints recdef_cong del: if_cong)
declare if_cong[fundef_cong]
declare let_cong[fundef_cong]
fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
where
"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))"
abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
where "a ^\<^sub>p k \<equiv> polypow k a"
function polynate :: "poly \<Rightarrow> poly"
where
"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)"
by pat_completeness auto
termination by (relation "measure polysize") auto
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"
definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
"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 *}
definition shift1 :: "poly \<Rightarrow> poly" where
"shift1 p \<equiv> CN 0\<^sub>p 0 p"
abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
"funpow \<equiv> compow"
partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> 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')))))))"
definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
"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 *}
primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, 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 \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> 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 *}
fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
where
"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)
definition isnpoly :: "poly \<Rightarrow> bool" where
"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 ab c' n' p' n0 n1)
from prems have th1: "isnpolyh (C ab) (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 ab +\<^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' ab n1 n0)
from prems have th1: "isnpolyh (C ab) (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 ab) (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'"
with "4.hyps"(3)[OF 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 "4.hyps"(4)[OF 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 "4.hyps"(1)[OF 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 "4.hyps"(2)[OF 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 field_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)
have "n' = n \<or> n < n' \<or> n' < n" by arith
thus ?case
proof (elim disjE)
assume [simp]: "n' = n"
from 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
assume "n < n'"
with 4 show ?thesis by auto
next
assume "n' < n"
with 4 show ?thesis by auto
qed
qed auto
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)
have "n' = n \<or> n < n' \<or> n' < n" by arith
thus ?case
proof (elim disjE)
assume [simp]: "n' = n"
from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
show ?thesis by (auto simp: Let_def)
qed simp_all
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)
{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
apply (cases "p +\<^sub>p p' = 0\<^sub>p")
apply (auto simp add: Let_def)
by blast
}
ultimately have ?case by blast}
ultimately show ?case by blast
qed simp_all
lemma polymul_properties:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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)
with "2.hyps"(1-3)[of n' n' n']
and "2.hyps"(4-6)[of "Suc n'" "Suc n'" n']
show ?case by (auto simp add: min_def)
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)
with "3.hyps"(1-3)[of n n n]
"3.hyps"(4-6)[of "Suc n" "Suc n" n]
show ?case by (auto simp add: min_def)
next
case (2 n0 n1) thus ?case by auto
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'"
{
case (1 n0 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
{ assume "n < n'"
with "4.hyps"(13-14)[OF np cnp', of n]
"4.hyps"(16)[OF nc cnp', of n] nn0 cnp
have ?case by (simp add: min_def)
} moreover {
assume "n' < n"
with "4.hyps"(1-2)[OF cnp np', of "n'"]
"4.hyps"(4)[OF cnp nc', of "Suc n'"] nn1 cnp'
have ?case
by (cases "Suc n' = n", simp_all add: min_def)
} moreover {
assume "n' = n"
with "4.hyps"(1-2)[OF cnp np', of n]
"4.hyps"(4)[OF cnp nc', of n] cnp cnp' nn1 nn0
have ?case
apply (auto intro!: polyadd_normh)
apply (simp_all add: min_def isnpolyh_mono[OF nn0])
done
}
ultimately show ?case by arith
next
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"(3,15,18) np np' m
have ?eq by auto }
moreover
{assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
from "4.hyps"(1,3)[of n n' n]
"4.hyps"(4,5)[of n "Suc n'" 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"(2,3)[OF norm(1,4), of n]
"4.hyps"(4,6)[OF norm(1,2), of 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"(1-3)[OF norm(1,4), of n]
"4.hyps"(4-6)[OF norm(1,2), of 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
from "4.hyps"(1-3)[OF norm(1,4) min1]
"4.hyps"(4-6)[OF 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' np np' by simp
with "4.hyps"(1-3)[OF norm(1,4) min1]
"4.hyps"(4-6)[OF norm(1,2) min2]
degreen_0[OF norm(3) mn']
have ?eq using nn' mn np np' by clarsimp}
ultimately have ?eq by blast}
ultimately show ?eq by blast}
{ 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"(1-3) [of n n n]
"4.hyps"(4-6)[of n "Suc n" 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: field_simps)
lemma polymul_normh:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero, power})"
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 forms *}
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 normal forms *}
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::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero})"
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
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, field_inverse_zero})) ^ 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::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero})"
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 ::{field_char_0, field_inverse_zero})"
by (induct p rule:polynate.induct, auto)
lemma polynate_norm[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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 funpow_swap1)
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 :: {field_char_0, field_inverse_zero}) = 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 :: {field_char_0, field_inverse_zero}) = 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: field_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 :: {field_char_0, field_inverse_zero})"
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] field_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{* Miscellaneous lemmas about indexes, decrementation, substitution etc ... *}
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
proof(induct p arbitrary: n rule: poly.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: poly.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: poly.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)
primrec 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 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::{field_char_0, field_inverse_zero, power})"
shows "p = 0\<^sub>p"
using nq eq
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
case less
note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
{assume nz: "maxindex p = 0"
then obtain c where "p = C c" using np by (cases p, auto)
with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
moreover
{assume nz: "maxindex p \<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> maxindex p" by auto
with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p" 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> maxindex p" by simp
with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" 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"
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 less(1)[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::{field_char_0, field_inverse_zero, power})) \<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{* consequences of unicity on the algorithms for polynomial normalization *}
lemma polyadd_commute: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero})"
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::{field_char_0, field_inverse_zero})"
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>{field_char_0, field_inverse_zero, power}"] by simp
declare polyneg_polyneg[simp]
lemma isnpolyh_polynate_id[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
and np:"isnpolyh p n0" shows "polynate p = p"
using isnpolyh_unique[where ?'a= "'a::{field_char_0, field_inverse_zero}", OF polynate_norm[of p, unfolded isnpoly_def] np] polynate[where ?'a = "'a::{field_char_0, field_inverse_zero}"] by simp
lemma polynate_idempotent[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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 ::{field_char_0, field_inverse_zero}. \<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::{field_char_0, field_inverse_zero})"
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 c c') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
next
case (2 c c' n' p')
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 c')
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 add: funpow_swap1)
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 (auto simp add: polyadd_eq_const_degree)
apply (metis head_nz)
apply (metis head_nz)
apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
lemma polymul_head_polyeq:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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
using norm
"4.hyps"(5)[OF norm(1,6)]
"4.hyps"(6)[OF norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
moreover {assume nn': "n'< n"
hence ?case using norm "4.hyps"(1) [OF norm(5,3)]
"4.hyps"(2)[OF norm(5,4)]
by (simp,cases n',simp,cases n,auto)}
moreover {assume nn': "n' = n"
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 "4.hyps"(1)[OF norm(5,3)]
"4.hyps"(2)[OF 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::{field_char_0, field_inverse_zero})"
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_properties:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
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 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 "degree s" arbitrary: s k k' r n1 rule: less_induct)
case less
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 ?ths = "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 ?b = "head s"
let ?p' = "funpow (degree s - n) shift1 p"
let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
let ?akk' = "a ^\<^sub>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
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 ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }
moreover
{assume sz: "s \<noteq> 0\<^sub>p"
{assume dn: "degree s < n"
hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }
moreover
{assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> 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
{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 degree_polysub_samehead[OF ns np' headsp' degsp']
have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
moreover
{assume deglt:"degree (s -\<^sub>p ?p') < degree s"
from polydivide_aux.simps sz dn' ba
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 less(1)[OF deglt nr, of k k' r]
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::{field_char_0, field_inverse_zero} 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::{field_char_0, field_inverse_zero} 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: field_simps)
hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - 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::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p)
+ Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - 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 (degree s - 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 (degree s - 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 ?ths by blast }
moreover
{assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0, field_inverse_zero}"]
have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} 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.simps sz dn' ba
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.simps spz by simp
finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
apply auto
apply (rule exI[where x="?xdn"])
apply (auto simp add: polymul_commute[of p])
done} }
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="degree s - 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="degree s - n"]
polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
ndp 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')) < degree s"
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.simps sz dn' ba
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 less(1)[OF dth nasbp', of "Suc k" k' 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::{field_char_0, field_inverse_zero} 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: field_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="degree s - 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: field_simps)}
hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} 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 ?ths 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 ?ths by blast }
moreover
{assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
{fix bs :: "'a::{field_char_0, field_inverse_zero} 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="degree s - 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::{field_char_0, field_inverse_zero} 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::{field_char_0, field_inverse_zero}", 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 ba dn' spz polydivide_aux.simps polydivide_aux.simps
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 ?ths 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 ?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]
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::{field_char_0, field_inverse_zero})"
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_def[where s="s" and p="p"]
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 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)
apply (simp_all, 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 *}
primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> 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 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>{field_char_0, field_inverse_zero})) = 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::{field_char_0, field_inverse_zero})"
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)
fun isweaknpoly :: "poly \<Rightarrow> bool"
where
"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