src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author haftmann Mon Feb 08 17:12:24 2010 +0100 (2010-02-08) changeset 35046 1266f04f42ec parent 34915 7894c7dab132 child 35054 a5db9779b026 permissions -rw-r--r--
tuned header
```     1 (*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header {* Implementation and verification of multivariate polynomials *}
```
```     6
```
```     7 theory Reflected_Multivariate_Polynomial
```
```     8 imports Complex_Main Abstract_Rat Polynomial_List
```
```     9 begin
```
```    10
```
```    11   (* Implementation *)
```
```    12
```
```    13 subsection{* Datatype of polynomial expressions *}
```
```    14
```
```    15 datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
```
```    16   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
```
```    17
```
```    18 ML{* @{term "Add"}*}
```
```    19 syntax "_poly0" :: "poly" ("0\<^sub>p")
```
```    20 translations "0\<^sub>p" \<rightleftharpoons> "C (0\<^sub>N)"
```
```    21 syntax "_poly" :: "int \<Rightarrow> poly" ("_\<^sub>p")
```
```    22 translations "i\<^sub>p" \<rightleftharpoons> "C (i\<^sub>N)"
```
```    23
```
```    24 subsection{* Boundedness, substitution and all that *}
```
```    25 consts polysize:: "poly \<Rightarrow> nat"
```
```    26 primrec
```
```    27   "polysize (C c) = 1"
```
```    28   "polysize (Bound n) = 1"
```
```    29   "polysize (Neg p) = 1 + polysize p"
```
```    30   "polysize (Add p q) = 1 + polysize p + polysize q"
```
```    31   "polysize (Sub p q) = 1 + polysize p + polysize q"
```
```    32   "polysize (Mul p q) = 1 + polysize p + polysize q"
```
```    33   "polysize (Pw p n) = 1 + polysize p"
```
```    34   "polysize (CN c n p) = 4 + polysize c + polysize p"
```
```    35
```
```    36 consts
```
```    37   polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *)
```
```    38   polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *)
```
```    39 primrec
```
```    40   "polybound0 (C c) = True"
```
```    41   "polybound0 (Bound n) = (n>0)"
```
```    42   "polybound0 (Neg a) = polybound0 a"
```
```    43   "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
```
```    44   "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)"
```
```    45   "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
```
```    46   "polybound0 (Pw p n) = (polybound0 p)"
```
```    47   "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
```
```    48 primrec
```
```    49   "polysubst0 t (C c) = (C c)"
```
```    50   "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"
```
```    51   "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
```
```    52   "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
```
```    53   "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
```
```    54   "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
```
```    55   "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
```
```    56   "polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
```
```    57                              else CN (polysubst0 t c) n (polysubst0 t p))"
```
```    58
```
```    59 consts
```
```    60   decrpoly:: "poly \<Rightarrow> poly"
```
```    61 recdef decrpoly "measure polysize"
```
```    62   "decrpoly (Bound n) = Bound (n - 1)"
```
```    63   "decrpoly (Neg a) = Neg (decrpoly a)"
```
```    64   "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
```
```    65   "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
```
```    66   "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
```
```    67   "decrpoly (Pw p n) = Pw (decrpoly p) n"
```
```    68   "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
```
```    69   "decrpoly a = a"
```
```    70
```
```    71 subsection{* Degrees and heads and coefficients *}
```
```    72
```
```    73 consts degree:: "poly \<Rightarrow> nat"
```
```    74 recdef degree "measure size"
```
```    75   "degree (CN c 0 p) = 1 + degree p"
```
```    76   "degree p = 0"
```
```    77 consts head:: "poly \<Rightarrow> poly"
```
```    78
```
```    79 recdef head "measure size"
```
```    80   "head (CN c 0 p) = head p"
```
```    81   "head p = p"
```
```    82   (* More general notions of degree and head *)
```
```    83 consts degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
```
```    84 recdef degreen "measure size"
```
```    85   "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
```
```    86   "degreen p = (\<lambda>m. 0)"
```
```    87
```
```    88 consts headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
```
```    89 recdef headn "measure size"
```
```    90   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
```
```    91   "headn p = (\<lambda>m. p)"
```
```    92
```
```    93 consts coefficients:: "poly \<Rightarrow> poly list"
```
```    94 recdef coefficients "measure size"
```
```    95   "coefficients (CN c 0 p) = c#(coefficients p)"
```
```    96   "coefficients p = [p]"
```
```    97
```
```    98 consts isconstant:: "poly \<Rightarrow> bool"
```
```    99 recdef isconstant "measure size"
```
```   100   "isconstant (CN c 0 p) = False"
```
```   101   "isconstant p = True"
```
```   102
```
```   103 consts behead:: "poly \<Rightarrow> poly"
```
```   104 recdef behead "measure size"
```
```   105   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
```
```   106   "behead p = 0\<^sub>p"
```
```   107
```
```   108 consts headconst:: "poly \<Rightarrow> Num"
```
```   109 recdef headconst "measure size"
```
```   110   "headconst (CN c n p) = headconst p"
```
```   111   "headconst (C n) = n"
```
```   112
```
```   113 subsection{* Operations for normalization *}
```
```   114 consts
```
```   115   polyadd :: "poly\<times>poly \<Rightarrow> poly"
```
```   116   polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   117   polysub :: "poly\<times>poly \<Rightarrow> poly"
```
```   118   polymul :: "poly\<times>poly \<Rightarrow> poly"
```
```   119   polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   120 syntax "_polyadd" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   121 translations "a +\<^sub>p b" \<rightleftharpoons> "polyadd (a,b)"
```
```   122 syntax "_polymul" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   123 translations "a *\<^sub>p b" \<rightleftharpoons> "polymul (a,b)"
```
```   124 syntax "_polysub" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   125 translations "a -\<^sub>p b" \<rightleftharpoons> "polysub (a,b)"
```
```   126 syntax "_polypow" :: "nat \<Rightarrow> poly \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   127 translations "a ^\<^sub>p k" \<rightleftharpoons> "polypow k a"
```
```   128
```
```   129 recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)"
```
```   130   "polyadd (C c, C c') = C (c+\<^sub>Nc')"
```
```   131   "polyadd (C c, CN c' n' p') = CN (polyadd (C c, c')) n' p'"
```
```   132   "polyadd (CN c n p, C c') = CN (polyadd (c, C c')) n p"
```
```   133 stupid:  "polyadd (CN c n p, CN c' n' p') =
```
```   134     (if n < n' then CN (polyadd(c,CN c' n' p')) n p
```
```   135      else if n'<n then CN (polyadd(CN c n p, c')) n' p'
```
```   136      else (let cc' = polyadd (c,c') ;
```
```   137                pp' = polyadd (p,p')
```
```   138            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
```
```   139   "polyadd (a, b) = Add a b"
```
```   140 (hints recdef_simp add: Let_def measure_def split_def inv_image_def)
```
```   141
```
```   142 (*
```
```   143 declare stupid [simp del, code del]
```
```   144
```
```   145 lemma [simp,code]: "polyadd (CN c n p, CN c' n' p') =
```
```   146     (if n < n' then CN (polyadd(c,CN c' n' p')) n p
```
```   147      else if n'<n then CN (polyadd(CN c n p, c')) n' p'
```
```   148      else (let cc' = polyadd (c,c') ;
```
```   149                pp' = polyadd (p,p')
```
```   150            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
```
```   151   by (simp add: Let_def stupid)
```
```   152 *)
```
```   153
```
```   154 recdef polyneg "measure size"
```
```   155   "polyneg (C c) = C (~\<^sub>N c)"
```
```   156   "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   157   "polyneg a = Neg a"
```
```   158
```
```   159 defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)"
```
```   160
```
```   161 recdef polymul "measure (\<lambda>(a,b). size a + size b)"
```
```   162   "polymul(C c, C c') = C (c*\<^sub>Nc')"
```
```   163   "polymul(C c, CN c' n' p') =
```
```   164       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul(C c,c')) n' (polymul(C c, p')))"
```
```   165   "polymul(CN c n p, C c') =
```
```   166       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul(c,C c')) n (polymul(p, C c')))"
```
```   167   "polymul(CN c n p, CN c' n' p') =
```
```   168   (if n<n' then CN (polymul(c,CN c' n' p')) n (polymul(p,CN c' n' p'))
```
```   169   else if n' < n
```
```   170   then CN (polymul(CN c n p,c')) n' (polymul(CN c n p,p'))
```
```   171   else polyadd(polymul(CN c n p, c'),CN 0\<^sub>p n' (polymul(CN c n p, p'))))"
```
```   172   "polymul (a,b) = Mul a b"
```
```   173 recdef polypow "measure id"
```
```   174   "polypow 0 = (\<lambda>p. 1\<^sub>p)"
```
```   175   "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in
```
```   176                     if even n then d else polymul(p,d))"
```
```   177
```
```   178 consts polynate :: "poly \<Rightarrow> poly"
```
```   179 recdef polynate "measure polysize"
```
```   180   "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
```
```   181   "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
```
```   182   "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
```
```   183   "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
```
```   184   "polynate (Neg p) = (~\<^sub>p (polynate p))"
```
```   185   "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
```
```   186   "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   187   "polynate (C c) = C (normNum c)"
```
```   188
```
```   189 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
```
```   190   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   191 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   192 | "poly_cmul y p = C y *\<^sub>p p"
```
```   193
```
```   194 constdefs monic:: "poly \<Rightarrow> (poly \<times> bool)"
```
```   195   "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))"
```
```   196
```
```   197 subsection{* Pseudo-division *}
```
```   198
```
```   199 constdefs shift1:: "poly \<Rightarrow> poly"
```
```   200   "shift1 p \<equiv> CN 0\<^sub>p 0 p"
```
```   201 consts funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   202
```
```   203 primrec
```
```   204   "funpow 0 f x = x"
```
```   205   "funpow (Suc n) f x = funpow n f (f x)"
```
```   206 function (tailrec) polydivide_aux :: "(poly \<times> nat \<times> poly \<times> nat \<times> poly) \<Rightarrow> (nat \<times> poly)"
```
```   207   where
```
```   208   "polydivide_aux (a,n,p,k,s) =
```
```   209   (if s = 0\<^sub>p then (k,s)
```
```   210   else (let b = head s; m = degree s in
```
```   211   (if m < n then (k,s) else
```
```   212   (let p'= funpow (m - n) shift1 p in
```
```   213   (if a = b then polydivide_aux (a,n,p,k,s -\<^sub>p p')
```
```   214   else polydivide_aux (a,n,p,Suc k, (a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
```
```   215   by pat_completeness auto
```
```   216
```
```   217
```
```   218 constdefs polydivide:: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)"
```
```   219   "polydivide s p \<equiv> polydivide_aux (head p,degree p,p,0, s)"
```
```   220
```
```   221 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```   222   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   223 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   224
```
```   225 fun poly_deriv :: "poly \<Rightarrow> poly" where
```
```   226   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   227 | "poly_deriv p = 0\<^sub>p"
```
```   228
```
```   229   (* Verification *)
```
```   230 lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
```
```   231 using Nat.gr0_conv_Suc
```
```   232 by clarsimp
```
```   233
```
```   234 subsection{* Semantics of the polynomial representation *}
```
```   235
```
```   236 consts Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{ring_char_0,power,division_by_zero,field}"
```
```   237 primrec
```
```   238   "Ipoly bs (C c) = INum c"
```
```   239   "Ipoly bs (Bound n) = bs!n"
```
```   240   "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   241   "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   242   "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   243   "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   244   "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
```
```   245   "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
```
```   246 syntax "_Ipoly" :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{ring_char_0,power,division_by_zero,field}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
```
```   247 translations "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>" \<rightleftharpoons> "Ipoly bs p"
```
```   248
```
```   249 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
```
```   250   by (simp add: INum_def)
```
```   251 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
```
```   252   by (simp  add: INum_def)
```
```   253
```
```   254 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
```
```   255
```
```   256 subsection {* Normal form and normalization *}
```
```   257
```
```   258 consts isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
```
```   259 recdef isnpolyh "measure size"
```
```   260   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
```
```   261   "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))"
```
```   262   "isnpolyh p = (\<lambda>k. False)"
```
```   263
```
```   264 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
```
```   265 by (induct p rule: isnpolyh.induct, auto)
```
```   266
```
```   267 constdefs isnpoly:: "poly \<Rightarrow> bool"
```
```   268   "isnpoly p \<equiv> isnpolyh p 0"
```
```   269
```
```   270 text{* polyadd preserves normal forms *}
```
```   271
```
```   272 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
```
```   273       \<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
```
```   274 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```   275   case (2 a b c' n' p' n0 n1)
```
```   276   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   277   from prems(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   278   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   279   with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
```
```   280   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   281   thus ?case using prems th3 by simp
```
```   282 next
```
```   283   case (3 c' n' p' a b n1 n0)
```
```   284   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   285   from prems(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   286   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   287   with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
```
```   288   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   289   thus ?case using prems th3 by simp
```
```   290 next
```
```   291   case (4 c n p c' n' p' n0 n1)
```
```   292   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
```
```   293   from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
```
```   294   from prems have ngen0: "n \<ge> n0" by simp
```
```   295   from prems have n'gen1: "n' \<ge> n1" by simp
```
```   296   have "n < n' \<or> n' < n \<or> n = n'" by auto
```
```   297   moreover {assume eq: "n = n'" hence eq': "\<not> n' < n \<and> \<not> n < n'" by simp
```
```   298     with prems(2)[rule_format, OF eq' nc nc']
```
```   299     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
```
```   300     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   301       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
```
```   302     from eq prems(1)[rule_format, OF eq' np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
```
```   303     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
```
```   304     from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
```
```   305   moreover {assume lt: "n < n'"
```
```   306     have "min n0 n1 \<le> n0" by simp
```
```   307     with prems have th1:"min n0 n1 \<le> n" by auto
```
```   308     from prems have th21: "isnpolyh c (Suc n)" by simp
```
```   309     from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
```
```   310     from lt have th23: "min (Suc n) n' = Suc n" by arith
```
```   311     from prems(4)[rule_format, OF lt th21 th22]
```
```   312     have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
```
```   313     with prems th1 have ?case by simp }
```
```   314   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
```
```   315     have "min n0 n1 \<le> n1"  by simp
```
```   316     with prems have th1:"min n0 n1 \<le> n'" by auto
```
```   317     from prems have th21: "isnpolyh c' (Suc n')" by simp_all
```
```   318     from prems have th22: "isnpolyh (CN c n p) n" by simp
```
```   319     from gt have th23: "min n (Suc n') = Suc n'" by arith
```
```   320     from prems(3)[rule_format, OF  gt' th22 th21]
```
```   321     have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
```
```   322     with prems th1 have ?case by simp}
```
```   323       ultimately show ?case by blast
```
```   324 qed auto
```
```   325
```
```   326 lemma polyadd[simp]: "Ipoly bs (polyadd (p,q)) = (Ipoly bs p) + (Ipoly bs q)"
```
```   327 by (induct p q rule: polyadd.induct, auto simp add: Let_def ring_simps right_distrib[symmetric] simp del: right_distrib)
```
```   328
```
```   329 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd(p,q))"
```
```   330   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   331
```
```   332 text{* The degree of addition and other general lemmas needed for the normal form of polymul*}
```
```   333
```
```   334 lemma polyadd_different_degreen:
```
```   335   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   336   degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)"
```
```   337 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   338   case (4 c n p c' n' p' m n0 n1)
```
```   339   thus ?case
```
```   340     apply (cases "n' < n", simp_all add: Let_def)
```
```   341     apply (cases "n = n'", simp_all)
```
```   342     apply (cases "n' = m", simp_all add: Let_def)
```
```   343     by (erule allE[where x="m"], erule allE[where x="Suc m"],
```
```   344            erule allE[where x="m"], erule allE[where x="Suc m"],
```
```   345            clarsimp,erule allE[where x="m"],erule allE[where x="Suc m"], simp)
```
```   346 qed simp_all
```
```   347
```
```   348 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   349   by (induct p arbitrary: n rule: headn.induct, auto)
```
```   350 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   351   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   352 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   353   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```   354
```
```   355 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   356   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   357
```
```   358 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   359   using degree_isnpolyh_Suc by auto
```
```   360 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   361   using degreen_0 by auto
```
```   362
```
```   363
```
```   364 lemma degreen_polyadd:
```
```   365   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
```
```   366   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   367   using np nq m
```
```   368 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   369   case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
```
```   370 next
```
```   371   case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
```
```   372 next
```
```   373   case (4 c n p c' n' p' n0 n1 m)
```
```   374   thus ?case
```
```   375     apply (cases "n < n'", simp_all add: Let_def)
```
```   376     apply (cases "n' < n", simp_all)
```
```   377     apply (erule allE[where x="n"],erule allE[where x="Suc n"],clarify)
```
```   378     apply (erule allE[where x="n'"],erule allE[where x="Suc n'"],clarify)
```
```   379     by (erule allE[where x="m"],erule allE[where x="m"], auto)
```
```   380 qed auto
```
```   381
```
```   382
```
```   383 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk>
```
```   384   \<Longrightarrow> degreen p m = degreen q m"
```
```   385 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   386   case (4 c n p c' n' p' m n0 n1 x)
```
```   387   hence z: "CN c n p +\<^sub>p CN c' n' p' = C x" by simp
```
```   388   {assume nn': "n' < n" hence ?case using prems by simp}
```
```   389   moreover
```
```   390   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
```
```   391     moreover {assume "n < n'" with prems have ?case by simp }
```
```   392     moreover {assume eq: "n = n'" hence ?case using prems
```
```   393         by (cases "p +\<^sub>p p' = 0\<^sub>p", auto simp add: Let_def) }
```
```   394     ultimately have ?case by blast}
```
```   395   ultimately show ?case by blast
```
```   396 qed simp_all
```
```   397
```
```   398 lemma polymul_properties:
```
```   399   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   400   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
```
```   401   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   402   and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
```
```   403   and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0
```
```   404                              else degreen p m + degreen q m)"
```
```   405   using np nq m
```
```   406 proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   407   case (2 a b c' n' p')
```
```   408   let ?c = "(a,b)"
```
```   409   { case (1 n0 n1)
```
```   410     hence n: "isnpolyh (C ?c) n'" "isnpolyh c' (Suc n')" "isnpolyh p' n'" "isnormNum ?c"
```
```   411       "isnpolyh (CN c' n' p') n1"
```
```   412       by simp_all
```
```   413     {assume "?c = 0\<^sub>N" hence ?case by auto}
```
```   414       moreover {assume cnz: "?c \<noteq> 0\<^sub>N"
```
```   415         from "2.hyps"(1)[rule_format,where xb="n'",  OF cnz n(1) n(3)]
```
```   416           "2.hyps"(2)[rule_format, where x="Suc n'"
```
```   417           and xa="Suc n'" and xb = "n'", OF cnz ] cnz n have ?case
```
```   418           by (auto simp add: min_def)}
```
```   419       ultimately show ?case by blast
```
```   420   next
```
```   421     case (2 n0 n1) thus ?case by auto
```
```   422   next
```
```   423     case (3 n0 n1) thus ?case  using "2.hyps" by auto }
```
```   424 next
```
```   425   case (3 c n p a b){
```
```   426     let ?c' = "(a,b)"
```
```   427     case (1 n0 n1)
```
```   428     hence n: "isnpolyh (C ?c') n" "isnpolyh c (Suc n)" "isnpolyh p n" "isnormNum ?c'"
```
```   429       "isnpolyh (CN c n p) n0"
```
```   430       by simp_all
```
```   431     {assume "?c' = 0\<^sub>N" hence ?case by auto}
```
```   432       moreover {assume cnz: "?c' \<noteq> 0\<^sub>N"
```
```   433         from "3.hyps"(1)[rule_format,where xb="n",  OF cnz n(3) n(1)]
```
```   434           "3.hyps"(2)[rule_format, where x="Suc n"
```
```   435           and xa="Suc n" and xb = "n", OF cnz ] cnz n have ?case
```
```   436           by (auto simp add: min_def)}
```
```   437       ultimately show ?case by blast
```
```   438   next
```
```   439     case (2 n0 n1) thus ?case apply auto done
```
```   440   next
```
```   441     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
```
```   442 next
```
```   443   case (4 c n p c' n' p')
```
```   444   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   445     {fix n0 n1
```
```   446       assume "isnpolyh ?cnp n0" and "isnpolyh ?cnp' n1"
```
```   447       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
```
```   448         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
```
```   449         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
```
```   450         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
```
```   451         by simp_all
```
```   452       have "n < n' \<or> n' < n \<or> n' = n" by auto
```
```   453       moreover
```
```   454       {assume nn': "n < n'"
```
```   455         with "4.hyps"(5)[rule_format, OF nn' np cnp', where xb ="n"]
```
```   456           "4.hyps"(6)[rule_format, OF nn' nc cnp', where xb="n"] nn' nn0 nn1 cnp
```
```   457         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   458           by (simp add: min_def) }
```
```   459       moreover
```
```   460
```
```   461       {assume nn': "n > n'" hence stupid: "n' < n \<and> \<not> n < n'" by arith
```
```   462         with "4.hyps"(3)[rule_format, OF stupid cnp np', where xb="n'"]
```
```   463           "4.hyps"(4)[rule_format, OF stupid cnp nc', where xb="Suc n'"]
```
```   464           nn' nn0 nn1 cnp'
```
```   465         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   466           by (cases "Suc n' = n", simp_all add: min_def)}
```
```   467       moreover
```
```   468       {assume nn': "n' = n" hence stupid: "\<not> n' < n \<and> \<not> n < n'" by arith
```
```   469         from "4.hyps"(1)[rule_format, OF stupid cnp np', where xb="n"]
```
```   470           "4.hyps"(2)[rule_format, OF stupid cnp nc', where xb="n"] nn' cnp cnp' nn1
```
```   471
```
```   472         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   473           by simp (rule polyadd_normh,simp_all add: min_def isnpolyh_mono[OF nn0]) }
```
```   474       ultimately show "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)" by blast }
```
```   475     note th = this
```
```   476     {fix n0 n1 m
```
```   477       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
```
```   478       and m: "m \<le> min n0 n1"
```
```   479       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   480       let ?d1 = "degreen ?cnp m"
```
```   481       let ?d2 = "degreen ?cnp' m"
```
```   482       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   483       have "n'<n \<or> n < n' \<or> n' = n" by auto
```
```   484       moreover
```
```   485       {assume "n' < n \<or> n < n'"
```
```   486         with "4.hyps" np np' m
```
```   487         have ?eq apply (cases "n' < n", simp_all)
```
```   488         apply (erule allE[where x="n"],erule allE[where x="n"],auto)
```
```   489         done }
```
```   490       moreover
```
```   491       {assume nn': "n' = n"  hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
```
```   492         from "4.hyps"(1)[rule_format, OF nn, where x="n" and xa ="n'" and xb="n"]
```
```   493           "4.hyps"(2)[rule_format, OF nn, where x="n" and xa ="Suc n'" and xb="n"]
```
```   494           np np' nn'
```
```   495         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   496           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   497           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   498           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
```
```   499         {assume mn: "m = n"
```
```   500           from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
```
```   501             "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"] norm nn' mn
```
```   502           have degs:  "degreen (?cnp *\<^sub>p c') n =
```
```   503             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   504             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
```
```   505           from degs norm
```
```   506           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
```
```   507           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   508             by simp
```
```   509           have nmin: "n \<le> min n n" by (simp add: min_def)
```
```   510           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   511           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
```
```   512           from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
```
```   513             "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"]
```
```   514             mn norm m nn' deg
```
```   515           have ?eq by simp}
```
```   516         moreover
```
```   517         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
```
```   518           from nn' m np have max1: "m \<le> max n n"  by simp
```
```   519           hence min1: "m \<le> min n n" by simp
```
```   520           hence min2: "m \<le> min n (Suc n)" by simp
```
```   521           {assume "c' = 0\<^sub>p"
```
```   522             from `c' = 0\<^sub>p` have ?eq
```
```   523               using "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   524             "4.hyps"(2)[rule_format, OF nn norm(1,2) min2] mn nn'
```
```   525               apply simp
```
```   526               done}
```
```   527           moreover
```
```   528           {assume cnz: "c' \<noteq> 0\<^sub>p"
```
```   529             from "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   530               "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
```
```   531               degreen_polyadd[OF norm(3,6) max1]
```
```   532
```
```   533             have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
```
```   534               \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   535               using mn nn' cnz np np' by simp
```
```   536             with "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   537               "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
```
```   538               degreen_0[OF norm(3) mn'] have ?eq using nn' mn cnz np np' by clarsimp}
```
```   539           ultimately have ?eq by blast }
```
```   540         ultimately have ?eq by blast}
```
```   541       ultimately show ?eq by blast}
```
```   542     note degth = this
```
```   543     { case (2 n0 n1)
```
```   544       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
```
```   545         and m: "m \<le> min n0 n1" by simp_all
```
```   546       hence mn: "m \<le> n" by simp
```
```   547       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   548       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   549         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
```
```   550         from "4.hyps"(1) [rule_format, OF nn, where x="n" and xa = "n" and xb="n"]
```
```   551           "4.hyps"(2) [rule_format, OF nn, where x="n" and xa = "Suc n" and xb="n"]
```
```   552           np np' C(2) mn
```
```   553         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   554           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   555           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   556           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   557           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   558             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   559           by (simp_all add: min_def)
```
```   560
```
```   561           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```   562           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   563             using norm by simp
```
```   564         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
```
```   565         have "False" by simp }
```
```   566       thus ?case using "4.hyps" by clarsimp}
```
```   567 qed auto
```
```   568
```
```   569 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
```
```   570 by(induct p q rule: polymul.induct, auto simp add: ring_simps)
```
```   571
```
```   572 lemma polymul_normh:
```
```   573     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   574   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   575   using polymul_properties(1)  by blast
```
```   576 lemma polymul_eq0_iff:
```
```   577   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   578   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) "
```
```   579   using polymul_properties(2)  by blast
```
```   580 lemma polymul_degreen:
```
```   581   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   582   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)"
```
```   583   using polymul_properties(3) by blast
```
```   584 lemma polymul_norm:
```
```   585   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   586   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
```
```   587   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   588
```
```   589 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   590   by (induct p arbitrary: n0 rule: headconst.induct, auto)
```
```   591
```
```   592 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   593   by (induct p arbitrary: n0, auto)
```
```   594
```
```   595 lemma monic_eqI: assumes np: "isnpolyh p n0"
```
```   596   shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{ring_char_0,power,division_by_zero,field})"
```
```   597   unfolding monic_def Let_def
```
```   598 proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   599   let ?h = "headconst p"
```
```   600   assume pz: "p \<noteq> 0\<^sub>p"
```
```   601   {assume hz: "INum ?h = (0::'a)"
```
```   602     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
```
```   603     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
```
```   604     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
```
```   605   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
```
```   606 qed
```
```   607
```
```   608
```
```   609
```
```   610
```
```   611 text{* polyneg is a negation and preserves normal form *}
```
```   612 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   613 by (induct p rule: polyneg.induct, auto)
```
```   614
```
```   615 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   616   by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
```
```   617 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   618   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```   619 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
```
```   620 by (induct p rule: polyneg.induct, auto simp add: polyneg0)
```
```   621
```
```   622 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   623   using isnpoly_def polyneg_normh by simp
```
```   624
```
```   625
```
```   626 text{* polysub is a substraction and preserves normalform *}
```
```   627 lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
```
```   628 by (simp add: polysub_def polyneg polyadd)
```
```   629 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
```
```   630 by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   631
```
```   632 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
```
```   633   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   634 lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   635   shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p"
```
```   636 unfolding polysub_def split_def fst_conv snd_conv
```
```   637 by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   638
```
```   639 lemma polysub_0:
```
```   640   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   641   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
```
```   642   unfolding polysub_def split_def fst_conv snd_conv
```
```   643   apply (induct p q arbitrary: n0 n1 rule:polyadd.induct, simp_all add: Nsub0[simplified Nsub_def])
```
```   644   apply (clarsimp simp add: Let_def)
```
```   645   apply (case_tac "n < n'", simp_all)
```
```   646   apply (case_tac "n' < n", simp_all)
```
```   647   apply (erule impE)+
```
```   648   apply (rule_tac x="Suc n" in exI, simp)
```
```   649   apply (rule_tac x="n" in exI, simp)
```
```   650   apply (erule impE)+
```
```   651   apply (rule_tac x="n" in exI, simp)
```
```   652   apply (rule_tac x="Suc n" in exI, simp)
```
```   653   apply (erule impE)+
```
```   654   apply (rule_tac x="Suc n" in exI, simp)
```
```   655   apply (rule_tac x="n" in exI, simp)
```
```   656   apply (erule impE)+
```
```   657   apply (rule_tac x="Suc n" in exI, simp)
```
```   658   apply clarsimp
```
```   659   done
```
```   660
```
```   661 text{* polypow is a power function and preserves normal forms *}
```
```   662 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{ring_char_0,division_by_zero,field})) ^ n"
```
```   663 proof(induct n rule: polypow.induct)
```
```   664   case 1 thus ?case by simp
```
```   665 next
```
```   666   case (2 n)
```
```   667   let ?q = "polypow ((Suc n) div 2) p"
```
```   668   let ?d = "polymul(?q,?q)"
```
```   669   have "odd (Suc n) \<or> even (Suc n)" by simp
```
```   670   moreover
```
```   671   {assume odd: "odd (Suc n)"
```
```   672     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
```
```   673     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
```
```   674     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
```
```   675       using "2.hyps" by simp
```
```   676     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   677       apply (simp only: power_add power_one_right) by simp
```
```   678     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
```
```   679       by (simp only: th)
```
```   680     finally have ?case
```
```   681     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
```
```   682   moreover
```
```   683   {assume even: "even (Suc n)"
```
```   684     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
```
```   685     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
```
```   686     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   687       using "2.hyps" apply (simp only: power_add) by simp
```
```   688     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
```
```   689   ultimately show ?case by blast
```
```   690 qed
```
```   691
```
```   692 lemma polypow_normh:
```
```   693     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   694   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   695 proof (induct k arbitrary: n rule: polypow.induct)
```
```   696   case (2 k n)
```
```   697   let ?q = "polypow (Suc k div 2) p"
```
```   698   let ?d = "polymul (?q,?q)"
```
```   699   from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
```
```   700   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
```
```   701   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
```
```   702   from dn on show ?case by (simp add: Let_def)
```
```   703 qed auto
```
```   704
```
```   705 lemma polypow_norm:
```
```   706   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   707   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   708   by (simp add: polypow_normh isnpoly_def)
```
```   709
```
```   710 text{* Finally the whole normalization*}
```
```   711
```
```   712 lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{ring_char_0,division_by_zero,field})"
```
```   713 by (induct p rule:polynate.induct, auto)
```
```   714
```
```   715 lemma polynate_norm[simp]:
```
```   716   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   717   shows "isnpoly (polynate p)"
```
```   718   by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
```
```   719
```
```   720 text{* shift1 *}
```
```   721
```
```   722
```
```   723 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   724 by (simp add: shift1_def polymul)
```
```   725
```
```   726 lemma shift1_isnpoly:
```
```   727   assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
```
```   728   using pn pnz by (simp add: shift1_def isnpoly_def )
```
```   729
```
```   730 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   731   by (simp add: shift1_def)
```
```   732 lemma funpow_shift1_isnpoly:
```
```   733   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   734   by (induct n arbitrary: p, auto simp add: shift1_isnpoly)
```
```   735
```
```   736 lemma funpow_isnpolyh:
```
```   737   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
```
```   738   shows "isnpolyh (funpow k f p) n"
```
```   739   using f np by (induct k arbitrary: p, auto)
```
```   740
```
```   741 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)"
```
```   742   by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
```
```   743
```
```   744 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   745   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   746
```
```   747 lemma funpow_shift1_1:
```
```   748   "(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)"
```
```   749   by (simp add: funpow_shift1)
```
```   750
```
```   751 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   752 by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: ring_simps)
```
```   753
```
```   754 lemma behead:
```
```   755   assumes np: "isnpolyh p n"
```
```   756   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})"
```
```   757   using np
```
```   758 proof (induct p arbitrary: n rule: behead.induct)
```
```   759   case (1 c p n) hence pn: "isnpolyh p n" by simp
```
```   760   from prems(2)[OF pn]
```
```   761   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   762   then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   763     by (simp_all add: th[symmetric] ring_simps power_Suc)
```
```   764 qed (auto simp add: Let_def)
```
```   765
```
```   766 lemma behead_isnpolyh:
```
```   767   assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
```
```   768   using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
```
```   769
```
```   770 subsection{* Miscilanious lemmas about indexes, decrementation, substitution  etc ... *}
```
```   771 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   772 proof(induct p arbitrary: n rule: polybound0.induct, auto)
```
```   773   case (goal1 c n p n')
```
```   774   hence "n = Suc (n - 1)" by simp
```
```   775   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
```
```   776   with prems(2) show ?case by simp
```
```   777 qed
```
```   778
```
```   779 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   780 by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
```
```   781
```
```   782 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
```
```   783
```
```   784 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   785   apply (induct p arbitrary: n0, auto)
```
```   786   apply (atomize)
```
```   787   apply (erule_tac x = "Suc nat" in allE)
```
```   788   apply auto
```
```   789   done
```
```   790
```
```   791 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   792  by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
```
```   793
```
```   794 lemma polybound0_I:
```
```   795   assumes nb: "polybound0 a"
```
```   796   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
```
```   797 using nb
```
```   798 by (induct a rule: polybound0.induct) auto
```
```   799 lemma polysubst0_I:
```
```   800   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
```
```   801   by (induct t) simp_all
```
```   802
```
```   803 lemma polysubst0_I':
```
```   804   assumes nb: "polybound0 a"
```
```   805   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
```
```   806   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
```
```   807
```
```   808 lemma decrpoly: assumes nb: "polybound0 t"
```
```   809   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
```
```   810   using nb by (induct t rule: decrpoly.induct, simp_all)
```
```   811
```
```   812 lemma polysubst0_polybound0: assumes nb: "polybound0 t"
```
```   813   shows "polybound0 (polysubst0 t a)"
```
```   814 using nb by (induct a rule: polysubst0.induct, auto)
```
```   815
```
```   816 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   817   by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
```
```   818
```
```   819 fun maxindex :: "poly \<Rightarrow> nat" where
```
```   820   "maxindex (Bound n) = n + 1"
```
```   821 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   822 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   823 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   824 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   825 | "maxindex (Neg p) = maxindex p"
```
```   826 | "maxindex (Pw p n) = maxindex p"
```
```   827 | "maxindex (C x) = 0"
```
```   828
```
```   829 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
```
```   830   "wf_bs bs p = (length bs \<ge> maxindex p)"
```
```   831
```
```   832 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
```
```   833 proof(induct p rule: coefficients.induct)
```
```   834   case (1 c p)
```
```   835   show ?case
```
```   836   proof
```
```   837     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   838     hence "x = c \<or> x \<in> set (coefficients p)" by simp
```
```   839     moreover
```
```   840     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
```
```   841     moreover
```
```   842     {assume H: "x \<in> set (coefficients p)"
```
```   843       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
```
```   844       with "1.hyps" H have "wf_bs bs x" by blast }
```
```   845     ultimately  show "wf_bs bs x" by blast
```
```   846   qed
```
```   847 qed simp_all
```
```   848
```
```   849 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   850 by (induct p rule: coefficients.induct, auto)
```
```   851
```
```   852 lemma length_exists: "\<exists>xs. length xs = n" by (rule exI[where x="replicate n x"], simp)
```
```   853
```
```   854 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
```
```   855   unfolding wf_bs_def by (induct p, auto simp add: nth_append)
```
```   856
```
```   857 lemma take_maxindex_wf: assumes wf: "wf_bs bs p"
```
```   858   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   859 proof-
```
```   860   let ?ip = "maxindex p"
```
```   861   let ?tbs = "take ?ip bs"
```
```   862   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
```
```   863   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
```
```   864   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
```
```   865   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
```
```   866 qed
```
```   867
```
```   868 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```   869   by (induct p, auto)
```
```   870
```
```   871 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```   872   unfolding wf_bs_def by simp
```
```   873
```
```   874 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
```
```   875   unfolding wf_bs_def by simp
```
```   876
```
```   877
```
```   878
```
```   879 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
```
```   880 by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
```
```   881 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```   882   by (induct p rule: coefficients.induct, simp_all)
```
```   883
```
```   884
```
```   885 lemma coefficients_head: "last (coefficients p) = head p"
```
```   886   by (induct p rule: coefficients.induct, auto)
```
```   887
```
```   888 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
```
```   889   unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
```
```   890
```
```   891 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
```
```   892   apply (rule exI[where x="replicate (n - length xs) z"])
```
```   893   by simp
```
```   894 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```   895 by (cases p, auto) (case_tac "nat", simp_all)
```
```   896
```
```   897 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```   898   unfolding wf_bs_def
```
```   899   apply (induct p q rule: polyadd.induct)
```
```   900   apply (auto simp add: Let_def)
```
```   901   done
```
```   902
```
```   903 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```   904
```
```   905  unfolding wf_bs_def
```
```   906   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```   907   apply (simp_all add: wf_bs_polyadd)
```
```   908   apply clarsimp
```
```   909   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```   910   apply auto
```
```   911   done
```
```   912
```
```   913 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```   914   unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
```
```   915
```
```   916 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```   917   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
```
```   918
```
```   919 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
```
```   920
```
```   921 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```   922 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
```
```   923 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
```
```   924
```
```   925 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
```
```   926 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```   927   case (1 c p n0)
```
```   928   have cp: "isnpolyh (CN c 0 p) n0" by fact
```
```   929   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```   930     by (auto simp add: isnpolyh_mono[where n'=0])
```
```   931   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
```
```   932 qed auto
```
```   933
```
```   934 lemma coefficients_isconst:
```
```   935   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
```
```   936   by (induct p arbitrary: n rule: coefficients.induct,
```
```   937     auto simp add: isnpolyh_Suc_const)
```
```   938
```
```   939 lemma polypoly_polypoly':
```
```   940   assumes np: "isnpolyh p n0"
```
```   941   shows "polypoly (x#bs) p = polypoly' bs p"
```
```   942 proof-
```
```   943   let ?cf = "set (coefficients p)"
```
```   944   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```   945   {fix q assume q: "q \<in> ?cf"
```
```   946     from q cn_norm have th: "isnpolyh q n0" by blast
```
```   947     from coefficients_isconst[OF np] q have "isconstant q" by blast
```
```   948     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
```
```   949   hence "\<forall>q \<in> ?cf. polybound0 q" ..
```
```   950   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
```
```   951     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```   952     by auto
```
```   953
```
```   954   thus ?thesis unfolding polypoly_def polypoly'_def by simp
```
```   955 qed
```
```   956
```
```   957 lemma polypoly_poly:
```
```   958   assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
```
```   959   using np
```
```   960 by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
```
```   961
```
```   962 lemma polypoly'_poly:
```
```   963   assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```   964   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
```
```   965
```
```   966
```
```   967 lemma polypoly_poly_polybound0:
```
```   968   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
```
```   969   shows "polypoly bs p = [Ipoly bs p]"
```
```   970   using np nb unfolding polypoly_def
```
```   971   by (cases p, auto, case_tac nat, auto)
```
```   972
```
```   973 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```   974   by (induct p rule: head.induct, auto)
```
```   975
```
```   976 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   977   by (cases p,auto)
```
```   978
```
```   979 lemma head_eq_headn0: "head p = headn p 0"
```
```   980   by (induct p rule: head.induct, simp_all)
```
```   981
```
```   982 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   983   by (simp add: head_eq_headn0)
```
```   984
```
```   985 lemma isnpolyh_zero_iff:
```
```   986   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})"
```
```   987   shows "p = 0\<^sub>p"
```
```   988 using nq eq
```
```   989 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```   990   case less
```
```   991   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```   992   {assume nz: "maxindex p = 0"
```
```   993     then obtain c where "p = C c" using np by (cases p, auto)
```
```   994     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
```
```   995   moreover
```
```   996   {assume nz: "maxindex p \<noteq> 0"
```
```   997     let ?h = "head p"
```
```   998     let ?hd = "decrpoly ?h"
```
```   999     let ?ihd = "maxindex ?hd"
```
```  1000     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
```
```  1001       by simp_all
```
```  1002     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
```
```  1003
```
```  1004     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```  1005     have mihn: "maxindex ?h \<le> maxindex p" by auto
```
```  1006     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
```
```  1007     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
```
```  1008       let ?ts = "take ?ihd bs"
```
```  1009       let ?rs = "drop ?ihd bs"
```
```  1010       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
```
```  1011       have bs_ts_eq: "?ts@ ?rs = bs" by simp
```
```  1012       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
```
```  1013       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
```
```  1014       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
```
```  1015       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
```
```  1016       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
```
```  1017       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
```
```  1018       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```  1019       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
```
```  1020       hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext)
```
```  1021       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```  1022         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```  1023       with coefficients_head[of p, symmetric]
```
```  1024       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
```
```  1025       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
```
```  1026       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
```
```  1027       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
```
```  1028     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
```
```  1029
```
```  1030     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
```
```  1031     hence "?h = 0\<^sub>p" by simp
```
```  1032     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
```
```  1033   ultimately show "p = 0\<^sub>p" by blast
```
```  1034 qed
```
```  1035
```
```  1036 lemma isnpolyh_unique:
```
```  1037   assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1038   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"
```
```  1039 proof(auto)
```
```  1040   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```  1041   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
```
```  1042   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1043     using wf_bs_polysub[where p=p and q=q] by auto
```
```  1044   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
```
```  1045   show "p = q" by blast
```
```  1046 qed
```
```  1047
```
```  1048
```
```  1049 text{* consequenses of unicity on the algorithms for polynomial normalization *}
```
```  1050
```
```  1051 lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1052   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1053   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
```
```  1054
```
```  1055 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
```
```  1056 lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
```
```  1057 lemma polyadd_0[simp]:
```
```  1058   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1059   and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
```
```  1060   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1061     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1062
```
```  1063 lemma polymul_1[simp]:
```
```  1064     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1065   and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
```
```  1066   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1067     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1068 lemma polymul_0[simp]:
```
```  1069   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1070   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"
```
```  1071   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1072     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1073
```
```  1074 lemma polymul_commute:
```
```  1075     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1076   and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1077   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1078 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
```
```  1079
```
```  1080 declare polyneg_polyneg[simp]
```
```  1081
```
```  1082 lemma isnpolyh_polynate_id[simp]:
```
```  1083   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1084   and np:"isnpolyh p n0" shows "polynate p = p"
```
```  1085   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
```
```  1086
```
```  1087 lemma polynate_idempotent[simp]:
```
```  1088     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1089   shows "polynate (polynate p) = polynate p"
```
```  1090   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1091
```
```  1092 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1093   unfolding poly_nate_def polypoly'_def ..
```
```  1094 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>)"
```
```  1095   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1096   unfolding poly_nate_polypoly' by (auto intro: ext)
```
```  1097
```
```  1098 subsection{* heads, degrees and all that *}
```
```  1099 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1100   by (induct p rule: degree.induct, simp_all)
```
```  1101
```
```  1102 lemma degree_polyneg: assumes n: "isnpolyh p n"
```
```  1103   shows "degree (polyneg p) = degree p"
```
```  1104   using n
```
```  1105   by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
```
```  1106
```
```  1107 lemma degree_polyadd:
```
```  1108   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1109   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1110 using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1111
```
```  1112
```
```  1113 lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1114   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1115 proof-
```
```  1116   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
```
```  1117   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1118 qed
```
```  1119
```
```  1120 lemma degree_polysub_samehead:
```
```  1121   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1122   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
```
```  1123   and d: "degree p = degree q"
```
```  1124   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1125 unfolding polysub_def split_def fst_conv snd_conv
```
```  1126 using np nq h d
```
```  1127 proof(induct p q rule:polyadd.induct)
```
```  1128   case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1129 next
```
```  1130   case (2 a b c' n' p')
```
```  1131   let ?c = "(a,b)"
```
```  1132   from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
```
```  1133   hence nz:"n' > 0" by (cases n', auto)
```
```  1134   hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1135   with prems show ?case by simp
```
```  1136 next
```
```  1137   case (3 c n p a' b')
```
```  1138   let ?c' = "(a',b')"
```
```  1139   from prems have "degree (C ?c') = degree (CN c n p)" by simp
```
```  1140   hence nz:"n > 0" by (cases n, auto)
```
```  1141   hence "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1142   with prems show ?case by simp
```
```  1143 next
```
```  1144   case (4 c n p c' n' p')
```
```  1145   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
```
```  1146     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
```
```  1147   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
```
```  1148   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1149     using H(1-2) degree_polyneg by auto
```
```  1150   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
```
```  1151   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
```
```  1152   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
```
```  1153   have "n = n' \<or> n < n' \<or> n > n'" by arith
```
```  1154   moreover
```
```  1155   {assume nn': "n = n'"
```
```  1156     have "n = 0 \<or> n >0" by arith
```
```  1157     moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
```
```  1158     moreover {assume nz: "n > 0"
```
```  1159       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
```
```  1160       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)}
```
```  1161     ultimately have ?case by blast}
```
```  1162   moreover
```
```  1163   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
```
```  1164     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
```
```  1165     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)
```
```  1166     hence "n > 0" by (cases n, simp_all)
```
```  1167     hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1168     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1169   moreover
```
```  1170   {assume nn': "n > n'"  hence np: "n > 0" by simp
```
```  1171     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
```
```  1172     from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
```
```  1173     from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
```
```  1174     with degcnpeq have "n' > 0" by (cases n', simp_all)
```
```  1175     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1176     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1177   ultimately show ?case  by blast
```
```  1178 qed auto
```
```  1179
```
```  1180 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1181 by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
```
```  1182
```
```  1183 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1184 proof(induct k arbitrary: n0 p)
```
```  1185   case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
```
```  1186   with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1187     and "head (shift1 p) = head p" by (simp_all add: shift1_head)
```
```  1188   thus ?case by simp
```
```  1189 qed auto
```
```  1190
```
```  1191 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1192   by (simp add: shift1_def)
```
```  1193 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1194   by (induct k arbitrary: p, auto simp add: shift1_degree)
```
```  1195
```
```  1196 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1197   by (induct n arbitrary: p, simp_all add: funpow_def)
```
```  1198
```
```  1199 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1200   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1201 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1202   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```  1203 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1204   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1205 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1206   by (induct p rule: head.induct, auto)
```
```  1207
```
```  1208 lemma polyadd_eq_const_degree:
```
```  1209   "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q"
```
```  1210   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1211
```
```  1212 lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1213   and deg: "degree p \<noteq> degree q"
```
```  1214   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1215 using np nq deg
```
```  1216 apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
```
```  1217 apply (case_tac n', simp, simp)
```
```  1218 apply (case_tac n, simp, simp)
```
```  1219 apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1220 apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
```
```  1221 apply (clarsimp simp add: polyadd_eq_const_degree)
```
```  1222 apply clarsimp
```
```  1223 apply (erule_tac impE,blast)
```
```  1224 apply (erule_tac impE,blast)
```
```  1225 apply clarsimp
```
```  1226 apply simp
```
```  1227 apply (case_tac n', simp_all)
```
```  1228 done
```
```  1229
```
```  1230 lemma polymul_head_polyeq:
```
```  1231    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1232   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"
```
```  1233 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1234   case (2 a b c' n' p' n0 n1)
```
```  1235   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)"  by (simp_all add: head_isnpolyh)
```
```  1236   thus ?case using prems by (cases n', auto)
```
```  1237 next
```
```  1238   case (3 c n p a' b' n0 n1)
```
```  1239   hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')"  by (simp_all add: head_isnpolyh)
```
```  1240   thus ?case using prems by (cases n, auto)
```
```  1241 next
```
```  1242   case (4 c n p c' n' p' n0 n1)
```
```  1243   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1244     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1245     by simp_all
```
```  1246   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1247   moreover
```
```  1248   {assume nn': "n < n'" hence ?case
```
```  1249       thm prems
```
```  1250       using norm
```
```  1251     prems(6)[rule_format, OF nn' norm(1,6)]
```
```  1252     prems(7)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
```
```  1253   moreover {assume nn': "n'< n"
```
```  1254     hence stupid: "n' < n \<and> \<not> n < n'" by simp
```
```  1255     hence ?case using norm prems(4) [rule_format, OF stupid norm(5,3)]
```
```  1256       prems(5)[rule_format, OF stupid norm(5,4)]
```
```  1257       by (simp,cases n',simp,cases n,auto)}
```
```  1258   moreover {assume nn': "n' = n"
```
```  1259     hence stupid: "\<not> n' < n \<and> \<not> n < n'" by simp
```
```  1260     from nn' polymul_normh[OF norm(5,4)]
```
```  1261     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1262     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1263     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1264     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1265     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1266     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1267     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"
```
```  1268       by (simp add: min_def)
```
```  1269     {assume np: "n > 0"
```
```  1270       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1271         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1272       have ?case by simp}
```
```  1273     moreover
```
```  1274     {moreover assume nz: "n = 0"
```
```  1275       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1276         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1277       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1278     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
```
```  1279     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
```
```  1280     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1281     have ?case   using norm prems(2)[rule_format, OF stupid norm(5,3)]
```
```  1282         prems(3)[rule_format, OF stupid norm(5,4)] nn' nz by simp }
```
```  1283     ultimately have ?case by (cases n) auto}
```
```  1284   ultimately show ?case by blast
```
```  1285 qed simp_all
```
```  1286
```
```  1287 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1288   by(induct p rule: degree.induct, auto)
```
```  1289
```
```  1290 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1291   by (induct p rule: head.induct, auto)
```
```  1292
```
```  1293 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p"
```
```  1294   by (cases n, simp_all)
```
```  1295 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1296   by (cases n, simp_all)
```
```  1297
```
```  1298 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)"
```
```  1299   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1300
```
```  1301 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1302   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```  1303
```
```  1304 lemma degree_polymul:
```
```  1305   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1306   and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1307   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1308   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1309
```
```  1310 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1311   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1312
```
```  1313 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
```
```  1314   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1315
```
```  1316 subsection {* Correctness of polynomial pseudo division *}
```
```  1317
```
```  1318 lemma polydivide_aux_real_domintros:
```
```  1319   assumes call1: "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a = head s\<rbrakk>
```
```  1320   \<Longrightarrow> polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)"
```
```  1321   and call2 : "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a \<noteq> head s\<rbrakk>
```
```  1322   \<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))"
```
```  1323   shows "polydivide_aux_dom (a, n, p, k, s)"
```
```  1324 proof (rule accpI, erule polydivide_aux_rel.cases)
```
```  1325   fix y aa ka na pa sa x xa xb
```
```  1326   assume eqs: "y = (aa, na, pa, ka, sa -\<^sub>p xb)" "(a, n, p, k, s) = (aa, na, pa, ka, sa)"
```
```  1327      and \<Gamma>1': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na"
```
```  1328     "xb = funpow (xa - na) shift1 pa" "aa = x"
```
```  1329
```
```  1330   hence \<Gamma>1: "s \<noteq> 0\<^sub>p" "a = head s" "xa = degree s" "\<not> degree s < n" "\<not> xa < na"
```
```  1331     "xb = funpow (xa - na) shift1 pa" "aa = x" by auto
```
```  1332
```
```  1333   with call1 have "polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)"
```
```  1334     by auto
```
```  1335   with eqs \<Gamma>1 show "polydivide_aux_dom y" by auto
```
```  1336 next
```
```  1337   fix y aa ka na pa sa x xa xb
```
```  1338   assume eqs: "y = (aa, na, pa, Suc ka, aa *\<^sub>p sa -\<^sub>p (x *\<^sub>p xb))"
```
```  1339     "(a, n, p, k, s) =(aa, na, pa, ka, sa)"
```
```  1340     and \<Gamma>2': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na"
```
```  1341     "xb = funpow (xa - na) shift1 pa" "aa \<noteq> x"
```
```  1342   hence \<Gamma>2: "s \<noteq> 0\<^sub>p" "a \<noteq> head s" "xa = degree s" "\<not> degree s < n"
```
```  1343     "xb = funpow (xa - na) shift1 pa" "aa \<noteq> x" by auto
```
```  1344   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
```
```  1345   with eqs \<Gamma>2'  show "polydivide_aux_dom y" by auto
```
```  1346 qed
```
```  1347
```
```  1348 lemma polydivide_aux_properties:
```
```  1349   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1350   and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
```
```  1351   and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1352   shows "polydivide_aux_dom (a,n,p,k,s) \<and>
```
```  1353   (polydivide_aux (a,n,p,k,s) = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1354           \<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)))"
```
```  1355   using ns
```
```  1356 proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1357   case less
```
```  1358   let ?D = "polydivide_aux_dom"
```
```  1359   let ?dths = "?D (a, n, p, k, s)"
```
```  1360   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1361   let ?qrths = "polydivide_aux (a, n, p, k, s) = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1362     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1363   let ?ths = "?dths \<and> ?qrths"
```
```  1364   let ?b = "head s"
```
```  1365   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1366   let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
```
```  1367   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1368   note ns = `isnpolyh s n1`
```
```  1369   from np have np0: "isnpolyh p 0"
```
```  1370     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
```
```  1371   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
```
```  1372   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
```
```  1373   from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
```
```  1374   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1375   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1376   {assume sz: "s = 0\<^sub>p"
```
```  1377     hence dom: ?dths apply - apply (rule polydivide_aux_real_domintros) apply simp_all done
```
```  1378     from polydivide_aux.psimps[OF dom] sz
```
```  1379     have ?qrths using np apply clarsimp by (rule exI[where x="0\<^sub>p"], simp)
```
```  1380     hence ?ths using dom by blast}
```
```  1381   moreover
```
```  1382   {assume sz: "s \<noteq> 0\<^sub>p"
```
```  1383     {assume dn: "degree s < n"
```
```  1384       with sz have dom:"?dths" by - (rule polydivide_aux_real_domintros,simp_all)
```
```  1385       from polydivide_aux.psimps[OF dom] sz dn
```
```  1386       have "?qrths" using ns ndp np by auto (rule exI[where x="0\<^sub>p"],simp)
```
```  1387       with dom have ?ths by blast}
```
```  1388     moreover
```
```  1389     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
```
```  1390       have degsp': "degree s = degree ?p'"
```
```  1391         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
```
```  1392       {assume ba: "?b = a"
```
```  1393         hence headsp': "head s = head ?p'" using ap headp' by simp
```
```  1394         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
```
```  1395         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1396         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
```
```  1397         moreover
```
```  1398         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1399           from  less(1)[OF deglt nr]
```
```  1400           have domsp: "?D (a, n, p, k, s -\<^sub>p ?p')" by blast
```
```  1401           have dom: ?dths apply (rule polydivide_aux_real_domintros)
```
```  1402             using ba dn' domsp by simp_all
```
```  1403           from polydivide_aux.psimps[OF dom] sz dn' ba
```
```  1404           have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
```
```  1405             by (simp add: Let_def)
```
```  1406           {assume h1: "polydivide_aux (a, n, p, k, s) = (k', r)"
```
```  1407             from less(1)[OF deglt nr, of k k' r]
```
```  1408               trans[OF eq[symmetric] h1]
```
```  1409             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1410               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
```
```  1411             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1412               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
```
```  1413             from nr obtain nr where nr': "isnpolyh r nr" by blast
```
```  1414             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
```
```  1415             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1416             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1417             from polyadd_normh[OF polymul_normh[OF np
```
```  1418               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1419             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
```
```  1420             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')) =
```
```  1421               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
```
```  1422             hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1423               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1424               by (simp add: ring_simps)
```
```  1425             hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1426               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p)
```
```  1427               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1428               by (auto simp only: funpow_shift1_1)
```
```  1429             hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1430               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p)
```
```  1431               + Ipoly bs q) + Ipoly bs r" by (simp add: ring_simps)
```
```  1432             hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1433               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
```
```  1434             with isnpolyh_unique[OF nakks' nqr']
```
```  1435             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1436               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
```
```  1437             hence ?qths using nq'
```
```  1438               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
```
```  1439               apply (rule_tac x="0" in exI) by simp
```
```  1440             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"
```
```  1441               by blast } hence ?qrths by blast
```
```  1442           with dom have ?ths by blast}
```
```  1443         moreover
```
```  1444         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1445           hence domsp: "?D (a, n, p, k, s -\<^sub>p ?p')"
```
```  1446             apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
```
```  1447           have dom: ?dths apply (rule polydivide_aux_real_domintros)
```
```  1448             using ba dn' domsp by simp_all
```
```  1449           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}"]
```
```  1450           have " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs ?p'" by simp
```
```  1451           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
```
```  1452             by (simp only: funpow_shift1_1) simp
```
```  1453           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
```
```  1454           {assume h1: "polydivide_aux (a,n,p,k,s) = (k',r)"
```
```  1455             from polydivide_aux.psimps[OF dom] sz dn' ba
```
```  1456             have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
```
```  1457               by (simp add: Let_def)
```
```  1458             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.psimps[OF domsp] spz by simp
```
```  1459             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
```
```  1460             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1461               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?qrths
```
```  1462               apply auto
```
```  1463               apply (rule exI[where x="?xdn"])
```
```  1464               apply (auto simp add: polymul_commute[of p])
```
```  1465               done}
```
```  1466           with dom have ?ths by blast}
```
```  1467         ultimately have ?ths by blast }
```
```  1468       moreover
```
```  1469       {assume ba: "?b \<noteq> a"
```
```  1470         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1471           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1472         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
```
```  1473         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1474           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1475             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1476             funpow_shift1_nz[OF pnz] by simp_all
```
```  1477         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1478           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1479         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1480           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1481             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1482           by (simp add: ap)
```
```  1483         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1484           head_nz[OF np] pnz sz ap[symmetric]
```
```  1485           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1486           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
```
```  1487           ndp dn
```
```  1488         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
```
```  1489           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1490         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1491           have th: "?D (a, n, p, Suc k, (a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))"
```
```  1492             using less(1)[OF dth nth] by blast
```
```  1493           have dom: ?dths using ba dn' th
```
```  1494             by - (rule polydivide_aux_real_domintros, simp_all)
```
```  1495           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
```
```  1496           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
```
```  1497           {assume h1:"polydivide_aux (a,n,p,k,s) = (k', r)"
```
```  1498             from h1  polydivide_aux.psimps[OF dom] sz dn' ba
```
```  1499             have eq:"polydivide_aux (a,n,p,Suc k,(a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1500               by (simp add: Let_def)
```
```  1501             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1502             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
```
```  1503               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1504               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
```
```  1505             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
```
```  1506             {fix bs:: "'a::{ring_char_0,division_by_zero,field} list"
```
```  1507
```
```  1508             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1509             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
```
```  1510             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"
```
```  1511               by (simp add: ring_simps power_Suc)
```
```  1512             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"
```
```  1513               by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1514             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"
```
```  1515               by (simp add: ring_simps)}
```
```  1516             hence ieq:"\<forall>(bs :: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1517               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
```
```  1518             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1519             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]]
```
```  1520             have nqw: "isnpolyh ?q 0" by simp
```
```  1521             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]]
```
```  1522             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
```
```  1523             from dr kk' nr h1 asth nqw have ?qrths apply simp
```
```  1524               apply (rule conjI)
```
```  1525               apply (rule exI[where x="nr"], simp)
```
```  1526               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1527               apply (rule exI[where x="0"], simp)
```
```  1528               done}
```
```  1529           hence ?qrths by blast
```
```  1530           with dom have ?ths by blast}
```
```  1531         moreover
```
```  1532         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1533           hence domsp: "?D (a, n, p, Suc k, a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p'))"
```
```  1534             apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
```
```  1535           have dom: ?dths using sz ba dn' domsp
```
```  1536             by - (rule polydivide_aux_real_domintros, simp_all)
```
```  1537           {fix bs :: "'a::{ring_char_0,division_by_zero,field} list"
```
```  1538             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1539           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
```
```  1540           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1541             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1542           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
```
```  1543         }
```
```  1544         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))" ..
```
```  1545           from hth
```
```  1546           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1547             using isnpolyh_unique[where ?'a = "'a::{ring_char_0,division_by_zero,field}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1548                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1549               simplified ap] by simp
```
```  1550           {assume h1: "polydivide_aux (a,n,p,k,s) = (k', r)"
```
```  1551           from h1 sz ba dn' spz polydivide_aux.psimps[OF dom] polydivide_aux.psimps[OF domsp]
```
```  1552           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
```
```  1553           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1554             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1555           have ?qrths apply (clarsimp simp add: Let_def)
```
```  1556             apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
```
```  1557             apply (rule exI[where x="0"], simp)
```
```  1558             done}
```
```  1559         hence ?qrths by blast
```
```  1560         with dom have ?ths by blast}
```
```  1561         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"]
```
```  1562           head_nz[OF np] pnz sz ap[symmetric]
```
```  1563           by (simp add: degree_eq_degreen0[symmetric]) blast }
```
```  1564       ultimately have ?ths by blast
```
```  1565     }
```
```  1566     ultimately have ?ths by blast}
```
```  1567   ultimately show ?ths by blast
```
```  1568 qed
```
```  1569
```
```  1570 lemma polydivide_properties:
```
```  1571   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1572   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1573   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1574   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
```
```  1575 proof-
```
```  1576   have trv: "head p = head p" "degree p = degree p" by simp_all
```
```  1577   from polydivide_aux_properties[OF np ns trv pnz, where k="0"]
```
```  1578   have d: "polydivide_aux_dom (head p, degree p, p, 0, s)" by blast
```
```  1579   from polydivide_def[where s="s" and p="p"] polydivide_aux.psimps[OF d]
```
```  1580   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
```
```  1581   then obtain k r where kr: "polydivide s p = (k,r)" by blast
```
```  1582   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s" and p="p"], symmetric] kr]
```
```  1583     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1584   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1585    (\<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
```
```  1586   with kr show ?thesis
```
```  1587     apply -
```
```  1588     apply (rule exI[where x="k"])
```
```  1589     apply (rule exI[where x="r"])
```
```  1590     apply simp
```
```  1591     done
```
```  1592 qed
```
```  1593
```
```  1594 subsection{* More about polypoly and pnormal etc *}
```
```  1595
```
```  1596 definition "isnonconstant p = (\<not> isconstant p)"
```
```  1597
```
```  1598 lemma last_map: "xs \<noteq> [] ==> last (map f xs) = f (last xs)" by (induct xs, auto)
```
```  1599
```
```  1600 lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p"
```
```  1601   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1602 proof
```
```  1603   let ?p = "polypoly bs p"
```
```  1604   assume H: "pnormal ?p"
```
```  1605   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1606
```
```  1607   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1608     pnormal_last_nonzero[OF H]
```
```  1609   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
```
```  1610 next
```
```  1611   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1612   let ?p = "polypoly bs p"
```
```  1613   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1614   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
```
```  1615   hence lg: "length ?p > 0" by simp
```
```  1616   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1617   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
```
```  1618   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1619 qed
```
```  1620
```
```  1621 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1622   unfolding isnonconstant_def
```
```  1623   apply (cases p, simp_all)
```
```  1624   apply (case_tac nat, auto)
```
```  1625   done
```
```  1626 lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
```
```  1627   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1628 proof
```
```  1629   let ?p = "polypoly bs p"
```
```  1630   assume nc: "nonconstant ?p"
```
```  1631   from isnonconstant_pnormal_iff[OF inc, of bs] nc
```
```  1632   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
```
```  1633 next
```
```  1634   let ?p = "polypoly bs p"
```
```  1635   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1636   from isnonconstant_pnormal_iff[OF inc, of bs] h
```
```  1637   have pn: "pnormal ?p" by blast
```
```  1638   {fix x assume H: "?p = [x]"
```
```  1639     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
```
```  1640     with isnonconstant_coefficients_length[OF inc] have False by arith}
```
```  1641   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
```
```  1642 qed
```
```  1643
```
```  1644 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  1645   unfolding pnormal_def
```
```  1646  apply (induct p rule: pnormalize.induct, simp_all)
```
```  1647  apply (case_tac "p=[]", simp_all)
```
```  1648  done
```
```  1649
```
```  1650 lemma degree_degree: assumes inc: "isnonconstant p"
```
```  1651   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1652 proof
```
```  1653   let  ?p = "polypoly bs p"
```
```  1654   assume H: "degree p = Polynomial_List.degree ?p"
```
```  1655   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
```
```  1656     unfolding polypoly_def by auto
```
```  1657   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1658   have lg:"length (pnormalize ?p) = length ?p"
```
```  1659     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  1660   hence "pnormal ?p" using pnormal_length[OF pz] by blast
```
```  1661   with isnonconstant_pnormal_iff[OF inc]
```
```  1662   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
```
```  1663 next
```
```  1664   let  ?p = "polypoly bs p"
```
```  1665   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1666   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
```
```  1667   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1668   show "degree p = Polynomial_List.degree ?p"
```
```  1669     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  1670 qed
```
```  1671
```
```  1672 section{* Swaps ; Division by a certain variable *}
```
```  1673 consts swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
```
```  1674 primrec
```
```  1675   "swap n m (C x) = C x"
```
```  1676   "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
```
```  1677   "swap n m (Neg t) = Neg (swap n m t)"
```
```  1678   "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  1679   "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  1680   "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  1681   "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  1682   "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)
```
```  1683   (swap n m p)"
```
```  1684
```
```  1685 lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1686   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1687 proof (induct t)
```
```  1688   case (Bound k) thus ?case using nbs mbs by simp
```
```  1689 next
```
```  1690   case (CN c k p) thus ?case using nbs mbs by simp
```
```  1691 qed simp_all
```
```  1692 lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
```
```  1693   by (induct t,simp_all)
```
```  1694
```
```  1695 lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
```
```  1696
```
```  1697 lemma swap_same_id[simp]: "swap n n t = t"
```
```  1698   by (induct t, simp_all)
```
```  1699
```
```  1700 definition "swapnorm n m t = polynate (swap n m t)"
```
```  1701
```
```  1702 lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1703   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"
```
```  1704   using swap[OF prems] swapnorm_def by simp
```
```  1705
```
```  1706 lemma swapnorm_isnpoly[simp]:
```
```  1707     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1708   shows "isnpoly (swapnorm n m p)"
```
```  1709   unfolding swapnorm_def by simp
```
```  1710
```
```  1711 definition "polydivideby n s p =
```
```  1712     (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
```
```  1713      in (k,swapnorm 0 n h,swapnorm 0 n r))"
```
```  1714
```
```  1715 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
```
```  1716
```
```  1717 consts isweaknpoly :: "poly \<Rightarrow> bool"
```
```  1718 recdef isweaknpoly "measure size"
```
```  1719   "isweaknpoly (C c) = True"
```
```  1720   "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  1721   "isweaknpoly p = False"
```
```  1722
```
```  1723 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  1724   by (induct p arbitrary: n0, auto)
```
```  1725
```
```  1726 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  1727   by (induct p, auto)
```
```  1728
```
`  1729 end`