src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author berghofe Sun Jan 10 18:43:45 2010 +0100 (2010-01-10) changeset 34915 7894c7dab132 parent 33268 02de0317f66f child 35046 1266f04f42ec permissions -rw-r--r--
Adapted to changes in induct method.
```     1 (*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header {* Implementation and verification of mutivariate polynomials Library *}
```
```     6
```
```     7
```
```     8 theory Reflected_Multivariate_Polynomial
```
```     9 imports Parity Abstract_Rat Efficient_Nat List Polynomial_List
```
```    10 begin
```
```    11
```
```    12   (* Impelementation *)
```
```    13
```
```    14 subsection{* Datatype of polynomial expressions *}
```
```    15
```
```    16 datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
```
```    17   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
```
```    18
```
```    19 ML{* @{term "Add"}*}
```
```    20 syntax "_poly0" :: "poly" ("0\<^sub>p")
```
```    21 translations "0\<^sub>p" \<rightleftharpoons> "C (0\<^sub>N)"
```
```    22 syntax "_poly" :: "int \<Rightarrow> poly" ("_\<^sub>p")
```
```    23 translations "i\<^sub>p" \<rightleftharpoons> "C (i\<^sub>N)"
```
```    24
```
```    25 subsection{* Boundedness, substitution and all that *}
```
```    26 consts polysize:: "poly \<Rightarrow> nat"
```
```    27 primrec
```
```    28   "polysize (C c) = 1"
```
```    29   "polysize (Bound n) = 1"
```
```    30   "polysize (Neg p) = 1 + polysize p"
```
```    31   "polysize (Add p q) = 1 + polysize p + polysize q"
```
```    32   "polysize (Sub p q) = 1 + polysize p + polysize q"
```
```    33   "polysize (Mul p q) = 1 + polysize p + polysize q"
```
```    34   "polysize (Pw p n) = 1 + polysize p"
```
```    35   "polysize (CN c n p) = 4 + polysize c + polysize p"
```
```    36
```
```    37 consts
```
```    38   polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *)
```
```    39   polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *)
```
```    40 primrec
```
```    41   "polybound0 (C c) = True"
```
```    42   "polybound0 (Bound n) = (n>0)"
```
```    43   "polybound0 (Neg a) = polybound0 a"
```
```    44   "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
```
```    45   "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)"
```
```    46   "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
```
```    47   "polybound0 (Pw p n) = (polybound0 p)"
```
```    48   "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
```
```    49 primrec
```
```    50   "polysubst0 t (C c) = (C c)"
```
```    51   "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"
```
```    52   "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
```
```    53   "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
```
```    54   "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
```
```    55   "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
```
```    56   "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
```
```    57   "polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
```
```    58                              else CN (polysubst0 t c) n (polysubst0 t p))"
```
```    59
```
```    60 consts
```
```    61   decrpoly:: "poly \<Rightarrow> poly"
```
```    62 recdef decrpoly "measure polysize"
```
```    63   "decrpoly (Bound n) = Bound (n - 1)"
```
```    64   "decrpoly (Neg a) = Neg (decrpoly a)"
```
```    65   "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
```
```    66   "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
```
```    67   "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
```
```    68   "decrpoly (Pw p n) = Pw (decrpoly p) n"
```
```    69   "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
```
```    70   "decrpoly a = a"
```
```    71
```
```    72 subsection{* Degrees and heads and coefficients *}
```
```    73
```
```    74 consts degree:: "poly \<Rightarrow> nat"
```
```    75 recdef degree "measure size"
```
```    76   "degree (CN c 0 p) = 1 + degree p"
```
```    77   "degree p = 0"
```
```    78 consts head:: "poly \<Rightarrow> poly"
```
```    79
```
```    80 recdef head "measure size"
```
```    81   "head (CN c 0 p) = head p"
```
```    82   "head p = p"
```
```    83   (* More general notions of degree and head *)
```
```    84 consts degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
```
```    85 recdef degreen "measure size"
```
```    86   "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
```
```    87   "degreen p = (\<lambda>m. 0)"
```
```    88
```
```    89 consts headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
```
```    90 recdef headn "measure size"
```
```    91   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
```
```    92   "headn p = (\<lambda>m. p)"
```
```    93
```
```    94 consts coefficients:: "poly \<Rightarrow> poly list"
```
```    95 recdef coefficients "measure size"
```
```    96   "coefficients (CN c 0 p) = c#(coefficients p)"
```
```    97   "coefficients p = [p]"
```
```    98
```
```    99 consts isconstant:: "poly \<Rightarrow> bool"
```
```   100 recdef isconstant "measure size"
```
```   101   "isconstant (CN c 0 p) = False"
```
```   102   "isconstant p = True"
```
```   103
```
```   104 consts behead:: "poly \<Rightarrow> poly"
```
```   105 recdef behead "measure size"
```
```   106   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
```
```   107   "behead p = 0\<^sub>p"
```
```   108
```
```   109 consts headconst:: "poly \<Rightarrow> Num"
```
```   110 recdef headconst "measure size"
```
```   111   "headconst (CN c n p) = headconst p"
```
```   112   "headconst (C n) = n"
```
```   113
```
```   114 subsection{* Operations for normalization *}
```
```   115 consts
```
```   116   polyadd :: "poly\<times>poly \<Rightarrow> poly"
```
```   117   polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   118   polysub :: "poly\<times>poly \<Rightarrow> poly"
```
```   119   polymul :: "poly\<times>poly \<Rightarrow> poly"
```
```   120   polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   121 syntax "_polyadd" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   122 translations "a +\<^sub>p b" \<rightleftharpoons> "polyadd (a,b)"
```
```   123 syntax "_polymul" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   124 translations "a *\<^sub>p b" \<rightleftharpoons> "polymul (a,b)"
```
```   125 syntax "_polysub" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   126 translations "a -\<^sub>p b" \<rightleftharpoons> "polysub (a,b)"
```
```   127 syntax "_polypow" :: "nat \<Rightarrow> poly \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   128 translations "a ^\<^sub>p k" \<rightleftharpoons> "polypow k a"
```
```   129
```
```   130 recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)"
```
```   131   "polyadd (C c, C c') = C (c+\<^sub>Nc')"
```
```   132   "polyadd (C c, CN c' n' p') = CN (polyadd (C c, c')) n' p'"
```
```   133   "polyadd (CN c n p, C c') = CN (polyadd (c, C c')) n p"
```
```   134 stupid:  "polyadd (CN c n p, CN c' n' p') =
```
```   135     (if n < n' then CN (polyadd(c,CN c' n' p')) n p
```
```   136      else if n'<n then CN (polyadd(CN c n p, c')) n' p'
```
```   137      else (let cc' = polyadd (c,c') ;
```
```   138                pp' = polyadd (p,p')
```
```   139            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
```
```   140   "polyadd (a, b) = Add a b"
```
```   141 (hints recdef_simp add: Let_def measure_def split_def inv_image_def)
```
```   142
```
```   143 (*
```
```   144 declare stupid [simp del, code del]
```
```   145
```
```   146 lemma [simp,code]: "polyadd (CN c n p, CN c' n' p') =
```
```   147     (if n < n' then CN (polyadd(c,CN c' n' p')) n p
```
```   148      else if n'<n then CN (polyadd(CN c n p, c')) n' p'
```
```   149      else (let cc' = polyadd (c,c') ;
```
```   150                pp' = polyadd (p,p')
```
```   151            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
```
```   152   by (simp add: Let_def stupid)
```
```   153 *)
```
```   154
```
```   155 recdef polyneg "measure size"
```
```   156   "polyneg (C c) = C (~\<^sub>N c)"
```
```   157   "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   158   "polyneg a = Neg a"
```
```   159
```
```   160 defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)"
```
```   161
```
```   162 recdef polymul "measure (\<lambda>(a,b). size a + size b)"
```
```   163   "polymul(C c, C c') = C (c*\<^sub>Nc')"
```
```   164   "polymul(C c, CN c' n' p') =
```
```   165       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul(C c,c')) n' (polymul(C c, p')))"
```
```   166   "polymul(CN c n p, C c') =
```
```   167       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul(c,C c')) n (polymul(p, C c')))"
```
```   168   "polymul(CN c n p, CN c' n' p') =
```
```   169   (if n<n' then CN (polymul(c,CN c' n' p')) n (polymul(p,CN c' n' p'))
```
```   170   else if n' < n
```
```   171   then CN (polymul(CN c n p,c')) n' (polymul(CN c n p,p'))
```
```   172   else polyadd(polymul(CN c n p, c'),CN 0\<^sub>p n' (polymul(CN c n p, p'))))"
```
```   173   "polymul (a,b) = Mul a b"
```
```   174 recdef polypow "measure id"
```
```   175   "polypow 0 = (\<lambda>p. 1\<^sub>p)"
```
```   176   "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in
```
```   177                     if even n then d else polymul(p,d))"
```
```   178
```
```   179 consts polynate :: "poly \<Rightarrow> poly"
```
```   180 recdef polynate "measure polysize"
```
```   181   "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
```
```   182   "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
```
```   183   "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
```
```   184   "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
```
```   185   "polynate (Neg p) = (~\<^sub>p (polynate p))"
```
```   186   "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
```
```   187   "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   188   "polynate (C c) = C (normNum c)"
```
```   189
```
```   190 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
```
```   191   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   192 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   193 | "poly_cmul y p = C y *\<^sub>p p"
```
```   194
```
```   195 constdefs monic:: "poly \<Rightarrow> (poly \<times> bool)"
```
```   196   "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))"
```
```   197
```
```   198 subsection{* Pseudo-division *}
```
```   199
```
```   200 constdefs shift1:: "poly \<Rightarrow> poly"
```
```   201   "shift1 p \<equiv> CN 0\<^sub>p 0 p"
```
```   202 consts funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   203
```
```   204 primrec
```
```   205   "funpow 0 f x = x"
```
```   206   "funpow (Suc n) f x = funpow n f (f x)"
```
```   207 function (tailrec) polydivide_aux :: "(poly \<times> nat \<times> poly \<times> nat \<times> poly) \<Rightarrow> (nat \<times> poly)"
```
```   208   where
```
```   209   "polydivide_aux (a,n,p,k,s) =
```
```   210   (if s = 0\<^sub>p then (k,s)
```
```   211   else (let b = head s; m = degree s in
```
```   212   (if m < n then (k,s) else
```
```   213   (let p'= funpow (m - n) shift1 p in
```
```   214   (if a = b then polydivide_aux (a,n,p,k,s -\<^sub>p p')
```
```   215   else polydivide_aux (a,n,p,Suc k, (a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
```
```   216   by pat_completeness auto
```
```   217
```
```   218
```
```   219 constdefs polydivide:: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)"
```
```   220   "polydivide s p \<equiv> polydivide_aux (head p,degree p,p,0, s)"
```
```   221
```
```   222 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```   223   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   224 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   225
```
```   226 fun poly_deriv :: "poly \<Rightarrow> poly" where
```
```   227   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   228 | "poly_deriv p = 0\<^sub>p"
```
```   229
```
```   230   (* Verification *)
```
```   231 lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
```
```   232 using Nat.gr0_conv_Suc
```
```   233 by clarsimp
```
```   234
```
```   235 subsection{* Semantics of the polynomial representation *}
```
```   236
```
```   237 consts Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{ring_char_0,power,division_by_zero,field}"
```
```   238 primrec
```
```   239   "Ipoly bs (C c) = INum c"
```
```   240   "Ipoly bs (Bound n) = bs!n"
```
```   241   "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   242   "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   243   "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   244   "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   245   "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
```
```   246   "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
```
```   247 syntax "_Ipoly" :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{ring_char_0,power,division_by_zero,field}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
```
```   248 translations "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>" \<rightleftharpoons> "Ipoly bs p"
```
```   249
```
```   250 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
```
```   251   by (simp add: INum_def)
```
```   252 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
```
```   253   by (simp  add: INum_def)
```
```   254
```
```   255 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
```
```   256
```
```   257 subsection {* Normal form and normalization *}
```
```   258
```
```   259 consts isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
```
```   260 recdef isnpolyh "measure size"
```
```   261   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
```
```   262   "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))"
```
```   263   "isnpolyh p = (\<lambda>k. False)"
```
```   264
```
```   265 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
```
```   266 by (induct p rule: isnpolyh.induct, auto)
```
```   267
```
```   268 constdefs isnpoly:: "poly \<Rightarrow> bool"
```
```   269   "isnpoly p \<equiv> isnpolyh p 0"
```
```   270
```
```   271 text{* polyadd preserves normal forms *}
```
```   272
```
```   273 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
```
```   274       \<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
```
```   275 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```   276   case (2 a b c' n' p' n0 n1)
```
```   277   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   278   from prems(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   279   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   280   with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
```
```   281   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   282   thus ?case using prems th3 by simp
```
```   283 next
```
```   284   case (3 c' n' p' a b n1 n0)
```
```   285   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   286   from prems(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   287   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   288   with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
```
```   289   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   290   thus ?case using prems th3 by simp
```
```   291 next
```
```   292   case (4 c n p c' n' p' n0 n1)
```
```   293   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
```
```   294   from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
```
```   295   from prems have ngen0: "n \<ge> n0" by simp
```
```   296   from prems have n'gen1: "n' \<ge> n1" by simp
```
```   297   have "n < n' \<or> n' < n \<or> n = n'" by auto
```
```   298   moreover {assume eq: "n = n'" hence eq': "\<not> n' < n \<and> \<not> n < n'" by simp
```
```   299     with prems(2)[rule_format, OF eq' nc nc']
```
```   300     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
```
```   301     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   302       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
```
```   303     from eq prems(1)[rule_format, OF eq' np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
```
```   304     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
```
```   305     from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
```
```   306   moreover {assume lt: "n < n'"
```
```   307     have "min n0 n1 \<le> n0" by simp
```
```   308     with prems have th1:"min n0 n1 \<le> n" by auto
```
```   309     from prems have th21: "isnpolyh c (Suc n)" by simp
```
```   310     from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
```
```   311     from lt have th23: "min (Suc n) n' = Suc n" by arith
```
```   312     from prems(4)[rule_format, OF lt th21 th22]
```
```   313     have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
```
```   314     with prems th1 have ?case by simp }
```
```   315   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
```
```   316     have "min n0 n1 \<le> n1"  by simp
```
```   317     with prems have th1:"min n0 n1 \<le> n'" by auto
```
```   318     from prems have th21: "isnpolyh c' (Suc n')" by simp_all
```
```   319     from prems have th22: "isnpolyh (CN c n p) n" by simp
```
```   320     from gt have th23: "min n (Suc n') = Suc n'" by arith
```
```   321     from prems(3)[rule_format, OF  gt' th22 th21]
```
```   322     have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
```
```   323     with prems th1 have ?case by simp}
```
```   324       ultimately show ?case by blast
```
```   325 qed auto
```
```   326
```
```   327 lemma polyadd[simp]: "Ipoly bs (polyadd (p,q)) = (Ipoly bs p) + (Ipoly bs q)"
```
```   328 by (induct p q rule: polyadd.induct, auto simp add: Let_def ring_simps right_distrib[symmetric] simp del: right_distrib)
```
```   329
```
```   330 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd(p,q))"
```
```   331   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   332
```
```   333 text{* The degree of addition and other general lemmas needed for the normal form of polymul*}
```
```   334
```
```   335 lemma polyadd_different_degreen:
```
```   336   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   337   degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)"
```
```   338 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   339   case (4 c n p c' n' p' m n0 n1)
```
```   340   thus ?case
```
```   341     apply (cases "n' < n", simp_all add: Let_def)
```
```   342     apply (cases "n = n'", simp_all)
```
```   343     apply (cases "n' = m", simp_all add: Let_def)
```
```   344     by (erule allE[where x="m"], erule allE[where x="Suc m"],
```
```   345            erule allE[where x="m"], erule allE[where x="Suc m"],
```
```   346            clarsimp,erule allE[where x="m"],erule allE[where x="Suc m"], simp)
```
```   347 qed simp_all
```
```   348
```
```   349 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   350   by (induct p arbitrary: n rule: headn.induct, auto)
```
```   351 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   352   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   353 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   354   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```   355
```
```   356 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   357   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   358
```
```   359 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   360   using degree_isnpolyh_Suc by auto
```
```   361 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   362   using degreen_0 by auto
```
```   363
```
```   364
```
```   365 lemma degreen_polyadd:
```
```   366   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
```
```   367   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   368   using np nq m
```
```   369 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   370   case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
```
```   371 next
```
```   372   case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
```
```   373 next
```
```   374   case (4 c n p c' n' p' n0 n1 m)
```
```   375   thus ?case
```
```   376     apply (cases "n < n'", simp_all add: Let_def)
```
```   377     apply (cases "n' < n", simp_all)
```
```   378     apply (erule allE[where x="n"],erule allE[where x="Suc n"],clarify)
```
```   379     apply (erule allE[where x="n'"],erule allE[where x="Suc n'"],clarify)
```
```   380     by (erule allE[where x="m"],erule allE[where x="m"], auto)
```
```   381 qed auto
```
```   382
```
```   383
```
```   384 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk>
```
```   385   \<Longrightarrow> degreen p m = degreen q m"
```
```   386 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   387   case (4 c n p c' n' p' m n0 n1 x)
```
```   388   hence z: "CN c n p +\<^sub>p CN c' n' p' = C x" by simp
```
```   389   {assume nn': "n' < n" hence ?case using prems by simp}
```
```   390   moreover
```
```   391   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
```
```   392     moreover {assume "n < n'" with prems have ?case by simp }
```
```   393     moreover {assume eq: "n = n'" hence ?case using prems
```
```   394         by (cases "p +\<^sub>p p' = 0\<^sub>p", auto simp add: Let_def) }
```
```   395     ultimately have ?case by blast}
```
```   396   ultimately show ?case by blast
```
```   397 qed simp_all
```
```   398
```
```   399 lemma polymul_properties:
```
```   400   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   401   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
```
```   402   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   403   and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
```
```   404   and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0
```
```   405                              else degreen p m + degreen q m)"
```
```   406   using np nq m
```
```   407 proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   408   case (2 a b c' n' p')
```
```   409   let ?c = "(a,b)"
```
```   410   { case (1 n0 n1)
```
```   411     hence n: "isnpolyh (C ?c) n'" "isnpolyh c' (Suc n')" "isnpolyh p' n'" "isnormNum ?c"
```
```   412       "isnpolyh (CN c' n' p') n1"
```
```   413       by simp_all
```
```   414     {assume "?c = 0\<^sub>N" hence ?case by auto}
```
```   415       moreover {assume cnz: "?c \<noteq> 0\<^sub>N"
```
```   416         from "2.hyps"(1)[rule_format,where xb="n'",  OF cnz n(1) n(3)]
```
```   417           "2.hyps"(2)[rule_format, where x="Suc n'"
```
```   418           and xa="Suc n'" and xb = "n'", OF cnz ] cnz n have ?case
```
```   419           by (auto simp add: min_def)}
```
```   420       ultimately show ?case by blast
```
```   421   next
```
```   422     case (2 n0 n1) thus ?case by auto
```
```   423   next
```
```   424     case (3 n0 n1) thus ?case  using "2.hyps" by auto }
```
```   425 next
```
```   426   case (3 c n p a b){
```
```   427     let ?c' = "(a,b)"
```
```   428     case (1 n0 n1)
```
```   429     hence n: "isnpolyh (C ?c') n" "isnpolyh c (Suc n)" "isnpolyh p n" "isnormNum ?c'"
```
```   430       "isnpolyh (CN c n p) n0"
```
```   431       by simp_all
```
```   432     {assume "?c' = 0\<^sub>N" hence ?case by auto}
```
```   433       moreover {assume cnz: "?c' \<noteq> 0\<^sub>N"
```
```   434         from "3.hyps"(1)[rule_format,where xb="n",  OF cnz n(3) n(1)]
```
```   435           "3.hyps"(2)[rule_format, where x="Suc n"
```
```   436           and xa="Suc n" and xb = "n", OF cnz ] cnz n have ?case
```
```   437           by (auto simp add: min_def)}
```
```   438       ultimately show ?case by blast
```
```   439   next
```
```   440     case (2 n0 n1) thus ?case apply auto done
```
```   441   next
```
```   442     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
```
```   443 next
```
```   444   case (4 c n p c' n' p')
```
```   445   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   446     {fix n0 n1
```
```   447       assume "isnpolyh ?cnp n0" and "isnpolyh ?cnp' n1"
```
```   448       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
```
```   449         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
```
```   450         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
```
```   451         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
```
```   452         by simp_all
```
```   453       have "n < n' \<or> n' < n \<or> n' = n" by auto
```
```   454       moreover
```
```   455       {assume nn': "n < n'"
```
```   456         with "4.hyps"(5)[rule_format, OF nn' np cnp', where xb ="n"]
```
```   457           "4.hyps"(6)[rule_format, OF nn' nc cnp', where xb="n"] nn' nn0 nn1 cnp
```
```   458         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   459           by (simp add: min_def) }
```
```   460       moreover
```
```   461
```
```   462       {assume nn': "n > n'" hence stupid: "n' < n \<and> \<not> n < n'" by arith
```
```   463         with "4.hyps"(3)[rule_format, OF stupid cnp np', where xb="n'"]
```
```   464           "4.hyps"(4)[rule_format, OF stupid cnp nc', where xb="Suc n'"]
```
```   465           nn' nn0 nn1 cnp'
```
```   466         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   467           by (cases "Suc n' = n", simp_all add: min_def)}
```
```   468       moreover
```
```   469       {assume nn': "n' = n" hence stupid: "\<not> n' < n \<and> \<not> n < n'" by arith
```
```   470         from "4.hyps"(1)[rule_format, OF stupid cnp np', where xb="n"]
```
```   471           "4.hyps"(2)[rule_format, OF stupid cnp nc', where xb="n"] nn' cnp cnp' nn1
```
```   472
```
```   473         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   474           by simp (rule polyadd_normh,simp_all add: min_def isnpolyh_mono[OF nn0]) }
```
```   475       ultimately show "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)" by blast }
```
```   476     note th = this
```
```   477     {fix n0 n1 m
```
```   478       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
```
```   479       and m: "m \<le> min n0 n1"
```
```   480       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   481       let ?d1 = "degreen ?cnp m"
```
```   482       let ?d2 = "degreen ?cnp' m"
```
```   483       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   484       have "n'<n \<or> n < n' \<or> n' = n" by auto
```
```   485       moreover
```
```   486       {assume "n' < n \<or> n < n'"
```
```   487         with "4.hyps" np np' m
```
```   488         have ?eq apply (cases "n' < n", simp_all)
```
```   489         apply (erule allE[where x="n"],erule allE[where x="n"],auto)
```
```   490         done }
```
```   491       moreover
```
```   492       {assume nn': "n' = n"  hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
```
```   493         from "4.hyps"(1)[rule_format, OF nn, where x="n" and xa ="n'" and xb="n"]
```
```   494           "4.hyps"(2)[rule_format, OF nn, where x="n" and xa ="Suc n'" and xb="n"]
```
```   495           np np' nn'
```
```   496         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   497           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   498           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   499           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
```
```   500         {assume mn: "m = n"
```
```   501           from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
```
```   502             "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"] norm nn' mn
```
```   503           have degs:  "degreen (?cnp *\<^sub>p c') n =
```
```   504             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   505             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
```
```   506           from degs norm
```
```   507           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
```
```   508           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   509             by simp
```
```   510           have nmin: "n \<le> min n n" by (simp add: min_def)
```
```   511           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   512           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
```
```   513           from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
```
```   514             "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"]
```
```   515             mn norm m nn' deg
```
```   516           have ?eq by simp}
```
```   517         moreover
```
```   518         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
```
```   519           from nn' m np have max1: "m \<le> max n n"  by simp
```
```   520           hence min1: "m \<le> min n n" by simp
```
```   521           hence min2: "m \<le> min n (Suc n)" by simp
```
```   522           {assume "c' = 0\<^sub>p"
```
```   523             from `c' = 0\<^sub>p` have ?eq
```
```   524               using "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   525             "4.hyps"(2)[rule_format, OF nn norm(1,2) min2] mn nn'
```
```   526               apply simp
```
```   527               done}
```
```   528           moreover
```
```   529           {assume cnz: "c' \<noteq> 0\<^sub>p"
```
```   530             from "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   531               "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
```
```   532               degreen_polyadd[OF norm(3,6) max1]
```
```   533
```
```   534             have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
```
```   535               \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   536               using mn nn' cnz np np' by simp
```
```   537             with "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   538               "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
```
```   539               degreen_0[OF norm(3) mn'] have ?eq using nn' mn cnz np np' by clarsimp}
```
```   540           ultimately have ?eq by blast }
```
```   541         ultimately have ?eq by blast}
```
```   542       ultimately show ?eq by blast}
```
```   543     note degth = this
```
```   544     { case (2 n0 n1)
```
```   545       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
```
```   546         and m: "m \<le> min n0 n1" by simp_all
```
```   547       hence mn: "m \<le> n" by simp
```
```   548       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   549       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   550         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
```
```   551         from "4.hyps"(1) [rule_format, OF nn, where x="n" and xa = "n" and xb="n"]
```
```   552           "4.hyps"(2) [rule_format, OF nn, where x="n" and xa = "Suc n" and xb="n"]
```
```   553           np np' C(2) mn
```
```   554         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   555           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   556           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   557           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   558           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   559             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   560           by (simp_all add: min_def)
```
```   561
```
```   562           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```   563           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   564             using norm by simp
```
```   565         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
```
```   566         have "False" by simp }
```
```   567       thus ?case using "4.hyps" by clarsimp}
```
```   568 qed auto
```
```   569
```
```   570 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
```
```   571 by(induct p q rule: polymul.induct, auto simp add: ring_simps)
```
```   572
```
```   573 lemma polymul_normh:
```
```   574     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   575   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   576   using polymul_properties(1)  by blast
```
```   577 lemma polymul_eq0_iff:
```
```   578   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   579   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) "
```
```   580   using polymul_properties(2)  by blast
```
```   581 lemma polymul_degreen:
```
```   582   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   583   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)"
```
```   584   using polymul_properties(3) by blast
```
```   585 lemma polymul_norm:
```
```   586   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   587   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
```
```   588   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   589
```
```   590 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   591   by (induct p arbitrary: n0 rule: headconst.induct, auto)
```
```   592
```
```   593 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   594   by (induct p arbitrary: n0, auto)
```
```   595
```
```   596 lemma monic_eqI: assumes np: "isnpolyh p n0"
```
```   597   shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{ring_char_0,power,division_by_zero,field})"
```
```   598   unfolding monic_def Let_def
```
```   599 proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   600   let ?h = "headconst p"
```
```   601   assume pz: "p \<noteq> 0\<^sub>p"
```
```   602   {assume hz: "INum ?h = (0::'a)"
```
```   603     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
```
```   604     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
```
```   605     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
```
```   606   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
```
```   607 qed
```
```   608
```
```   609
```
```   610
```
```   611
```
```   612 text{* polyneg is a negation and preserves normal form *}
```
```   613 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   614 by (induct p rule: polyneg.induct, auto)
```
```   615
```
```   616 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   617   by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
```
```   618 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   619   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```   620 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
```
```   621 by (induct p rule: polyneg.induct, auto simp add: polyneg0)
```
```   622
```
```   623 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   624   using isnpoly_def polyneg_normh by simp
```
```   625
```
```   626
```
```   627 text{* polysub is a substraction and preserves normalform *}
```
```   628 lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
```
```   629 by (simp add: polysub_def polyneg polyadd)
```
```   630 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
```
```   631 by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   632
```
```   633 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
```
```   634   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   635 lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   636   shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p"
```
```   637 unfolding polysub_def split_def fst_conv snd_conv
```
```   638 by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   639
```
```   640 lemma polysub_0:
```
```   641   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   642   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
```
```   643   unfolding polysub_def split_def fst_conv snd_conv
```
```   644   apply (induct p q arbitrary: n0 n1 rule:polyadd.induct, simp_all add: Nsub0[simplified Nsub_def])
```
```   645   apply (clarsimp simp add: Let_def)
```
```   646   apply (case_tac "n < n'", simp_all)
```
```   647   apply (case_tac "n' < n", simp_all)
```
```   648   apply (erule impE)+
```
```   649   apply (rule_tac x="Suc n" in exI, simp)
```
```   650   apply (rule_tac x="n" in exI, simp)
```
```   651   apply (erule impE)+
```
```   652   apply (rule_tac x="n" in exI, simp)
```
```   653   apply (rule_tac x="Suc n" in exI, simp)
```
```   654   apply (erule impE)+
```
```   655   apply (rule_tac x="Suc n" in exI, simp)
```
```   656   apply (rule_tac x="n" in exI, simp)
```
```   657   apply (erule impE)+
```
```   658   apply (rule_tac x="Suc n" in exI, simp)
```
```   659   apply clarsimp
```
```   660   done
```
```   661
```
```   662 text{* polypow is a power function and preserves normal forms *}
```
```   663 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{ring_char_0,division_by_zero,field})) ^ n"
```
```   664 proof(induct n rule: polypow.induct)
```
```   665   case 1 thus ?case by simp
```
```   666 next
```
```   667   case (2 n)
```
```   668   let ?q = "polypow ((Suc n) div 2) p"
```
```   669   let ?d = "polymul(?q,?q)"
```
```   670   have "odd (Suc n) \<or> even (Suc n)" by simp
```
```   671   moreover
```
```   672   {assume odd: "odd (Suc n)"
```
```   673     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
```
```   674     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
```
```   675     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
```
```   676       using "2.hyps" by simp
```
```   677     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   678       apply (simp only: power_add power_one_right) by simp
```
```   679     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
```
```   680       by (simp only: th)
```
```   681     finally have ?case
```
```   682     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
```
```   683   moreover
```
```   684   {assume even: "even (Suc n)"
```
```   685     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
```
```   686     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
```
```   687     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   688       using "2.hyps" apply (simp only: power_add) by simp
```
```   689     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
```
```   690   ultimately show ?case by blast
```
```   691 qed
```
```   692
```
```   693 lemma polypow_normh:
```
```   694     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   695   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   696 proof (induct k arbitrary: n rule: polypow.induct)
```
```   697   case (2 k n)
```
```   698   let ?q = "polypow (Suc k div 2) p"
```
```   699   let ?d = "polymul (?q,?q)"
```
```   700   from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
```
```   701   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
```
```   702   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
```
```   703   from dn on show ?case by (simp add: Let_def)
```
```   704 qed auto
```
```   705
```
```   706 lemma polypow_norm:
```
```   707   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   708   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   709   by (simp add: polypow_normh isnpoly_def)
```
```   710
```
```   711 text{* Finally the whole normalization*}
```
```   712
```
```   713 lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{ring_char_0,division_by_zero,field})"
```
```   714 by (induct p rule:polynate.induct, auto)
```
```   715
```
```   716 lemma polynate_norm[simp]:
```
```   717   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```   718   shows "isnpoly (polynate p)"
```
```   719   by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
```
```   720
```
```   721 text{* shift1 *}
```
```   722
```
```   723
```
```   724 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   725 by (simp add: shift1_def polymul)
```
```   726
```
```   727 lemma shift1_isnpoly:
```
```   728   assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
```
```   729   using pn pnz by (simp add: shift1_def isnpoly_def )
```
```   730
```
```   731 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   732   by (simp add: shift1_def)
```
```   733 lemma funpow_shift1_isnpoly:
```
```   734   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   735   by (induct n arbitrary: p, auto simp add: shift1_isnpoly)
```
```   736
```
```   737 lemma funpow_isnpolyh:
```
```   738   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
```
```   739   shows "isnpolyh (funpow k f p) n"
```
```   740   using f np by (induct k arbitrary: p, auto)
```
```   741
```
```   742 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)"
```
```   743   by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
```
```   744
```
```   745 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   746   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   747
```
```   748 lemma funpow_shift1_1:
```
```   749   "(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)"
```
```   750   by (simp add: funpow_shift1)
```
```   751
```
```   752 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   753 by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: ring_simps)
```
```   754
```
```   755 lemma behead:
```
```   756   assumes np: "isnpolyh p n"
```
```   757   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})"
```
```   758   using np
```
```   759 proof (induct p arbitrary: n rule: behead.induct)
```
```   760   case (1 c p n) hence pn: "isnpolyh p n" by simp
```
```   761   from prems(2)[OF pn]
```
```   762   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   763   then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   764     by (simp_all add: th[symmetric] ring_simps power_Suc)
```
```   765 qed (auto simp add: Let_def)
```
```   766
```
```   767 lemma behead_isnpolyh:
```
```   768   assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
```
```   769   using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
```
```   770
```
```   771 subsection{* Miscilanious lemmas about indexes, decrementation, substitution  etc ... *}
```
```   772 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   773 proof(induct p arbitrary: n rule: polybound0.induct, auto)
```
```   774   case (goal1 c n p n')
```
```   775   hence "n = Suc (n - 1)" by simp
```
```   776   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
```
```   777   with prems(2) show ?case by simp
```
```   778 qed
```
```   779
```
```   780 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   781 by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
```
```   782
```
```   783 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
```
```   784
```
```   785 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   786   apply (induct p arbitrary: n0, auto)
```
```   787   apply (atomize)
```
```   788   apply (erule_tac x = "Suc nat" in allE)
```
```   789   apply auto
```
```   790   done
```
```   791
```
```   792 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   793  by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
```
```   794
```
```   795 lemma polybound0_I:
```
```   796   assumes nb: "polybound0 a"
```
```   797   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
```
```   798 using nb
```
```   799 by (induct a rule: polybound0.induct) auto
```
```   800 lemma polysubst0_I:
```
```   801   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
```
```   802   by (induct t) simp_all
```
```   803
```
```   804 lemma polysubst0_I':
```
```   805   assumes nb: "polybound0 a"
```
```   806   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
```
```   807   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
```
```   808
```
```   809 lemma decrpoly: assumes nb: "polybound0 t"
```
```   810   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
```
```   811   using nb by (induct t rule: decrpoly.induct, simp_all)
```
```   812
```
```   813 lemma polysubst0_polybound0: assumes nb: "polybound0 t"
```
```   814   shows "polybound0 (polysubst0 t a)"
```
```   815 using nb by (induct a rule: polysubst0.induct, auto)
```
```   816
```
```   817 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   818   by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
```
```   819
```
```   820 fun maxindex :: "poly \<Rightarrow> nat" where
```
```   821   "maxindex (Bound n) = n + 1"
```
```   822 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   823 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   824 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   825 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   826 | "maxindex (Neg p) = maxindex p"
```
```   827 | "maxindex (Pw p n) = maxindex p"
```
```   828 | "maxindex (C x) = 0"
```
```   829
```
```   830 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
```
```   831   "wf_bs bs p = (length bs \<ge> maxindex p)"
```
```   832
```
```   833 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
```
```   834 proof(induct p rule: coefficients.induct)
```
```   835   case (1 c p)
```
```   836   show ?case
```
```   837   proof
```
```   838     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   839     hence "x = c \<or> x \<in> set (coefficients p)" by simp
```
```   840     moreover
```
```   841     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
```
```   842     moreover
```
```   843     {assume H: "x \<in> set (coefficients p)"
```
```   844       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
```
```   845       with "1.hyps" H have "wf_bs bs x" by blast }
```
```   846     ultimately  show "wf_bs bs x" by blast
```
```   847   qed
```
```   848 qed simp_all
```
```   849
```
```   850 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   851 by (induct p rule: coefficients.induct, auto)
```
```   852
```
```   853 lemma length_exists: "\<exists>xs. length xs = n" by (rule exI[where x="replicate n x"], simp)
```
```   854
```
```   855 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
```
```   856   unfolding wf_bs_def by (induct p, auto simp add: nth_append)
```
```   857
```
```   858 lemma take_maxindex_wf: assumes wf: "wf_bs bs p"
```
```   859   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   860 proof-
```
```   861   let ?ip = "maxindex p"
```
```   862   let ?tbs = "take ?ip bs"
```
```   863   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
```
```   864   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
```
```   865   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
```
```   866   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
```
```   867 qed
```
```   868
```
```   869 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```   870   by (induct p, auto)
```
```   871
```
```   872 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```   873   unfolding wf_bs_def by simp
```
```   874
```
```   875 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
```
```   876   unfolding wf_bs_def by simp
```
```   877
```
```   878
```
```   879
```
```   880 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
```
```   881 by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
```
```   882 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```   883   by (induct p rule: coefficients.induct, simp_all)
```
```   884
```
```   885
```
```   886 lemma coefficients_head: "last (coefficients p) = head p"
```
```   887   by (induct p rule: coefficients.induct, auto)
```
```   888
```
```   889 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
```
```   890   unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
```
```   891
```
```   892 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
```
```   893   apply (rule exI[where x="replicate (n - length xs) z"])
```
```   894   by simp
```
```   895 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```   896 by (cases p, auto) (case_tac "nat", simp_all)
```
```   897
```
```   898 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```   899   unfolding wf_bs_def
```
```   900   apply (induct p q rule: polyadd.induct)
```
```   901   apply (auto simp add: Let_def)
```
```   902   done
```
```   903
```
```   904 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```   905
```
```   906  unfolding wf_bs_def
```
```   907   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```   908   apply (simp_all add: wf_bs_polyadd)
```
```   909   apply clarsimp
```
```   910   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```   911   apply auto
```
```   912   done
```
```   913
```
```   914 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```   915   unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
```
```   916
```
```   917 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```   918   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
```
```   919
```
```   920 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
```
```   921
```
```   922 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```   923 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
```
```   924 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
```
```   925
```
```   926 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
```
```   927 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```   928   case (1 c p n0)
```
```   929   have cp: "isnpolyh (CN c 0 p) n0" by fact
```
```   930   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```   931     by (auto simp add: isnpolyh_mono[where n'=0])
```
```   932   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
```
```   933 qed auto
```
```   934
```
```   935 lemma coefficients_isconst:
```
```   936   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
```
```   937   by (induct p arbitrary: n rule: coefficients.induct,
```
```   938     auto simp add: isnpolyh_Suc_const)
```
```   939
```
```   940 lemma polypoly_polypoly':
```
```   941   assumes np: "isnpolyh p n0"
```
```   942   shows "polypoly (x#bs) p = polypoly' bs p"
```
```   943 proof-
```
```   944   let ?cf = "set (coefficients p)"
```
```   945   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```   946   {fix q assume q: "q \<in> ?cf"
```
```   947     from q cn_norm have th: "isnpolyh q n0" by blast
```
```   948     from coefficients_isconst[OF np] q have "isconstant q" by blast
```
```   949     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
```
```   950   hence "\<forall>q \<in> ?cf. polybound0 q" ..
```
```   951   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
```
```   952     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```   953     by auto
```
```   954
```
```   955   thus ?thesis unfolding polypoly_def polypoly'_def by simp
```
```   956 qed
```
```   957
```
```   958 lemma polypoly_poly:
```
```   959   assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
```
```   960   using np
```
```   961 by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
```
```   962
```
```   963 lemma polypoly'_poly:
```
```   964   assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```   965   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
```
```   966
```
```   967
```
```   968 lemma polypoly_poly_polybound0:
```
```   969   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
```
```   970   shows "polypoly bs p = [Ipoly bs p]"
```
```   971   using np nb unfolding polypoly_def
```
```   972   by (cases p, auto, case_tac nat, auto)
```
```   973
```
```   974 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```   975   by (induct p rule: head.induct, auto)
```
```   976
```
```   977 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   978   by (cases p,auto)
```
```   979
```
```   980 lemma head_eq_headn0: "head p = headn p 0"
```
```   981   by (induct p rule: head.induct, simp_all)
```
```   982
```
```   983 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   984   by (simp add: head_eq_headn0)
```
```   985
```
```   986 lemma isnpolyh_zero_iff:
```
```   987   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})"
```
```   988   shows "p = 0\<^sub>p"
```
```   989 using nq eq
```
```   990 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```   991   case less
```
```   992   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```   993   {assume nz: "maxindex p = 0"
```
```   994     then obtain c where "p = C c" using np by (cases p, auto)
```
```   995     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
```
```   996   moreover
```
```   997   {assume nz: "maxindex p \<noteq> 0"
```
```   998     let ?h = "head p"
```
```   999     let ?hd = "decrpoly ?h"
```
```  1000     let ?ihd = "maxindex ?hd"
```
```  1001     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
```
```  1002       by simp_all
```
```  1003     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
```
```  1004
```
```  1005     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```  1006     have mihn: "maxindex ?h \<le> maxindex p" by auto
```
```  1007     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
```
```  1008     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
```
```  1009       let ?ts = "take ?ihd bs"
```
```  1010       let ?rs = "drop ?ihd bs"
```
```  1011       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
```
```  1012       have bs_ts_eq: "?ts@ ?rs = bs" by simp
```
```  1013       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
```
```  1014       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
```
```  1015       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
```
```  1016       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
```
```  1017       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
```
```  1018       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
```
```  1019       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```  1020       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
```
```  1021       hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext)
```
```  1022       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```  1023         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```  1024       with coefficients_head[of p, symmetric]
```
```  1025       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
```
```  1026       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
```
```  1027       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
```
```  1028       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
```
```  1029     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
```
```  1030
```
```  1031     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
```
```  1032     hence "?h = 0\<^sub>p" by simp
```
```  1033     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
```
```  1034   ultimately show "p = 0\<^sub>p" by blast
```
```  1035 qed
```
```  1036
```
```  1037 lemma isnpolyh_unique:
```
```  1038   assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1039   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"
```
```  1040 proof(auto)
```
```  1041   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```  1042   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
```
```  1043   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1044     using wf_bs_polysub[where p=p and q=q] by auto
```
```  1045   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
```
```  1046   show "p = q" by blast
```
```  1047 qed
```
```  1048
```
```  1049
```
```  1050 text{* consequenses of unicity on the algorithms for polynomial normalization *}
```
```  1051
```
```  1052 lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1053   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1054   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
```
```  1055
```
```  1056 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
```
```  1057 lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
```
```  1058 lemma polyadd_0[simp]:
```
```  1059   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1060   and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
```
```  1061   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1062     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1063
```
```  1064 lemma polymul_1[simp]:
```
```  1065     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1066   and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
```
```  1067   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1068     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1069 lemma polymul_0[simp]:
```
```  1070   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1071   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"
```
```  1072   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1073     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1074
```
```  1075 lemma polymul_commute:
```
```  1076     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1077   and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1078   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1079 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
```
```  1080
```
```  1081 declare polyneg_polyneg[simp]
```
```  1082
```
```  1083 lemma isnpolyh_polynate_id[simp]:
```
```  1084   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1085   and np:"isnpolyh p n0" shows "polynate p = p"
```
```  1086   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
```
```  1087
```
```  1088 lemma polynate_idempotent[simp]:
```
```  1089     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1090   shows "polynate (polynate p) = polynate p"
```
```  1091   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1092
```
```  1093 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1094   unfolding poly_nate_def polypoly'_def ..
```
```  1095 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>)"
```
```  1096   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1097   unfolding poly_nate_polypoly' by (auto intro: ext)
```
```  1098
```
```  1099 subsection{* heads, degrees and all that *}
```
```  1100 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1101   by (induct p rule: degree.induct, simp_all)
```
```  1102
```
```  1103 lemma degree_polyneg: assumes n: "isnpolyh p n"
```
```  1104   shows "degree (polyneg p) = degree p"
```
```  1105   using n
```
```  1106   by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
```
```  1107
```
```  1108 lemma degree_polyadd:
```
```  1109   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1110   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1111 using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1112
```
```  1113
```
```  1114 lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1115   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1116 proof-
```
```  1117   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
```
```  1118   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1119 qed
```
```  1120
```
```  1121 lemma degree_polysub_samehead:
```
```  1122   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1123   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
```
```  1124   and d: "degree p = degree q"
```
```  1125   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1126 unfolding polysub_def split_def fst_conv snd_conv
```
```  1127 using np nq h d
```
```  1128 proof(induct p q rule:polyadd.induct)
```
```  1129   case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1130 next
```
```  1131   case (2 a b c' n' p')
```
```  1132   let ?c = "(a,b)"
```
```  1133   from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
```
```  1134   hence nz:"n' > 0" by (cases n', auto)
```
```  1135   hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1136   with prems show ?case by simp
```
```  1137 next
```
```  1138   case (3 c n p a' b')
```
```  1139   let ?c' = "(a',b')"
```
```  1140   from prems have "degree (C ?c') = degree (CN c n p)" by simp
```
```  1141   hence nz:"n > 0" by (cases n, auto)
```
```  1142   hence "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1143   with prems show ?case by simp
```
```  1144 next
```
```  1145   case (4 c n p c' n' p')
```
```  1146   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
```
```  1147     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
```
```  1148   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
```
```  1149   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1150     using H(1-2) degree_polyneg by auto
```
```  1151   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
```
```  1152   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
```
```  1153   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
```
```  1154   have "n = n' \<or> n < n' \<or> n > n'" by arith
```
```  1155   moreover
```
```  1156   {assume nn': "n = n'"
```
```  1157     have "n = 0 \<or> n >0" by arith
```
```  1158     moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
```
```  1159     moreover {assume nz: "n > 0"
```
```  1160       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
```
```  1161       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)}
```
```  1162     ultimately have ?case by blast}
```
```  1163   moreover
```
```  1164   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
```
```  1165     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
```
```  1166     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)
```
```  1167     hence "n > 0" by (cases n, simp_all)
```
```  1168     hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1169     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1170   moreover
```
```  1171   {assume nn': "n > n'"  hence np: "n > 0" by simp
```
```  1172     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
```
```  1173     from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
```
```  1174     from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
```
```  1175     with degcnpeq have "n' > 0" by (cases n', simp_all)
```
```  1176     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1177     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1178   ultimately show ?case  by blast
```
```  1179 qed auto
```
```  1180
```
```  1181 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1182 by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
```
```  1183
```
```  1184 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1185 proof(induct k arbitrary: n0 p)
```
```  1186   case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
```
```  1187   with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1188     and "head (shift1 p) = head p" by (simp_all add: shift1_head)
```
```  1189   thus ?case by simp
```
```  1190 qed auto
```
```  1191
```
```  1192 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1193   by (simp add: shift1_def)
```
```  1194 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1195   by (induct k arbitrary: p, auto simp add: shift1_degree)
```
```  1196
```
```  1197 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1198   by (induct n arbitrary: p, simp_all add: funpow_def)
```
```  1199
```
```  1200 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1201   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1202 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1203   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```  1204 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1205   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1206 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1207   by (induct p rule: head.induct, auto)
```
```  1208
```
```  1209 lemma polyadd_eq_const_degree:
```
```  1210   "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q"
```
```  1211   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1212
```
```  1213 lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1214   and deg: "degree p \<noteq> degree q"
```
```  1215   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1216 using np nq deg
```
```  1217 apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
```
```  1218 apply (case_tac n', simp, simp)
```
```  1219 apply (case_tac n, simp, simp)
```
```  1220 apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1221 apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
```
```  1222 apply (clarsimp simp add: polyadd_eq_const_degree)
```
```  1223 apply clarsimp
```
```  1224 apply (erule_tac impE,blast)
```
```  1225 apply (erule_tac impE,blast)
```
```  1226 apply clarsimp
```
```  1227 apply simp
```
```  1228 apply (case_tac n', simp_all)
```
```  1229 done
```
```  1230
```
```  1231 lemma polymul_head_polyeq:
```
```  1232    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1233   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"
```
```  1234 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1235   case (2 a b c' n' p' n0 n1)
```
```  1236   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)"  by (simp_all add: head_isnpolyh)
```
```  1237   thus ?case using prems by (cases n', auto)
```
```  1238 next
```
```  1239   case (3 c n p a' b' n0 n1)
```
```  1240   hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')"  by (simp_all add: head_isnpolyh)
```
```  1241   thus ?case using prems by (cases n, auto)
```
```  1242 next
```
```  1243   case (4 c n p c' n' p' n0 n1)
```
```  1244   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1245     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1246     by simp_all
```
```  1247   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1248   moreover
```
```  1249   {assume nn': "n < n'" hence ?case
```
```  1250       thm prems
```
```  1251       using norm
```
```  1252     prems(6)[rule_format, OF nn' norm(1,6)]
```
```  1253     prems(7)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
```
```  1254   moreover {assume nn': "n'< n"
```
```  1255     hence stupid: "n' < n \<and> \<not> n < n'" by simp
```
```  1256     hence ?case using norm prems(4) [rule_format, OF stupid norm(5,3)]
```
```  1257       prems(5)[rule_format, OF stupid norm(5,4)]
```
```  1258       by (simp,cases n',simp,cases n,auto)}
```
```  1259   moreover {assume nn': "n' = n"
```
```  1260     hence stupid: "\<not> n' < n \<and> \<not> n < n'" by simp
```
```  1261     from nn' polymul_normh[OF norm(5,4)]
```
```  1262     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1263     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1264     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1265     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1266     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1267     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1268     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"
```
```  1269       by (simp add: min_def)
```
```  1270     {assume np: "n > 0"
```
```  1271       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1272         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1273       have ?case by simp}
```
```  1274     moreover
```
```  1275     {moreover assume nz: "n = 0"
```
```  1276       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1277         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1278       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1279     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
```
```  1280     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
```
```  1281     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1282     have ?case   using norm prems(2)[rule_format, OF stupid norm(5,3)]
```
```  1283         prems(3)[rule_format, OF stupid norm(5,4)] nn' nz by simp }
```
```  1284     ultimately have ?case by (cases n) auto}
```
```  1285   ultimately show ?case by blast
```
```  1286 qed simp_all
```
```  1287
```
```  1288 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1289   by(induct p rule: degree.induct, auto)
```
```  1290
```
```  1291 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1292   by (induct p rule: head.induct, auto)
```
```  1293
```
```  1294 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p"
```
```  1295   by (cases n, simp_all)
```
```  1296 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1297   by (cases n, simp_all)
```
```  1298
```
```  1299 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)"
```
```  1300   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1301
```
```  1302 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1303   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```  1304
```
```  1305 lemma degree_polymul:
```
```  1306   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1307   and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1308   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1309   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1310
```
```  1311 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1312   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1313
```
```  1314 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
```
```  1315   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1316
```
```  1317 subsection {* Correctness of polynomial pseudo division *}
```
```  1318
```
```  1319 lemma polydivide_aux_real_domintros:
```
```  1320   assumes call1: "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a = head s\<rbrakk>
```
```  1321   \<Longrightarrow> polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)"
```
```  1322   and call2 : "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a \<noteq> head s\<rbrakk>
```
```  1323   \<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))"
```
```  1324   shows "polydivide_aux_dom (a, n, p, k, s)"
```
```  1325 proof (rule accpI, erule polydivide_aux_rel.cases)
```
```  1326   fix y aa ka na pa sa x xa xb
```
```  1327   assume eqs: "y = (aa, na, pa, ka, sa -\<^sub>p xb)" "(a, n, p, k, s) = (aa, na, pa, ka, sa)"
```
```  1328      and \<Gamma>1': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na"
```
```  1329     "xb = funpow (xa - na) shift1 pa" "aa = x"
```
```  1330
```
```  1331   hence \<Gamma>1: "s \<noteq> 0\<^sub>p" "a = head s" "xa = degree s" "\<not> degree s < n" "\<not> xa < na"
```
```  1332     "xb = funpow (xa - na) shift1 pa" "aa = x" by auto
```
```  1333
```
```  1334   with call1 have "polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)"
```
```  1335     by auto
```
```  1336   with eqs \<Gamma>1 show "polydivide_aux_dom y" by auto
```
```  1337 next
```
```  1338   fix y aa ka na pa sa x xa xb
```
```  1339   assume eqs: "y = (aa, na, pa, Suc ka, aa *\<^sub>p sa -\<^sub>p (x *\<^sub>p xb))"
```
```  1340     "(a, n, p, k, s) =(aa, na, pa, ka, sa)"
```
```  1341     and \<Gamma>2': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na"
```
```  1342     "xb = funpow (xa - na) shift1 pa" "aa \<noteq> x"
```
```  1343   hence \<Gamma>2: "s \<noteq> 0\<^sub>p" "a \<noteq> head s" "xa = degree s" "\<not> degree s < n"
```
```  1344     "xb = funpow (xa - na) shift1 pa" "aa \<noteq> x" by auto
```
```  1345   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
```
```  1346   with eqs \<Gamma>2'  show "polydivide_aux_dom y" by auto
```
```  1347 qed
```
```  1348
```
```  1349 lemma polydivide_aux_properties:
```
```  1350   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1351   and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
```
```  1352   and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1353   shows "polydivide_aux_dom (a,n,p,k,s) \<and>
```
```  1354   (polydivide_aux (a,n,p,k,s) = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1355           \<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)))"
```
```  1356   using ns
```
```  1357 proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1358   case less
```
```  1359   let ?D = "polydivide_aux_dom"
```
```  1360   let ?dths = "?D (a, n, p, k, s)"
```
```  1361   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1362   let ?qrths = "polydivide_aux (a, n, p, k, s) = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1363     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1364   let ?ths = "?dths \<and> ?qrths"
```
```  1365   let ?b = "head s"
```
```  1366   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1367   let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
```
```  1368   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1369   note ns = `isnpolyh s n1`
```
```  1370   from np have np0: "isnpolyh p 0"
```
```  1371     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
```
```  1372   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
```
```  1373   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
```
```  1374   from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
```
```  1375   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1376   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1377   {assume sz: "s = 0\<^sub>p"
```
```  1378     hence dom: ?dths apply - apply (rule polydivide_aux_real_domintros) apply simp_all done
```
```  1379     from polydivide_aux.psimps[OF dom] sz
```
```  1380     have ?qrths using np apply clarsimp by (rule exI[where x="0\<^sub>p"], simp)
```
```  1381     hence ?ths using dom by blast}
```
```  1382   moreover
```
```  1383   {assume sz: "s \<noteq> 0\<^sub>p"
```
```  1384     {assume dn: "degree s < n"
```
```  1385       with sz have dom:"?dths" by - (rule polydivide_aux_real_domintros,simp_all)
```
```  1386       from polydivide_aux.psimps[OF dom] sz dn
```
```  1387       have "?qrths" using ns ndp np by auto (rule exI[where x="0\<^sub>p"],simp)
```
```  1388       with dom have ?ths by blast}
```
```  1389     moreover
```
```  1390     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
```
```  1391       have degsp': "degree s = degree ?p'"
```
```  1392         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
```
```  1393       {assume ba: "?b = a"
```
```  1394         hence headsp': "head s = head ?p'" using ap headp' by simp
```
```  1395         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
```
```  1396         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1397         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
```
```  1398         moreover
```
```  1399         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1400           from  less(1)[OF deglt nr]
```
```  1401           have domsp: "?D (a, n, p, k, s -\<^sub>p ?p')" by blast
```
```  1402           have dom: ?dths apply (rule polydivide_aux_real_domintros)
```
```  1403             using ba dn' domsp by simp_all
```
```  1404           from polydivide_aux.psimps[OF dom] sz dn' ba
```
```  1405           have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
```
```  1406             by (simp add: Let_def)
```
```  1407           {assume h1: "polydivide_aux (a, n, p, k, s) = (k', r)"
```
```  1408             from less(1)[OF deglt nr, of k k' r]
```
```  1409               trans[OF eq[symmetric] h1]
```
```  1410             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1411               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
```
```  1412             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1413               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
```
```  1414             from nr obtain nr where nr': "isnpolyh r nr" by blast
```
```  1415             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
```
```  1416             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1417             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1418             from polyadd_normh[OF polymul_normh[OF np
```
```  1419               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1420             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
```
```  1421             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')) =
```
```  1422               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
```
```  1423             hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1424               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1425               by (simp add: ring_simps)
```
```  1426             hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1427               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p)
```
```  1428               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1429               by (auto simp only: funpow_shift1_1)
```
```  1430             hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1431               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p)
```
```  1432               + Ipoly bs q) + Ipoly bs r" by (simp add: ring_simps)
```
```  1433             hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1434               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
```
```  1435             with isnpolyh_unique[OF nakks' nqr']
```
```  1436             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1437               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
```
```  1438             hence ?qths using nq'
```
```  1439               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
```
```  1440               apply (rule_tac x="0" in exI) by simp
```
```  1441             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"
```
```  1442               by blast } hence ?qrths by blast
```
```  1443           with dom have ?ths by blast}
```
```  1444         moreover
```
```  1445         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1446           hence domsp: "?D (a, n, p, k, s -\<^sub>p ?p')"
```
```  1447             apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
```
```  1448           have dom: ?dths apply (rule polydivide_aux_real_domintros)
```
```  1449             using ba dn' domsp by simp_all
```
```  1450           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}"]
```
```  1451           have " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs ?p'" by simp
```
```  1452           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
```
```  1453             by (simp only: funpow_shift1_1) simp
```
```  1454           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
```
```  1455           {assume h1: "polydivide_aux (a,n,p,k,s) = (k',r)"
```
```  1456             from polydivide_aux.psimps[OF dom] sz dn' ba
```
```  1457             have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
```
```  1458               by (simp add: Let_def)
```
```  1459             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.psimps[OF domsp] spz by simp
```
```  1460             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
```
```  1461             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1462               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?qrths
```
```  1463               apply auto
```
```  1464               apply (rule exI[where x="?xdn"])
```
```  1465               apply (auto simp add: polymul_commute[of p])
```
```  1466               done}
```
```  1467           with dom have ?ths by blast}
```
```  1468         ultimately have ?ths by blast }
```
```  1469       moreover
```
```  1470       {assume ba: "?b \<noteq> a"
```
```  1471         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1472           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1473         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
```
```  1474         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1475           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1476             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1477             funpow_shift1_nz[OF pnz] by simp_all
```
```  1478         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1479           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1480         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1481           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1482             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1483           by (simp add: ap)
```
```  1484         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1485           head_nz[OF np] pnz sz ap[symmetric]
```
```  1486           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1487           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
```
```  1488           ndp dn
```
```  1489         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
```
```  1490           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1491         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1492           have th: "?D (a, n, p, Suc k, (a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))"
```
```  1493             using less(1)[OF dth nth] by blast
```
```  1494           have dom: ?dths using ba dn' th
```
```  1495             by - (rule polydivide_aux_real_domintros, simp_all)
```
```  1496           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
```
```  1497           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
```
```  1498           {assume h1:"polydivide_aux (a,n,p,k,s) = (k', r)"
```
```  1499             from h1  polydivide_aux.psimps[OF dom] sz dn' ba
```
```  1500             have eq:"polydivide_aux (a,n,p,Suc k,(a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1501               by (simp add: Let_def)
```
```  1502             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1503             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
```
```  1504               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1505               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
```
```  1506             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
```
```  1507             {fix bs:: "'a::{ring_char_0,division_by_zero,field} list"
```
```  1508
```
```  1509             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1510             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
```
```  1511             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"
```
```  1512               by (simp add: ring_simps power_Suc)
```
```  1513             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"
```
```  1514               by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1515             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"
```
```  1516               by (simp add: ring_simps)}
```
```  1517             hence ieq:"\<forall>(bs :: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1518               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
```
```  1519             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1520             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]]
```
```  1521             have nqw: "isnpolyh ?q 0" by simp
```
```  1522             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]]
```
```  1523             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
```
```  1524             from dr kk' nr h1 asth nqw have ?qrths apply simp
```
```  1525               apply (rule conjI)
```
```  1526               apply (rule exI[where x="nr"], simp)
```
```  1527               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1528               apply (rule exI[where x="0"], simp)
```
```  1529               done}
```
```  1530           hence ?qrths by blast
```
```  1531           with dom have ?ths by blast}
```
```  1532         moreover
```
```  1533         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1534           hence domsp: "?D (a, n, p, Suc k, a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p'))"
```
```  1535             apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
```
```  1536           have dom: ?dths using sz ba dn' domsp
```
```  1537             by - (rule polydivide_aux_real_domintros, simp_all)
```
```  1538           {fix bs :: "'a::{ring_char_0,division_by_zero,field} list"
```
```  1539             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1540           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
```
```  1541           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1542             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1543           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
```
```  1544         }
```
```  1545         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))" ..
```
```  1546           from hth
```
```  1547           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1548             using isnpolyh_unique[where ?'a = "'a::{ring_char_0,division_by_zero,field}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1549                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1550               simplified ap] by simp
```
```  1551           {assume h1: "polydivide_aux (a,n,p,k,s) = (k', r)"
```
```  1552           from h1 sz ba dn' spz polydivide_aux.psimps[OF dom] polydivide_aux.psimps[OF domsp]
```
```  1553           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
```
```  1554           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1555             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1556           have ?qrths apply (clarsimp simp add: Let_def)
```
```  1557             apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
```
```  1558             apply (rule exI[where x="0"], simp)
```
```  1559             done}
```
```  1560         hence ?qrths by blast
```
```  1561         with dom have ?ths by blast}
```
```  1562         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"]
```
```  1563           head_nz[OF np] pnz sz ap[symmetric]
```
```  1564           by (simp add: degree_eq_degreen0[symmetric]) blast }
```
```  1565       ultimately have ?ths by blast
```
```  1566     }
```
```  1567     ultimately have ?ths by blast}
```
```  1568   ultimately show ?ths by blast
```
```  1569 qed
```
```  1570
```
```  1571 lemma polydivide_properties:
```
```  1572   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1573   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1574   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1575   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
```
```  1576 proof-
```
```  1577   have trv: "head p = head p" "degree p = degree p" by simp_all
```
```  1578   from polydivide_aux_properties[OF np ns trv pnz, where k="0"]
```
```  1579   have d: "polydivide_aux_dom (head p, degree p, p, 0, s)" by blast
```
```  1580   from polydivide_def[where s="s" and p="p"] polydivide_aux.psimps[OF d]
```
```  1581   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
```
```  1582   then obtain k r where kr: "polydivide s p = (k,r)" by blast
```
```  1583   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s" and p="p"], symmetric] kr]
```
```  1584     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1585   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1586    (\<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
```
```  1587   with kr show ?thesis
```
```  1588     apply -
```
```  1589     apply (rule exI[where x="k"])
```
```  1590     apply (rule exI[where x="r"])
```
```  1591     apply simp
```
```  1592     done
```
```  1593 qed
```
```  1594
```
```  1595 subsection{* More about polypoly and pnormal etc *}
```
```  1596
```
```  1597 definition "isnonconstant p = (\<not> isconstant p)"
```
```  1598
```
```  1599 lemma last_map: "xs \<noteq> [] ==> last (map f xs) = f (last xs)" by (induct xs, auto)
```
```  1600
```
```  1601 lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p"
```
```  1602   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1603 proof
```
```  1604   let ?p = "polypoly bs p"
```
```  1605   assume H: "pnormal ?p"
```
```  1606   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1607
```
```  1608   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1609     pnormal_last_nonzero[OF H]
```
```  1610   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
```
```  1611 next
```
```  1612   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1613   let ?p = "polypoly bs p"
```
```  1614   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1615   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
```
```  1616   hence lg: "length ?p > 0" by simp
```
```  1617   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1618   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
```
```  1619   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1620 qed
```
```  1621
```
```  1622 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1623   unfolding isnonconstant_def
```
```  1624   apply (cases p, simp_all)
```
```  1625   apply (case_tac nat, auto)
```
```  1626   done
```
```  1627 lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
```
```  1628   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1629 proof
```
```  1630   let ?p = "polypoly bs p"
```
```  1631   assume nc: "nonconstant ?p"
```
```  1632   from isnonconstant_pnormal_iff[OF inc, of bs] nc
```
```  1633   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
```
```  1634 next
```
```  1635   let ?p = "polypoly bs p"
```
```  1636   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1637   from isnonconstant_pnormal_iff[OF inc, of bs] h
```
```  1638   have pn: "pnormal ?p" by blast
```
```  1639   {fix x assume H: "?p = [x]"
```
```  1640     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
```
```  1641     with isnonconstant_coefficients_length[OF inc] have False by arith}
```
```  1642   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
```
```  1643 qed
```
```  1644
```
```  1645 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  1646   unfolding pnormal_def
```
```  1647  apply (induct p rule: pnormalize.induct, simp_all)
```
```  1648  apply (case_tac "p=[]", simp_all)
```
```  1649  done
```
```  1650
```
```  1651 lemma degree_degree: assumes inc: "isnonconstant p"
```
```  1652   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1653 proof
```
```  1654   let  ?p = "polypoly bs p"
```
```  1655   assume H: "degree p = Polynomial_List.degree ?p"
```
```  1656   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
```
```  1657     unfolding polypoly_def by auto
```
```  1658   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1659   have lg:"length (pnormalize ?p) = length ?p"
```
```  1660     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  1661   hence "pnormal ?p" using pnormal_length[OF pz] by blast
```
```  1662   with isnonconstant_pnormal_iff[OF inc]
```
```  1663   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
```
```  1664 next
```
```  1665   let  ?p = "polypoly bs p"
```
```  1666   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1667   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
```
```  1668   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1669   show "degree p = Polynomial_List.degree ?p"
```
```  1670     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  1671 qed
```
```  1672
```
```  1673 section{* Swaps ; Division by a certain variable *}
```
```  1674 consts swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
```
```  1675 primrec
```
```  1676   "swap n m (C x) = C x"
```
```  1677   "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
```
```  1678   "swap n m (Neg t) = Neg (swap n m t)"
```
```  1679   "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  1680   "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  1681   "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  1682   "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  1683   "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)
```
```  1684   (swap n m p)"
```
```  1685
```
```  1686 lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1687   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1688 proof (induct t)
```
```  1689   case (Bound k) thus ?case using nbs mbs by simp
```
```  1690 next
```
```  1691   case (CN c k p) thus ?case using nbs mbs by simp
```
```  1692 qed simp_all
```
```  1693 lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
```
```  1694   by (induct t,simp_all)
```
```  1695
```
```  1696 lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
```
```  1697
```
```  1698 lemma swap_same_id[simp]: "swap n n t = t"
```
```  1699   by (induct t, simp_all)
```
```  1700
```
```  1701 definition "swapnorm n m t = polynate (swap n m t)"
```
```  1702
```
```  1703 lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1704   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"
```
```  1705   using swap[OF prems] swapnorm_def by simp
```
```  1706
```
```  1707 lemma swapnorm_isnpoly[simp]:
```
```  1708     assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
```
```  1709   shows "isnpoly (swapnorm n m p)"
```
```  1710   unfolding swapnorm_def by simp
```
```  1711
```
```  1712 definition "polydivideby n s p =
```
```  1713     (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
```
```  1714      in (k,swapnorm 0 n h,swapnorm 0 n r))"
```
```  1715
```
```  1716 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
```
```  1717
```
```  1718 consts isweaknpoly :: "poly \<Rightarrow> bool"
```
```  1719 recdef isweaknpoly "measure size"
```
```  1720   "isweaknpoly (C c) = True"
```
```  1721   "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  1722   "isweaknpoly p = False"
```
```  1723
```
```  1724 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  1725   by (induct p arbitrary: n0, auto)
```
```  1726
```
```  1727 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  1728   by (induct p, auto)
```
```  1729
```
`  1730 end`