src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
author wenzelm
Sun Sep 13 20:20:16 2015 +0200 (2015-09-13)
changeset 61166 5976fe402824
parent 60698 29e8bdc41f90
child 61586 5197a2ecb658
permissions -rw-r--r--
renamed method "goals" to "goal_cases" to emphasize its meaning;
chaieb@33154
     1
(*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
chaieb@33154
     2
    Author:     Amine Chaieb
chaieb@33154
     3
*)
chaieb@33154
     4
wenzelm@60533
     5
section \<open>Implementation and verification of multivariate polynomials\<close>
chaieb@33154
     6
chaieb@33154
     7
theory Reflected_Multivariate_Polynomial
haftmann@54220
     8
imports Complex_Main Rat_Pair Polynomial_List
chaieb@33154
     9
begin
chaieb@33154
    10
wenzelm@60698
    11
subsection \<open>Datatype of polynomial expressions\<close>
chaieb@33154
    12
blanchet@58310
    13
datatype poly = C Num | Bound nat | Add poly poly | Sub poly poly
chaieb@33154
    14
  | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
chaieb@33154
    15
wenzelm@35054
    16
abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
wenzelm@50282
    17
abbreviation poly_p :: "int \<Rightarrow> poly" ("'((_)')\<^sub>p") where "(i)\<^sub>p \<equiv> C (i)\<^sub>N"
chaieb@33154
    18
wenzelm@52658
    19
wenzelm@60533
    20
subsection\<open>Boundedness, substitution and all that\<close>
wenzelm@52658
    21
wenzelm@52658
    22
primrec polysize:: "poly \<Rightarrow> nat"
wenzelm@52658
    23
where
chaieb@33154
    24
  "polysize (C c) = 1"
haftmann@39246
    25
| "polysize (Bound n) = 1"
haftmann@39246
    26
| "polysize (Neg p) = 1 + polysize p"
haftmann@39246
    27
| "polysize (Add p q) = 1 + polysize p + polysize q"
haftmann@39246
    28
| "polysize (Sub p q) = 1 + polysize p + polysize q"
haftmann@39246
    29
| "polysize (Mul p q) = 1 + polysize p + polysize q"
haftmann@39246
    30
| "polysize (Pw p n) = 1 + polysize p"
haftmann@39246
    31
| "polysize (CN c n p) = 4 + polysize c + polysize p"
chaieb@33154
    32
wenzelm@60533
    33
primrec polybound0:: "poly \<Rightarrow> bool" -- \<open>a poly is INDEPENDENT of Bound 0\<close>
wenzelm@52658
    34
where
wenzelm@56000
    35
  "polybound0 (C c) \<longleftrightarrow> True"
wenzelm@56000
    36
| "polybound0 (Bound n) \<longleftrightarrow> n > 0"
wenzelm@56000
    37
| "polybound0 (Neg a) \<longleftrightarrow> polybound0 a"
wenzelm@56000
    38
| "polybound0 (Add a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
wenzelm@56000
    39
| "polybound0 (Sub a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
wenzelm@56000
    40
| "polybound0 (Mul a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
wenzelm@56000
    41
| "polybound0 (Pw p n) \<longleftrightarrow> polybound0 p"
wenzelm@56000
    42
| "polybound0 (CN c n p) \<longleftrightarrow> n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p"
haftmann@39246
    43
wenzelm@60533
    44
primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" -- \<open>substitute a poly into a poly for Bound 0\<close>
wenzelm@52658
    45
where
wenzelm@56000
    46
  "polysubst0 t (C c) = C c"
wenzelm@56000
    47
| "polysubst0 t (Bound n) = (if n = 0 then t else Bound n)"
haftmann@39246
    48
| "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
haftmann@39246
    49
| "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
wenzelm@52803
    50
| "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
haftmann@39246
    51
| "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
haftmann@39246
    52
| "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
wenzelm@56000
    53
| "polysubst0 t (CN c n p) =
wenzelm@56000
    54
    (if n = 0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
wenzelm@56000
    55
     else CN (polysubst0 t c) n (polysubst0 t p))"
chaieb@33154
    56
wenzelm@52803
    57
fun decrpoly:: "poly \<Rightarrow> poly"
krauss@41808
    58
where
chaieb@33154
    59
  "decrpoly (Bound n) = Bound (n - 1)"
krauss@41808
    60
| "decrpoly (Neg a) = Neg (decrpoly a)"
krauss@41808
    61
| "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
krauss@41808
    62
| "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
krauss@41808
    63
| "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
krauss@41808
    64
| "decrpoly (Pw p n) = Pw (decrpoly p) n"
krauss@41808
    65
| "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
krauss@41808
    66
| "decrpoly a = a"
chaieb@33154
    67
wenzelm@52658
    68
wenzelm@60698
    69
subsection \<open>Degrees and heads and coefficients\<close>
chaieb@33154
    70
wenzelm@56207
    71
fun degree :: "poly \<Rightarrow> nat"
krauss@41808
    72
where
chaieb@33154
    73
  "degree (CN c 0 p) = 1 + degree p"
krauss@41808
    74
| "degree p = 0"
chaieb@33154
    75
wenzelm@56207
    76
fun head :: "poly \<Rightarrow> poly"
krauss@41808
    77
where
chaieb@33154
    78
  "head (CN c 0 p) = head p"
krauss@41808
    79
| "head p = p"
krauss@41808
    80
wenzelm@60698
    81
text \<open>More general notions of degree and head.\<close>
wenzelm@60698
    82
wenzelm@56207
    83
fun degreen :: "poly \<Rightarrow> nat \<Rightarrow> nat"
krauss@41808
    84
where
wenzelm@56000
    85
  "degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
wenzelm@56000
    86
| "degreen p = (\<lambda>m. 0)"
chaieb@33154
    87
wenzelm@56207
    88
fun headn :: "poly \<Rightarrow> nat \<Rightarrow> poly"
krauss@41808
    89
where
krauss@41808
    90
  "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
krauss@41808
    91
| "headn p = (\<lambda>m. p)"
chaieb@33154
    92
wenzelm@56207
    93
fun coefficients :: "poly \<Rightarrow> poly list"
krauss@41808
    94
where
wenzelm@56000
    95
  "coefficients (CN c 0 p) = c # coefficients p"
krauss@41808
    96
| "coefficients p = [p]"
chaieb@33154
    97
wenzelm@56207
    98
fun isconstant :: "poly \<Rightarrow> bool"
krauss@41808
    99
where
krauss@41808
   100
  "isconstant (CN c 0 p) = False"
krauss@41808
   101
| "isconstant p = True"
chaieb@33154
   102
wenzelm@56207
   103
fun behead :: "poly \<Rightarrow> poly"
krauss@41808
   104
where
krauss@41808
   105
  "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
krauss@41808
   106
| "behead p = 0\<^sub>p"
krauss@41808
   107
wenzelm@56207
   108
fun headconst :: "poly \<Rightarrow> Num"
krauss@41808
   109
where
chaieb@33154
   110
  "headconst (CN c n p) = headconst p"
krauss@41808
   111
| "headconst (C n) = n"
chaieb@33154
   112
wenzelm@52658
   113
wenzelm@60698
   114
subsection \<open>Operations for normalization\<close>
krauss@41812
   115
krauss@41812
   116
declare if_cong[fundef_cong del]
krauss@41812
   117
declare let_cong[fundef_cong del]
krauss@41812
   118
krauss@41812
   119
fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
krauss@41812
   120
where
wenzelm@56000
   121
  "polyadd (C c) (C c') = C (c +\<^sub>N c')"
wenzelm@52803
   122
| "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
krauss@41812
   123
| "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
krauss@41812
   124
| "polyadd (CN c n p) (CN c' n' p') =
krauss@41812
   125
    (if n < n' then CN (polyadd c (CN c' n' p')) n p
wenzelm@56000
   126
     else if n' < n then CN (polyadd (CN c n p) c') n' p'
wenzelm@56000
   127
     else
wenzelm@56000
   128
      let
wenzelm@56000
   129
        cc' = polyadd c c';
wenzelm@56000
   130
        pp' = polyadd p p'
wenzelm@56000
   131
      in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
krauss@41812
   132
| "polyadd a b = Add a b"
krauss@41812
   133
chaieb@33154
   134
krauss@41808
   135
fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
krauss@41808
   136
where
chaieb@33154
   137
  "polyneg (C c) = C (~\<^sub>N c)"
krauss@41808
   138
| "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
krauss@41808
   139
| "polyneg a = Neg a"
chaieb@33154
   140
krauss@41814
   141
definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
wenzelm@52658
   142
  where "p -\<^sub>p q = polyadd p (polyneg q)"
krauss@41813
   143
krauss@41813
   144
fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
krauss@41813
   145
where
wenzelm@56043
   146
  "polymul (C c) (C c') = C (c *\<^sub>N c')"
wenzelm@52803
   147
| "polymul (C c) (CN c' n' p') =
wenzelm@56000
   148
    (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
wenzelm@52803
   149
| "polymul (CN c n p) (C c') =
wenzelm@56000
   150
    (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
wenzelm@52803
   151
| "polymul (CN c n p) (CN c' n' p') =
wenzelm@56000
   152
    (if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
wenzelm@56000
   153
     else if n' < n then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
wenzelm@56000
   154
     else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
krauss@41813
   155
| "polymul a b = Mul a b"
krauss@41808
   156
krauss@41812
   157
declare if_cong[fundef_cong]
krauss@41812
   158
declare let_cong[fundef_cong]
krauss@41812
   159
krauss@41808
   160
fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
krauss@41808
   161
where
wenzelm@50282
   162
  "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
wenzelm@56000
   163
| "polypow n =
wenzelm@56000
   164
    (\<lambda>p.
wenzelm@56000
   165
      let
wenzelm@56000
   166
        q = polypow (n div 2) p;
wenzelm@56000
   167
        d = polymul q q
wenzelm@56000
   168
      in if even n then d else polymul p d)"
chaieb@33154
   169
wenzelm@35054
   170
abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
wenzelm@35054
   171
  where "a ^\<^sub>p k \<equiv> polypow k a"
chaieb@33154
   172
krauss@41808
   173
function polynate :: "poly \<Rightarrow> poly"
krauss@41808
   174
where
wenzelm@50282
   175
  "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
wenzelm@56000
   176
| "polynate (Add p q) = polynate p +\<^sub>p polynate q"
wenzelm@56000
   177
| "polynate (Sub p q) = polynate p -\<^sub>p polynate q"
wenzelm@56000
   178
| "polynate (Mul p q) = polynate p *\<^sub>p polynate q"
wenzelm@56000
   179
| "polynate (Neg p) = ~\<^sub>p (polynate p)"
wenzelm@56000
   180
| "polynate (Pw p n) = polynate p ^\<^sub>p n"
krauss@41808
   181
| "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
krauss@41808
   182
| "polynate (C c) = C (normNum c)"
wenzelm@60698
   183
  by pat_completeness auto
krauss@41808
   184
termination by (relation "measure polysize") auto
chaieb@33154
   185
wenzelm@52658
   186
fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
wenzelm@52658
   187
where
chaieb@33154
   188
  "poly_cmul y (C x) = C (y *\<^sub>N x)"
chaieb@33154
   189
| "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
chaieb@33154
   190
| "poly_cmul y p = C y *\<^sub>p p"
chaieb@33154
   191
wenzelm@56009
   192
definition monic :: "poly \<Rightarrow> poly \<times> bool"
wenzelm@56000
   193
where
wenzelm@56000
   194
  "monic p =
wenzelm@56000
   195
    (let h = headconst p
wenzelm@56000
   196
     in if h = 0\<^sub>N then (p, False) else (C (Ninv h) *\<^sub>p p, 0>\<^sub>N h))"
chaieb@33154
   197
wenzelm@52658
   198
wenzelm@60533
   199
subsection \<open>Pseudo-division\<close>
chaieb@33154
   200
wenzelm@52658
   201
definition shift1 :: "poly \<Rightarrow> poly"
wenzelm@56000
   202
  where "shift1 p = CN 0\<^sub>p 0 p"
chaieb@33154
   203
wenzelm@56009
   204
abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@52658
   205
  where "funpow \<equiv> compow"
haftmann@39246
   206
krauss@41403
   207
partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
wenzelm@52658
   208
where
wenzelm@52803
   209
  "polydivide_aux a n p k s =
wenzelm@56000
   210
    (if s = 0\<^sub>p then (k, s)
wenzelm@52803
   211
     else
wenzelm@56000
   212
      let
wenzelm@56000
   213
        b = head s;
wenzelm@56000
   214
        m = degree s
wenzelm@56000
   215
      in
wenzelm@56000
   216
        if m < n then (k,s)
wenzelm@56000
   217
        else
wenzelm@56000
   218
          let p' = funpow (m - n) shift1 p
wenzelm@56000
   219
          in
wenzelm@56000
   220
            if a = b then polydivide_aux a n p k (s -\<^sub>p p')
wenzelm@56000
   221
            else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))"
chaieb@33154
   222
wenzelm@56000
   223
definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
wenzelm@56000
   224
  where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s"
chaieb@33154
   225
wenzelm@52658
   226
fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
wenzelm@52658
   227
where
chaieb@33154
   228
  "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
chaieb@33154
   229
| "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
chaieb@33154
   230
wenzelm@52658
   231
fun poly_deriv :: "poly \<Rightarrow> poly"
wenzelm@52658
   232
where
chaieb@33154
   233
  "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
chaieb@33154
   234
| "poly_deriv p = 0\<^sub>p"
chaieb@33154
   235
wenzelm@52658
   236
wenzelm@60698
   237
subsection \<open>Semantics of the polynomial representation\<close>
chaieb@33154
   238
haftmann@59867
   239
primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,field,power}"
wenzelm@56000
   240
where
chaieb@33154
   241
  "Ipoly bs (C c) = INum c"
haftmann@39246
   242
| "Ipoly bs (Bound n) = bs!n"
haftmann@39246
   243
| "Ipoly bs (Neg a) = - Ipoly bs a"
haftmann@39246
   244
| "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
haftmann@39246
   245
| "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
haftmann@39246
   246
| "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
wenzelm@56000
   247
| "Ipoly bs (Pw t n) = Ipoly bs t ^ n"
wenzelm@56000
   248
| "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p"
haftmann@39246
   249
wenzelm@60698
   250
abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field,power}"  ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
wenzelm@35054
   251
  where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
chaieb@33154
   252
wenzelm@56009
   253
lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i"
chaieb@33154
   254
  by (simp add: INum_def)
wenzelm@56000
   255
wenzelm@52803
   256
lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
chaieb@33154
   257
  by (simp  add: INum_def)
chaieb@33154
   258
chaieb@33154
   259
lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
chaieb@33154
   260
wenzelm@52658
   261
wenzelm@60533
   262
subsection \<open>Normal form and normalization\<close>
chaieb@33154
   263
krauss@41808
   264
fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
krauss@41808
   265
where
chaieb@33154
   266
  "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
wenzelm@56000
   267
| "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)"
krauss@41808
   268
| "isnpolyh p = (\<lambda>k. False)"
chaieb@33154
   269
wenzelm@56000
   270
lemma isnpolyh_mono: "n' \<le> n \<Longrightarrow> isnpolyh p n \<Longrightarrow> isnpolyh p n'"
wenzelm@52658
   271
  by (induct p rule: isnpolyh.induct) auto
chaieb@33154
   272
wenzelm@52658
   273
definition isnpoly :: "poly \<Rightarrow> bool"
wenzelm@56000
   274
  where "isnpoly p = isnpolyh p 0"
chaieb@33154
   275
wenzelm@60698
   276
text \<open>polyadd preserves normal forms\<close>
chaieb@33154
   277
wenzelm@56000
   278
lemma polyadd_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
wenzelm@52803
   279
proof (induct p q arbitrary: n0 n1 rule: polyadd.induct)
krauss@41812
   280
  case (2 ab c' n' p' n0 n1)
wenzelm@56009
   281
  from 2 have  th1: "isnpolyh (C ab) (Suc n')"
wenzelm@56009
   282
    by simp
wenzelm@60698
   283
  from 2(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1"
wenzelm@56009
   284
    by simp_all
wenzelm@56009
   285
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
wenzelm@56009
   286
    by simp
wenzelm@56009
   287
  with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')"
wenzelm@56009
   288
    by simp
wenzelm@56009
   289
  from nplen1 have n01len1: "min n0 n1 \<le> n'"
wenzelm@56009
   290
    by simp
wenzelm@56009
   291
  then show ?case using 2 th3
wenzelm@56009
   292
    by simp
chaieb@33154
   293
next
krauss@41812
   294
  case (3 c' n' p' ab n1 n0)
wenzelm@56009
   295
  from 3 have  th1: "isnpolyh (C ab) (Suc n')"
wenzelm@56009
   296
    by simp
wenzelm@56009
   297
  from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
wenzelm@56009
   298
    by simp_all
wenzelm@56009
   299
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
wenzelm@56009
   300
    by simp
wenzelm@56009
   301
  with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')"
wenzelm@56009
   302
    by simp
wenzelm@56009
   303
  from nplen1 have n01len1: "min n0 n1 \<le> n'"
wenzelm@56009
   304
    by simp
wenzelm@56009
   305
  then show ?case using 3 th3
wenzelm@56009
   306
    by simp
chaieb@33154
   307
next
chaieb@33154
   308
  case (4 c n p c' n' p' n0 n1)
wenzelm@56009
   309
  then have nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n"
wenzelm@56009
   310
    by simp_all
wenzelm@56009
   311
  from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'"
wenzelm@56009
   312
    by simp_all
wenzelm@56009
   313
  from 4 have ngen0: "n \<ge> n0"
wenzelm@56009
   314
    by simp
wenzelm@56009
   315
  from 4 have n'gen1: "n' \<ge> n1"
wenzelm@56009
   316
    by simp
wenzelm@60698
   317
  consider (eq) "n = n'" | (lt) "n < n'" | (gt) "n > n'"
wenzelm@60698
   318
    by arith
wenzelm@60698
   319
  then show ?case
wenzelm@60698
   320
  proof cases
wenzelm@60698
   321
    case eq
wenzelm@52803
   322
    with "4.hyps"(3)[OF nc nc']
wenzelm@56009
   323
    have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)"
wenzelm@56009
   324
      by auto
wenzelm@56009
   325
    then have ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
wenzelm@56009
   326
      using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1
wenzelm@56009
   327
      by auto
wenzelm@56009
   328
    from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n"
wenzelm@56009
   329
      by simp
wenzelm@56009
   330
    have minle: "min n0 n1 \<le> n'"
wenzelm@56009
   331
      using ngen0 n'gen1 eq by simp
wenzelm@60698
   332
    from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' show ?thesis
wenzelm@56009
   333
      by (simp add: Let_def)
wenzelm@60698
   334
  next
wenzelm@60698
   335
    case lt
wenzelm@56009
   336
    have "min n0 n1 \<le> n0"
wenzelm@56009
   337
      by simp
wenzelm@56009
   338
    with 4 lt have th1:"min n0 n1 \<le> n"
wenzelm@56009
   339
      by auto
wenzelm@56009
   340
    from 4 have th21: "isnpolyh c (Suc n)"
wenzelm@56009
   341
      by simp
wenzelm@56009
   342
    from 4 have th22: "isnpolyh (CN c' n' p') n'"
wenzelm@56009
   343
      by simp
wenzelm@56009
   344
    from lt have th23: "min (Suc n) n' = Suc n"
wenzelm@56009
   345
      by arith
wenzelm@56009
   346
    from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
wenzelm@56009
   347
      using th23 by simp
wenzelm@60698
   348
    with 4 lt th1 show ?thesis
wenzelm@56009
   349
      by simp
wenzelm@60698
   350
  next
wenzelm@60698
   351
    case gt
wenzelm@56009
   352
    then have gt': "n' < n \<and> \<not> n < n'"
wenzelm@56009
   353
      by simp
wenzelm@56009
   354
    have "min n0 n1 \<le> n1"
wenzelm@56009
   355
      by simp
wenzelm@56009
   356
    with 4 gt have th1: "min n0 n1 \<le> n'"
wenzelm@56009
   357
      by auto
wenzelm@56009
   358
    from 4 have th21: "isnpolyh c' (Suc n')"
wenzelm@56009
   359
      by simp_all
wenzelm@56009
   360
    from 4 have th22: "isnpolyh (CN c n p) n"
wenzelm@56009
   361
      by simp
wenzelm@56009
   362
    from gt have th23: "min n (Suc n') = Suc n'"
wenzelm@56009
   363
      by arith
wenzelm@56009
   364
    from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
wenzelm@56009
   365
      using th23 by simp
wenzelm@60698
   366
    with 4 gt th1 show ?thesis
wenzelm@56009
   367
      by simp
wenzelm@60698
   368
  qed
chaieb@33154
   369
qed auto
chaieb@33154
   370
krauss@41812
   371
lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
wenzelm@52658
   372
  by (induct p q rule: polyadd.induct)
hoelzl@58776
   373
     (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left_NO_MATCH)
chaieb@33154
   374
wenzelm@56009
   375
lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)"
wenzelm@60698
   376
  using polyadd_normh[of p 0 q 0] isnpoly_def by simp
chaieb@33154
   377
wenzelm@60698
   378
text \<open>The degree of addition and other general lemmas needed for the normal form of polymul.\<close>
chaieb@33154
   379
wenzelm@52803
   380
lemma polyadd_different_degreen:
wenzelm@56009
   381
  assumes "isnpolyh p n0"
wenzelm@56009
   382
    and "isnpolyh q n1"
wenzelm@56009
   383
    and "degreen p m \<noteq> degreen q m"
wenzelm@56009
   384
    and "m \<le> min n0 n1"
wenzelm@56009
   385
  shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
wenzelm@56009
   386
  using assms
chaieb@33154
   387
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
chaieb@33154
   388
  case (4 c n p c' n' p' m n0 n1)
wenzelm@60698
   389
  show ?case
wenzelm@60698
   390
  proof (cases "n = n'")
wenzelm@60698
   391
    case True
wenzelm@60698
   392
    with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   393
    show ?thesis by (auto simp: Let_def)
krauss@41763
   394
  next
wenzelm@60698
   395
    case False
krauss@41763
   396
    with 4 show ?thesis by auto
krauss@41763
   397
  qed
krauss@41763
   398
qed auto
chaieb@33154
   399
wenzelm@56009
   400
lemma headnz[simp]: "isnpolyh p n \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
wenzelm@52658
   401
  by (induct p arbitrary: n rule: headn.induct) auto
wenzelm@56009
   402
chaieb@33154
   403
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
wenzelm@52658
   404
  by (induct p arbitrary: n rule: degree.induct) auto
wenzelm@56009
   405
chaieb@33154
   406
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
wenzelm@52658
   407
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
   408
chaieb@33154
   409
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
wenzelm@52658
   410
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
   411
chaieb@33154
   412
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
chaieb@33154
   413
  using degree_isnpolyh_Suc by auto
wenzelm@56009
   414
chaieb@33154
   415
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
chaieb@33154
   416
  using degreen_0 by auto
chaieb@33154
   417
chaieb@33154
   418
chaieb@33154
   419
lemma degreen_polyadd:
wenzelm@56009
   420
  assumes np: "isnpolyh p n0"
wenzelm@56009
   421
    and nq: "isnpolyh q n1"
wenzelm@56009
   422
    and m: "m \<le> max n0 n1"
chaieb@33154
   423
  shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
chaieb@33154
   424
  using np nq m
chaieb@33154
   425
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
wenzelm@52803
   426
  case (2 c c' n' p' n0 n1)
wenzelm@56009
   427
  then show ?case
wenzelm@56009
   428
    by (cases n') simp_all
chaieb@33154
   429
next
wenzelm@52803
   430
  case (3 c n p c' n0 n1)
wenzelm@56009
   431
  then show ?case
wenzelm@56009
   432
    by (cases n) auto
chaieb@33154
   433
next
wenzelm@52803
   434
  case (4 c n p c' n' p' n0 n1 m)
wenzelm@60698
   435
  show ?case
wenzelm@60698
   436
  proof (cases "n = n'")
wenzelm@60698
   437
    case True
wenzelm@60698
   438
    with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   439
    show ?thesis by (auto simp: Let_def)
wenzelm@60698
   440
  next
wenzelm@60698
   441
    case False
wenzelm@60698
   442
    then show ?thesis by simp
wenzelm@60698
   443
  qed
chaieb@33154
   444
qed auto
chaieb@33154
   445
wenzelm@56009
   446
lemma polyadd_eq_const_degreen:
wenzelm@56009
   447
  assumes "isnpolyh p n0"
wenzelm@56009
   448
    and "isnpolyh q n1"
wenzelm@56009
   449
    and "polyadd p q = C c"
wenzelm@56009
   450
  shows "degreen p m = degreen q m"
wenzelm@56009
   451
  using assms
chaieb@33154
   452
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
wenzelm@52803
   453
  case (4 c n p c' n' p' m n0 n1 x)
wenzelm@60698
   454
  consider "n = n'" | "n > n' \<or> n < n'" by arith
wenzelm@60698
   455
  then show ?case
wenzelm@60698
   456
  proof cases
wenzelm@60698
   457
    case 1
wenzelm@60698
   458
    with 4 show ?thesis
wenzelm@60698
   459
      by (cases "p +\<^sub>p p' = 0\<^sub>p") (auto simp add: Let_def)
wenzelm@60698
   460
  next
wenzelm@60698
   461
    case 2
wenzelm@60698
   462
    with 4 show ?thesis by auto
wenzelm@60698
   463
  qed
chaieb@33154
   464
qed simp_all
chaieb@33154
   465
chaieb@33154
   466
lemma polymul_properties:
haftmann@59867
   467
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@52658
   468
    and np: "isnpolyh p n0"
wenzelm@52658
   469
    and nq: "isnpolyh q n1"
wenzelm@52658
   470
    and m: "m \<le> min n0 n1"
wenzelm@52803
   471
  shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@56009
   472
    and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
wenzelm@56009
   473
    and "degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
chaieb@33154
   474
  using np nq m
wenzelm@52658
   475
proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
wenzelm@52803
   476
  case (2 c c' n' p')
wenzelm@56009
   477
  {
wenzelm@56009
   478
    case (1 n0 n1)
wenzelm@56009
   479
    with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
krauss@41811
   480
    show ?case by (auto simp add: min_def)
chaieb@33154
   481
  next
wenzelm@56009
   482
    case (2 n0 n1)
wenzelm@56009
   483
    then show ?case by auto
chaieb@33154
   484
  next
wenzelm@56009
   485
    case (3 n0 n1)
wenzelm@60698
   486
    then show ?case using "2.hyps" by auto
wenzelm@56009
   487
  }
chaieb@33154
   488
next
krauss@41813
   489
  case (3 c n p c')
wenzelm@56009
   490
  {
wenzelm@56009
   491
    case (1 n0 n1)
wenzelm@56009
   492
    with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
krauss@41811
   493
    show ?case by (auto simp add: min_def)
chaieb@33154
   494
  next
wenzelm@56009
   495
    case (2 n0 n1)
wenzelm@56009
   496
    then show ?case by auto
chaieb@33154
   497
  next
wenzelm@56009
   498
    case (3 n0 n1)
wenzelm@56009
   499
    then show ?case  using "3.hyps" by auto
wenzelm@56009
   500
  }
chaieb@33154
   501
next
chaieb@33154
   502
  case (4 c n p c' n' p')
chaieb@33154
   503
  let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
wenzelm@56009
   504
  {
wenzelm@56009
   505
    case (1 n0 n1)
wenzelm@56009
   506
    then have cnp: "isnpolyh ?cnp n"
wenzelm@56009
   507
      and cnp': "isnpolyh ?cnp' n'"
wenzelm@56009
   508
      and np: "isnpolyh p n"
wenzelm@56009
   509
      and nc: "isnpolyh c (Suc n)"
wenzelm@56009
   510
      and np': "isnpolyh p' n'"
wenzelm@56009
   511
      and nc': "isnpolyh c' (Suc n')"
wenzelm@56009
   512
      and nn0: "n \<ge> n0"
wenzelm@56009
   513
      and nn1: "n' \<ge> n1"
wenzelm@56009
   514
      by simp_all
krauss@41811
   515
    {
wenzelm@56009
   516
      assume "n < n'"
wenzelm@56009
   517
      with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
wenzelm@56009
   518
      have ?case by (simp add: min_def)
wenzelm@56009
   519
    } moreover {
wenzelm@56009
   520
      assume "n' < n"
wenzelm@56009
   521
      with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
wenzelm@56009
   522
      have ?case by (cases "Suc n' = n") (simp_all add: min_def)
wenzelm@56009
   523
    } moreover {
wenzelm@56009
   524
      assume "n' = n"
wenzelm@56009
   525
      with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
wenzelm@56009
   526
      have ?case
wenzelm@56009
   527
        apply (auto intro!: polyadd_normh)
wenzelm@56009
   528
        apply (simp_all add: min_def isnpolyh_mono[OF nn0])
wenzelm@56009
   529
        done
wenzelm@56009
   530
    }
wenzelm@56009
   531
    ultimately show ?case by arith
wenzelm@56009
   532
  next
wenzelm@56009
   533
    fix n0 n1 m
wenzelm@56009
   534
    assume np: "isnpolyh ?cnp n0"
wenzelm@56009
   535
    assume np':"isnpolyh ?cnp' n1"
wenzelm@56009
   536
    assume m: "m \<le> min n0 n1"
wenzelm@56009
   537
    let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
wenzelm@56009
   538
    let ?d1 = "degreen ?cnp m"
wenzelm@56009
   539
    let ?d2 = "degreen ?cnp' m"
wenzelm@56009
   540
    let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
wenzelm@56043
   541
    have "n' < n \<or> n < n' \<or> n' = n" by auto
wenzelm@56009
   542
    moreover
wenzelm@56009
   543
    {
wenzelm@56009
   544
      assume "n' < n \<or> n < n'"
wenzelm@56009
   545
      with "4.hyps"(3,6,18) np np' m have ?eq
wenzelm@56009
   546
        by auto
wenzelm@56009
   547
    }
wenzelm@56009
   548
    moreover
wenzelm@56009
   549
    {
wenzelm@56009
   550
      assume nn': "n' = n"
wenzelm@56009
   551
      then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
wenzelm@56009
   552
      from "4.hyps"(16,18)[of n n' n]
wenzelm@56009
   553
        "4.hyps"(13,14)[of n "Suc n'" n]
wenzelm@56009
   554
        np np' nn'
wenzelm@56043
   555
      have norm:
wenzelm@56043
   556
        "isnpolyh ?cnp n"
wenzelm@56043
   557
        "isnpolyh c' (Suc n)"
wenzelm@56043
   558
        "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@56043
   559
        "isnpolyh p' n"
wenzelm@56043
   560
        "isnpolyh (?cnp *\<^sub>p p') n"
wenzelm@56043
   561
        "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56043
   562
        "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
wenzelm@56043
   563
        "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
wenzelm@56043
   564
        by (auto simp add: min_def)
wenzelm@56009
   565
      {
wenzelm@56009
   566
        assume mn: "m = n"
wenzelm@56009
   567
        from "4.hyps"(17,18)[OF norm(1,4), of n]
wenzelm@56009
   568
          "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
wenzelm@56009
   569
        have degs:
wenzelm@56009
   570
          "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
wenzelm@56009
   571
          "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
wenzelm@56009
   572
          by (simp_all add: min_def)
wenzelm@56009
   573
        from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   574
          by simp
wenzelm@56009
   575
        then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   576
          by simp
wenzelm@56009
   577
        have nmin: "n \<le> min n n"
wenzelm@56009
   578
          by (simp add: min_def)
wenzelm@56009
   579
        from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
wenzelm@56009
   580
        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 =
wenzelm@56009
   581
            degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   582
          by simp
wenzelm@56009
   583
        from "4.hyps"(16-18)[OF norm(1,4), of n]
wenzelm@56009
   584
          "4.hyps"(13-15)[OF norm(1,2), of n]
wenzelm@56009
   585
          mn norm m nn' deg
wenzelm@56009
   586
        have ?eq by simp
krauss@41811
   587
      }
chaieb@33154
   588
      moreover
wenzelm@56009
   589
      {
wenzelm@56009
   590
        assume mn: "m \<noteq> n"
wenzelm@56009
   591
        then have mn': "m < n"
wenzelm@56009
   592
          using m np by auto
wenzelm@56009
   593
        from nn' m np have max1: "m \<le> max n n"
wenzelm@56009
   594
          by simp
wenzelm@56009
   595
        then have min1: "m \<le> min n n"
wenzelm@56009
   596
          by simp
wenzelm@56009
   597
        then have min2: "m \<le> min n (Suc n)"
wenzelm@56009
   598
          by simp
wenzelm@56009
   599
        from "4.hyps"(16-18)[OF norm(1,4) min1]
wenzelm@56009
   600
          "4.hyps"(13-15)[OF norm(1,2) min2]
wenzelm@56009
   601
          degreen_polyadd[OF norm(3,6) max1]
wenzelm@56009
   602
        have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le>
wenzelm@56009
   603
            max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
wenzelm@56009
   604
          using mn nn' np np' by simp
wenzelm@56009
   605
        with "4.hyps"(16-18)[OF norm(1,4) min1]
wenzelm@56009
   606
          "4.hyps"(13-15)[OF norm(1,2) min2]
wenzelm@56009
   607
          degreen_0[OF norm(3) mn']
wenzelm@56009
   608
        have ?eq using nn' mn np np' by clarsimp
wenzelm@56009
   609
      }
wenzelm@56009
   610
      ultimately have ?eq by blast
wenzelm@56009
   611
    }
wenzelm@56009
   612
    ultimately show ?eq by blast
wenzelm@56009
   613
  }
wenzelm@56009
   614
  {
wenzelm@56009
   615
    case (2 n0 n1)
wenzelm@56009
   616
    then have np: "isnpolyh ?cnp n0"
wenzelm@56009
   617
      and np': "isnpolyh ?cnp' n1"
wenzelm@56043
   618
      and m: "m \<le> min n0 n1"
wenzelm@56043
   619
      by simp_all
wenzelm@56009
   620
    then have mn: "m \<le> n" by simp
wenzelm@56009
   621
    let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
wenzelm@56009
   622
    {
wenzelm@56009
   623
      assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
wenzelm@56009
   624
      then have nn: "\<not> n' < n \<and> \<not> n < n'"
wenzelm@56009
   625
        by simp
wenzelm@56009
   626
      from "4.hyps"(16-18) [of n n n]
wenzelm@56009
   627
        "4.hyps"(13-15)[of n "Suc n" n]
wenzelm@56009
   628
        np np' C(2) mn
wenzelm@56009
   629
      have norm:
wenzelm@56009
   630
        "isnpolyh ?cnp n"
wenzelm@56009
   631
        "isnpolyh c' (Suc n)"
wenzelm@56009
   632
        "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@56009
   633
        "isnpolyh p' n"
wenzelm@56009
   634
        "isnpolyh (?cnp *\<^sub>p p') n"
wenzelm@56009
   635
        "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   636
        "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
wenzelm@56009
   637
        "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
wenzelm@56009
   638
        "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
wenzelm@56009
   639
        "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
wenzelm@56009
   640
        by (simp_all add: min_def)
wenzelm@56009
   641
      from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   642
        by simp
wenzelm@56009
   643
      have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   644
        using norm by simp
wenzelm@56009
   645
      from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
wenzelm@56009
   646
      have False by simp
wenzelm@56009
   647
    }
wenzelm@56009
   648
    then show ?case using "4.hyps" by clarsimp
wenzelm@56009
   649
  }
chaieb@33154
   650
qed auto
chaieb@33154
   651
wenzelm@56009
   652
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
wenzelm@52658
   653
  by (induct p q rule: polymul.induct) (auto simp add: field_simps)
chaieb@33154
   654
wenzelm@52803
   655
lemma polymul_normh:
haftmann@59867
   656
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56009
   657
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@52803
   658
  using polymul_properties(1) by blast
wenzelm@52658
   659
wenzelm@52803
   660
lemma polymul_eq0_iff:
haftmann@59867
   661
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56009
   662
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
wenzelm@52803
   663
  using polymul_properties(2) by blast
wenzelm@52658
   664
wenzelm@56207
   665
lemma polymul_degreen:
haftmann@59867
   666
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56009
   667
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow>
wenzelm@56009
   668
    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)"
wenzelm@56207
   669
  by (fact polymul_properties(3))
wenzelm@52658
   670
wenzelm@52803
   671
lemma polymul_norm:
haftmann@59867
   672
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56009
   673
  shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
chaieb@33154
   674
  using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   675
chaieb@33154
   676
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   677
  by (induct p arbitrary: n0 rule: headconst.induct) auto
chaieb@33154
   678
chaieb@33154
   679
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
wenzelm@52658
   680
  by (induct p arbitrary: n0) auto
chaieb@33154
   681
wenzelm@52658
   682
lemma monic_eqI:
wenzelm@52803
   683
  assumes np: "isnpolyh p n0"
wenzelm@52658
   684
  shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
haftmann@59867
   685
    (Ipoly bs p ::'a::{field_char_0,field, power})"
chaieb@33154
   686
  unfolding monic_def Let_def
wenzelm@52658
   687
proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
chaieb@33154
   688
  let ?h = "headconst p"
chaieb@33154
   689
  assume pz: "p \<noteq> 0\<^sub>p"
wenzelm@56000
   690
  {
wenzelm@56000
   691
    assume hz: "INum ?h = (0::'a)"
wenzelm@56043
   692
    from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N"
wenzelm@56043
   693
      by simp_all
wenzelm@56043
   694
    from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N"
wenzelm@56043
   695
      by simp
wenzelm@56043
   696
    with headconst_zero[OF np] have "p = 0\<^sub>p"
wenzelm@56043
   697
      by blast
wenzelm@56043
   698
    with pz have False
wenzelm@56043
   699
      by blast
wenzelm@56043
   700
  }
wenzelm@56043
   701
  then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
wenzelm@56043
   702
    by blast
chaieb@33154
   703
qed
chaieb@33154
   704
chaieb@33154
   705
wenzelm@60698
   706
text \<open>polyneg is a negation and preserves normal forms\<close>
chaieb@33154
   707
chaieb@33154
   708
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
wenzelm@52658
   709
  by (induct p rule: polyneg.induct) auto
chaieb@33154
   710
wenzelm@56009
   711
lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   712
  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
wenzelm@56009
   713
chaieb@33154
   714
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
wenzelm@52658
   715
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
wenzelm@56009
   716
wenzelm@56009
   717
lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n"
wenzelm@56009
   718
  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)
chaieb@33154
   719
chaieb@33154
   720
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
chaieb@33154
   721
  using isnpoly_def polyneg_normh by simp
chaieb@33154
   722
chaieb@33154
   723
wenzelm@60698
   724
text \<open>polysub is a substraction and preserves normal forms\<close>
krauss@41404
   725
wenzelm@56009
   726
lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
wenzelm@52658
   727
  by (simp add: polysub_def)
wenzelm@56009
   728
wenzelm@56009
   729
lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
wenzelm@52658
   730
  by (simp add: polysub_def polyneg_normh polyadd_normh)
chaieb@33154
   731
wenzelm@56009
   732
lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
wenzelm@52803
   733
  using polyadd_norm polyneg_norm by (simp add: polysub_def)
wenzelm@56009
   734
wenzelm@52658
   735
lemma polysub_same_0[simp]:
haftmann@59867
   736
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
krauss@41814
   737
  shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
wenzelm@52658
   738
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
   739
  by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
chaieb@33154
   740
wenzelm@52803
   741
lemma polysub_0:
haftmann@59867
   742
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56009
   743
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q"
chaieb@33154
   744
  unfolding polysub_def split_def fst_conv snd_conv
krauss@41763
   745
  by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
wenzelm@52658
   746
    (auto simp: Nsub0[simplified Nsub_def] Let_def)
chaieb@33154
   747
wenzelm@60698
   748
text \<open>polypow is a power function and preserves normal forms\<close>
krauss@41404
   749
wenzelm@56009
   750
lemma polypow[simp]:
haftmann@59867
   751
  "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field}) ^ n"
wenzelm@52658
   752
proof (induct n rule: polypow.induct)
wenzelm@52658
   753
  case 1
wenzelm@56043
   754
  then show ?case
wenzelm@56043
   755
    by simp
chaieb@33154
   756
next
chaieb@33154
   757
  case (2 n)
chaieb@33154
   758
  let ?q = "polypow ((Suc n) div 2) p"
krauss@41813
   759
  let ?d = "polymul ?q ?q"
wenzelm@56043
   760
  have "odd (Suc n) \<or> even (Suc n)"
wenzelm@56043
   761
    by simp
wenzelm@52803
   762
  moreover
wenzelm@56043
   763
  {
wenzelm@56043
   764
    assume odd: "odd (Suc n)"
wenzelm@56000
   765
    have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
wenzelm@52658
   766
      by arith
wenzelm@56043
   767
    from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)"
wenzelm@56043
   768
      by (simp add: Let_def)
wenzelm@56043
   769
    also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
chaieb@33154
   770
      using "2.hyps" by simp
chaieb@33154
   771
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
wenzelm@52658
   772
      by (simp only: power_add power_one_right) simp
wenzelm@56000
   773
    also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
chaieb@33154
   774
      by (simp only: th)
haftmann@58710
   775
    finally have ?case unfolding numeral_2_eq_2 [symmetric]
haftmann@58834
   776
    using odd_two_times_div_two_nat [OF odd] by simp
wenzelm@56043
   777
  }
wenzelm@52803
   778
  moreover
wenzelm@56043
   779
  {
wenzelm@56043
   780
    assume even: "even (Suc n)"
wenzelm@56043
   781
    from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
wenzelm@56043
   782
      by (simp add: Let_def)
haftmann@58710
   783
    also have "\<dots> = (Ipoly bs p) ^ (2 * (Suc n div 2))"
haftmann@58710
   784
      using "2.hyps" by (simp only: mult_2 power_add) simp
haftmann@58710
   785
    finally have ?case using even_two_times_div_two [OF even]
haftmann@58710
   786
      by simp
wenzelm@56043
   787
  }
chaieb@33154
   788
  ultimately show ?case by blast
chaieb@33154
   789
qed
chaieb@33154
   790
wenzelm@52803
   791
lemma polypow_normh:
haftmann@59867
   792
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
chaieb@33154
   793
  shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
chaieb@33154
   794
proof (induct k arbitrary: n rule: polypow.induct)
wenzelm@56043
   795
  case 1
wenzelm@56043
   796
  then show ?case by auto
wenzelm@56043
   797
next
chaieb@33154
   798
  case (2 k n)
chaieb@33154
   799
  let ?q = "polypow (Suc k div 2) p"
krauss@41813
   800
  let ?d = "polymul ?q ?q"
wenzelm@56043
   801
  from 2 have th1: "isnpolyh ?q n" and th2: "isnpolyh p n"
wenzelm@56043
   802
    by blast+
wenzelm@56043
   803
  from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n"
wenzelm@56043
   804
    by simp
wenzelm@56043
   805
  from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n"
wenzelm@56043
   806
    by simp
haftmann@58710
   807
  from dn on show ?case by (simp, unfold Let_def) auto
haftmann@58710
   808
    
wenzelm@56043
   809
qed
chaieb@33154
   810
wenzelm@52803
   811
lemma polypow_norm:
haftmann@59867
   812
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
chaieb@33154
   813
  shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
chaieb@33154
   814
  by (simp add: polypow_normh isnpoly_def)
chaieb@33154
   815
wenzelm@60698
   816
text \<open>Finally the whole normalization\<close>
chaieb@33154
   817
wenzelm@52658
   818
lemma polynate [simp]:
haftmann@59867
   819
  "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field})"
wenzelm@52658
   820
  by (induct p rule:polynate.induct) auto
chaieb@33154
   821
wenzelm@52803
   822
lemma polynate_norm[simp]:
haftmann@59867
   823
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
chaieb@33154
   824
  shows "isnpoly (polynate p)"
wenzelm@52658
   825
  by (induct p rule: polynate.induct)
wenzelm@52658
   826
     (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
wenzelm@52658
   827
      simp_all add: isnpoly_def)
chaieb@33154
   828
wenzelm@60698
   829
text \<open>shift1\<close>
chaieb@33154
   830
chaieb@33154
   831
chaieb@33154
   832
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
wenzelm@52658
   833
  by (simp add: shift1_def)
chaieb@33154
   834
wenzelm@52803
   835
lemma shift1_isnpoly:
wenzelm@56207
   836
  assumes "isnpoly p"
wenzelm@56207
   837
    and "p \<noteq> 0\<^sub>p"
wenzelm@52658
   838
  shows "isnpoly (shift1 p) "
wenzelm@56207
   839
  using assms by (simp add: shift1_def isnpoly_def)
chaieb@33154
   840
chaieb@33154
   841
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
chaieb@33154
   842
  by (simp add: shift1_def)
wenzelm@56043
   843
wenzelm@56043
   844
lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)"
haftmann@39246
   845
  by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
chaieb@33154
   846
wenzelm@52803
   847
lemma funpow_isnpolyh:
wenzelm@56207
   848
  assumes "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
wenzelm@56207
   849
    and "isnpolyh p n"
chaieb@33154
   850
  shows "isnpolyh (funpow k f p) n"
wenzelm@56207
   851
  using assms by (induct k arbitrary: p) auto
chaieb@33154
   852
wenzelm@52658
   853
lemma funpow_shift1:
haftmann@59867
   854
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field}) =
wenzelm@52658
   855
    Ipoly bs (Mul (Pw (Bound 0) n) p)"
wenzelm@52658
   856
  by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
chaieb@33154
   857
wenzelm@56043
   858
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
chaieb@33154
   859
  using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
chaieb@33154
   860
wenzelm@52803
   861
lemma funpow_shift1_1:
haftmann@59867
   862
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field}) =
wenzelm@52658
   863
    Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
chaieb@33154
   864
  by (simp add: funpow_shift1)
chaieb@33154
   865
chaieb@33154
   866
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
wenzelm@45129
   867
  by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
chaieb@33154
   868
chaieb@33154
   869
lemma behead:
wenzelm@56207
   870
  assumes "isnpolyh p n"
wenzelm@52658
   871
  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
haftmann@59867
   872
    (Ipoly bs p :: 'a :: {field_char_0,field})"
wenzelm@56207
   873
  using assms
chaieb@33154
   874
proof (induct p arbitrary: n rule: behead.induct)
wenzelm@56009
   875
  case (1 c p n)
wenzelm@56009
   876
  then have pn: "isnpolyh p n" by simp
wenzelm@52803
   877
  from 1(1)[OF pn]
wenzelm@52803
   878
  have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
wenzelm@52658
   879
  then show ?case using "1.hyps"
wenzelm@52658
   880
    apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
wenzelm@52658
   881
    apply (simp_all add: th[symmetric] field_simps)
wenzelm@52658
   882
    done
chaieb@33154
   883
qed (auto simp add: Let_def)
chaieb@33154
   884
chaieb@33154
   885
lemma behead_isnpolyh:
wenzelm@56207
   886
  assumes "isnpolyh p n"
wenzelm@52658
   887
  shows "isnpolyh (behead p) n"
wenzelm@56207
   888
  using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
wenzelm@52658
   889
chaieb@33154
   890
wenzelm@60533
   891
subsection \<open>Miscellaneous lemmas about indexes, decrementation, substitution  etc ...\<close>
wenzelm@52658
   892
chaieb@33154
   893
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
wenzelm@61166
   894
proof (induct p arbitrary: n rule: poly.induct, auto, goal_cases)
wenzelm@60580
   895
  case prems: (1 c n p n')
wenzelm@56009
   896
  then have "n = Suc (n - 1)"
wenzelm@56009
   897
    by simp
wenzelm@56009
   898
  then have "isnpolyh p (Suc (n - 1))"
wenzelm@60533
   899
    using \<open>isnpolyh p n\<close> by simp
wenzelm@60580
   900
  with prems(2) show ?case
wenzelm@56009
   901
    by simp
chaieb@33154
   902
qed
chaieb@33154
   903
chaieb@33154
   904
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
wenzelm@52658
   905
  by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   906
wenzelm@52658
   907
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   908
  by (induct p) auto
chaieb@33154
   909
chaieb@33154
   910
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
wenzelm@52658
   911
  apply (induct p arbitrary: n0)
wenzelm@52658
   912
  apply auto
wenzelm@56043
   913
  apply atomize
blanchet@58259
   914
  apply (rename_tac nat a b, erule_tac x = "Suc nat" in allE)
chaieb@33154
   915
  apply auto
chaieb@33154
   916
  done
chaieb@33154
   917
chaieb@33154
   918
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
wenzelm@52658
   919
  by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
chaieb@33154
   920
chaieb@33154
   921
lemma polybound0_I:
wenzelm@56207
   922
  assumes "polybound0 a"
wenzelm@56009
   923
  shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
wenzelm@56207
   924
  using assms by (induct a rule: poly.induct) auto
wenzelm@52658
   925
wenzelm@56009
   926
lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
chaieb@33154
   927
  by (induct t) simp_all
chaieb@33154
   928
chaieb@33154
   929
lemma polysubst0_I':
wenzelm@56207
   930
  assumes "polybound0 a"
wenzelm@56009
   931
  shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
wenzelm@56207
   932
  by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"])
chaieb@33154
   933
wenzelm@52658
   934
lemma decrpoly:
wenzelm@56207
   935
  assumes "polybound0 t"
wenzelm@56043
   936
  shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
wenzelm@56207
   937
  using assms by (induct t rule: decrpoly.induct) simp_all
chaieb@33154
   938
wenzelm@52658
   939
lemma polysubst0_polybound0:
wenzelm@56207
   940
  assumes "polybound0 t"
chaieb@33154
   941
  shows "polybound0 (polysubst0 t a)"
wenzelm@56207
   942
  using assms by (induct a rule: poly.induct) auto
chaieb@33154
   943
chaieb@33154
   944
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
wenzelm@52658
   945
  by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   946
wenzelm@56043
   947
primrec maxindex :: "poly \<Rightarrow> nat"
wenzelm@56043
   948
where
chaieb@33154
   949
  "maxindex (Bound n) = n + 1"
chaieb@33154
   950
| "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
chaieb@33154
   951
| "maxindex (Add p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   952
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   953
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   954
| "maxindex (Neg p) = maxindex p"
chaieb@33154
   955
| "maxindex (Pw p n) = maxindex p"
chaieb@33154
   956
| "maxindex (C x) = 0"
chaieb@33154
   957
wenzelm@52658
   958
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
wenzelm@56000
   959
  where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
chaieb@33154
   960
wenzelm@56043
   961
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
wenzelm@52658
   962
proof (induct p rule: coefficients.induct)
wenzelm@52803
   963
  case (1 c p)
wenzelm@52803
   964
  show ?case
chaieb@33154
   965
  proof
wenzelm@56009
   966
    fix x
wenzelm@56009
   967
    assume xc: "x \<in> set (coefficients (CN c 0 p))"
wenzelm@56009
   968
    then have "x = c \<or> x \<in> set (coefficients p)"
wenzelm@56009
   969
      by simp
wenzelm@52803
   970
    moreover
wenzelm@56009
   971
    {
wenzelm@56009
   972
      assume "x = c"
wenzelm@56009
   973
      then have "wf_bs bs x"
wenzelm@56043
   974
        using "1.prems" unfolding wf_bs_def by simp
wenzelm@56009
   975
    }
wenzelm@56009
   976
    moreover
wenzelm@56009
   977
    {
wenzelm@56009
   978
      assume H: "x \<in> set (coefficients p)"
wenzelm@56009
   979
      from "1.prems" have "wf_bs bs p"
wenzelm@56009
   980
        unfolding wf_bs_def by simp
wenzelm@56009
   981
      with "1.hyps" H have "wf_bs bs x"
wenzelm@56009
   982
        by blast
wenzelm@56009
   983
    }
wenzelm@56066
   984
    ultimately show "wf_bs bs x"
wenzelm@56009
   985
      by blast
chaieb@33154
   986
  qed
chaieb@33154
   987
qed simp_all
chaieb@33154
   988
wenzelm@56043
   989
lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
wenzelm@52658
   990
  by (induct p rule: coefficients.induct) auto
chaieb@33154
   991
wenzelm@56000
   992
lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
wenzelm@52658
   993
  unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
chaieb@33154
   994
wenzelm@52658
   995
lemma take_maxindex_wf:
wenzelm@52803
   996
  assumes wf: "wf_bs bs p"
chaieb@33154
   997
  shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
wenzelm@56009
   998
proof -
chaieb@33154
   999
  let ?ip = "maxindex p"
chaieb@33154
  1000
  let ?tbs = "take ?ip bs"
wenzelm@56009
  1001
  from wf have "length ?tbs = ?ip"
wenzelm@56009
  1002
    unfolding wf_bs_def by simp
wenzelm@56009
  1003
  then have wf': "wf_bs ?tbs p"
wenzelm@56009
  1004
    unfolding wf_bs_def by  simp
wenzelm@56043
  1005
  have eq: "bs = ?tbs @ drop ?ip bs"
wenzelm@56009
  1006
    by simp
wenzelm@56009
  1007
  from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
wenzelm@56009
  1008
    using eq by simp
chaieb@33154
  1009
qed
chaieb@33154
  1010
chaieb@33154
  1011
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
wenzelm@52658
  1012
  by (induct p) auto
chaieb@33154
  1013
chaieb@33154
  1014
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
chaieb@33154
  1015
  unfolding wf_bs_def by simp
chaieb@33154
  1016
wenzelm@56207
  1017
lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"
chaieb@33154
  1018
  unfolding wf_bs_def by simp
chaieb@33154
  1019
wenzelm@56207
  1020
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p"
wenzelm@52658
  1021
  by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
wenzelm@56043
  1022
chaieb@33154
  1023
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
wenzelm@52658
  1024
  by (induct p rule: coefficients.induct) simp_all
chaieb@33154
  1025
chaieb@33154
  1026
lemma coefficients_head: "last (coefficients p) = head p"
wenzelm@52658
  1027
  by (induct p rule: coefficients.induct) auto
chaieb@33154
  1028
wenzelm@56207
  1029
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p"
wenzelm@52658
  1030
  unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
chaieb@33154
  1031
wenzelm@56043
  1032
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
wenzelm@56207
  1033
  apply (rule exI[where x="replicate (n - length xs) z" for z])
wenzelm@52658
  1034
  apply simp
wenzelm@52658
  1035
  done
wenzelm@52658
  1036
wenzelm@56043
  1037
lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
wenzelm@52658
  1038
  apply (cases p)
wenzelm@52658
  1039
  apply auto
blanchet@58259
  1040
  apply (rename_tac nat a, case_tac "nat")
wenzelm@52658
  1041
  apply simp_all
wenzelm@52658
  1042
  done
chaieb@33154
  1043
chaieb@33154
  1044
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
wenzelm@56066
  1045
  unfolding wf_bs_def by (induct p q rule: polyadd.induct) (auto simp add: Let_def)
chaieb@33154
  1046
chaieb@33154
  1047
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
wenzelm@52803
  1048
  unfolding wf_bs_def
wenzelm@52803
  1049
  apply (induct p q arbitrary: bs rule: polymul.induct)
chaieb@33154
  1050
  apply (simp_all add: wf_bs_polyadd)
chaieb@33154
  1051
  apply clarsimp
chaieb@33154
  1052
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
chaieb@33154
  1053
  apply auto
chaieb@33154
  1054
  done
chaieb@33154
  1055
chaieb@33154
  1056
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
wenzelm@52658
  1057
  unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
chaieb@33154
  1058
chaieb@33154
  1059
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
wenzelm@56043
  1060
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@56043
  1061
  using wf_bs_polyadd wf_bs_polyneg by blast
chaieb@33154
  1062
wenzelm@52658
  1063
wenzelm@60533
  1064
subsection \<open>Canonicity of polynomial representation, see lemma isnpolyh_unique\<close>
chaieb@33154
  1065
chaieb@33154
  1066
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
wenzelm@56043
  1067
definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
wenzelm@56043
  1068
definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
chaieb@33154
  1069
wenzelm@56043
  1070
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
chaieb@33154
  1071
proof (induct p arbitrary: n0 rule: coefficients.induct)
chaieb@33154
  1072
  case (1 c p n0)
wenzelm@56009
  1073
  have cp: "isnpolyh (CN c 0 p) n0"
wenzelm@56009
  1074
    by fact
wenzelm@56009
  1075
  then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
chaieb@33154
  1076
    by (auto simp add: isnpolyh_mono[where n'=0])
wenzelm@56009
  1077
  from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
wenzelm@56009
  1078
    by simp
chaieb@33154
  1079
qed auto
chaieb@33154
  1080
wenzelm@56043
  1081
lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
wenzelm@56043
  1082
  by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
chaieb@33154
  1083
chaieb@33154
  1084
lemma polypoly_polypoly':
chaieb@33154
  1085
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1086
  shows "polypoly (x # bs) p = polypoly' bs p"
wenzelm@56043
  1087
proof -
chaieb@33154
  1088
  let ?cf = "set (coefficients p)"
chaieb@33154
  1089
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
wenzelm@56043
  1090
  {
wenzelm@56043
  1091
    fix q
wenzelm@56043
  1092
    assume q: "q \<in> ?cf"
wenzelm@56043
  1093
    from q cn_norm have th: "isnpolyh q n0"
wenzelm@56043
  1094
      by blast
wenzelm@56043
  1095
    from coefficients_isconst[OF np] q have "isconstant q"
wenzelm@56043
  1096
      by blast
wenzelm@56043
  1097
    with isconstant_polybound0[OF th] have "polybound0 q"
wenzelm@56043
  1098
      by blast
wenzelm@56043
  1099
  }
wenzelm@56009
  1100
  then have "\<forall>q \<in> ?cf. polybound0 q" ..
wenzelm@56043
  1101
  then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
chaieb@33154
  1102
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
chaieb@33154
  1103
    by auto
wenzelm@56043
  1104
  then show ?thesis
wenzelm@56043
  1105
    unfolding polypoly_def polypoly'_def by simp
chaieb@33154
  1106
qed
chaieb@33154
  1107
chaieb@33154
  1108
lemma polypoly_poly:
wenzelm@56043
  1109
  assumes "isnpolyh p n0"
wenzelm@56043
  1110
  shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
wenzelm@56043
  1111
  using assms
wenzelm@52658
  1112
  by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
chaieb@33154
  1113
wenzelm@52803
  1114
lemma polypoly'_poly:
wenzelm@56043
  1115
  assumes "isnpolyh p n0"
wenzelm@52658
  1116
  shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
wenzelm@56043
  1117
  using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
chaieb@33154
  1118
chaieb@33154
  1119
chaieb@33154
  1120
lemma polypoly_poly_polybound0:
wenzelm@56043
  1121
  assumes "isnpolyh p n0"
wenzelm@56043
  1122
    and "polybound0 p"
chaieb@33154
  1123
  shows "polypoly bs p = [Ipoly bs p]"
wenzelm@56043
  1124
  using assms
wenzelm@56043
  1125
  unfolding polypoly_def
wenzelm@52658
  1126
  apply (cases p)
wenzelm@52658
  1127
  apply auto
blanchet@58259
  1128
  apply (rename_tac nat a, case_tac nat)
wenzelm@52658
  1129
  apply auto
wenzelm@52658
  1130
  done
chaieb@33154
  1131
wenzelm@52803
  1132
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
wenzelm@52658
  1133
  by (induct p rule: head.induct) auto
chaieb@33154
  1134
wenzelm@56043
  1135
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  1136
  by (cases p) auto
chaieb@33154
  1137
chaieb@33154
  1138
lemma head_eq_headn0: "head p = headn p 0"
wenzelm@52658
  1139
  by (induct p rule: head.induct) simp_all
chaieb@33154
  1140
wenzelm@56043
  1141
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
chaieb@33154
  1142
  by (simp add: head_eq_headn0)
chaieb@33154
  1143
wenzelm@52803
  1144
lemma isnpolyh_zero_iff:
wenzelm@52658
  1145
  assumes nq: "isnpolyh p n0"
haftmann@59867
  1146
    and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0,field, power})"
chaieb@33154
  1147
  shows "p = 0\<^sub>p"
wenzelm@52658
  1148
  using nq eq
berghofe@34915
  1149
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
berghofe@34915
  1150
  case less
wenzelm@60533
  1151
  note np = \<open>isnpolyh p n0\<close> and zp = \<open>\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)\<close>
wenzelm@56000
  1152
  {
wenzelm@56000
  1153
    assume nz: "maxindex p = 0"
wenzelm@56000
  1154
    then obtain c where "p = C c"
wenzelm@56000
  1155
      using np by (cases p) auto
wenzelm@56000
  1156
    with zp np have "p = 0\<^sub>p"
wenzelm@56000
  1157
      unfolding wf_bs_def by simp
wenzelm@56000
  1158
  }
chaieb@33154
  1159
  moreover
wenzelm@56000
  1160
  {
wenzelm@56000
  1161
    assume nz: "maxindex p \<noteq> 0"
chaieb@33154
  1162
    let ?h = "head p"
chaieb@33154
  1163
    let ?hd = "decrpoly ?h"
chaieb@33154
  1164
    let ?ihd = "maxindex ?hd"
wenzelm@56000
  1165
    from head_isnpolyh[OF np] head_polybound0[OF np]
wenzelm@56000
  1166
    have h: "isnpolyh ?h n0" "polybound0 ?h"
chaieb@33154
  1167
      by simp_all
wenzelm@56000
  1168
    then have nhd: "isnpolyh ?hd (n0 - 1)"
wenzelm@56000
  1169
      using decrpoly_normh by blast
wenzelm@52803
  1170
chaieb@33154
  1171
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
wenzelm@56000
  1172
    have mihn: "maxindex ?h \<le> maxindex p"
wenzelm@56000
  1173
      by auto
wenzelm@56000
  1174
    with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
wenzelm@56000
  1175
      by auto
wenzelm@56000
  1176
    {
wenzelm@56000
  1177
      fix bs :: "'a list"
wenzelm@56000
  1178
      assume bs: "wf_bs bs ?hd"
chaieb@33154
  1179
      let ?ts = "take ?ihd bs"
chaieb@33154
  1180
      let ?rs = "drop ?ihd bs"
wenzelm@56000
  1181
      have ts: "wf_bs ?ts ?hd"
wenzelm@56000
  1182
        using bs unfolding wf_bs_def by simp
wenzelm@56000
  1183
      have bs_ts_eq: "?ts @ ?rs = bs"
wenzelm@56000
  1184
        by simp
wenzelm@56000
  1185
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
wenzelm@56000
  1186
        by simp
wenzelm@56000
  1187
      from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
wenzelm@56000
  1188
        by simp
wenzelm@56000
  1189
      with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
wenzelm@56000
  1190
        by blast
wenzelm@56000
  1191
      then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
wenzelm@56000
  1192
        unfolding wf_bs_def by simp
wenzelm@56000
  1193
      with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
wenzelm@56000
  1194
        by blast
wenzelm@56000
  1195
      then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
wenzelm@56000
  1196
        by simp
chaieb@33154
  1197
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
wenzelm@56000
  1198
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
wenzelm@56000
  1199
        by simp
wenzelm@56000
  1200
      then have "poly (polypoly' (?ts @ xs) p) = poly []"
wenzelm@56000
  1201
        by auto
wenzelm@56000
  1202
      then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
wenzelm@60537
  1203
        using poly_zero[where ?'a='a] by (simp add: polypoly'_def)
chaieb@33154
  1204
      with coefficients_head[of p, symmetric]
wenzelm@56000
  1205
      have th0: "Ipoly (?ts @ xs) ?hd = 0"
wenzelm@56000
  1206
        by simp
wenzelm@56000
  1207
      from bs have wf'': "wf_bs ?ts ?hd"
wenzelm@56000
  1208
        unfolding wf_bs_def by simp
wenzelm@56000
  1209
      with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
wenzelm@56000
  1210
        by simp
wenzelm@56000
  1211
      with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
wenzelm@56000
  1212
        by simp
wenzelm@56000
  1213
    }
wenzelm@56000
  1214
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
wenzelm@56000
  1215
      by blast
wenzelm@56000
  1216
    from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
wenzelm@56000
  1217
      by blast
wenzelm@56000
  1218
    then have "?h = 0\<^sub>p" by simp
wenzelm@56000
  1219
    with head_nz[OF np] have "p = 0\<^sub>p" by simp
wenzelm@56000
  1220
  }
wenzelm@56000
  1221
  ultimately show "p = 0\<^sub>p"
wenzelm@56000
  1222
    by blast
chaieb@33154
  1223
qed
chaieb@33154
  1224
wenzelm@52803
  1225
lemma isnpolyh_unique:
wenzelm@56000
  1226
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1227
    and nq: "isnpolyh q n1"
haftmann@59867
  1228
  shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0,field,power})) \<longleftrightarrow> p = q"
wenzelm@56000
  1229
proof auto
wenzelm@56000
  1230
  assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
wenzelm@56000
  1231
  then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
wenzelm@56000
  1232
    by simp
wenzelm@56000
  1233
  then have "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
chaieb@33154
  1234
    using wf_bs_polysub[where p=p and q=q] by auto
wenzelm@56000
  1235
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
wenzelm@56000
  1236
    by blast
chaieb@33154
  1237
qed
chaieb@33154
  1238
chaieb@33154
  1239
wenzelm@60698
  1240
text \<open>consequences of unicity on the algorithms for polynomial normalization\<close>
chaieb@33154
  1241
wenzelm@52658
  1242
lemma polyadd_commute:
haftmann@59867
  1243
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@52658
  1244
    and np: "isnpolyh p n0"
wenzelm@52658
  1245
    and nq: "isnpolyh q n1"
wenzelm@52658
  1246
  shows "p +\<^sub>p q = q +\<^sub>p p"
wenzelm@56000
  1247
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
wenzelm@56000
  1248
  by simp
chaieb@33154
  1249
wenzelm@56000
  1250
lemma zero_normh: "isnpolyh 0\<^sub>p n"
wenzelm@56000
  1251
  by simp
wenzelm@56000
  1252
wenzelm@56000
  1253
lemma one_normh: "isnpolyh (1)\<^sub>p n"
wenzelm@56000
  1254
  by simp
wenzelm@52658
  1255
wenzelm@52803
  1256
lemma polyadd_0[simp]:
haftmann@59867
  1257
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@52658
  1258
    and np: "isnpolyh p n0"
wenzelm@56000
  1259
  shows "p +\<^sub>p 0\<^sub>p = p"
wenzelm@56000
  1260
    and "0\<^sub>p +\<^sub>p p = p"
wenzelm@52803
  1261
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
chaieb@33154
  1262
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
chaieb@33154
  1263
wenzelm@52803
  1264
lemma polymul_1[simp]:
haftmann@59867
  1265
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@52658
  1266
    and np: "isnpolyh p n0"
wenzelm@56000
  1267
  shows "p *\<^sub>p (1)\<^sub>p = p"
wenzelm@56000
  1268
    and "(1)\<^sub>p *\<^sub>p p = p"
wenzelm@52803
  1269
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
chaieb@33154
  1270
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
wenzelm@52658
  1271
wenzelm@52803
  1272
lemma polymul_0[simp]:
haftmann@59867
  1273
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@52658
  1274
    and np: "isnpolyh p n0"
wenzelm@56000
  1275
  shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
wenzelm@56000
  1276
    and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
wenzelm@52803
  1277
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
chaieb@33154
  1278
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
chaieb@33154
  1279
wenzelm@52803
  1280
lemma polymul_commute:
haftmann@59867
  1281
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56000
  1282
    and np: "isnpolyh p n0"
wenzelm@52658
  1283
    and nq: "isnpolyh q n1"
chaieb@33154
  1284
  shows "p *\<^sub>p q = q *\<^sub>p p"
wenzelm@56043
  1285
  using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
haftmann@59867
  1286
    where ?'a = "'a::{field_char_0,field, power}"]
wenzelm@52658
  1287
  by simp
chaieb@33154
  1288
wenzelm@52658
  1289
declare polyneg_polyneg [simp]
wenzelm@52803
  1290
wenzelm@52803
  1291
lemma isnpolyh_polynate_id [simp]:
haftmann@59867
  1292
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56000
  1293
    and np: "isnpolyh p n0"
wenzelm@52658
  1294
  shows "polynate p = p"
haftmann@59867
  1295
  using isnpolyh_unique[where ?'a= "'a::{field_char_0,field}",
wenzelm@56043
  1296
      OF polynate_norm[of p, unfolded isnpoly_def] np]
haftmann@59867
  1297
    polynate[where ?'a = "'a::{field_char_0,field}"]
wenzelm@52658
  1298
  by simp
chaieb@33154
  1299
wenzelm@52803
  1300
lemma polynate_idempotent[simp]:
haftmann@59867
  1301
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
chaieb@33154
  1302
  shows "polynate (polynate p) = polynate p"
chaieb@33154
  1303
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
chaieb@33154
  1304
chaieb@33154
  1305
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
chaieb@33154
  1306
  unfolding poly_nate_def polypoly'_def ..
wenzelm@52658
  1307
wenzelm@52658
  1308
lemma poly_nate_poly:
haftmann@59867
  1309
  "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0,field}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
chaieb@33154
  1310
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
wenzelm@52658
  1311
  unfolding poly_nate_polypoly' by auto
wenzelm@52658
  1312
chaieb@33154
  1313
wenzelm@60698
  1314
subsection \<open>heads, degrees and all that\<close>
wenzelm@52658
  1315
chaieb@33154
  1316
lemma degree_eq_degreen0: "degree p = degreen p 0"
wenzelm@52658
  1317
  by (induct p rule: degree.induct) simp_all
chaieb@33154
  1318
wenzelm@52658
  1319
lemma degree_polyneg:
wenzelm@56043
  1320
  assumes "isnpolyh p n"
chaieb@33154
  1321
  shows "degree (polyneg p) = degree p"
wenzelm@56043
  1322
  apply (induct p rule: polyneg.induct)
wenzelm@56043
  1323
  using assms
wenzelm@56043
  1324
  apply simp_all
wenzelm@52658
  1325
  apply (case_tac na)
wenzelm@52658
  1326
  apply auto
wenzelm@52658
  1327
  done
chaieb@33154
  1328
chaieb@33154
  1329
lemma degree_polyadd:
wenzelm@56043
  1330
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1331
    and nq: "isnpolyh q n1"
chaieb@33154
  1332
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
wenzelm@52658
  1333
  using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
chaieb@33154
  1334
chaieb@33154
  1335
wenzelm@52658
  1336
lemma degree_polysub:
wenzelm@52658
  1337
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1338
    and nq: "isnpolyh q n1"
chaieb@33154
  1339
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1340
proof-
wenzelm@56043
  1341
  from nq have nq': "isnpolyh (~\<^sub>p q) n1"
wenzelm@56043
  1342
    using polyneg_normh by simp
wenzelm@56043
  1343
  from degree_polyadd[OF np nq'] show ?thesis
wenzelm@56043
  1344
    by (simp add: polysub_def degree_polyneg[OF nq])
chaieb@33154
  1345
qed
chaieb@33154
  1346
wenzelm@52803
  1347
lemma degree_polysub_samehead:
haftmann@59867
  1348
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56043
  1349
    and np: "isnpolyh p n0"
wenzelm@56043
  1350
    and nq: "isnpolyh q n1"
wenzelm@56043
  1351
    and h: "head p = head q"
wenzelm@52658
  1352
    and d: "degree p = degree q"
chaieb@33154
  1353
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
wenzelm@52658
  1354
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
  1355
  using np nq h d
wenzelm@52658
  1356
proof (induct p q rule: polyadd.induct)
wenzelm@52658
  1357
  case (1 c c')
wenzelm@56009
  1358
  then show ?case
wenzelm@56009
  1359
    by (simp add: Nsub_def Nsub0[simplified Nsub_def])
chaieb@33154
  1360
next
wenzelm@52803
  1361
  case (2 c c' n' p')
wenzelm@56009
  1362
  from 2 have "degree (C c) = degree (CN c' n' p')"
wenzelm@56009
  1363
    by simp
wenzelm@56009
  1364
  then have nz: "n' > 0"
wenzelm@56009
  1365
    by (cases n') auto
wenzelm@56009
  1366
  then have "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1367
    by (cases n') auto
wenzelm@56009
  1368
  with 2 show ?case
wenzelm@56009
  1369
    by simp
chaieb@33154
  1370
next
wenzelm@52803
  1371
  case (3 c n p c')
wenzelm@56009
  1372
  then have "degree (C c') = degree (CN c n p)"
wenzelm@56009
  1373
    by simp
wenzelm@56009
  1374
  then have nz: "n > 0"
wenzelm@56009
  1375
    by (cases n) auto
wenzelm@56009
  1376
  then have "head (CN c n p) = CN c n p"
wenzelm@56009
  1377
    by (cases n) auto
wenzelm@41807
  1378
  with 3 show ?case by simp
chaieb@33154
  1379
next
chaieb@33154
  1380
  case (4 c n p c' n' p')
wenzelm@56009
  1381
  then have H:
wenzelm@56009
  1382
    "isnpolyh (CN c n p) n0"
wenzelm@56009
  1383
    "isnpolyh (CN c' n' p') n1"
wenzelm@56009
  1384
    "head (CN c n p) = head (CN c' n' p')"
wenzelm@56009
  1385
    "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@56009
  1386
    by simp_all
wenzelm@56009
  1387
  then have degc: "degree c = 0" and degc': "degree c' = 0"
wenzelm@56009
  1388
    by simp_all
wenzelm@56009
  1389
  then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
chaieb@33154
  1390
    using H(1-2) degree_polyneg by auto
wenzelm@56009
  1391
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
wenzelm@56009
  1392
    by simp_all
wenzelm@56009
  1393
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
wenzelm@56009
  1394
  have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
wenzelm@56009
  1395
    by simp
wenzelm@56009
  1396
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
wenzelm@56009
  1397
    by auto
wenzelm@56009
  1398
  have "n = n' \<or> n < n' \<or> n > n'"
wenzelm@56009
  1399
    by arith
chaieb@33154
  1400
  moreover
wenzelm@56009
  1401
  {
wenzelm@56009
  1402
    assume nn': "n = n'"
wenzelm@56009
  1403
    have "n = 0 \<or> n > 0" by arith
wenzelm@56066
  1404
    moreover
wenzelm@56066
  1405
    {
wenzelm@56009
  1406
      assume nz: "n = 0"
wenzelm@56009
  1407
      then have ?case using 4 nn'
wenzelm@56009
  1408
        by (auto simp add: Let_def degcmc')
wenzelm@56009
  1409
    }
wenzelm@56066
  1410
    moreover
wenzelm@56066
  1411
    {
wenzelm@56009
  1412
      assume nz: "n > 0"
wenzelm@56009
  1413
      with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
wenzelm@56009
  1414
        by (cases n, auto)+
wenzelm@56009
  1415
      then have ?case
wenzelm@56009
  1416
        using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
wenzelm@56009
  1417
        using polysub_same_0[OF c'nh, simplified polysub_def]
wenzelm@56009
  1418
        using nn' 4 by (simp add: Let_def)
wenzelm@56009
  1419
    }
wenzelm@56009
  1420
    ultimately have ?case by blast
wenzelm@56009
  1421
  }
chaieb@33154
  1422
  moreover
wenzelm@56009
  1423
  {
wenzelm@56009
  1424
    assume nn': "n < n'"
wenzelm@56009
  1425
    then have n'p: "n' > 0"
wenzelm@56009
  1426
      by simp
wenzelm@56009
  1427
    then have headcnp':"head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1428
      by (cases n') simp_all
wenzelm@56009
  1429
    have degcnp': "degree (CN c' n' p') = 0"
wenzelm@56009
  1430
      and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@52658
  1431
      using 4 nn' by (cases n', simp_all)
wenzelm@56009
  1432
    then have "n > 0"
wenzelm@56009
  1433
      by (cases n) simp_all
wenzelm@56009
  1434
    then have headcnp: "head (CN c n p) = CN c n p"
wenzelm@56009
  1435
      by (cases n) auto
wenzelm@56009
  1436
    from H(3) headcnp headcnp' nn' have ?case
wenzelm@56009
  1437
      by auto
wenzelm@56009
  1438
  }
chaieb@33154
  1439
  moreover
wenzelm@56009
  1440
  {
wenzelm@56009
  1441
    assume nn': "n > n'"
wenzelm@56009
  1442
    then have np: "n > 0" by simp
wenzelm@56009
  1443
    then have headcnp:"head (CN c n p) = CN c n p"
wenzelm@56009
  1444
      by (cases n) simp_all
wenzelm@56009
  1445
    from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
wenzelm@56009
  1446
      by simp
wenzelm@56009
  1447
    from np have degcnp: "degree (CN c n p) = 0"
wenzelm@56009
  1448
      by (cases n) simp_all
wenzelm@56009
  1449
    with degcnpeq have "n' > 0"
wenzelm@56009
  1450
      by (cases n') simp_all
wenzelm@56009
  1451
    then have headcnp': "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1452
      by (cases n') auto
wenzelm@56009
  1453
    from H(3) headcnp headcnp' nn' have ?case by auto
wenzelm@56009
  1454
  }
wenzelm@56009
  1455
  ultimately show ?case by blast
krauss@41812
  1456
qed auto
wenzelm@52803
  1457
chaieb@33154
  1458
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
wenzelm@52658
  1459
  by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
chaieb@33154
  1460
chaieb@33154
  1461
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
wenzelm@52658
  1462
proof (induct k arbitrary: n0 p)
wenzelm@52658
  1463
  case 0
wenzelm@56198
  1464
  then show ?case
wenzelm@56198
  1465
    by auto
wenzelm@52658
  1466
next
wenzelm@52658
  1467
  case (Suc k n0 p)
wenzelm@56066
  1468
  then have "isnpolyh (shift1 p) 0"
wenzelm@56066
  1469
    by (simp add: shift1_isnpolyh)
wenzelm@41807
  1470
  with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
wenzelm@56066
  1471
    and "head (shift1 p) = head p"
wenzelm@56066
  1472
    by (simp_all add: shift1_head)
wenzelm@56066
  1473
  then show ?case
wenzelm@56066
  1474
    by (simp add: funpow_swap1)
wenzelm@52658
  1475
qed
chaieb@33154
  1476
chaieb@33154
  1477
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
chaieb@33154
  1478
  by (simp add: shift1_def)
wenzelm@56009
  1479
chaieb@33154
  1480
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
wenzelm@46991
  1481
  by (induct k arbitrary: p) (auto simp add: shift1_degree)
chaieb@33154
  1482
chaieb@33154
  1483
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
wenzelm@52658
  1484
  by (induct n arbitrary: p) simp_all
chaieb@33154
  1485
chaieb@33154
  1486
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
wenzelm@52658
  1487
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1488
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
wenzelm@52658
  1489
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
  1490
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
wenzelm@52658
  1491
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1492
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
wenzelm@52658
  1493
  by (induct p rule: head.induct) auto
chaieb@33154
  1494
wenzelm@52803
  1495
lemma polyadd_eq_const_degree:
wenzelm@52658
  1496
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
chaieb@33154
  1497
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
chaieb@33154
  1498
wenzelm@52658
  1499
lemma polyadd_head:
wenzelm@52658
  1500
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1501
    and nq: "isnpolyh q n1"
wenzelm@52658
  1502
    and deg: "degree p \<noteq> degree q"
chaieb@33154
  1503
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
wenzelm@52658
  1504
  using np nq deg
wenzelm@52658
  1505
  apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
wenzelm@52658
  1506
  apply simp_all
wenzelm@52658
  1507
  apply (case_tac n', simp, simp)
wenzelm@52658
  1508
  apply (case_tac n, simp, simp)
wenzelm@52658
  1509
  apply (case_tac n, case_tac n', simp add: Let_def)
haftmann@54489
  1510
  apply (auto simp add: polyadd_eq_const_degree)[2]
wenzelm@52658
  1511
  apply (metis head_nz)
wenzelm@52658
  1512
  apply (metis head_nz)
wenzelm@52658
  1513
  apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
wenzelm@52658
  1514
  done
chaieb@33154
  1515
wenzelm@52803
  1516
lemma polymul_head_polyeq:
haftmann@59867
  1517
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56066
  1518
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> q \<noteq> 0\<^sub>p \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
chaieb@33154
  1519
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
krauss@41813
  1520
  case (2 c c' n' p' n0 n1)
wenzelm@56009
  1521
  then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
wenzelm@56009
  1522
    by (simp_all add: head_isnpolyh)
wenzelm@56009
  1523
  then show ?case
wenzelm@56009
  1524
    using 2 by (cases n') auto
wenzelm@52803
  1525
next
wenzelm@52803
  1526
  case (3 c n p c' n0 n1)
wenzelm@56009
  1527
  then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
wenzelm@56009
  1528
    by (simp_all add: head_isnpolyh)
wenzelm@56066
  1529
  then show ?case
wenzelm@56066
  1530
    using 3 by (cases n) auto
chaieb@33154
  1531
next
chaieb@33154
  1532
  case (4 c n p c' n' p' n0 n1)
wenzelm@56066
  1533
  then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
chaieb@33154
  1534
    "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
chaieb@33154
  1535
    by simp_all
chaieb@33154
  1536
  have "n < n' \<or> n' < n \<or> n = n'" by arith
wenzelm@52803
  1537
  moreover
wenzelm@56009
  1538
  {
wenzelm@56009
  1539
    assume nn': "n < n'"
wenzelm@56009
  1540
    then have ?case
wenzelm@52658
  1541
      using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
wenzelm@52658
  1542
      apply simp
wenzelm@52658
  1543
      apply (cases n)
wenzelm@52658
  1544
      apply simp
wenzelm@52658
  1545
      apply (cases n')
wenzelm@52658
  1546
      apply simp_all
wenzelm@56009
  1547
      done
wenzelm@56009
  1548
  }
wenzelm@56009
  1549
  moreover {
wenzelm@56009
  1550
    assume nn': "n'< n"
wenzelm@56009
  1551
    then have ?case
wenzelm@52803
  1552
      using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
wenzelm@52658
  1553
      apply simp
wenzelm@52658
  1554
      apply (cases n')
wenzelm@52658
  1555
      apply simp
wenzelm@52658
  1556
      apply (cases n)
wenzelm@52658
  1557
      apply auto
wenzelm@56009
  1558
      done
wenzelm@56009
  1559
  }
wenzelm@56066
  1560
  moreover
wenzelm@56066
  1561
  {
wenzelm@56009
  1562
    assume nn': "n' = n"
wenzelm@52803
  1563
    from nn' polymul_normh[OF norm(5,4)]
chaieb@33154
  1564
    have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
wenzelm@52803
  1565
    from nn' polymul_normh[OF norm(5,3)] norm
chaieb@33154
  1566
    have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
chaieb@33154
  1567
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
wenzelm@52803
  1568
    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
wenzelm@52803
  1569
    from polyadd_normh[OF ncnpc' ncnpp0']
wenzelm@52803
  1570
    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"
chaieb@33154
  1571
      by (simp add: min_def)
wenzelm@56009
  1572
    {
wenzelm@56009
  1573
      assume np: "n > 0"
chaieb@33154
  1574
      with nn' head_isnpolyh_Suc'[OF np nth]
wenzelm@33268
  1575
        head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
wenzelm@56009
  1576
      have ?case by simp
wenzelm@56009
  1577
    }
chaieb@33154
  1578
    moreover
wenzelm@56009
  1579
    {
wenzelm@56009
  1580
      assume nz: "n = 0"
chaieb@33154
  1581
      from polymul_degreen[OF norm(5,4), where m="0"]
wenzelm@33268
  1582
        polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
chaieb@33154
  1583
      norm(5,6) degree_npolyhCN[OF norm(6)]
wenzelm@56066
  1584
    have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
wenzelm@56066
  1585
      by simp
wenzelm@56066
  1586
    then have dth': "degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
wenzelm@56066
  1587
      by simp
chaieb@33154
  1588
    from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
wenzelm@56066
  1589
    have ?case
wenzelm@56066
  1590
      using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
wenzelm@56066
  1591
      by simp
wenzelm@56009
  1592
    }
wenzelm@56066
  1593
    ultimately have ?case
wenzelm@56066
  1594
      by (cases n) auto
wenzelm@56009
  1595
  }
chaieb@33154
  1596
  ultimately show ?case by blast
chaieb@33154
  1597
qed simp_all
chaieb@33154
  1598
chaieb@33154
  1599
lemma degree_coefficients: "degree p = length (coefficients p) - 1"
wenzelm@52658
  1600
  by (induct p rule: degree.induct) auto
chaieb@33154
  1601
chaieb@33154
  1602
lemma degree_head[simp]: "degree (head p) = 0"
wenzelm@52658
  1603
  by (induct p rule: head.induct) auto
chaieb@33154
  1604
krauss@41812
  1605
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
wenzelm@52658
  1606
  by (cases n) simp_all
wenzelm@56066
  1607
chaieb@33154
  1608
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
wenzelm@52658
  1609
  by (cases n) simp_all
chaieb@33154
  1610
wenzelm@52658
  1611
lemma polyadd_different_degree:
wenzelm@56066
  1612
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow>
wenzelm@52658
  1613
    degree (polyadd p q) = max (degree p) (degree q)"
chaieb@33154
  1614
  using polyadd_different_degreen degree_eq_degreen0 by simp
chaieb@33154
  1615
chaieb@33154
  1616
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
wenzelm@52658
  1617
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
chaieb@33154
  1618
chaieb@33154
  1619
lemma degree_polymul:
haftmann@59867
  1620
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@52658
  1621
    and np: "isnpolyh p n0"
wenzelm@52658
  1622
    and nq: "isnpolyh q n1"
chaieb@33154
  1623
  shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
chaieb@33154
  1624
  using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
chaieb@33154
  1625
chaieb@33154
  1626
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
wenzelm@52658
  1627
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1628
wenzelm@56207
  1629
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head (polyneg p) = polyneg (head p)"
wenzelm@52658
  1630
  by (induct p arbitrary: n rule: degree.induct) auto
wenzelm@52658
  1631
chaieb@33154
  1632
wenzelm@60533
  1633
subsection \<open>Correctness of polynomial pseudo division\<close>
chaieb@33154
  1634
chaieb@33154
  1635
lemma polydivide_aux_properties:
haftmann@59867
  1636
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@52658
  1637
    and np: "isnpolyh p n0"
wenzelm@52658
  1638
    and ns: "isnpolyh s n1"
wenzelm@52658
  1639
    and ap: "head p = a"
wenzelm@56198
  1640
    and ndp: "degree p = n"
wenzelm@56198
  1641
    and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@56198
  1642
  shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1643
    (\<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)"
chaieb@33154
  1644
  using ns
wenzelm@52658
  1645
proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
berghofe@34915
  1646
  case less
chaieb@33154
  1647
  let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@56066
  1648
  let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and>
wenzelm@56066
  1649
    (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
chaieb@33154
  1650
  let ?b = "head s"
berghofe@34915
  1651
  let ?p' = "funpow (degree s - n) shift1 p"
wenzelm@50282
  1652
  let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
chaieb@33154
  1653
  let ?akk' = "a ^\<^sub>p (k' - k)"
wenzelm@60533
  1654
  note ns = \<open>isnpolyh s n1\<close>
wenzelm@52803
  1655
  from np have np0: "isnpolyh p 0"
wenzelm@52803
  1656
    using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
wenzelm@52803
  1657
  have np': "isnpolyh ?p' 0"
wenzelm@52803
  1658
    using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
wenzelm@52803
  1659
    by simp
wenzelm@52803
  1660
  have headp': "head ?p' = head p"
wenzelm@52803
  1661
    using funpow_shift1_head[OF np pnz] by simp
wenzelm@52803
  1662
  from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
wenzelm@52803
  1663
    by (simp add: isnpoly_def)
wenzelm@52803
  1664
  from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
chaieb@33154
  1665
  have nakk':"isnpolyh ?akk' 0" by blast
wenzelm@56066
  1666
  {
wenzelm@56066
  1667
    assume sz: "s = 0\<^sub>p"
wenzelm@56066
  1668
    then have ?ths
wenzelm@56066
  1669
      using np polydivide_aux.simps
wenzelm@52658
  1670
      apply clarsimp
wenzelm@52658
  1671
      apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1672
      apply simp
wenzelm@56066
  1673
      done
wenzelm@56066
  1674
  }
chaieb@33154
  1675
  moreover
wenzelm@56066
  1676
  {
wenzelm@56066
  1677
    assume sz: "s \<noteq> 0\<^sub>p"
wenzelm@56066
  1678
    {
wenzelm@56066
  1679
      assume dn: "degree s < n"
wenzelm@56066
  1680
      then have "?ths"
wenzelm@56066
  1681
        using ns ndp np polydivide_aux.simps
wenzelm@52658
  1682
        apply auto
wenzelm@52658
  1683
        apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1684
        apply simp
wenzelm@56066
  1685
        done
wenzelm@56066
  1686
    }
wenzelm@52803
  1687
    moreover
wenzelm@56066
  1688
    {
wenzelm@56066
  1689
      assume dn': "\<not> degree s < n"
wenzelm@56066
  1690
      then have dn: "degree s \<ge> n"
wenzelm@56066
  1691
        by arith
wenzelm@52803
  1692
      have degsp': "degree s = degree ?p'"
wenzelm@56066
  1693
        using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
wenzelm@56066
  1694
        by simp
wenzelm@56066
  1695
      {
wenzelm@56066
  1696
        assume ba: "?b = a"
wenzelm@56066
  1697
        then have headsp': "head s = head ?p'"
wenzelm@52803
  1698
          using ap headp' by simp
wenzelm@52803
  1699
        have nr: "isnpolyh (s -\<^sub>p ?p') 0"
wenzelm@52803
  1700
          using polysub_normh[OF ns np'] by simp
berghofe@34915
  1701
        from degree_polysub_samehead[OF ns np' headsp' degsp']
wenzelm@56207
  1702
        have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p"
wenzelm@56207
  1703
          by simp
wenzelm@52803
  1704
        moreover
wenzelm@56066
  1705
        {
wenzelm@56066
  1706
          assume deglt:"degree (s -\<^sub>p ?p') < degree s"
krauss@41403
  1707
          from polydivide_aux.simps sz dn' ba
krauss@41403
  1708
          have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1709
            by (simp add: Let_def)
wenzelm@56066
  1710
          {
wenzelm@56066
  1711
            assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1712
            from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
wenzelm@52803
  1713
            have kk': "k \<le> k'"
wenzelm@56066
  1714
              and nr: "\<exists>nr. isnpolyh r nr"
wenzelm@52803
  1715
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@56066
  1716
              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"
wenzelm@52803
  1717
              by auto
wenzelm@52803
  1718
            from q1 obtain q n1 where nq: "isnpolyh q n1"
wenzelm@56066
  1719
              and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
wenzelm@56066
  1720
              by blast
wenzelm@56066
  1721
            from nr obtain nr where nr': "isnpolyh r nr"
wenzelm@56066
  1722
              by blast
wenzelm@52803
  1723
            from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
wenzelm@52803
  1724
              by simp
wenzelm@33268
  1725
            from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
wenzelm@33268
  1726
            have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
wenzelm@52803
  1727
            from polyadd_normh[OF polymul_normh[OF np
wenzelm@33268
  1728
              polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
wenzelm@52803
  1729
            have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
wenzelm@52803
  1730
              by simp
haftmann@59867
  1731
            from asp have "\<forall>bs :: 'a::{field_char_0,field} list.
wenzelm@56066
  1732
              Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
wenzelm@56066
  1733
              by simp
haftmann@59867
  1734
            then have "\<forall>bs :: 'a::{field_char_0,field} list.
wenzelm@56066
  1735
              Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
wenzelm@52803
  1736
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
haftmann@36349
  1737
              by (simp add: field_simps)
haftmann@59867
  1738
            then have "\<forall>bs :: 'a::{field_char_0,field} list.
wenzelm@56066
  1739
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1740
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
wenzelm@52803
  1741
              Ipoly bs p * Ipoly bs q + Ipoly bs r"
wenzelm@52803
  1742
              by (auto simp only: funpow_shift1_1)
haftmann@59867
  1743
            then have "\<forall>bs:: 'a::{field_char_0,field} list.
wenzelm@56066
  1744
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1745
              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
wenzelm@52803
  1746
              Ipoly bs q) + Ipoly bs r"
wenzelm@52803
  1747
              by (simp add: field_simps)
haftmann@59867
  1748
            then have "\<forall>bs:: 'a::{field_char_0,field} list.
wenzelm@56066
  1749
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1750
              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)"
wenzelm@52803
  1751
              by simp
wenzelm@33268
  1752
            with isnpolyh_unique[OF nakks' nqr']
wenzelm@52803
  1753
            have "a ^\<^sub>p (k' - k) *\<^sub>p s =
wenzelm@52803
  1754
              p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
wenzelm@52803
  1755
              by blast
wenzelm@56066
  1756
            then have ?qths using nq'
wenzelm@50282
  1757
              apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
wenzelm@52803
  1758
              apply (rule_tac x="0" in exI)
wenzelm@52803
  1759
              apply simp
wenzelm@52803
  1760
              done
wenzelm@56066
  1761
            with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1762
              (\<exists>nr. isnpolyh r nr) \<and> ?qths"
wenzelm@52803
  1763
              by blast
wenzelm@52803
  1764
          }
wenzelm@56066
  1765
          then have ?ths by blast
wenzelm@52803
  1766
        }
wenzelm@52803
  1767
        moreover
wenzelm@56066
  1768
        {
wenzelm@56066
  1769
          assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
haftmann@59867
  1770
          from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0,field}"]
haftmann@59867
  1771
          have "\<forall>bs:: 'a::{field_char_0,field} list. Ipoly bs s = Ipoly bs ?p'"
wenzelm@52803
  1772
            by simp
haftmann@59867
  1773
          then have "\<forall>bs:: 'a::{field_char_0,field} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
wenzelm@52658
  1774
            using np nxdn
wenzelm@52658
  1775
            apply simp
wenzelm@52658
  1776
            apply (simp only: funpow_shift1_1)
wenzelm@52658
  1777
            apply simp
wenzelm@52658
  1778
            done
wenzelm@56066
  1779
          then have sp': "s = ?xdn *\<^sub>p p"
wenzelm@56066
  1780
            using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
wenzelm@52658
  1781
            by blast
wenzelm@56066
  1782
          {
wenzelm@56207
  1783
            assume h1: "polydivide_aux a n p k s = (k', r)"
krauss@41403
  1784
            from polydivide_aux.simps sz dn' ba
krauss@41403
  1785
            have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1786
              by (simp add: Let_def)
wenzelm@52803
  1787
            also have "\<dots> = (k,0\<^sub>p)"
wenzelm@52803
  1788
              using polydivide_aux.simps spz by simp
wenzelm@56066
  1789
            finally have "(k', r) = (k, 0\<^sub>p)"
wenzelm@56066
  1790
              using h1 by simp
berghofe@34915
  1791
            with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
krauss@41403
  1792
              polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
wenzelm@33268
  1793
              apply auto
wenzelm@52803
  1794
              apply (rule exI[where x="?xdn"])
berghofe@34915
  1795
              apply (auto simp add: polymul_commute[of p])
wenzelm@52803
  1796
              done
wenzelm@52803
  1797
          }
wenzelm@52803
  1798
        }
wenzelm@52803
  1799
        ultimately have ?ths by blast
wenzelm@52803
  1800
      }
chaieb@33154
  1801
      moreover
wenzelm@56066
  1802
      {
wenzelm@56066
  1803
        assume ba: "?b \<noteq> a"
wenzelm@52803
  1804
        from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1805
          polymul_normh[OF head_isnpolyh[OF ns] np']]
wenzelm@52803
  1806
        have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1807
          by (simp add: min_def)
wenzelm@33268
  1808
        have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
wenzelm@52803
  1809
          using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1810
            polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
wenzelm@52803
  1811
            funpow_shift1_nz[OF pnz]
wenzelm@52803
  1812
          by simp_all
wenzelm@33268
  1813
        from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
berghofe@34915
  1814
          polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@52803
  1815
        have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
wenzelm@33268
  1816
          using head_head[OF ns] funpow_shift1_head[OF np pnz]
wenzelm@33268
  1817
            polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
wenzelm@33268
  1818
          by (simp add: ap)
wenzelm@33268
  1819
        from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
wenzelm@33268
  1820
          head_nz[OF np] pnz sz ap[symmetric]
berghofe@34915
  1821
          funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@52803
  1822
          polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
berghofe@34915
  1823
          ndp dn
wenzelm@52803
  1824
        have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
wenzelm@33268
  1825
          by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
wenzelm@56066
  1826
        {
wenzelm@56066
  1827
          assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
wenzelm@52803
  1828
          from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
wenzelm@52803
  1829
            polymul_normh[OF head_isnpolyh[OF ns]np']] ap
wenzelm@52803
  1830
          have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1831
            by simp
wenzelm@56066
  1832
          {
wenzelm@56066
  1833
            assume h1:"polydivide_aux a n p k s = (k', r)"
krauss@41403
  1834
            from h1 polydivide_aux.simps sz dn' ba
krauss@41403
  1835
            have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
wenzelm@33268
  1836
              by (simp add: Let_def)
berghofe@34915
  1837
            with less(1)[OF dth nasbp', of "Suc k" k' r]
wenzelm@52803
  1838
            obtain q nq nr where kk': "Suc k \<le> k'"
wenzelm@52803
  1839
              and nr: "isnpolyh r nr"
wenzelm@52803
  1840
              and nq: "isnpolyh q nq"
wenzelm@33268
  1841
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@52803
  1842
              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"
wenzelm@52803
  1843
              by auto
wenzelm@56066
  1844
            from kk' have kk'': "Suc (k' - Suc k) = k' - k"
wenzelm@56066
  1845
              by arith
wenzelm@52803
  1846
            {
haftmann@59867
  1847
              fix bs :: "'a::{field_char_0,field} list"
wenzelm@52803
  1848
              from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
wenzelm@52803
  1849
              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)"
wenzelm@52803
  1850
                by simp
wenzelm@56066
  1851
              then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
wenzelm@52803
  1852
                Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
wenzelm@52803
  1853
                by (simp add: field_simps)
wenzelm@56066
  1854
              then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
wenzelm@52803
  1855
                Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
wenzelm@52803
  1856
                by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@56066
  1857
              then have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1858
                Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
wenzelm@52803
  1859
                by (simp add: field_simps)
wenzelm@52803
  1860
            }
haftmann@59867
  1861
            then have ieq:"\<forall>bs :: 'a::{field_char_0,field} list.
wenzelm@56207
  1862
                Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@56207
  1863
                Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
wenzelm@52803
  1864
              by auto
wenzelm@33268
  1865
            let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
wenzelm@56207
  1866
            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]]
wenzelm@52803
  1867
            have nqw: "isnpolyh ?q 0"
wenzelm@52803
  1868
              by simp
wenzelm@33268
  1869
            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]]
wenzelm@52803
  1870
            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"
wenzelm@52803
  1871
              by blast
wenzelm@52803
  1872
            from dr kk' nr h1 asth nqw have ?ths
wenzelm@52803
  1873
              apply simp
wenzelm@33268
  1874
              apply (rule conjI)
wenzelm@33268
  1875
              apply (rule exI[where x="nr"], simp)
wenzelm@33268
  1876
              apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
wenzelm@33268
  1877
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1878
              done
wenzelm@52803
  1879
          }
wenzelm@56066
  1880
          then have ?ths by blast
wenzelm@52803
  1881
        }
wenzelm@52803
  1882
        moreover
wenzelm@56066
  1883
        {
wenzelm@56066
  1884
          assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
wenzelm@52803
  1885
          {
haftmann@59867
  1886
            fix bs :: "'a::{field_char_0,field} list"
wenzelm@33268
  1887
            from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
wenzelm@52803
  1888
            have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
wenzelm@52803
  1889
              by simp
wenzelm@56066
  1890
            then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
wenzelm@52803
  1891
              by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@56066
  1892
            then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
wenzelm@52803
  1893
              by simp
wenzelm@52803
  1894
          }
haftmann@59867
  1895
          then have hth: "\<forall>bs :: 'a::{field_char_0,field} list.
wenzelm@56066
  1896
            Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
wenzelm@52803
  1897
          from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
haftmann@59867
  1898
            using isnpolyh_unique[where ?'a = "'a::{field_char_0,field}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
chaieb@33154
  1899
                    polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
wenzelm@56066
  1900
              simplified ap]
wenzelm@56066
  1901
            by simp
wenzelm@56066
  1902
          {
wenzelm@56066
  1903
            assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1904
            from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
wenzelm@56066
  1905
            have "(k', r) = (Suc k, 0\<^sub>p)"
wenzelm@56066
  1906
              by (simp add: Let_def)
wenzelm@52803
  1907
            with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
wenzelm@52803
  1908
              polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
wenzelm@52803
  1909
            have ?ths
wenzelm@52803
  1910
              apply (clarsimp simp add: Let_def)
wenzelm@52803
  1911
              apply (rule exI[where x="?b *\<^sub>p ?xdn"])
wenzelm@52803
  1912
              apply simp
wenzelm@52803
  1913
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1914
              done
wenzelm@52803
  1915
          }
wenzelm@56066
  1916
          then have ?ths by blast
wenzelm@52803
  1917
        }
wenzelm@52658
  1918
        ultimately have ?ths
wenzelm@56066
  1919
          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"]
wenzelm@52658
  1920
            head_nz[OF np] pnz sz ap[symmetric]
wenzelm@56066
  1921
          by (auto simp add: degree_eq_degreen0[symmetric])
wenzelm@52803
  1922
      }
chaieb@33154
  1923
      ultimately have ?ths by blast
chaieb@33154
  1924
    }
wenzelm@52803
  1925
    ultimately have ?ths by blast
wenzelm@52803
  1926
  }
chaieb@33154
  1927
  ultimately show ?ths by blast
chaieb@33154
  1928
qed
chaieb@33154
  1929
wenzelm@52803
  1930
lemma polydivide_properties:
haftmann@59867
  1931
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
wenzelm@56066
  1932
    and np: "isnpolyh p n0"
wenzelm@56066
  1933
    and ns: "isnpolyh s n1"
wenzelm@56066
  1934
    and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@56066
  1935
  shows "\<exists>k r. polydivide s p = (k, r) \<and>
wenzelm@52803
  1936
    (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1937
    (\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@52803
  1938
proof -
wenzelm@52803
  1939
  have trv: "head p = head p" "degree p = degree p"
wenzelm@52803
  1940
    by simp_all
wenzelm@52803
  1941
  from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
wenzelm@52803
  1942
    by auto
wenzelm@52803
  1943
  then obtain k r where kr: "polydivide s p = (k,r)"
wenzelm@52803
  1944
    by blast
wenzelm@56000
  1945
  from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
chaieb@33154
  1946
    polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
chaieb@33154
  1947
  have "(degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@52803
  1948
    (\<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)"
wenzelm@52803
  1949
    by blast
wenzelm@52803
  1950
  with kr show ?thesis
chaieb@33154
  1951
    apply -
chaieb@33154
  1952
    apply (rule exI[where x="k"])
chaieb@33154
  1953
    apply (rule exI[where x="r"])
chaieb@33154
  1954
    apply simp
chaieb@33154
  1955
    done
chaieb@33154
  1956
qed
chaieb@33154
  1957
wenzelm@52658
  1958
wenzelm@60533
  1959
subsection \<open>More about polypoly and pnormal etc\<close>
chaieb@33154
  1960
wenzelm@56000
  1961
definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
chaieb@33154
  1962
wenzelm@52658
  1963
lemma isnonconstant_pnormal_iff:
wenzelm@56198
  1964
  assumes "isnonconstant p"
wenzelm@52803
  1965
  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1966
proof
wenzelm@52803
  1967
  let ?p = "polypoly bs p"
chaieb@33154
  1968
  assume H: "pnormal ?p"
wenzelm@56066
  1969
  have csz: "coefficients p \<noteq> []"
wenzelm@56198
  1970
    using assms by (cases p) auto
wenzelm@56066
  1971
  from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] pnormal_last_nonzero[OF H]
wenzelm@56066
  1972
  show "Ipoly bs (head p) \<noteq> 0"
wenzelm@56066
  1973
    by (simp add: polypoly_def)
chaieb@33154
  1974
next
chaieb@33154
  1975
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1976
  let ?p = "polypoly bs p"
wenzelm@56066
  1977
  have csz: "coefficients p \<noteq> []"
wenzelm@56198
  1978
    using assms by (cases p) auto
wenzelm@56066
  1979
  then have pz: "?p \<noteq> []"
wenzelm@56066
  1980
    by (simp add: polypoly_def)
wenzelm@56066
  1981
  then have lg: "length ?p > 0"
wenzelm@56066
  1982
    by simp
wenzelm@52803
  1983
  from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
wenzelm@56066
  1984
  have lz: "last ?p \<noteq> 0"
wenzelm@56066
  1985
    by (simp add: polypoly_def)
chaieb@33154
  1986
  from pnormal_last_length[OF lg lz] show "pnormal ?p" .
chaieb@33154
  1987
qed
chaieb@33154
  1988
chaieb@33154
  1989
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
chaieb@33154
  1990
  unfolding isnonconstant_def
wenzelm@52658
  1991
  apply (cases p)
wenzelm@52658
  1992
  apply simp_all
blanchet@58259
  1993
  apply (rename_tac nat a, case_tac nat)
wenzelm@52658
  1994
  apply auto
chaieb@33154
  1995
  done
wenzelm@52658
  1996
wenzelm@52658
  1997
lemma isnonconstant_nonconstant:
wenzelm@56198
  1998
  assumes "isnonconstant p"
chaieb@33154
  1999
  shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  2000
proof
chaieb@33154
  2001
  let ?p = "polypoly bs p"
chaieb@33154
  2002
  assume nc: "nonconstant ?p"
wenzelm@56198
  2003
  from isnonconstant_pnormal_iff[OF assms, of bs] nc
wenzelm@56066
  2004
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56066
  2005
    unfolding nonconstant_def by blast
chaieb@33154
  2006
next
chaieb@33154
  2007
  let ?p = "polypoly bs p"
chaieb@33154
  2008
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56198
  2009
  from isnonconstant_pnormal_iff[OF assms, of bs] h
wenzelm@56066
  2010
  have pn: "pnormal ?p"
wenzelm@56066
  2011
    by blast
wenzelm@56009
  2012
  {
wenzelm@56009
  2013
    fix x
wenzelm@56009
  2014
    assume H: "?p = [x]"
wenzelm@56009
  2015
    from H have "length (coefficients p) = 1"
wenzelm@56009
  2016
      unfolding polypoly_def by auto
wenzelm@56198
  2017
    with isnonconstant_coefficients_length[OF assms]
wenzelm@56198
  2018
    have False by arith
wenzelm@56009
  2019
  }
wenzelm@56009
  2020
  then show "nonconstant ?p"
wenzelm@56009
  2021
    using pn unfolding nonconstant_def by blast
chaieb@33154
  2022
qed
chaieb@33154
  2023
wenzelm@56066
  2024
lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
wenzelm@52658
  2025
  apply (induct p)
wenzelm@52658
  2026
  apply (simp_all add: pnormal_def)
wenzelm@52658
  2027
  apply (case_tac "p = []")
wenzelm@52658
  2028
  apply simp_all
wenzelm@52658
  2029
  done
chaieb@33154
  2030
wenzelm@52658
  2031
lemma degree_degree:
wenzelm@56207
  2032
  assumes "isnonconstant p"
chaieb@33154
  2033
  shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  2034
proof
wenzelm@52803
  2035
  let ?p = "polypoly bs p"
chaieb@33154
  2036
  assume H: "degree p = Polynomial_List.degree ?p"
wenzelm@56207
  2037
  from isnonconstant_coefficients_length[OF assms] have pz: "?p \<noteq> []"
chaieb@33154
  2038
    unfolding polypoly_def by auto
wenzelm@56207
  2039
  from H degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
wenzelm@56066
  2040
  have lg: "length (pnormalize ?p) = length ?p"
chaieb@33154
  2041
    unfolding Polynomial_List.degree_def polypoly_def by simp
wenzelm@56066
  2042
  then have "pnormal ?p"
wenzelm@56066
  2043
    using pnormal_length[OF pz] by blast
wenzelm@56207
  2044
  with isnonconstant_pnormal_iff[OF assms] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56066
  2045
    by blast
chaieb@33154
  2046
next
wenzelm@56066
  2047
  let ?p = "polypoly bs p"
chaieb@33154
  2048
  assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56207
  2049
  with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
wenzelm@56066
  2050
    by blast
wenzelm@56207
  2051
  with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
wenzelm@52803
  2052
  show "degree p = Polynomial_List.degree ?p"
chaieb@33154
  2053
    unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
chaieb@33154
  2054
qed
chaieb@33154
  2055
wenzelm@52658
  2056
wenzelm@60533
  2057
section \<open>Swaps ; Division by a certain variable\<close>
wenzelm@52658
  2058
wenzelm@56066
  2059
primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
wenzelm@56066
  2060
where
chaieb@33154
  2061
  "swap n m (C x) = C x"
wenzelm@56198
  2062
| "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
haftmann@39246
  2063
| "swap n m (Neg t) = Neg (swap n m t)"
haftmann@39246
  2064
| "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
haftmann@39246
  2065
| "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
haftmann@39246
  2066
| "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
haftmann@39246
  2067
| "swap n m (Pw t k) = Pw (swap n m t) k"
wenzelm@56066
  2068
| "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
chaieb@33154
  2069
wenzelm@52658
  2070
lemma swap:
wenzelm@56066
  2071
  assumes "n < length bs"
wenzelm@56066
  2072
    and "m < length bs"
chaieb@33154
  2073
  shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  2074
proof (induct t)
wenzelm@52658
  2075
  case (Bound k)
wenzelm@56066
  2076
  then show ?case
wenzelm@56066
  2077
    using assms by simp
chaieb@33154
  2078
next
wenzelm@52658
  2079
  case (CN c k p)
wenzelm@56066
  2080
  then show ?case
wenzelm@56066
  2081
    using assms by simp
chaieb@33154
  2082
qed simp_all
chaieb@33154
  2083
wenzelm@52658
  2084
lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
wenzelm@52658
  2085
  by (induct t) simp_all
wenzelm@52658
  2086
wenzelm@52658
  2087
lemma swap_commute: "swap n m p = swap m n p"
wenzelm@52658
  2088
  by (induct p) simp_all
chaieb@33154
  2089
chaieb@33154
  2090
lemma swap_same_id[simp]: "swap n n t = t"
wenzelm@52658
  2091
  by (induct t) simp_all
chaieb@33154
  2092
chaieb@33154
  2093
definition "swapnorm n m t = polynate (swap n m t)"
chaieb@33154
  2094
wenzelm@52658
  2095
lemma swapnorm:
wenzelm@52658
  2096
  assumes nbs: "n < length bs"
wenzelm@52658
  2097
    and mbs: "m < length bs"
haftmann@59867
  2098
  shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field})) =
wenzelm@52658
  2099
    Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
wenzelm@41807
  2100
  using swap[OF assms] swapnorm_def by simp
chaieb@33154
  2101
wenzelm@52658
  2102
lemma swapnorm_isnpoly [simp]:
haftmann@59867
  2103
  assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
chaieb@33154
  2104
  shows "isnpoly (swapnorm n m p)"
chaieb@33154
  2105
  unfolding swapnorm_def by simp
chaieb@33154
  2106
wenzelm@52803
  2107
definition "polydivideby n s p =
wenzelm@56000
  2108
  (let
wenzelm@56000
  2109
    ss = swapnorm 0 n s;
wenzelm@56000
  2110
    sp = swapnorm 0 n p;
wenzelm@56000
  2111
    h = head sp;
wenzelm@56000
  2112
    (k, r) = polydivide ss sp
wenzelm@56000
  2113
   in (k, swapnorm 0 n h, swapnorm 0 n r))"
chaieb@33154
  2114
wenzelm@56000
  2115
lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  2116
  by (induct p) simp_all
chaieb@33154
  2117
krauss@41808
  2118
fun isweaknpoly :: "poly \<Rightarrow> bool"
krauss@41808
  2119
where
chaieb@33154
  2120
  "isweaknpoly (C c) = True"
krauss@41808
  2121
| "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
krauss@41808
  2122
| "isweaknpoly p = False"
chaieb@33154
  2123
wenzelm@52803
  2124
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
wenzelm@52658
  2125
  by (induct p arbitrary: n0) auto
chaieb@33154
  2126
wenzelm@52803
  2127
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
wenzelm@52658
  2128
  by (induct p) auto
chaieb@33154
  2129
chaieb@33154
  2130
end