src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
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more robust syntax that survives collapse of \<^isub> and \<^sub>;
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(*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
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    Author:     Amine Chaieb
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*)
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header {* Implementation and verification of multivariate polynomials *}
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theory Reflected_Multivariate_Polynomial
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imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
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begin
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  (* Implementation *)
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subsection{* Datatype of polynomial expressions *} 
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datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
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  | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
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abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
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abbreviation poly_p :: "int \<Rightarrow> poly" ("'((_)')\<^sub>p") where "(i)\<^sub>p \<equiv> C (i)\<^sub>N"
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subsection{* Boundedness, substitution and all that *}
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primrec polysize:: "poly \<Rightarrow> nat" where
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  "polysize (C c) = 1"
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| "polysize (Bound n) = 1"
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| "polysize (Neg p) = 1 + polysize p"
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| "polysize (Add p q) = 1 + polysize p + polysize q"
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| "polysize (Sub p q) = 1 + polysize p + polysize q"
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| "polysize (Mul p q) = 1 + polysize p + polysize q"
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| "polysize (Pw p n) = 1 + polysize p"
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| "polysize (CN c n p) = 4 + polysize c + polysize p"
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primrec polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *) where
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  "polybound0 (C c) = True"
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| "polybound0 (Bound n) = (n>0)"
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| "polybound0 (Neg a) = polybound0 a"
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| "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
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| "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)" 
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| "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
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| "polybound0 (Pw p n) = (polybound0 p)"
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| "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
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primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *) where
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  "polysubst0 t (C c) = (C c)"
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| "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"
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| "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
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| "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
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| "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)" 
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| "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
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| "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
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| "polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
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                             else CN (polysubst0 t c) n (polysubst0 t p))"
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fun decrpoly:: "poly \<Rightarrow> poly" 
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where
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  "decrpoly (Bound n) = Bound (n - 1)"
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| "decrpoly (Neg a) = Neg (decrpoly a)"
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| "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
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| "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
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| "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
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| "decrpoly (Pw p n) = Pw (decrpoly p) n"
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| "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
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| "decrpoly a = a"
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subsection{* Degrees and heads and coefficients *}
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fun degree:: "poly \<Rightarrow> nat"
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where
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  "degree (CN c 0 p) = 1 + degree p"
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| "degree p = 0"
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fun head:: "poly \<Rightarrow> poly"
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where
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  "head (CN c 0 p) = head p"
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| "head p = p"
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(* More general notions of degree and head *)
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fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
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 |"degreen p = (\<lambda>m. 0)"
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fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
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where
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  "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
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| "headn p = (\<lambda>m. p)"
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fun coefficients:: "poly \<Rightarrow> poly list"
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where
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  "coefficients (CN c 0 p) = c#(coefficients p)"
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| "coefficients p = [p]"
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fun isconstant:: "poly \<Rightarrow> bool"
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where
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  "isconstant (CN c 0 p) = False"
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| "isconstant p = True"
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fun behead:: "poly \<Rightarrow> poly"
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where
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  "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
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| "behead p = 0\<^sub>p"
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fun headconst:: "poly \<Rightarrow> Num"
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where
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  "headconst (CN c n p) = headconst p"
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| "headconst (C n) = n"
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subsection{* Operations for normalization *}
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declare if_cong[fundef_cong del]
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declare let_cong[fundef_cong del]
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fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
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where
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  "polyadd (C c) (C c') = C (c+\<^sub>Nc')"
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|  "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
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| "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
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| "polyadd (CN c n p) (CN c' n' p') =
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    (if n < n' then CN (polyadd c (CN c' n' p')) n p
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     else if n'<n then CN (polyadd (CN c n p) c') n' p'
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     else (let cc' = polyadd c c' ; 
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               pp' = polyadd p p'
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           in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
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| "polyadd a b = Add a b"
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fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
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where
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  "polyneg (C c) = C (~\<^sub>N c)"
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| "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
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| "polyneg a = Neg a"
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definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
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where
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  "p -\<^sub>p q = polyadd p (polyneg q)"
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fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
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where
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  "polymul (C c) (C c') = C (c*\<^sub>Nc')"
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| "polymul (C c) (CN c' n' p') = 
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      (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
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| "polymul (CN c n p) (C c') = 
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      (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
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| "polymul (CN c n p) (CN c' n' p') = 
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  (if n<n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
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  else if n' < n 
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  then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
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  else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
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| "polymul a b = Mul a b"
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declare if_cong[fundef_cong]
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declare let_cong[fundef_cong]
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fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
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where
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   156
  "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
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4eb43410d2fa recdef -> fun; curried
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   157
| "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul q q in 
4eb43410d2fa recdef -> fun; curried
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   158
                    if even n then d else polymul p d)"
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   159
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a5db9779b026 modernized some syntax translations;
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   160
abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
a5db9779b026 modernized some syntax translations;
wenzelm
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diff changeset
   161
  where "a ^\<^sub>p k \<equiv> polypow k a"
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   162
41808
9f436d00248f recdef -> function
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   163
function polynate :: "poly \<Rightarrow> poly"
9f436d00248f recdef -> function
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   164
where
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fe4d4bb9f4c2 more robust syntax that survives collapse of \<^isub> and \<^sub>;
wenzelm
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   165
  "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
41808
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   166
| "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
9f436d00248f recdef -> function
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   167
| "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
9f436d00248f recdef -> function
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   168
| "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
9f436d00248f recdef -> function
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   169
| "polynate (Neg p) = (~\<^sub>p (polynate p))"
9f436d00248f recdef -> function
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   170
| "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
9f436d00248f recdef -> function
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   171
| "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
9f436d00248f recdef -> function
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   172
| "polynate (C c) = C (normNum c)"
9f436d00248f recdef -> function
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   173
by pat_completeness auto
9f436d00248f recdef -> function
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   174
termination by (relation "measure polysize") auto
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   175
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   176
fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
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   177
  "poly_cmul y (C x) = C (y *\<^sub>N x)"
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   178
| "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
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   179
| "poly_cmul y p = C y *\<^sub>p p"
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   180
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d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
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   181
definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
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   182
  "monic p \<equiv> (let h = headconst p in if h = 0\<^sub>N then (p,False) else ((C (Ninv h)) *\<^sub>p p, 0>\<^sub>N h))"
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   183
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chaieb
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   184
subsection{* Pseudo-division *}
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   185
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d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
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   186
definition shift1 :: "poly \<Rightarrow> poly" where
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   187
  "shift1 p \<equiv> CN 0\<^sub>p 0 p"
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   188
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   189
abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
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   190
  "funpow \<equiv> compow"
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haftmann
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diff changeset
   191
41403
7eba049f7310 partial_function (tailrec) replaces function (tailrec);
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   192
partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
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chaieb
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   193
  where
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7eba049f7310 partial_function (tailrec) replaces function (tailrec);
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   194
  "polydivide_aux a n p k s = 
33154
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   195
  (if s = 0\<^sub>p then (k,s)
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parents:
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   196
  else (let b = head s; m = degree s in
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   197
  (if m < n then (k,s) else 
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   198
  (let p'= funpow (m - n) shift1 p in 
41403
7eba049f7310 partial_function (tailrec) replaces function (tailrec);
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   199
  (if a = b then polydivide_aux a n p k (s -\<^sub>p p') 
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   200
  else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
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chaieb
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diff changeset
   201
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
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   202
definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
41403
7eba049f7310 partial_function (tailrec) replaces function (tailrec);
krauss
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diff changeset
   203
  "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
33154
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diff changeset
   204
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   205
fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
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   206
  "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
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chaieb
parents:
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   207
| "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
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chaieb
parents:
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   208
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chaieb
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   209
fun poly_deriv :: "poly \<Rightarrow> poly" where
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   210
  "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
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   211
| "poly_deriv p = 0\<^sub>p"
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   212
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chaieb
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   213
subsection{* Semantics of the polynomial representation *}
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chaieb
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   214
39246
9e58f0499f57 modernized primrec
haftmann
parents: 36409
diff changeset
   215
primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
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chaieb
parents:
diff changeset
   216
  "Ipoly bs (C c) = INum c"
39246
9e58f0499f57 modernized primrec
haftmann
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diff changeset
   217
| "Ipoly bs (Bound n) = bs!n"
9e58f0499f57 modernized primrec
haftmann
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diff changeset
   218
| "Ipoly bs (Neg a) = - Ipoly bs a"
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haftmann
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diff changeset
   219
| "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
9e58f0499f57 modernized primrec
haftmann
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diff changeset
   220
| "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
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haftmann
parents: 36409
diff changeset
   221
| "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
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haftmann
parents: 36409
diff changeset
   222
| "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
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haftmann
parents: 36409
diff changeset
   223
| "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
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haftmann
parents: 36409
diff changeset
   224
35054
a5db9779b026 modernized some syntax translations;
wenzelm
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   225
abbreviation
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   226
  Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
35054
a5db9779b026 modernized some syntax translations;
wenzelm
parents: 35046
diff changeset
   227
  where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
33154
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parents:
diff changeset
   228
daa6ddece9f0 Multivariate polynomials library over fields
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   229
lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i" 
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chaieb
parents:
diff changeset
   230
  by (simp add: INum_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   231
lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   232
  by (simp  add: INum_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   233
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   234
lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
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chaieb
parents:
diff changeset
   235
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
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   236
subsection {* Normal form and normalization *}
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   237
41808
9f436d00248f recdef -> function
krauss
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diff changeset
   238
fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
9f436d00248f recdef -> function
krauss
parents: 41763
diff changeset
   239
where
33154
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chaieb
parents:
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   240
  "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
41808
9f436d00248f recdef -> function
krauss
parents: 41763
diff changeset
   241
| "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))"
9f436d00248f recdef -> function
krauss
parents: 41763
diff changeset
   242
| "isnpolyh p = (\<lambda>k. False)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   243
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   244
lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   245
by (induct p rule: isnpolyh.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   246
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35054
diff changeset
   247
definition isnpoly :: "poly \<Rightarrow> bool" where
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   248
  "isnpoly p \<equiv> isnpolyh p 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   249
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   250
text{* polyadd preserves normal forms *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   251
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   252
lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> 
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   253
      \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   254
proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   255
  case (2 ab c' n' p' n0 n1)
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
   256
  from 2 have  th1: "isnpolyh (C ab) (Suc n')" by simp 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   257
  from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   258
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
   259
  with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   260
  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   261
  thus ?case using 2 th3 by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   262
next
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   263
  case (3 c' n' p' ab n1 n0)
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
   264
  from 3 have  th1: "isnpolyh (C ab) (Suc n')" by simp 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   265
  from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   266
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
   267
  with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   268
  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   269
  thus ?case using 3 th3 by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   270
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   271
  case (4 c n p c' n' p' n0 n1)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   272
  hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   273
  from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all 
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   274
  from 4 have ngen0: "n \<ge> n0" by simp
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   275
  from 4 have n'gen1: "n' \<ge> n1" by simp 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   276
  have "n < n' \<or> n' < n \<or> n = n'" by auto
41763
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   277
  moreover {assume eq: "n = n'"
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   278
    with "4.hyps"(3)[OF nc nc'] 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   279
    have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   280
    hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   281
      using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   282
    from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   283
    have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
   284
    from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   285
  moreover {assume lt: "n < n'"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   286
    have "min n0 n1 \<le> n0" by simp
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
   287
    with 4 lt have th1:"min n0 n1 \<le> n" by auto 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   288
    from 4 have th21: "isnpolyh c (Suc n)" by simp
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   289
    from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   290
    from lt have th23: "min (Suc n) n' = Suc n" by arith
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   291
    from "4.hyps"(1)[OF th21 th22]
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   292
    have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
   293
    with 4 lt th1 have ?case by simp } 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   294
  moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   295
    have "min n0 n1 \<le> n1"  by simp
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
   296
    with 4 gt have th1:"min n0 n1 \<le> n'" by auto
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   297
    from 4 have th21: "isnpolyh c' (Suc n')" by simp_all
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   298
    from 4 have th22: "isnpolyh (CN c n p) n" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   299
    from gt have th23: "min n (Suc n') = Suc n'" by arith
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   300
    from "4.hyps"(2)[OF th22 th21]
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   301
    have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   302
    with 4 gt th1 have ?case by simp}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   303
      ultimately show ?case by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   304
qed auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   305
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   306
lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 46991
diff changeset
   307
by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   308
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   309
lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   310
  using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   311
41404
aae9f912cca8 dropped duplicate unused lemmas;
krauss
parents: 41403
diff changeset
   312
text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   313
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   314
lemma polyadd_different_degreen: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   315
  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   316
  degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   317
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   318
  case (4 c n p c' n' p' m n0 n1)
41763
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   319
  have "n' = n \<or> n < n' \<or> n' < n" by arith
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   320
  thus ?case
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   321
  proof (elim disjE)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   322
    assume [simp]: "n' = n"
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   323
    from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
41763
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   324
    show ?thesis by (auto simp: Let_def)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   325
  next
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   326
    assume "n < n'"
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   327
    with 4 show ?thesis by auto
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   328
  next
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   329
    assume "n' < n"
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   330
    with 4 show ?thesis by auto
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   331
  qed
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   332
qed auto
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   333
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   334
lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   335
  by (induct p arbitrary: n rule: headn.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   336
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   337
  by (induct p arbitrary: n rule: degree.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   338
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   339
  by (induct p arbitrary: n rule: degreen.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   340
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   341
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   342
  by (induct p arbitrary: n rule: degree.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   343
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   344
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   345
  using degree_isnpolyh_Suc by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   346
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   347
  using degreen_0 by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   348
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   349
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   350
lemma degreen_polyadd:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   351
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   352
  shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   353
  using np nq m
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   354
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   355
  case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   356
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   357
  case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   358
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   359
  case (4 c n p c' n' p' n0 n1 m) 
41763
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   360
  have "n' = n \<or> n < n' \<or> n' < n" by arith
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   361
  thus ?case
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   362
  proof (elim disjE)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   363
    assume [simp]: "n' = n"
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   364
    from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
41763
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   365
    show ?thesis by (auto simp: Let_def)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   366
  qed simp_all
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   367
qed auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   368
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
   369
lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   370
  \<Longrightarrow> degreen p m = degreen q m"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   371
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   372
  case (4 c n p c' n' p' m n0 n1 x) 
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   373
  {assume nn': "n' < n" hence ?case using 4 by simp}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   374
  moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   375
  {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   376
    moreover {assume "n < n'" with 4 have ?case by simp }
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   377
    moreover {assume eq: "n = n'" hence ?case using 4 
41763
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   378
        apply (cases "p +\<^sub>p p' = 0\<^sub>p")
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   379
        apply (auto simp add: Let_def)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   380
        by blast
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   381
      }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   382
    ultimately have ?case by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   383
  ultimately show ?case by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   384
qed simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   385
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   386
lemma polymul_properties:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   387
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   388
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   389
  shows "isnpolyh (p *\<^sub>p q) (min n0 n1)" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   390
  and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   391
  and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   392
                             else degreen p m + degreen q m)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   393
  using np nq m
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   394
proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   395
  case (2 c c' n' p') 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   396
  { case (1 n0 n1) 
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   397
    with "2.hyps"(4-6)[of n' n' n']
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   398
      and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   399
    show ?case by (auto simp add: min_def)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   400
  next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   401
    case (2 n0 n1) thus ?case by auto 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   402
  next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   403
    case (3 n0 n1) thus ?case  using "2.hyps" by auto } 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   404
next
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   405
  case (3 c n p c')
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   406
  { case (1 n0 n1) 
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   407
    with "3.hyps"(4-6)[of n n n]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   408
      "3.hyps"(1-3)[of "Suc n" "Suc n" n]
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   409
    show ?case by (auto simp add: min_def)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   410
  next
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   411
    case (2 n0 n1) thus ?case by auto
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   412
  next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   413
    case (3 n0 n1) thus ?case  using "3.hyps" by auto } 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   414
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   415
  case (4 c n p c' n' p')
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   416
  let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   417
    {
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   418
      case (1 n0 n1)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   419
      hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   420
        and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   421
        and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   422
        and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   423
        by simp_all
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   424
      { assume "n < n'"
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   425
        with "4.hyps"(4-5)[OF np cnp', of n]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   426
          "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   427
        have ?case by (simp add: min_def)
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   428
      } moreover {
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   429
        assume "n' < n"
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   430
        with "4.hyps"(16-17)[OF cnp np', of "n'"]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   431
          "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   432
        have ?case
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   433
          by (cases "Suc n' = n", simp_all add: min_def)
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   434
      } moreover {
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   435
        assume "n' = n"
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   436
        with "4.hyps"(16-17)[OF cnp np', of n]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   437
          "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   438
        have ?case
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   439
          apply (auto intro!: polyadd_normh)
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   440
          apply (simp_all add: min_def isnpolyh_mono[OF nn0])
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   441
          done
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   442
      }
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   443
      ultimately show ?case by arith
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   444
    next
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   445
      fix n0 n1 m
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   446
      assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   447
      and m: "m \<le> min n0 n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   448
      let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   449
      let ?d1 = "degreen ?cnp m"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   450
      let ?d2 = "degreen ?cnp' m"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   451
      let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   452
      have "n'<n \<or> n < n' \<or> n' = n" by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   453
      moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   454
      {assume "n' < n \<or> n < n'"
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   455
        with "4.hyps"(3,6,18) np np' m 
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   456
        have ?eq by auto }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   457
      moreover
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   458
      {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   459
        from "4.hyps"(16,18)[of n n' n]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   460
          "4.hyps"(13,14)[of n "Suc n'" n]
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   461
          np np' nn'
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   462
        have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   463
          "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   464
          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   465
          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   466
        {assume mn: "m = n" 
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   467
          from "4.hyps"(17,18)[OF norm(1,4), of n]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   468
            "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   469
          have degs:  "degreen (?cnp *\<^sub>p c') n = 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   470
            (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   471
            "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   472
          from degs norm
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   473
          have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   474
          hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   475
            by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   476
          have nmin: "n \<le> min n n" by (simp add: min_def)
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   477
          from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   478
          have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n = degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   479
          from "4.hyps"(16-18)[OF norm(1,4), of n]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   480
            "4.hyps"(13-15)[OF norm(1,2), of n]
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   481
            mn norm m nn' deg
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   482
          have ?eq by simp}
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   483
        moreover
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   484
        {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   485
          from nn' m np have max1: "m \<le> max n n"  by simp 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   486
          hence min1: "m \<le> min n n" by simp     
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   487
          hence min2: "m \<le> min n (Suc n)" by simp
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   488
          from "4.hyps"(16-18)[OF norm(1,4) min1]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   489
            "4.hyps"(13-15)[OF norm(1,2) min2]
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   490
            degreen_polyadd[OF norm(3,6) max1]
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   491
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   492
          have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m 
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   493
            \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   494
            using mn nn' np np' by simp
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   495
          with "4.hyps"(16-18)[OF norm(1,4) min1]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   496
            "4.hyps"(13-15)[OF norm(1,2) min2]
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   497
            degreen_0[OF norm(3) mn']
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   498
          have ?eq using nn' mn np np' by clarsimp}
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   499
        ultimately have ?eq by blast}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   500
      ultimately show ?eq by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   501
    { case (2 n0 n1)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   502
      hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1" 
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   503
        and m: "m \<le> min n0 n1" by simp_all
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   504
      hence mn: "m \<le> n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   505
      let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   506
      {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   507
        hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   508
        from "4.hyps"(16-18) [of n n n]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   509
          "4.hyps"(13-15)[of n "Suc n" n]
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   510
          np np' C(2) mn
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   511
        have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   512
          "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   513
          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   514
          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   515
          "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   516
            "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   517
          by (simp_all add: min_def)
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   518
            
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   519
          from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   520
          have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   521
            using norm by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   522
        from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   523
        have "False" by simp }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   524
      thus ?case using "4.hyps" by clarsimp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   525
qed auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   526
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   527
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
36349
39be26d1bc28 class division_ring_inverse_zero
haftmann
parents: 35416
diff changeset
   528
by(induct p q rule: polymul.induct, auto simp add: field_simps)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   529
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   530
lemma polymul_normh: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   531
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   532
  shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   533
  using polymul_properties(1)  by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   534
lemma polymul_eq0_iff: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   535
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   536
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p) "
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   537
  using polymul_properties(2)  by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   538
lemma polymul_degreen:  
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   539
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   540
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 else degreen p m + degreen q m)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   541
  using polymul_properties(3) by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   542
lemma polymul_norm:   
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   543
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   544
  shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul p q)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   545
  using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   546
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   547
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   548
  by (induct p arbitrary: n0 rule: headconst.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   549
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   550
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   551
  by (induct p arbitrary: n0, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   552
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   553
lemma monic_eqI: assumes np: "isnpolyh p n0" 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   554
  shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   555
  unfolding monic_def Let_def
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   556
proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   557
  let ?h = "headconst p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   558
  assume pz: "p \<noteq> 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   559
  {assume hz: "INum ?h = (0::'a)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   560
    from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   561
    from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   562
    with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   563
  thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   564
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   565
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   566
41404
aae9f912cca8 dropped duplicate unused lemmas;
krauss
parents: 41403
diff changeset
   567
text{* polyneg is a negation and preserves normal forms *}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   568
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   569
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   570
by (induct p rule: polyneg.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   571
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   572
lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   573
  by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   574
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   575
  by (induct p arbitrary: n0 rule: polyneg.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   576
lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   577
by (induct p rule: polyneg.induct, auto simp add: polyneg0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   578
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   579
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   580
  using isnpoly_def polyneg_normh by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   581
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   582
41404
aae9f912cca8 dropped duplicate unused lemmas;
krauss
parents: 41403
diff changeset
   583
text{* polysub is a substraction and preserves normal forms *}
aae9f912cca8 dropped duplicate unused lemmas;
krauss
parents: 41403
diff changeset
   584
41814
3848eb635eab modernized specification; curried
krauss
parents: 41813
diff changeset
   585
lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   586
by (simp add: polysub_def polyneg polyadd)
41814
3848eb635eab modernized specification; curried
krauss
parents: 41813
diff changeset
   587
lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   588
by (simp add: polysub_def polyneg_normh polyadd_normh)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   589
41814
3848eb635eab modernized specification; curried
krauss
parents: 41813
diff changeset
   590
lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   591
  using polyadd_norm polyneg_norm by (simp add: polysub_def) 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   592
lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
41814
3848eb635eab modernized specification; curried
krauss
parents: 41813
diff changeset
   593
  shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   594
unfolding polysub_def split_def fst_conv snd_conv
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   595
by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   596
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   597
lemma polysub_0: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   598
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   599
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   600
  unfolding polysub_def split_def fst_conv snd_conv
41763
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   601
  by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
   602
  (auto simp: Nsub0[simplified Nsub_def] Let_def)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   603
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   604
text{* polypow is a power function and preserves normal forms *}
41404
aae9f912cca8 dropped duplicate unused lemmas;
krauss
parents: 41403
diff changeset
   605
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   606
lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   607
proof(induct n rule: polypow.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   608
  case 1 thus ?case by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   609
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   610
  case (2 n)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   611
  let ?q = "polypow ((Suc n) div 2) p"
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   612
  let ?d = "polymul ?q ?q"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   613
  have "odd (Suc n) \<or> even (Suc n)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   614
  moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   615
  {assume odd: "odd (Suc n)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   616
    have th: "(Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat))))) = Suc n div 2 + Suc n div 2 + 1" by arith
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   617
    from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   618
    also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   619
      using "2.hyps" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   620
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   621
      apply (simp only: power_add power_one_right) by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   622
    also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   623
      by (simp only: th)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   624
    finally have ?case 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   625
    using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   626
  moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   627
  {assume even: "even (Suc n)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   628
    have th: "(Suc (Suc (0\<Colon>nat))) * (Suc n div Suc (Suc (0\<Colon>nat))) = Suc n div 2 + Suc n div 2" by arith
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   629
    from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   630
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   631
      using "2.hyps" apply (simp only: power_add) by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   632
    finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   633
  ultimately show ?case by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   634
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   635
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   636
lemma polypow_normh: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   637
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   638
  shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   639
proof (induct k arbitrary: n rule: polypow.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   640
  case (2 k n)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   641
  let ?q = "polypow (Suc k div 2) p"
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   642
  let ?d = "polymul ?q ?q"
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   643
  from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   644
  from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
   645
  from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   646
  from dn on show ?case by (simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   647
qed auto 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   648
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   649
lemma polypow_norm:   
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   650
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   651
  shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   652
  by (simp add: polypow_normh isnpoly_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   653
41404
aae9f912cca8 dropped duplicate unused lemmas;
krauss
parents: 41403
diff changeset
   654
text{* Finally the whole normalization *}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   655
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   656
lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   657
by (induct p rule:polynate.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   658
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   659
lemma polynate_norm[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   660
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   661
  shows "isnpoly (polynate p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   662
  by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   663
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   664
text{* shift1 *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   665
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   666
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   667
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   668
by (simp add: shift1_def polymul)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   669
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   670
lemma shift1_isnpoly: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   671
  assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   672
  using pn pnz by (simp add: shift1_def isnpoly_def )
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   673
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   674
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   675
  by (simp add: shift1_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   676
lemma funpow_shift1_isnpoly: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   677
  "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 36409
diff changeset
   678
  by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   679
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   680
lemma funpow_isnpolyh: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   681
  assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   682
  shows "isnpolyh (funpow k f p) n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   683
  using f np by (induct k arbitrary: p, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   684
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   685
lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   686
  by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   687
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   688
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   689
  using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   690
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   691
lemma funpow_shift1_1: 
50282
fe4d4bb9f4c2 more robust syntax that survives collapse of \<^isub> and \<^sub>;
wenzelm
parents: 49962
diff changeset
   692
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   693
  by (simp add: funpow_shift1)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   694
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   695
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
45129
1fce03e3e8ad tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents: 41842
diff changeset
   696
  by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   697
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   698
lemma behead:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   699
  assumes np: "isnpolyh p n"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   700
  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   701
  using np
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   702
proof (induct p arbitrary: n rule: behead.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   703
  case (1 c p n) hence pn: "isnpolyh p n" by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   704
  from 1(1)[OF pn] 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   705
  have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" . 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   706
  then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
36349
39be26d1bc28 class division_ring_inverse_zero
haftmann
parents: 35416
diff changeset
   707
    by (simp_all add: th[symmetric] field_simps power_Suc)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   708
qed (auto simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   709
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   710
lemma behead_isnpolyh:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   711
  assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   712
  using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   713
41404
aae9f912cca8 dropped duplicate unused lemmas;
krauss
parents: 41403
diff changeset
   714
subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   715
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 36409
diff changeset
   716
proof(induct p arbitrary: n rule: poly.induct, auto)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   717
  case (goal1 c n p n')
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   718
  hence "n = Suc (n - 1)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   719
  hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
   720
  with goal1(2) show ?case by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   721
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   722
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   723
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   724
by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   725
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   726
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   727
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   728
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   729
  apply (induct p arbitrary: n0, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   730
  apply (atomize)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   731
  apply (erule_tac x = "Suc nat" in allE)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   732
  apply auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   733
  done
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   734
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   735
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   736
 by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   737
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   738
lemma polybound0_I:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   739
  assumes nb: "polybound0 a"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   740
  shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   741
using nb
39246
9e58f0499f57 modernized primrec
haftmann
parents: 36409
diff changeset
   742
by (induct a rule: poly.induct) auto 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   743
lemma polysubst0_I:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   744
  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   745
  by (induct t) simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   746
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   747
lemma polysubst0_I':
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   748
  assumes nb: "polybound0 a"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   749
  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   750
  by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   751
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   752
lemma decrpoly: assumes nb: "polybound0 t"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   753
  shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   754
  using nb by (induct t rule: decrpoly.induct, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   755
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   756
lemma polysubst0_polybound0: assumes nb: "polybound0 t"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   757
  shows "polybound0 (polysubst0 t a)"
39246
9e58f0499f57 modernized primrec
haftmann
parents: 36409
diff changeset
   758
using nb by (induct a rule: poly.induct, auto)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   759
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   760
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   761
  by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   762
39246
9e58f0499f57 modernized primrec
haftmann
parents: 36409
diff changeset
   763
primrec maxindex :: "poly \<Rightarrow> nat" where
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   764
  "maxindex (Bound n) = n + 1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   765
| "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   766
| "maxindex (Add p q) = max (maxindex p) (maxindex q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   767
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   768
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   769
| "maxindex (Neg p) = maxindex p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   770
| "maxindex (Pw p n) = maxindex p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   771
| "maxindex (C x) = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   772
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   773
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   774
  "wf_bs bs p = (length bs \<ge> maxindex p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   775
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   776
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   777
proof(induct p rule: coefficients.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   778
  case (1 c p) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   779
  show ?case 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   780
  proof
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   781
    fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   782
    hence "x = c \<or> x \<in> set (coefficients p)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   783
    moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   784
    {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   785
    moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   786
    {assume H: "x \<in> set (coefficients p)" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   787
      from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   788
      with "1.hyps" H have "wf_bs bs x" by blast }
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   789
    ultimately  show "wf_bs bs x" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   790
  qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   791
qed simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   792
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   793
lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   794
by (induct p rule: coefficients.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   795
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   796
lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   797
  unfolding wf_bs_def by (induct p, auto simp add: nth_append)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   798
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   799
lemma take_maxindex_wf: assumes wf: "wf_bs bs p" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   800
  shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   801
proof-
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   802
  let ?ip = "maxindex p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   803
  let ?tbs = "take ?ip bs"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   804
  from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   805
  hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   806
  have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   807
  from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   808
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   809
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   810
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   811
  by (induct p, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   812
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   813
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   814
  unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   815
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   816
lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   817
  unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   818
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   819
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   820
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   821
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   822
by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   823
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   824
  by (induct p rule: coefficients.induct, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   825
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   826
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   827
lemma coefficients_head: "last (coefficients p) = head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   828
  by (induct p rule: coefficients.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   829
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   830
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   831
  unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   832
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   833
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   834
  apply (rule exI[where x="replicate (n - length xs) z"])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   835
  by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   836
lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   837
by (cases p, auto) (case_tac "nat", simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   838
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   839
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   840
  unfolding wf_bs_def 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   841
  apply (induct p q rule: polyadd.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   842
  apply (auto simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   843
  done
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   844
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   845
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
41811
7e338ccabff0 strengthened polymul.induct
krauss
parents: 41810
diff changeset
   846
  unfolding wf_bs_def 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   847
  apply (induct p q arbitrary: bs rule: polymul.induct) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   848
  apply (simp_all add: wf_bs_polyadd)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   849
  apply clarsimp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   850
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   851
  apply auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   852
  done
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   853
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   854
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   855
  unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   856
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   857
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   858
  unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   859
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   860
subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   861
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   862
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   863
definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   864
definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   865
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   866
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   867
proof (induct p arbitrary: n0 rule: coefficients.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   868
  case (1 c p n0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   869
  have cp: "isnpolyh (CN c 0 p) n0" by fact
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   870
  hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   871
    by (auto simp add: isnpolyh_mono[where n'=0])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   872
  from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   873
qed auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   874
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   875
lemma coefficients_isconst:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   876
  "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   877
  by (induct p arbitrary: n rule: coefficients.induct, 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   878
    auto simp add: isnpolyh_Suc_const)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   879
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   880
lemma polypoly_polypoly':
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   881
  assumes np: "isnpolyh p n0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   882
  shows "polypoly (x#bs) p = polypoly' bs p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   883
proof-
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   884
  let ?cf = "set (coefficients p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   885
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   886
  {fix q assume q: "q \<in> ?cf"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   887
    from q cn_norm have th: "isnpolyh q n0" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   888
    from coefficients_isconst[OF np] q have "isconstant q" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   889
    with isconstant_polybound0[OF th] have "polybound0 q" by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   890
  hence "\<forall>q \<in> ?cf. polybound0 q" ..
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   891
  hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   892
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   893
    by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   894
  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   895
  thus ?thesis unfolding polypoly_def polypoly'_def by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   896
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   897
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   898
lemma polypoly_poly:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   899
  assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   900
  using np 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   901
by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   902
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   903
lemma polypoly'_poly: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   904
  assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   905
  using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   906
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   907
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   908
lemma polypoly_poly_polybound0:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   909
  assumes np: "isnpolyh p n0" and nb: "polybound0 p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   910
  shows "polypoly bs p = [Ipoly bs p]"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   911
  using np nb unfolding polypoly_def 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   912
  by (cases p, auto, case_tac nat, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   913
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   914
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   915
  by (induct p rule: head.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   916
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   917
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   918
  by (cases p,auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   919
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   920
lemma head_eq_headn0: "head p = headn p 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   921
  by (induct p rule: head.induct, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   922
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   923
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   924
  by (simp add: head_eq_headn0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   925
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   926
lemma isnpolyh_zero_iff: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   927
  assumes nq: "isnpolyh p n0" and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0, field_inverse_zero, power})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   928
  shows "p = 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   929
using nq eq
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   930
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   931
  case less
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   932
  note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   933
  {assume nz: "maxindex p = 0"
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   934
    then obtain c where "p = C c" using np by (cases p, auto)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   935
    with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   936
  moreover
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   937
  {assume nz: "maxindex p \<noteq> 0"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   938
    let ?h = "head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   939
    let ?hd = "decrpoly ?h"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   940
    let ?ihd = "maxindex ?hd"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   941
    from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   942
      by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   943
    hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   944
    
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   945
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   946
    have mihn: "maxindex ?h \<le> maxindex p" by auto
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   947
    with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   948
    {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   949
      let ?ts = "take ?ihd bs"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   950
      let ?rs = "drop ?ihd bs"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   951
      have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   952
      have bs_ts_eq: "?ts@ ?rs = bs" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   953
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   954
      from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   955
      with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   956
      hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   957
      with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   958
      hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   959
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   960
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   961
      hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   962
      hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   963
        using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   964
      with coefficients_head[of p, symmetric]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   965
      have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   966
      from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   967
      with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   968
      with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   969
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   970
    
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 33268
diff changeset
   971
    from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   972
    hence "?h = 0\<^sub>p" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   973
    with head_nz[OF np] have "p = 0\<^sub>p" by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   974
  ultimately show "p = 0\<^sub>p" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   975
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   976
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   977
lemma isnpolyh_unique:  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   978
  assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   979
  shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0, field_inverse_zero, power})) \<longleftrightarrow>  p = q"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   980
proof(auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   981
  assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   982
  hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   983
  hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   984
    using wf_bs_polysub[where p=p and q=q] by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   985
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   986
  show "p = q" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   987
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   988
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   989
41404
aae9f912cca8 dropped duplicate unused lemmas;
krauss
parents: 41403
diff changeset
   990
text{* consequences of unicity on the algorithms for polynomial normalization *}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   991
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   992
lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   993
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   994
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   995
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   996
lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
50282
fe4d4bb9f4c2 more robust syntax that survives collapse of \<^isub> and \<^sub>;
wenzelm
parents: 49962
diff changeset
   997
lemma one_normh: "isnpolyh (1)\<^sub>p n" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   998
lemma polyadd_0[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   999
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1000
  and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1001
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1002
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1003
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1004
lemma polymul_1[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1005
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
50282
fe4d4bb9f4c2 more robust syntax that survives collapse of \<^isub> and \<^sub>;
wenzelm
parents: 49962
diff changeset
  1006
  and np: "isnpolyh p n0" shows "p *\<^sub>p (1)\<^sub>p = p" and "(1)\<^sub>p *\<^sub>p p = p"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1007
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1008
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1009
lemma polymul_0[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1010
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1011
  and np: "isnpolyh p n0" shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1012
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1013
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1014
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1015
lemma polymul_commute: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1016
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1017
  and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1018
  shows "p *\<^sub>p q = q *\<^sub>p p"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1019
using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a\<Colon>{field_char_0, field_inverse_zero, power}"] by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1020
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1021
declare polyneg_polyneg[simp]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1022
  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1023
lemma isnpolyh_polynate_id[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1024
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1025
  and np:"isnpolyh p n0" shows "polynate p = p"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1026
  using isnpolyh_unique[where ?'a= "'a::{field_char_0, field_inverse_zero}", OF polynate_norm[of p, unfolded isnpoly_def] np] polynate[where ?'a = "'a::{field_char_0, field_inverse_zero}"] by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1027
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1028
lemma polynate_idempotent[simp]: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1029
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1030
  shows "polynate (polynate p) = polynate p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1031
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1032
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1033
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1034
  unfolding poly_nate_def polypoly'_def ..
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1035
lemma poly_nate_poly: shows "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0, field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1036
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1037
  unfolding poly_nate_polypoly' by (auto intro: ext)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1038
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1039
subsection{* heads, degrees and all that *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1040
lemma degree_eq_degreen0: "degree p = degreen p 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1041
  by (induct p rule: degree.induct, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1042
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1043
lemma degree_polyneg: assumes n: "isnpolyh p n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1044
  shows "degree (polyneg p) = degree p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1045
  using n
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1046
  by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1047
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1048
lemma degree_polyadd:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1049
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1050
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1051
using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1052
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1053
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1054
lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1055
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1056
proof-
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1057
  from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1058
  from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1059
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1060
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1061
lemma degree_polysub_samehead: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1062
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1063
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1064
  and d: "degree p = degree q"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1065
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1066
unfolding polysub_def split_def fst_conv snd_conv
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1067
using np nq h d
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1068
proof(induct p q rule:polyadd.induct)
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
  1069
  case (1 c c') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1070
next
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
  1071
  case (2 c c' n' p') 
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
  1072
  from 2 have "degree (C c) = degree (CN c' n' p')" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1073
  hence nz:"n' > 0" by (cases n', auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1074
  hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
  1075
  with 2 show ?case by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1076
next
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
  1077
  case (3 c n p c') 
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
  1078
  hence "degree (C c') = degree (CN c n p)" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1079
  hence nz:"n > 0" by (cases n, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1080
  hence "head (CN c n p) = CN c n p" by (cases n, auto)
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
  1081
  with 3 show ?case by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1082
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1083
  case (4 c n p c' n' p')
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1084
  hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1085
    "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1086
  hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1087
  hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1088
    using H(1-2) degree_polyneg by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1089
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1090
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1091
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1092
  have "n = n' \<or> n < n' \<or> n > n'" by arith
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1093
  moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1094
  {assume nn': "n = n'"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1095
    have "n = 0 \<or> n >0" by arith
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
  1096
    moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1097
    moreover {assume nz: "n > 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1098
      with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
  1099
      hence ?case using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] polysub_same_0[OF c'nh, simplified polysub_def] using nn' 4 by (simp add: Let_def)}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1100
    ultimately have ?case by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1101
  moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1102
  {assume nn': "n < n'" hence n'p: "n' > 0" by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1103
    hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
  1104
    have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" using 4 nn' by (cases n', simp_all)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1105
    hence "n > 0" by (cases n, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1106
    hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1107
    from H(3) headcnp headcnp' nn' have ?case by auto}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1108
  moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1109
  {assume nn': "n > n'"  hence np: "n > 0" by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1110
    hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
  1111
    from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1112
    from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1113
    with degcnpeq have "n' > 0" by (cases n', simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1114
    hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1115
    from H(3) headcnp headcnp' nn' have ?case by auto}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1116
  ultimately show ?case  by blast
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
  1117
qed auto
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1118
 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1119
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1120
by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1121
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1122
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1123
proof(induct k arbitrary: n0 p)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1124
  case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
41807
ab5d2d81f9fb tuned proofs -- eliminated prems;
wenzelm
parents: 41763
diff changeset
  1125
  with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1126
    and "head (shift1 p) = head p" by (simp_all add: shift1_head) 
39246
9e58f0499f57 modernized primrec
haftmann
parents: 36409
diff changeset
  1127
  thus ?case by (simp add: funpow_swap1)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1128
qed auto  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1129
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1130
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1131
  by (simp add: shift1_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1132
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
46991
196f2d9406c4 tuned proofs;
wenzelm
parents: 45129
diff changeset
  1133
  by (induct k arbitrary: p) (auto simp add: shift1_degree)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1134
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1135
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
46991
196f2d9406c4 tuned proofs;
wenzelm
parents: 45129
diff changeset
  1136
  by (induct n arbitrary: p) (simp_all add: funpow.simps)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1137
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1138
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1139
  by (induct p arbitrary: n rule: degree.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1140
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1141
  by (induct p arbitrary: n rule: degreen.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1142
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1143
  by (induct p arbitrary: n rule: degree.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1144
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1145
  by (induct p rule: head.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1146
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1147
lemma polyadd_eq_const_degree: 
41812
d46c2908a838 recdef -> fun; curried
krauss
parents: 41811
diff changeset
  1148
  "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> \<Longrightarrow> degree p = degree q" 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1149
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1150
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1151
lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1152
  and deg: "degree p \<noteq> degree q"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1153
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1154
using np nq deg
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1155
apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1156
apply (case_tac n', simp, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1157
apply (case_tac n, simp, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1158
apply (case_tac n, case_tac n', simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1159
apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
41763
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
  1160
apply (auto simp add: polyadd_eq_const_degree)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
  1161
apply (metis head_nz)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
  1162
apply (metis head_nz)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
  1163
apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
8ce56536fda7 strengthened induction rule;
krauss
parents: 41413
diff changeset
  1164
by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1165
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1166
lemma polymul_head_polyeq: 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
  1167
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1168
  shows "\<lbrakk>isnpolyh p n0; isnpolyh q n1 ; p \<noteq> 0\<^sub>p ; q \<noteq> 0\<^sub>p \<rbrakk> \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1169
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1170
  case (2 c c' n' p' n0 n1)
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1171
  hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"  by (simp_all add: head_isnpolyh)
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
  1172
  thus ?case using 2 by (cases n', auto) 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1173
next 
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1174
  case (3 c n p c' n0 n1) 
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1175
  hence "isnpolyh (head (CN c n p)) n0" "isnormNum c'"  by (simp_all add: head_isnpolyh)
41815
9a0cacbcd825 eliminated global prems
krauss
parents: 41814
diff changeset
  1176
  thus ?case using 3 by (cases n, auto)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1177
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1178
  case (4 c n p c' n' p' n0 n1)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1179
  hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1180
    "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1181
    by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1182
  have "n < n' \<or> n' < n \<or> n = n'" by arith
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1183
  moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1184
  {assume nn': "n < n'" hence ?case 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1185
      using norm 
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1186
    "4.hyps"(2)[OF norm(1,6)]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1187
    "4.hyps"(1)[OF norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1188
  moreover {assume nn': "n'< n"
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1189
    hence ?case using norm "4.hyps"(6) [OF norm(5,3)]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1190
      "4.hyps"(5)[OF norm(5,4)] 
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1191
      by (simp,cases n',simp,cases n,auto)}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1192
  moreover {assume nn': "n' = n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1193
    from nn' polymul_normh[OF norm(5,4)] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1194
    have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1195
    from nn' polymul_normh[OF norm(5,3)] norm 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1196
    have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1197
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1198
    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1199
    from polyadd_normh[OF ncnpc' ncnpp0'] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1200
    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" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1201
      by (simp add: min_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1202
    {assume np: "n > 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1203
      with nn' head_isnpolyh_Suc'[OF np nth]
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
  1204
        head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1205
      have ?case by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1206
    moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1207
    {moreover assume nz: "n = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1208
      from polymul_degreen[OF norm(5,4), where m="0"]
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
  1209
        polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1210
      norm(5,6) degree_npolyhCN[OF norm(6)]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1211
    have dth:"degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1212
    hence dth':"degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1213
    from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
41813
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1214
    have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
4eb43410d2fa recdef -> fun; curried
krauss
parents: 41812
diff changeset
  1215
        "4.hyps"(5)[OF norm(5,4)] nn' nz by simp }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1216
    ultimately have ?case by (cases n) auto}