author | nipkow |
Tue, 23 Feb 2016 16:25:08 +0100 | |
changeset 62390 | 842917225d56 |
parent 61586 | 5197a2ecb658 |
child 67123 | 3fe40ff1b921 |
permissions | -rw-r--r-- |
33154 | 1 |
(* Title: HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy |
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Author: Amine Chaieb |
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*) |
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section \<open>Implementation and verification of multivariate polynomials\<close> |
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theory Reflected_Multivariate_Polynomial |
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imports Complex_Main Rat_Pair Polynomial_List |
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begin |
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subsection \<open>Datatype of polynomial expressions\<close> |
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|
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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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||
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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\<open>Boundedness, substitution and all that\<close> |
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primrec polysize:: "poly \<Rightarrow> nat" |
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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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|
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primrec polybound0:: "poly \<Rightarrow> bool" \<comment> \<open>a poly is INDEPENDENT of Bound 0\<close> |
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where |
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"polybound0 (C c) \<longleftrightarrow> True" |
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| "polybound0 (Bound n) \<longleftrightarrow> n > 0" |
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| "polybound0 (Neg a) \<longleftrightarrow> polybound0 a" |
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| "polybound0 (Add a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b" |
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| "polybound0 (Sub a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b" |
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| "polybound0 (Mul a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b" |
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| "polybound0 (Pw p n) \<longleftrightarrow> polybound0 p" |
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| "polybound0 (CN c n p) \<longleftrightarrow> n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p" |
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|
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primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" \<comment> \<open>substitute a poly into a poly for Bound 0\<close> |
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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) = |
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(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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|
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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 \<open>Degrees and heads and coefficients\<close> |
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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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|
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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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||
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text \<open>More general notions of degree and head.\<close> |
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||
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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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|
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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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|
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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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|
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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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|
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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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||
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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 \<open>Operations for normalization\<close> |
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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>N c')" |
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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 |
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let |
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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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||
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|
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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)" |
139 |
| "polyneg a = Neg a" |
|
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|
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definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60) |
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where "p -\<^sub>p q = polyadd p (polyneg q)" |
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|
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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>N c')" |
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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'))" |
52803 | 149 |
| "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')))" |
52803 | 151 |
| "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 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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|
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declare if_cong[fundef_cong] |
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declare let_cong[fundef_cong] |
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||
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fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly" |
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where |
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"polypow 0 = (\<lambda>p. (1)\<^sub>p)" |
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| "polypow n = |
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(\<lambda>p. |
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let |
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q = polypow (n div 2) p; |
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d = polymul q q |
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in if even n then d else polymul p d)" |
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|
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abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60) |
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where "a ^\<^sub>p k \<equiv> polypow k a" |
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function polynate :: "poly \<Rightarrow> poly" |
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where |
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"polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p" |
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| "polynate (Add p q) = polynate p +\<^sub>p polynate q" |
177 |
| "polynate (Sub p q) = polynate p -\<^sub>p polynate q" |
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| "polynate (Mul p q) = polynate p *\<^sub>p polynate q" |
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| "polynate (Neg p) = ~\<^sub>p (polynate p)" |
|
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| "polynate (Pw p n) = polynate p ^\<^sub>p n" |
|
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| "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))" |
182 |
| "polynate (C c) = C (normNum c)" |
|
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by pat_completeness auto |
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termination by (relation "measure polysize") auto |
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|
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fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" |
187 |
where |
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"poly_cmul y (C x) = C (y *\<^sub>N x)" |
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| "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)" |
|
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| "poly_cmul y p = C y *\<^sub>p p" |
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||
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definition monic :: "poly \<Rightarrow> poly \<times> bool" |
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where |
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"monic p = |
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195 |
(let h = headconst p |
|
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in if h = 0\<^sub>N then (p, False) else (C (Ninv h) *\<^sub>p p, 0>\<^sub>N h))" |
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|
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subsection \<open>Pseudo-division\<close> |
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|
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definition shift1 :: "poly \<Rightarrow> poly" |
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where "shift1 p = CN 0\<^sub>p 0 p" |
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|
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abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" |
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where "funpow \<equiv> compow" |
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partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly" |
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where |
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"polydivide_aux a n p k s = |
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(if s = 0\<^sub>p then (k, s) |
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else |
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let |
213 |
b = head s; |
|
214 |
m = degree s |
|
215 |
in |
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216 |
if m < n then (k,s) |
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else |
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let p' = funpow (m - n) shift1 p |
|
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in |
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220 |
if a = b then polydivide_aux a n p k (s -\<^sub>p p') |
|
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else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))" |
|
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|
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definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> nat \<times> poly" |
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where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s" |
|
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|
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fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" |
227 |
where |
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"poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)" |
229 |
| "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p" |
|
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||
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fun poly_deriv :: "poly \<Rightarrow> poly" |
232 |
where |
|
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"poly_deriv (CN c 0 p) = poly_deriv_aux 1 p" |
234 |
| "poly_deriv p = 0\<^sub>p" |
|
235 |
||
52658 | 236 |
|
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subsection \<open>Semantics of the polynomial representation\<close> |
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239 |
primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,field,power}" |
56000 | 240 |
where |
33154 | 241 |
"Ipoly bs (C c) = INum c" |
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| "Ipoly bs (Bound n) = bs!n" |
243 |
| "Ipoly bs (Neg a) = - Ipoly bs a" |
|
244 |
| "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b" |
|
245 |
| "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b" |
|
246 |
| "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b" |
|
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| "Ipoly bs (Pw t n) = Ipoly bs t ^ n" |
248 |
| "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p" |
|
39246 | 249 |
|
60698 | 250 |
abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field,power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>") |
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where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p" |
33154 | 252 |
|
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lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i" |
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by (simp add: INum_def) |
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|
52803 | 256 |
lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j" |
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by (simp add: INum_def) |
258 |
||
259 |
lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat |
|
260 |
||
52658 | 261 |
|
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subsection \<open>Normal form and normalization\<close> |
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|
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fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool" |
265 |
where |
|
33154 | 266 |
"isnpolyh (C c) = (\<lambda>k. isnormNum c)" |
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| "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)" |
41808 | 268 |
| "isnpolyh p = (\<lambda>k. False)" |
33154 | 269 |
|
56000 | 270 |
lemma isnpolyh_mono: "n' \<le> n \<Longrightarrow> isnpolyh p n \<Longrightarrow> isnpolyh p n'" |
52658 | 271 |
by (induct p rule: isnpolyh.induct) auto |
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|
52658 | 273 |
definition isnpoly :: "poly \<Rightarrow> bool" |
56000 | 274 |
where "isnpoly p = isnpolyh p 0" |
33154 | 275 |
|
60698 | 276 |
text \<open>polyadd preserves normal forms\<close> |
33154 | 277 |
|
56000 | 278 |
lemma polyadd_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)" |
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proof (induct p q arbitrary: n0 n1 rule: polyadd.induct) |
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case (2 ab c' n' p' n0 n1) |
56009 | 281 |
from 2 have th1: "isnpolyh (C ab) (Suc n')" |
282 |
by simp |
|
60698 | 283 |
from 2(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" |
56009 | 284 |
by simp_all |
285 |
with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" |
|
286 |
by simp |
|
287 |
with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" |
|
288 |
by simp |
|
289 |
from nplen1 have n01len1: "min n0 n1 \<le> n'" |
|
290 |
by simp |
|
291 |
then show ?case using 2 th3 |
|
292 |
by simp |
|
33154 | 293 |
next |
41812 | 294 |
case (3 c' n' p' ab n1 n0) |
56009 | 295 |
from 3 have th1: "isnpolyh (C ab) (Suc n')" |
296 |
by simp |
|
297 |
from 3(2) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" |
|
298 |
by simp_all |
|
299 |
with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" |
|
300 |
by simp |
|
301 |
with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" |
|
302 |
by simp |
|
303 |
from nplen1 have n01len1: "min n0 n1 \<le> n'" |
|
304 |
by simp |
|
305 |
then show ?case using 3 th3 |
|
306 |
by simp |
|
33154 | 307 |
next |
308 |
case (4 c n p c' n' p' n0 n1) |
|
56009 | 309 |
then have nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" |
310 |
by simp_all |
|
311 |
from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" |
|
312 |
by simp_all |
|
313 |
from 4 have ngen0: "n \<ge> n0" |
|
314 |
by simp |
|
315 |
from 4 have n'gen1: "n' \<ge> n1" |
|
316 |
by simp |
|
60698 | 317 |
consider (eq) "n = n'" | (lt) "n < n'" | (gt) "n > n'" |
318 |
by arith |
|
319 |
then show ?case |
|
320 |
proof cases |
|
321 |
case eq |
|
52803 | 322 |
with "4.hyps"(3)[OF nc nc'] |
56009 | 323 |
have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" |
324 |
by auto |
|
325 |
then have ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)" |
|
326 |
using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 |
|
327 |
by auto |
|
328 |
from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" |
|
329 |
by simp |
|
330 |
have minle: "min n0 n1 \<le> n'" |
|
331 |
using ngen0 n'gen1 eq by simp |
|
60698 | 332 |
from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' show ?thesis |
56009 | 333 |
by (simp add: Let_def) |
60698 | 334 |
next |
335 |
case lt |
|
56009 | 336 |
have "min n0 n1 \<le> n0" |
337 |
by simp |
|
338 |
with 4 lt have th1:"min n0 n1 \<le> n" |
|
339 |
by auto |
|
340 |
from 4 have th21: "isnpolyh c (Suc n)" |
|
341 |
by simp |
|
342 |
from 4 have th22: "isnpolyh (CN c' n' p') n'" |
|
343 |
by simp |
|
344 |
from lt have th23: "min (Suc n) n' = Suc n" |
|
345 |
by arith |
|
346 |
from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" |
|
347 |
using th23 by simp |
|
60698 | 348 |
with 4 lt th1 show ?thesis |
56009 | 349 |
by simp |
60698 | 350 |
next |
351 |
case gt |
|
56009 | 352 |
then have gt': "n' < n \<and> \<not> n < n'" |
353 |
by simp |
|
354 |
have "min n0 n1 \<le> n1" |
|
355 |
by simp |
|
356 |
with 4 gt have th1: "min n0 n1 \<le> n'" |
|
357 |
by auto |
|
358 |
from 4 have th21: "isnpolyh c' (Suc n')" |
|
359 |
by simp_all |
|
360 |
from 4 have th22: "isnpolyh (CN c n p) n" |
|
361 |
by simp |
|
362 |
from gt have th23: "min n (Suc n') = Suc n'" |
|
363 |
by arith |
|
364 |
from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')" |
|
365 |
using th23 by simp |
|
60698 | 366 |
with 4 gt th1 show ?thesis |
56009 | 367 |
by simp |
60698 | 368 |
qed |
33154 | 369 |
qed auto |
370 |
||
41812 | 371 |
lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q" |
52658 | 372 |
by (induct p q rule: polyadd.induct) |
58776
95e58e04e534
use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
hoelzl
parents:
58710
diff
changeset
|
373 |
(auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left_NO_MATCH) |
33154 | 374 |
|
56009 | 375 |
lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)" |
60698 | 376 |
using polyadd_normh[of p 0 q 0] isnpoly_def by simp |
33154 | 377 |
|
60698 | 378 |
text \<open>The degree of addition and other general lemmas needed for the normal form of polymul.\<close> |
33154 | 379 |
|
52803 | 380 |
lemma polyadd_different_degreen: |
56009 | 381 |
assumes "isnpolyh p n0" |
382 |
and "isnpolyh q n1" |
|
383 |
and "degreen p m \<noteq> degreen q m" |
|
384 |
and "m \<le> min n0 n1" |
|
385 |
shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)" |
|
386 |
using assms |
|
33154 | 387 |
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct) |
388 |
case (4 c n p c' n' p' m n0 n1) |
|
60698 | 389 |
show ?case |
390 |
proof (cases "n = n'") |
|
391 |
case True |
|
392 |
with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7) |
|
41763 | 393 |
show ?thesis by (auto simp: Let_def) |
394 |
next |
|
60698 | 395 |
case False |
41763 | 396 |
with 4 show ?thesis by auto |
397 |
qed |
|
398 |
qed auto |
|
33154 | 399 |
|
56009 | 400 |
lemma headnz[simp]: "isnpolyh p n \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> headn p m \<noteq> 0\<^sub>p" |
52658 | 401 |
by (induct p arbitrary: n rule: headn.induct) auto |
56009 | 402 |
|
33154 | 403 |
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0" |
52658 | 404 |
by (induct p arbitrary: n rule: degree.induct) auto |
56009 | 405 |
|
33154 | 406 |
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0" |
52658 | 407 |
by (induct p arbitrary: n rule: degreen.induct) auto |
33154 | 408 |
|
409 |
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0" |
|
52658 | 410 |
by (induct p arbitrary: n rule: degree.induct) auto |
33154 | 411 |
|
412 |
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0" |
|
413 |
using degree_isnpolyh_Suc by auto |
|
56009 | 414 |
|
33154 | 415 |
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0" |
416 |
using degreen_0 by auto |
|
417 |
||
418 |
||
419 |
lemma degreen_polyadd: |
|
56009 | 420 |
assumes np: "isnpolyh p n0" |
421 |
and nq: "isnpolyh q n1" |
|
422 |
and m: "m \<le> max n0 n1" |
|
33154 | 423 |
shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)" |
424 |
using np nq m |
|
425 |
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct) |
|
52803 | 426 |
case (2 c c' n' p' n0 n1) |
56009 | 427 |
then show ?case |
428 |
by (cases n') simp_all |
|
33154 | 429 |
next |
52803 | 430 |
case (3 c n p c' n0 n1) |
56009 | 431 |
then show ?case |
432 |
by (cases n) auto |
|
33154 | 433 |
next |
52803 | 434 |
case (4 c n p c' n' p' n0 n1 m) |
60698 | 435 |
show ?case |
436 |
proof (cases "n = n'") |
|
437 |
case True |
|
438 |
with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7) |
|
41763 | 439 |
show ?thesis by (auto simp: Let_def) |
60698 | 440 |
next |
441 |
case False |
|
442 |
then show ?thesis by simp |
|
443 |
qed |
|
33154 | 444 |
qed auto |
445 |
||
56009 | 446 |
lemma polyadd_eq_const_degreen: |
447 |
assumes "isnpolyh p n0" |
|
448 |
and "isnpolyh q n1" |
|
449 |
and "polyadd p q = C c" |
|
450 |
shows "degreen p m = degreen q m" |
|
451 |
using assms |
|
33154 | 452 |
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct) |
52803 | 453 |
case (4 c n p c' n' p' m n0 n1 x) |
60698 | 454 |
consider "n = n'" | "n > n' \<or> n < n'" by arith |
455 |
then show ?case |
|
456 |
proof cases |
|
457 |
case 1 |
|
458 |
with 4 show ?thesis |
|
459 |
by (cases "p +\<^sub>p p' = 0\<^sub>p") (auto simp add: Let_def) |
|
460 |
next |
|
461 |
case 2 |
|
462 |
with 4 show ?thesis by auto |
|
463 |
qed |
|
33154 | 464 |
qed simp_all |
465 |
||
466 |
lemma polymul_properties: |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
467 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
52658 | 468 |
and np: "isnpolyh p n0" |
469 |
and nq: "isnpolyh q n1" |
|
470 |
and m: "m \<le> min n0 n1" |
|
52803 | 471 |
shows "isnpolyh (p *\<^sub>p q) (min n0 n1)" |
56009 | 472 |
and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p" |
473 |
and "degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)" |
|
33154 | 474 |
using np nq m |
52658 | 475 |
proof (induct p q arbitrary: n0 n1 m rule: polymul.induct) |
52803 | 476 |
case (2 c c' n' p') |
56009 | 477 |
{ |
478 |
case (1 n0 n1) |
|
479 |
with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n'] |
|
41811 | 480 |
show ?case by (auto simp add: min_def) |
33154 | 481 |
next |
56009 | 482 |
case (2 n0 n1) |
483 |
then show ?case by auto |
|
33154 | 484 |
next |
56009 | 485 |
case (3 n0 n1) |
60698 | 486 |
then show ?case using "2.hyps" by auto |
56009 | 487 |
} |
33154 | 488 |
next |
41813 | 489 |
case (3 c n p c') |
56009 | 490 |
{ |
491 |
case (1 n0 n1) |
|
492 |
with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n] |
|
41811 | 493 |
show ?case by (auto simp add: min_def) |
33154 | 494 |
next |
56009 | 495 |
case (2 n0 n1) |
496 |
then show ?case by auto |
|
33154 | 497 |
next |
56009 | 498 |
case (3 n0 n1) |
499 |
then show ?case using "3.hyps" by auto |
|
500 |
} |
|
33154 | 501 |
next |
502 |
case (4 c n p c' n' p') |
|
503 |
let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'" |
|
56009 | 504 |
{ |
505 |
case (1 n0 n1) |
|
506 |
then have cnp: "isnpolyh ?cnp n" |
|
507 |
and cnp': "isnpolyh ?cnp' n'" |
|
508 |
and np: "isnpolyh p n" |
|
509 |
and nc: "isnpolyh c (Suc n)" |
|
510 |
and np': "isnpolyh p' n'" |
|
511 |
and nc': "isnpolyh c' (Suc n')" |
|
512 |
and nn0: "n \<ge> n0" |
|
513 |
and nn1: "n' \<ge> n1" |
|
514 |
by simp_all |
|
41811 | 515 |
{ |
56009 | 516 |
assume "n < n'" |
517 |
with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp |
|
518 |
have ?case by (simp add: min_def) |
|
519 |
} moreover { |
|
520 |
assume "n' < n" |
|
521 |
with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp' |
|
522 |
have ?case by (cases "Suc n' = n") (simp_all add: min_def) |
|
523 |
} moreover { |
|
524 |
assume "n' = n" |
|
525 |
with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0 |
|
526 |
have ?case |
|
527 |
apply (auto intro!: polyadd_normh) |
|
528 |
apply (simp_all add: min_def isnpolyh_mono[OF nn0]) |
|
529 |
done |
|
530 |
} |
|
531 |
ultimately show ?case by arith |
|
532 |
next |
|
533 |
fix n0 n1 m |
|
534 |
assume np: "isnpolyh ?cnp n0" |
|
535 |
assume np':"isnpolyh ?cnp' n1" |
|
536 |
assume m: "m \<le> min n0 n1" |
|
537 |
let ?d = "degreen (?cnp *\<^sub>p ?cnp') m" |
|
538 |
let ?d1 = "degreen ?cnp m" |
|
539 |
let ?d2 = "degreen ?cnp' m" |
|
540 |
let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0 else ?d1 + ?d2)" |
|
56043 | 541 |
have "n' < n \<or> n < n' \<or> n' = n" by auto |
56009 | 542 |
moreover |
543 |
{ |
|
544 |
assume "n' < n \<or> n < n'" |
|
545 |
with "4.hyps"(3,6,18) np np' m have ?eq |
|
546 |
by auto |
|
547 |
} |
|
548 |
moreover |
|
549 |
{ |
|
550 |
assume nn': "n' = n" |
|
551 |
then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith |
|
552 |
from "4.hyps"(16,18)[of n n' n] |
|
553 |
"4.hyps"(13,14)[of n "Suc n'" n] |
|
554 |
np np' nn' |
|
56043 | 555 |
have norm: |
556 |
"isnpolyh ?cnp n" |
|
557 |
"isnpolyh c' (Suc n)" |
|
558 |
"isnpolyh (?cnp *\<^sub>p c') n" |
|
559 |
"isnpolyh p' n" |
|
560 |
"isnpolyh (?cnp *\<^sub>p p') n" |
|
561 |
"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" |
|
562 |
"?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p" |
|
563 |
"?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" |
|
564 |
by (auto simp add: min_def) |
|
56009 | 565 |
{ |
566 |
assume mn: "m = n" |
|
567 |
from "4.hyps"(17,18)[OF norm(1,4), of n] |
|
568 |
"4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn |
|
569 |
have degs: |
|
570 |
"degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)" |
|
571 |
"degreen (?cnp *\<^sub>p p') n = ?d1 + degreen p' n" |
|
572 |
by (simp_all add: min_def) |
|
573 |
from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" |
|
574 |
by simp |
|
575 |
then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" |
|
576 |
by simp |
|
577 |
have nmin: "n \<le> min n n" |
|
578 |
by (simp add: min_def) |
|
579 |
from polyadd_different_degreen[OF norm(3,6) neq nmin] th1 |
|
580 |
have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n = |
|
581 |
degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" |
|
582 |
by simp |
|
583 |
from "4.hyps"(16-18)[OF norm(1,4), of n] |
|
584 |
"4.hyps"(13-15)[OF norm(1,2), of n] |
|
585 |
mn norm m nn' deg |
|
586 |
have ?eq by simp |
|
41811 | 587 |
} |
33154 | 588 |
moreover |
56009 | 589 |
{ |
590 |
assume mn: "m \<noteq> n" |
|
591 |
then have mn': "m < n" |
|
592 |
using m np by auto |
|
593 |
from nn' m np have max1: "m \<le> max n n" |
|
594 |
by simp |
|
595 |
then have min1: "m \<le> min n n" |
|
596 |
by simp |
|
597 |
then have min2: "m \<le> min n (Suc n)" |
|
598 |
by simp |
|
599 |
from "4.hyps"(16-18)[OF norm(1,4) min1] |
|
600 |
"4.hyps"(13-15)[OF norm(1,2) min2] |
|
601 |
degreen_polyadd[OF norm(3,6) max1] |
|
602 |
have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le> |
|
603 |
max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)" |
|
604 |
using mn nn' np np' by simp |
|
605 |
with "4.hyps"(16-18)[OF norm(1,4) min1] |
|
606 |
"4.hyps"(13-15)[OF norm(1,2) min2] |
|
607 |
degreen_0[OF norm(3) mn'] |
|
608 |
have ?eq using nn' mn np np' by clarsimp |
|
609 |
} |
|
610 |
ultimately have ?eq by blast |
|
611 |
} |
|
612 |
ultimately show ?eq by blast |
|
613 |
} |
|
614 |
{ |
|
615 |
case (2 n0 n1) |
|
616 |
then have np: "isnpolyh ?cnp n0" |
|
617 |
and np': "isnpolyh ?cnp' n1" |
|
56043 | 618 |
and m: "m \<le> min n0 n1" |
619 |
by simp_all |
|
56009 | 620 |
then have mn: "m \<le> n" by simp |
621 |
let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')" |
|
622 |
{ |
|
623 |
assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n" |
|
624 |
then have nn: "\<not> n' < n \<and> \<not> n < n'" |
|
625 |
by simp |
|
626 |
from "4.hyps"(16-18) [of n n n] |
|
627 |
"4.hyps"(13-15)[of n "Suc n" n] |
|
628 |
np np' C(2) mn |
|
629 |
have norm: |
|
630 |
"isnpolyh ?cnp n" |
|
631 |
"isnpolyh c' (Suc n)" |
|
632 |
"isnpolyh (?cnp *\<^sub>p c') n" |
|
633 |
"isnpolyh p' n" |
|
634 |
"isnpolyh (?cnp *\<^sub>p p') n" |
|
635 |
"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" |
|
636 |
"?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p" |
|
637 |
"?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" |
|
638 |
"degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)" |
|
639 |
"degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n" |
|
640 |
by (simp_all add: min_def) |
|
641 |
from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" |
|
642 |
by simp |
|
643 |
have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" |
|
644 |
using norm by simp |
|
645 |
from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq |
|
646 |
have False by simp |
|
647 |
} |
|
648 |
then show ?case using "4.hyps" by clarsimp |
|
649 |
} |
|
33154 | 650 |
qed auto |
651 |
||
56009 | 652 |
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q" |
52658 | 653 |
by (induct p q rule: polymul.induct) (auto simp add: field_simps) |
33154 | 654 |
|
52803 | 655 |
lemma polymul_normh: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
656 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56009 | 657 |
shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)" |
52803 | 658 |
using polymul_properties(1) by blast |
52658 | 659 |
|
52803 | 660 |
lemma polymul_eq0_iff: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
661 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56009 | 662 |
shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p" |
52803 | 663 |
using polymul_properties(2) by blast |
52658 | 664 |
|
56207 | 665 |
lemma polymul_degreen: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
666 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56009 | 667 |
shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow> |
668 |
degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)" |
|
56207 | 669 |
by (fact polymul_properties(3)) |
52658 | 670 |
|
52803 | 671 |
lemma polymul_norm: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
672 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56009 | 673 |
shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)" |
33154 | 674 |
using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp |
675 |
||
676 |
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p" |
|
52658 | 677 |
by (induct p arbitrary: n0 rule: headconst.induct) auto |
33154 | 678 |
|
679 |
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)" |
|
52658 | 680 |
by (induct p arbitrary: n0) auto |
33154 | 681 |
|
52658 | 682 |
lemma monic_eqI: |
52803 | 683 |
assumes np: "isnpolyh p n0" |
52658 | 684 |
shows "INum (headconst p) * Ipoly bs (fst (monic p)) = |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
685 |
(Ipoly bs p ::'a::{field_char_0,field, power})" |
33154 | 686 |
unfolding monic_def Let_def |
52658 | 687 |
proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np]) |
33154 | 688 |
let ?h = "headconst p" |
689 |
assume pz: "p \<noteq> 0\<^sub>p" |
|
56000 | 690 |
{ |
691 |
assume hz: "INum ?h = (0::'a)" |
|
56043 | 692 |
from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" |
693 |
by simp_all |
|
694 |
from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" |
|
695 |
by simp |
|
696 |
with headconst_zero[OF np] have "p = 0\<^sub>p" |
|
697 |
by blast |
|
698 |
with pz have False |
|
699 |
by blast |
|
700 |
} |
|
701 |
then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" |
|
702 |
by blast |
|
33154 | 703 |
qed |
704 |
||
705 |
||
60698 | 706 |
text \<open>polyneg is a negation and preserves normal forms\<close> |
33154 | 707 |
|
708 |
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p" |
|
52658 | 709 |
by (induct p rule: polyneg.induct) auto |
33154 | 710 |
|
56009 | 711 |
lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" |
52658 | 712 |
by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def) |
56009 | 713 |
|
33154 | 714 |
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p" |
52658 | 715 |
by (induct p arbitrary: n0 rule: polyneg.induct) auto |
56009 | 716 |
|
717 |
lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n" |
|
718 |
by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0) |
|
33154 | 719 |
|
720 |
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)" |
|
721 |
using isnpoly_def polyneg_normh by simp |
|
722 |
||
723 |
||
60698 | 724 |
text \<open>polysub is a substraction and preserves normal forms\<close> |
41404 | 725 |
|
56009 | 726 |
lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q" |
52658 | 727 |
by (simp add: polysub_def) |
56009 | 728 |
|
729 |
lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)" |
|
52658 | 730 |
by (simp add: polysub_def polyneg_normh polyadd_normh) |
33154 | 731 |
|
56009 | 732 |
lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)" |
52803 | 733 |
using polyadd_norm polyneg_norm by (simp add: polysub_def) |
56009 | 734 |
|
52658 | 735 |
lemma polysub_same_0[simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
736 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
41814 | 737 |
shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p" |
52658 | 738 |
unfolding polysub_def split_def fst_conv snd_conv |
739 |
by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def]) |
|
33154 | 740 |
|
52803 | 741 |
lemma polysub_0: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
742 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56009 | 743 |
shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q" |
33154 | 744 |
unfolding polysub_def split_def fst_conv snd_conv |
41763 | 745 |
by (induct p q arbitrary: n0 n1 rule:polyadd.induct) |
52658 | 746 |
(auto simp: Nsub0[simplified Nsub_def] Let_def) |
33154 | 747 |
|
60698 | 748 |
text \<open>polypow is a power function and preserves normal forms\<close> |
41404 | 749 |
|
56009 | 750 |
lemma polypow[simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
751 |
"Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field}) ^ n" |
52658 | 752 |
proof (induct n rule: polypow.induct) |
753 |
case 1 |
|
56043 | 754 |
then show ?case |
755 |
by simp |
|
33154 | 756 |
next |
757 |
case (2 n) |
|
758 |
let ?q = "polypow ((Suc n) div 2) p" |
|
41813 | 759 |
let ?d = "polymul ?q ?q" |
56043 | 760 |
have "odd (Suc n) \<or> even (Suc n)" |
761 |
by simp |
|
52803 | 762 |
moreover |
56043 | 763 |
{ |
764 |
assume odd: "odd (Suc n)" |
|
56000 | 765 |
have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1" |
52658 | 766 |
by arith |
56043 | 767 |
from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" |
768 |
by (simp add: Let_def) |
|
769 |
also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)" |
|
33154 | 770 |
using "2.hyps" by simp |
771 |
also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)" |
|
52658 | 772 |
by (simp only: power_add power_one_right) simp |
56000 | 773 |
also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))" |
33154 | 774 |
by (simp only: th) |
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58310
diff
changeset
|
775 |
finally have ?case unfolding numeral_2_eq_2 [symmetric] |
58834 | 776 |
using odd_two_times_div_two_nat [OF odd] by simp |
56043 | 777 |
} |
52803 | 778 |
moreover |
56043 | 779 |
{ |
780 |
assume even: "even (Suc n)" |
|
781 |
from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" |
|
782 |
by (simp add: Let_def) |
|
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58310
diff
changeset
|
783 |
also have "\<dots> = (Ipoly bs p) ^ (2 * (Suc n div 2))" |
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58310
diff
changeset
|
784 |
using "2.hyps" by (simp only: mult_2 power_add) simp |
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58310
diff
changeset
|
785 |
finally have ?case using even_two_times_div_two [OF even] |
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58310
diff
changeset
|
786 |
by simp |
56043 | 787 |
} |
33154 | 788 |
ultimately show ?case by blast |
789 |
qed |
|
790 |
||
52803 | 791 |
lemma polypow_normh: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
792 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
33154 | 793 |
shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n" |
794 |
proof (induct k arbitrary: n rule: polypow.induct) |
|
56043 | 795 |
case 1 |
796 |
then show ?case by auto |
|
797 |
next |
|
33154 | 798 |
case (2 k n) |
799 |
let ?q = "polypow (Suc k div 2) p" |
|
41813 | 800 |
let ?d = "polymul ?q ?q" |
56043 | 801 |
from 2 have th1: "isnpolyh ?q n" and th2: "isnpolyh p n" |
802 |
by blast+ |
|
803 |
from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" |
|
804 |
by simp |
|
805 |
from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" |
|
806 |
by simp |
|
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58310
diff
changeset
|
807 |
from dn on show ?case by (simp, unfold Let_def) auto |
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58310
diff
changeset
|
808 |
|
56043 | 809 |
qed |
33154 | 810 |
|
52803 | 811 |
lemma polypow_norm: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
812 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
33154 | 813 |
shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)" |
814 |
by (simp add: polypow_normh isnpoly_def) |
|
815 |
||
60698 | 816 |
text \<open>Finally the whole normalization\<close> |
33154 | 817 |
|
52658 | 818 |
lemma polynate [simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
819 |
"Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field})" |
52658 | 820 |
by (induct p rule:polynate.induct) auto |
33154 | 821 |
|
52803 | 822 |
lemma polynate_norm[simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
823 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
33154 | 824 |
shows "isnpoly (polynate p)" |
52658 | 825 |
by (induct p rule: polynate.induct) |
826 |
(simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm, |
|
827 |
simp_all add: isnpoly_def) |
|
33154 | 828 |
|
60698 | 829 |
text \<open>shift1\<close> |
33154 | 830 |
|
831 |
||
832 |
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)" |
|
52658 | 833 |
by (simp add: shift1_def) |
33154 | 834 |
|
52803 | 835 |
lemma shift1_isnpoly: |
56207 | 836 |
assumes "isnpoly p" |
837 |
and "p \<noteq> 0\<^sub>p" |
|
52658 | 838 |
shows "isnpoly (shift1 p) " |
56207 | 839 |
using assms by (simp add: shift1_def isnpoly_def) |
33154 | 840 |
|
841 |
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p" |
|
842 |
by (simp add: shift1_def) |
|
56043 | 843 |
|
844 |
lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)" |
|
39246 | 845 |
by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1) |
33154 | 846 |
|
52803 | 847 |
lemma funpow_isnpolyh: |
56207 | 848 |
assumes "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n" |
849 |
and "isnpolyh p n" |
|
33154 | 850 |
shows "isnpolyh (funpow k f p) n" |
56207 | 851 |
using assms by (induct k arbitrary: p) auto |
33154 | 852 |
|
52658 | 853 |
lemma funpow_shift1: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
854 |
"(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field}) = |
52658 | 855 |
Ipoly bs (Mul (Pw (Bound 0) n) p)" |
856 |
by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1) |
|
33154 | 857 |
|
56043 | 858 |
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0" |
33154 | 859 |
using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def) |
860 |
||
52803 | 861 |
lemma funpow_shift1_1: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
862 |
"(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field}) = |
52658 | 863 |
Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)" |
33154 | 864 |
by (simp add: funpow_shift1) |
865 |
||
866 |
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
|
867 |
by (induct p rule: poly_cmul.induct) (auto simp add: field_simps) |
33154 | 868 |
|
869 |
lemma behead: |
|
56207 | 870 |
assumes "isnpolyh p n" |
52658 | 871 |
shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
872 |
(Ipoly bs p :: 'a :: {field_char_0,field})" |
56207 | 873 |
using assms |
33154 | 874 |
proof (induct p arbitrary: n rule: behead.induct) |
56009 | 875 |
case (1 c p n) |
876 |
then have pn: "isnpolyh p n" by simp |
|
52803 | 877 |
from 1(1)[OF pn] |
878 |
have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" . |
|
52658 | 879 |
then show ?case using "1.hyps" |
880 |
apply (simp add: Let_def,cases "behead p = 0\<^sub>p") |
|
881 |
apply (simp_all add: th[symmetric] field_simps) |
|
882 |
done |
|
33154 | 883 |
qed (auto simp add: Let_def) |
884 |
||
885 |
lemma behead_isnpolyh: |
|
56207 | 886 |
assumes "isnpolyh p n" |
52658 | 887 |
shows "isnpolyh (behead p) n" |
56207 | 888 |
using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono) |
52658 | 889 |
|
33154 | 890 |
|
60533 | 891 |
subsection \<open>Miscellaneous lemmas about indexes, decrementation, substitution etc ...\<close> |
52658 | 892 |
|
33154 | 893 |
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p" |
61166
5976fe402824
renamed method "goals" to "goal_cases" to emphasize its meaning;
wenzelm
parents:
60698
diff
changeset
|
894 |
proof (induct p arbitrary: n rule: poly.induct, auto, goal_cases) |
60580 | 895 |
case prems: (1 c n p n') |
56009 | 896 |
then have "n = Suc (n - 1)" |
897 |
by simp |
|
898 |
then have "isnpolyh p (Suc (n - 1))" |
|
60533 | 899 |
using \<open>isnpolyh p n\<close> by simp |
60580 | 900 |
with prems(2) show ?case |
56009 | 901 |
by simp |
33154 | 902 |
qed |
903 |
||
904 |
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p" |
|
52658 | 905 |
by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0) |
33154 | 906 |
|
52658 | 907 |
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" |
908 |
by (induct p) auto |
|
33154 | 909 |
|
910 |
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)" |
|
52658 | 911 |
apply (induct p arbitrary: n0) |
912 |
apply auto |
|
56043 | 913 |
apply atomize |
58259 | 914 |
apply (rename_tac nat a b, erule_tac x = "Suc nat" in allE) |
33154 | 915 |
apply auto |
916 |
done |
|
917 |
||
918 |
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)" |
|
52658 | 919 |
by (induct p arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0) |
33154 | 920 |
|
921 |
lemma polybound0_I: |
|
56207 | 922 |
assumes "polybound0 a" |
56009 | 923 |
shows "Ipoly (b # bs) a = Ipoly (b' # bs) a" |
56207 | 924 |
using assms by (induct a rule: poly.induct) auto |
52658 | 925 |
|
56009 | 926 |
lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t" |
33154 | 927 |
by (induct t) simp_all |
928 |
||
929 |
lemma polysubst0_I': |
|
56207 | 930 |
assumes "polybound0 a" |
56009 | 931 |
shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t" |
56207 | 932 |
by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"]) |
33154 | 933 |
|
52658 | 934 |
lemma decrpoly: |
56207 | 935 |
assumes "polybound0 t" |
56043 | 936 |
shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)" |
56207 | 937 |
using assms by (induct t rule: decrpoly.induct) simp_all |
33154 | 938 |
|
52658 | 939 |
lemma polysubst0_polybound0: |
56207 | 940 |
assumes "polybound0 t" |
33154 | 941 |
shows "polybound0 (polysubst0 t a)" |
56207 | 942 |
using assms by (induct a rule: poly.induct) auto |
33154 | 943 |
|
944 |
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p" |
|
52658 | 945 |
by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0) |
33154 | 946 |
|
56043 | 947 |
primrec maxindex :: "poly \<Rightarrow> nat" |
948 |
where |
|
33154 | 949 |
"maxindex (Bound n) = n + 1" |
950 |
| "maxindex (CN c n p) = max (n + 1) (max (maxindex c) (maxindex p))" |
|
951 |
| "maxindex (Add p q) = max (maxindex p) (maxindex q)" |
|
952 |
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)" |
|
953 |
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)" |
|
954 |
| "maxindex (Neg p) = maxindex p" |
|
955 |
| "maxindex (Pw p n) = maxindex p" |
|
956 |
| "maxindex (C x) = 0" |
|
957 |
||
52658 | 958 |
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" |
56000 | 959 |
where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p" |
33154 | 960 |
|
56043 | 961 |
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c" |
52658 | 962 |
proof (induct p rule: coefficients.induct) |
52803 | 963 |
case (1 c p) |
964 |
show ?case |
|
33154 | 965 |
proof |
56009 | 966 |
fix x |
967 |
assume xc: "x \<in> set (coefficients (CN c 0 p))" |
|
968 |
then have "x = c \<or> x \<in> set (coefficients p)" |
|
969 |
by simp |
|
52803 | 970 |
moreover |
56009 | 971 |
{ |
972 |
assume "x = c" |
|
973 |
then have "wf_bs bs x" |
|
56043 | 974 |
using "1.prems" unfolding wf_bs_def by simp |
56009 | 975 |
} |
976 |
moreover |
|
977 |
{ |
|
978 |
assume H: "x \<in> set (coefficients p)" |
|
979 |
from "1.prems" have "wf_bs bs p" |
|
980 |
unfolding wf_bs_def by simp |
|
981 |
with "1.hyps" H have "wf_bs bs x" |
|
982 |
by blast |
|
983 |
} |
|
56066 | 984 |
ultimately show "wf_bs bs x" |
56009 | 985 |
by blast |
33154 | 986 |
qed |
987 |
qed simp_all |
|
988 |
||
56043 | 989 |
lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p" |
52658 | 990 |
by (induct p rule: coefficients.induct) auto |
33154 | 991 |
|
56000 | 992 |
lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p" |
52658 | 993 |
unfolding wf_bs_def by (induct p) (auto simp add: nth_append) |
33154 | 994 |
|
52658 | 995 |
lemma take_maxindex_wf: |
52803 | 996 |
assumes wf: "wf_bs bs p" |
33154 | 997 |
shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p" |
56009 | 998 |
proof - |
33154 | 999 |
let ?ip = "maxindex p" |
1000 |
let ?tbs = "take ?ip bs" |
|
56009 | 1001 |
from wf have "length ?tbs = ?ip" |
1002 |
unfolding wf_bs_def by simp |
|
1003 |
then have wf': "wf_bs ?tbs p" |
|
1004 |
unfolding wf_bs_def by simp |
|
56043 | 1005 |
have eq: "bs = ?tbs @ drop ?ip bs" |
56009 | 1006 |
by simp |
1007 |
from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis |
|
1008 |
using eq by simp |
|
33154 | 1009 |
qed |
1010 |
||
1011 |
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1" |
|
52658 | 1012 |
by (induct p) auto |
33154 | 1013 |
|
1014 |
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p" |
|
1015 |
unfolding wf_bs_def by simp |
|
1016 |
||
56207 | 1017 |
lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p" |
33154 | 1018 |
unfolding wf_bs_def by simp |
1019 |
||
56207 | 1020 |
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p" |
52658 | 1021 |
by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def) |
56043 | 1022 |
|
33154 | 1023 |
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []" |
52658 | 1024 |
by (induct p rule: coefficients.induct) simp_all |
33154 | 1025 |
|
1026 |
lemma coefficients_head: "last (coefficients p) = head p" |
|
52658 | 1027 |
by (induct p rule: coefficients.induct) auto |
33154 | 1028 |
|
56207 | 1029 |
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p" |
52658 | 1030 |
unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto |
33154 | 1031 |
|
56043 | 1032 |
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n" |
56207 | 1033 |
apply (rule exI[where x="replicate (n - length xs) z" for z]) |
52658 | 1034 |
apply simp |
1035 |
done |
|
1036 |
||
56043 | 1037 |
lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p" |
52658 | 1038 |
apply (cases p) |
1039 |
apply auto |
|
58259 | 1040 |
apply (rename_tac nat a, case_tac "nat") |
52658 | 1041 |
apply simp_all |
1042 |
done |
|
33154 | 1043 |
|
1044 |
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)" |
|
56066 | 1045 |
unfolding wf_bs_def by (induct p q rule: polyadd.induct) (auto simp add: Let_def) |
33154 | 1046 |
|
1047 |
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)" |
|
52803 | 1048 |
unfolding wf_bs_def |
1049 |
apply (induct p q arbitrary: bs rule: polymul.induct) |
|
33154 | 1050 |
apply (simp_all add: wf_bs_polyadd) |
1051 |
apply clarsimp |
|
1052 |
apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format]) |
|
1053 |
apply auto |
|
1054 |
done |
|
1055 |
||
1056 |
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)" |
|
52658 | 1057 |
unfolding wf_bs_def by (induct p rule: polyneg.induct) auto |
33154 | 1058 |
|
1059 |
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)" |
|
56043 | 1060 |
unfolding polysub_def split_def fst_conv snd_conv |
1061 |
using wf_bs_polyadd wf_bs_polyneg by blast |
|
33154 | 1062 |
|
52658 | 1063 |
|
60533 | 1064 |
subsection \<open>Canonicity of polynomial representation, see lemma isnpolyh_unique\<close> |
33154 | 1065 |
|
1066 |
definition "polypoly bs p = map (Ipoly bs) (coefficients p)" |
|
56043 | 1067 |
definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)" |
1068 |
definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))" |
|
33154 | 1069 |
|
56043 | 1070 |
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0" |
33154 | 1071 |
proof (induct p arbitrary: n0 rule: coefficients.induct) |
1072 |
case (1 c p n0) |
|
56009 | 1073 |
have cp: "isnpolyh (CN c 0 p) n0" |
1074 |
by fact |
|
1075 |
then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0" |
|
33154 | 1076 |
by (auto simp add: isnpolyh_mono[where n'=0]) |
56009 | 1077 |
from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case |
1078 |
by simp |
|
33154 | 1079 |
qed auto |
1080 |
||
56043 | 1081 |
lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q" |
1082 |
by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const) |
|
33154 | 1083 |
|
1084 |
lemma polypoly_polypoly': |
|
1085 |
assumes np: "isnpolyh p n0" |
|
56043 | 1086 |
shows "polypoly (x # bs) p = polypoly' bs p" |
1087 |
proof - |
|
33154 | 1088 |
let ?cf = "set (coefficients p)" |
1089 |
from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" . |
|
56043 | 1090 |
{ |
1091 |
fix q |
|
1092 |
assume q: "q \<in> ?cf" |
|
1093 |
from q cn_norm have th: "isnpolyh q n0" |
|
1094 |
by blast |
|
1095 |
from coefficients_isconst[OF np] q have "isconstant q" |
|
1096 |
by blast |
|
1097 |
with isconstant_polybound0[OF th] have "polybound0 q" |
|
1098 |
by blast |
|
1099 |
} |
|
56009 | 1100 |
then have "\<forall>q \<in> ?cf. polybound0 q" .. |
56043 | 1101 |
then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)" |
33154 | 1102 |
using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs] |
1103 |
by auto |
|
56043 | 1104 |
then show ?thesis |
1105 |
unfolding polypoly_def polypoly'_def by simp |
|
33154 | 1106 |
qed |
1107 |
||
1108 |
lemma polypoly_poly: |
|
56043 | 1109 |
assumes "isnpolyh p n0" |
1110 |
shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x" |
|
1111 |
using assms |
|
52658 | 1112 |
by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def) |
33154 | 1113 |
|
52803 | 1114 |
lemma polypoly'_poly: |
56043 | 1115 |
assumes "isnpolyh p n0" |
52658 | 1116 |
shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x" |
56043 | 1117 |
using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] . |
33154 | 1118 |
|
1119 |
||
1120 |
lemma polypoly_poly_polybound0: |
|
56043 | 1121 |
assumes "isnpolyh p n0" |
1122 |
and "polybound0 p" |
|
33154 | 1123 |
shows "polypoly bs p = [Ipoly bs p]" |
56043 | 1124 |
using assms |
1125 |
unfolding polypoly_def |
|
52658 | 1126 |
apply (cases p) |
1127 |
apply auto |
|
58259 | 1128 |
apply (rename_tac nat a, case_tac nat) |
52658 | 1129 |
apply auto |
1130 |
done |
|
33154 | 1131 |
|
52803 | 1132 |
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" |
52658 | 1133 |
by (induct p rule: head.induct) auto |
33154 | 1134 |
|
56043 | 1135 |
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" |
52658 | 1136 |
by (cases p) auto |
33154 | 1137 |
|
1138 |
lemma head_eq_headn0: "head p = headn p 0" |
|
52658 | 1139 |
by (induct p rule: head.induct) simp_all |
33154 | 1140 |
|
56043 | 1141 |
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" |
33154 | 1142 |
by (simp add: head_eq_headn0) |
1143 |
||
52803 | 1144 |
lemma isnpolyh_zero_iff: |
52658 | 1145 |
assumes nq: "isnpolyh p n0" |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1146 |
and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0,field, power})" |
33154 | 1147 |
shows "p = 0\<^sub>p" |
52658 | 1148 |
using nq eq |
34915 | 1149 |
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct) |
1150 |
case less |
|
60533 | 1151 |
note np = \<open>isnpolyh p n0\<close> and zp = \<open>\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)\<close> |
56000 | 1152 |
{ |
1153 |
assume nz: "maxindex p = 0" |
|
1154 |
then obtain c where "p = C c" |
|
1155 |
using np by (cases p) auto |
|
1156 |
with zp np have "p = 0\<^sub>p" |
|
1157 |
unfolding wf_bs_def by simp |
|
1158 |
} |
|
33154 | 1159 |
moreover |
56000 | 1160 |
{ |
1161 |
assume nz: "maxindex p \<noteq> 0" |
|
33154 | 1162 |
let ?h = "head p" |
1163 |
let ?hd = "decrpoly ?h" |
|
1164 |
let ?ihd = "maxindex ?hd" |
|
56000 | 1165 |
from head_isnpolyh[OF np] head_polybound0[OF np] |
1166 |
have h: "isnpolyh ?h n0" "polybound0 ?h" |
|
33154 | 1167 |
by simp_all |
56000 | 1168 |
then have nhd: "isnpolyh ?hd (n0 - 1)" |
1169 |
using decrpoly_normh by blast |
|
52803 | 1170 |
|
33154 | 1171 |
from maxindex_coefficients[of p] coefficients_head[of p, symmetric] |
56000 | 1172 |
have mihn: "maxindex ?h \<le> maxindex p" |
1173 |
by auto |
|
1174 |
with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p" |
|
1175 |
by auto |
|
1176 |
{ |
|
1177 |
fix bs :: "'a list" |
|
1178 |
assume bs: "wf_bs bs ?hd" |
|
33154 | 1179 |
let ?ts = "take ?ihd bs" |
1180 |
let ?rs = "drop ?ihd bs" |
|
56000 | 1181 |
have ts: "wf_bs ?ts ?hd" |
1182 |
using bs unfolding wf_bs_def by simp |
|
1183 |
have bs_ts_eq: "?ts @ ?rs = bs" |
|
1184 |
by simp |
|
1185 |
from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h" |
|
1186 |
by simp |
|
1187 |
from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p" |
|
1188 |
by simp |
|
1189 |
with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p" |
|
1190 |
by blast |
|
1191 |
then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p" |
|
1192 |
unfolding wf_bs_def by simp |
|
1193 |
with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0" |
|
1194 |
by blast |
|
1195 |
then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0" |
|
1196 |
by simp |
|
33154 | 1197 |
with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a] |
56000 | 1198 |
have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x" |
1199 |
by simp |
|
1200 |
then have "poly (polypoly' (?ts @ xs) p) = poly []" |
|
1201 |
by auto |
|
1202 |
then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0" |
|
60537 | 1203 |
using poly_zero[where ?'a='a] by (simp add: polypoly'_def) |
33154 | 1204 |
with coefficients_head[of p, symmetric] |
56000 | 1205 |
have th0: "Ipoly (?ts @ xs) ?hd = 0" |
1206 |
by simp |
|
1207 |
from bs have wf'': "wf_bs ?ts ?hd" |
|
1208 |
unfolding wf_bs_def by simp |
|
1209 |
with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" |
|
1210 |
by simp |
|
1211 |
with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" |
|
1212 |
by simp |
|
1213 |
} |
|
1214 |
then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" |
|
1215 |
by blast |
|
1216 |
from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" |
|
1217 |
by blast |
|
1218 |
then have "?h = 0\<^sub>p" by simp |
|
1219 |
with head_nz[OF np] have "p = 0\<^sub>p" by simp |
|
1220 |
} |
|
1221 |
ultimately show "p = 0\<^sub>p" |
|
1222 |
by blast |
|
33154 | 1223 |
qed |
1224 |
||
52803 | 1225 |
lemma isnpolyh_unique: |
56000 | 1226 |
assumes np: "isnpolyh p n0" |
52658 | 1227 |
and nq: "isnpolyh q n1" |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1228 |
shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0,field,power})) \<longleftrightarrow> p = q" |
56000 | 1229 |
proof auto |
1230 |
assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>" |
|
1231 |
then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" |
|
1232 |
by simp |
|
1233 |
then have "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" |
|
33154 | 1234 |
using wf_bs_polysub[where p=p and q=q] by auto |
56000 | 1235 |
with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q" |
1236 |
by blast |
|
33154 | 1237 |
qed |
1238 |
||
1239 |
||
60698 | 1240 |
text \<open>consequences of unicity on the algorithms for polynomial normalization\<close> |
33154 | 1241 |
|
52658 | 1242 |
lemma polyadd_commute: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1243 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
52658 | 1244 |
and np: "isnpolyh p n0" |
1245 |
and nq: "isnpolyh q n1" |
|
1246 |
shows "p +\<^sub>p q = q +\<^sub>p p" |
|
56000 | 1247 |
using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] |
1248 |
by simp |
|
33154 | 1249 |
|
56000 | 1250 |
lemma zero_normh: "isnpolyh 0\<^sub>p n" |
1251 |
by simp |
|
1252 |
||
1253 |
lemma one_normh: "isnpolyh (1)\<^sub>p n" |
|
1254 |
by simp |
|
52658 | 1255 |
|
52803 | 1256 |
lemma polyadd_0[simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1257 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
52658 | 1258 |
and np: "isnpolyh p n0" |
56000 | 1259 |
shows "p +\<^sub>p 0\<^sub>p = p" |
1260 |
and "0\<^sub>p +\<^sub>p p = p" |
|
52803 | 1261 |
using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] |
33154 | 1262 |
isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all |
1263 |
||
52803 | 1264 |
lemma polymul_1[simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1265 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
52658 | 1266 |
and np: "isnpolyh p n0" |
56000 | 1267 |
shows "p *\<^sub>p (1)\<^sub>p = p" |
1268 |
and "(1)\<^sub>p *\<^sub>p p = p" |
|
52803 | 1269 |
using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] |
33154 | 1270 |
isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all |
52658 | 1271 |
|
52803 | 1272 |
lemma polymul_0[simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1273 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
52658 | 1274 |
and np: "isnpolyh p n0" |
56000 | 1275 |
shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" |
1276 |
and "0\<^sub>p *\<^sub>p p = 0\<^sub>p" |
|
52803 | 1277 |
using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] |
33154 | 1278 |
isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all |
1279 |
||
52803 | 1280 |
lemma polymul_commute: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1281 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56000 | 1282 |
and np: "isnpolyh p n0" |
52658 | 1283 |
and nq: "isnpolyh q n1" |
33154 | 1284 |
shows "p *\<^sub>p q = q *\<^sub>p p" |
56043 | 1285 |
using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1286 |
where ?'a = "'a::{field_char_0,field, power}"] |
52658 | 1287 |
by simp |
33154 | 1288 |
|
52658 | 1289 |
declare polyneg_polyneg [simp] |
52803 | 1290 |
|
1291 |
lemma isnpolyh_polynate_id [simp]: |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1292 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56000 | 1293 |
and np: "isnpolyh p n0" |
52658 | 1294 |
shows "polynate p = p" |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1295 |
using isnpolyh_unique[where ?'a= "'a::{field_char_0,field}", |
56043 | 1296 |
OF polynate_norm[of p, unfolded isnpoly_def] np] |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1297 |
polynate[where ?'a = "'a::{field_char_0,field}"] |
52658 | 1298 |
by simp |
33154 | 1299 |
|
52803 | 1300 |
lemma polynate_idempotent[simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1301 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
33154 | 1302 |
shows "polynate (polynate p) = polynate p" |
1303 |
using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] . |
|
1304 |
||
1305 |
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)" |
|
1306 |
unfolding poly_nate_def polypoly'_def .. |
|
52658 | 1307 |
|
1308 |
lemma poly_nate_poly: |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1309 |
"poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0,field}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)" |
33154 | 1310 |
using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p] |
52658 | 1311 |
unfolding poly_nate_polypoly' by auto |
1312 |
||
33154 | 1313 |
|
60698 | 1314 |
subsection \<open>heads, degrees and all that\<close> |
52658 | 1315 |
|
33154 | 1316 |
lemma degree_eq_degreen0: "degree p = degreen p 0" |
52658 | 1317 |
by (induct p rule: degree.induct) simp_all |
33154 | 1318 |
|
52658 | 1319 |
lemma degree_polyneg: |
56043 | 1320 |
assumes "isnpolyh p n" |
33154 | 1321 |
shows "degree (polyneg p) = degree p" |
56043 | 1322 |
apply (induct p rule: polyneg.induct) |
1323 |
using assms |
|
1324 |
apply simp_all |
|
52658 | 1325 |
apply (case_tac na) |
1326 |
apply auto |
|
1327 |
done |
|
33154 | 1328 |
|
1329 |
lemma degree_polyadd: |
|
56043 | 1330 |
assumes np: "isnpolyh p n0" |
1331 |
and nq: "isnpolyh q n1" |
|
33154 | 1332 |
shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)" |
52658 | 1333 |
using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp |
33154 | 1334 |
|
1335 |
||
52658 | 1336 |
lemma degree_polysub: |
1337 |
assumes np: "isnpolyh p n0" |
|
1338 |
and nq: "isnpolyh q n1" |
|
33154 | 1339 |
shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)" |
1340 |
proof- |
|
56043 | 1341 |
from nq have nq': "isnpolyh (~\<^sub>p q) n1" |
1342 |
using polyneg_normh by simp |
|
1343 |
from degree_polyadd[OF np nq'] show ?thesis |
|
1344 |
by (simp add: polysub_def degree_polyneg[OF nq]) |
|
33154 | 1345 |
qed |
1346 |
||
52803 | 1347 |
lemma degree_polysub_samehead: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1348 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56043 | 1349 |
and np: "isnpolyh p n0" |
1350 |
and nq: "isnpolyh q n1" |
|
1351 |
and h: "head p = head q" |
|
52658 | 1352 |
and d: "degree p = degree q" |
33154 | 1353 |
shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)" |
52658 | 1354 |
unfolding polysub_def split_def fst_conv snd_conv |
1355 |
using np nq h d |
|
1356 |
proof (induct p q rule: polyadd.induct) |
|
1357 |
case (1 c c') |
|
56009 | 1358 |
then show ?case |
1359 |
by (simp add: Nsub_def Nsub0[simplified Nsub_def]) |
|
33154 | 1360 |
next |
52803 | 1361 |
case (2 c c' n' p') |
56009 | 1362 |
from 2 have "degree (C c) = degree (CN c' n' p')" |
1363 |
by simp |
|
1364 |
then have nz: "n' > 0" |
|
1365 |
by (cases n') auto |
|
1366 |
then have "head (CN c' n' p') = CN c' n' p'" |
|
1367 |
by (cases n') auto |
|
1368 |
with 2 show ?case |
|
1369 |
by simp |
|
33154 | 1370 |
next |
52803 | 1371 |
case (3 c n p c') |
56009 | 1372 |
then have "degree (C c') = degree (CN c n p)" |
1373 |
by simp |
|
1374 |
then have nz: "n > 0" |
|
1375 |
by (cases n) auto |
|
1376 |
then have "head (CN c n p) = CN c n p" |
|
1377 |
by (cases n) auto |
|
41807 | 1378 |
with 3 show ?case by simp |
33154 | 1379 |
next |
1380 |
case (4 c n p c' n' p') |
|
56009 | 1381 |
then have H: |
1382 |
"isnpolyh (CN c n p) n0" |
|
1383 |
"isnpolyh (CN c' n' p') n1" |
|
1384 |
"head (CN c n p) = head (CN c' n' p')" |
|
1385 |
"degree (CN c n p) = degree (CN c' n' p')" |
|
1386 |
by simp_all |
|
1387 |
then have degc: "degree c = 0" and degc': "degree c' = 0" |
|
1388 |
by simp_all |
|
1389 |
then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0" |
|
33154 | 1390 |
using H(1-2) degree_polyneg by auto |
56009 | 1391 |
from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')" |
1392 |
by simp_all |
|
1393 |
from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' |
|
1394 |
have degcmc': "degree (c +\<^sub>p ~\<^sub>pc') = 0" |
|
1395 |
by simp |
|
1396 |
from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" |
|
1397 |
by auto |
|
1398 |
have "n = n' \<or> n < n' \<or> n > n'" |
|
1399 |
by arith |
|
33154 | 1400 |
moreover |
56009 | 1401 |
{ |
1402 |
assume nn': "n = n'" |
|
1403 |
have "n = 0 \<or> n > 0" by arith |
|
56066 | 1404 |
moreover |
1405 |
{ |
|
56009 | 1406 |
assume nz: "n = 0" |
1407 |
then have ?case using 4 nn' |
|
1408 |
by (auto simp add: Let_def degcmc') |
|
1409 |
} |
|
56066 | 1410 |
moreover |
1411 |
{ |
|
56009 | 1412 |
assume nz: "n > 0" |
1413 |
with nn' H(3) have cc': "c = c'" and pp': "p = p'" |
|
1414 |
by (cases n, auto)+ |
|
1415 |
then have ?case |
|
1416 |
using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] |
|
1417 |
using polysub_same_0[OF c'nh, simplified polysub_def] |
|
1418 |
using nn' 4 by (simp add: Let_def) |
|
1419 |
} |
|
1420 |
ultimately have ?case by blast |
|
1421 |
} |
|
33154 | 1422 |
moreover |
56009 | 1423 |
{ |
1424 |
assume nn': "n < n'" |
|
1425 |
then have n'p: "n' > 0" |
|
1426 |
by simp |
|
1427 |
then have headcnp':"head (CN c' n' p') = CN c' n' p'" |
|
1428 |
by (cases n') simp_all |
|
1429 |
have degcnp': "degree (CN c' n' p') = 0" |
|
1430 |
and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" |
|
52658 | 1431 |
using 4 nn' by (cases n', simp_all) |
56009 | 1432 |
then have "n > 0" |
1433 |
by (cases n) simp_all |
|
1434 |
then have headcnp: "head (CN c n p) = CN c n p" |
|
1435 |
by (cases n) auto |
|
1436 |
from H(3) headcnp headcnp' nn' have ?case |
|
1437 |
by auto |
|
1438 |
} |
|
33154 | 1439 |
moreover |
56009 | 1440 |
{ |
1441 |
assume nn': "n > n'" |
|
1442 |
then have np: "n > 0" by simp |
|
1443 |
then have headcnp:"head (CN c n p) = CN c n p" |
|
1444 |
by (cases n) simp_all |
|
1445 |
from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" |
|
1446 |
by simp |
|
1447 |
from np have degcnp: "degree (CN c n p) = 0" |
|
1448 |
by (cases n) simp_all |
|
1449 |
with degcnpeq have "n' > 0" |
|
1450 |
by (cases n') simp_all |
|
1451 |
then have headcnp': "head (CN c' n' p') = CN c' n' p'" |
|
1452 |
by (cases n') auto |
|
1453 |
from H(3) headcnp headcnp' nn' have ?case by auto |
|
1454 |
} |
|
1455 |
ultimately show ?case by blast |
|
41812 | 1456 |
qed auto |
52803 | 1457 |
|
33154 | 1458 |
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p" |
52658 | 1459 |
by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def) |
33154 | 1460 |
|
1461 |
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p" |
|
52658 | 1462 |
proof (induct k arbitrary: n0 p) |
1463 |
case 0 |
|
56198 | 1464 |
then show ?case |
1465 |
by auto |
|
52658 | 1466 |
next |
1467 |
case (Suc k n0 p) |
|
56066 | 1468 |
then have "isnpolyh (shift1 p) 0" |
1469 |
by (simp add: shift1_isnpolyh) |
|
41807 | 1470 |
with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)" |
56066 | 1471 |
and "head (shift1 p) = head p" |
1472 |
by (simp_all add: shift1_head) |
|
1473 |
then show ?case |
|
1474 |
by (simp add: funpow_swap1) |
|
52658 | 1475 |
qed |
33154 | 1476 |
|
1477 |
lemma shift1_degree: "degree (shift1 p) = 1 + degree p" |
|
1478 |
by (simp add: shift1_def) |
|
56009 | 1479 |
|
33154 | 1480 |
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p " |
46991 | 1481 |
by (induct k arbitrary: p) (auto simp add: shift1_degree) |
33154 | 1482 |
|
1483 |
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p" |
|
52658 | 1484 |
by (induct n arbitrary: p) simp_all |
33154 | 1485 |
|
1486 |
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p" |
|
52658 | 1487 |
by (induct p arbitrary: n rule: degree.induct) auto |
33154 | 1488 |
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p" |
52658 | 1489 |
by (induct p arbitrary: n rule: degreen.induct) auto |
33154 | 1490 |
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p" |
52658 | 1491 |
by (induct p arbitrary: n rule: degree.induct) auto |
33154 | 1492 |
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p" |
52658 | 1493 |
by (induct p rule: head.induct) auto |
33154 | 1494 |
|
52803 | 1495 |
lemma polyadd_eq_const_degree: |
52658 | 1496 |
"isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q" |
33154 | 1497 |
using polyadd_eq_const_degreen degree_eq_degreen0 by simp |
1498 |
||
52658 | 1499 |
lemma polyadd_head: |
1500 |
assumes np: "isnpolyh p n0" |
|
1501 |
and nq: "isnpolyh q n1" |
|
1502 |
and deg: "degree p \<noteq> degree q" |
|
33154 | 1503 |
shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)" |
52658 | 1504 |
using np nq deg |
1505 |
apply (induct p q arbitrary: n0 n1 rule: polyadd.induct) |
|
1506 |
apply simp_all |
|
1507 |
apply (case_tac n', simp, simp) |
|
1508 |
apply (case_tac n, simp, simp) |
|
1509 |
apply (case_tac n, case_tac n', simp add: Let_def) |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54220
diff
changeset
|
1510 |
apply (auto simp add: polyadd_eq_const_degree)[2] |
52658 | 1511 |
apply (metis head_nz) |
1512 |
apply (metis head_nz) |
|
1513 |
apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq) |
|
1514 |
done |
|
33154 | 1515 |
|
52803 | 1516 |
lemma polymul_head_polyeq: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1517 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56066 | 1518 |
shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> q \<noteq> 0\<^sub>p \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q" |
33154 | 1519 |
proof (induct p q arbitrary: n0 n1 rule: polymul.induct) |
41813 | 1520 |
case (2 c c' n' p' n0 n1) |
56009 | 1521 |
then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c" |
1522 |
by (simp_all add: head_isnpolyh) |
|
1523 |
then show ?case |
|
1524 |
using 2 by (cases n') auto |
|
52803 | 1525 |
next |
1526 |
case (3 c n p c' n0 n1) |
|
56009 | 1527 |
then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'" |
1528 |
by (simp_all add: head_isnpolyh) |
|
56066 | 1529 |
then show ?case |
1530 |
using 3 by (cases n) auto |
|
33154 | 1531 |
next |
1532 |
case (4 c n p c' n' p' n0 n1) |
|
56066 | 1533 |
then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')" |
33154 | 1534 |
"isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'" |
1535 |
by simp_all |
|
1536 |
have "n < n' \<or> n' < n \<or> n = n'" by arith |
|
52803 | 1537 |
moreover |
56009 | 1538 |
{ |
1539 |
assume nn': "n < n'" |
|
1540 |
then have ?case |
|
52658 | 1541 |
using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)] |
1542 |
apply simp |
|
1543 |
apply (cases n) |
|
1544 |
apply simp |
|
1545 |
apply (cases n') |
|
1546 |
apply simp_all |
|
56009 | 1547 |
done |
1548 |
} |
|
1549 |
moreover { |
|
1550 |
assume nn': "n'< n" |
|
1551 |
then have ?case |
|
52803 | 1552 |
using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] |
52658 | 1553 |
apply simp |
1554 |
apply (cases n') |
|
1555 |
apply simp |
|
1556 |
apply (cases n) |
|
1557 |
apply auto |
|
56009 | 1558 |
done |
1559 |
} |
|
56066 | 1560 |
moreover |
1561 |
{ |
|
56009 | 1562 |
assume nn': "n' = n" |
52803 | 1563 |
from nn' polymul_normh[OF norm(5,4)] |
33154 | 1564 |
have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def) |
52803 | 1565 |
from nn' polymul_normh[OF norm(5,3)] norm |
33154 | 1566 |
have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp |
1567 |
from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6) |
|
52803 | 1568 |
have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp |
1569 |
from polyadd_normh[OF ncnpc' ncnpp0'] |
|
1570 |
have nth: "isnpolyh ((CN c n p *\<^sub>p c') +\<^sub>p (CN 0\<^sub>p n (CN c n p *\<^sub>p p'))) n" |
|
33154 | 1571 |
by (simp add: min_def) |
56009 | 1572 |
{ |
1573 |
assume np: "n > 0" |
|
33154 | 1574 |
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
|
1575 |
head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']] |
56009 | 1576 |
have ?case by simp |
1577 |
} |
|
33154 | 1578 |
moreover |
56009 | 1579 |
{ |
1580 |
assume nz: "n = 0" |
|
33154 | 1581 |
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
|
1582 |
polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0 |
33154 | 1583 |
norm(5,6) degree_npolyhCN[OF norm(6)] |
56066 | 1584 |
have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" |
1585 |
by simp |
|
1586 |
then have dth': "degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" |
|
1587 |
by simp |
|
33154 | 1588 |
from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth |
56066 | 1589 |
have ?case |
1590 |
using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz |
|
1591 |
by simp |
|
56009 | 1592 |
} |
56066 | 1593 |
ultimately have ?case |
1594 |
by (cases n) auto |
|
56009 | 1595 |
} |
33154 | 1596 |
ultimately show ?case by blast |
1597 |
qed simp_all |
|
1598 |
||
1599 |
lemma degree_coefficients: "degree p = length (coefficients p) - 1" |
|
52658 | 1600 |
by (induct p rule: degree.induct) auto |
33154 | 1601 |
|
1602 |
lemma degree_head[simp]: "degree (head p) = 0" |
|
52658 | 1603 |
by (induct p rule: head.induct) auto |
33154 | 1604 |
|
41812 | 1605 |
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p" |
52658 | 1606 |
by (cases n) simp_all |
56066 | 1607 |
|
33154 | 1608 |
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge> degree p" |
52658 | 1609 |
by (cases n) simp_all |
33154 | 1610 |
|
52658 | 1611 |
lemma polyadd_different_degree: |
56066 | 1612 |
"isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow> |
52658 | 1613 |
degree (polyadd p q) = max (degree p) (degree q)" |
33154 | 1614 |
using polyadd_different_degreen degree_eq_degreen0 by simp |
1615 |
||
1616 |
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m" |
|
52658 | 1617 |
by (induct p arbitrary: n0 rule: polyneg.induct) auto |
33154 | 1618 |
|
1619 |
lemma degree_polymul: |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1620 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
52658 | 1621 |
and np: "isnpolyh p n0" |
1622 |
and nq: "isnpolyh q n1" |
|
33154 | 1623 |
shows "degree (p *\<^sub>p q) \<le> degree p + degree q" |
1624 |
using polymul_degreen[OF np nq, where m="0"] degree_eq_degreen0 by simp |
|
1625 |
||
1626 |
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p" |
|
52658 | 1627 |
by (induct p arbitrary: n rule: degree.induct) auto |
33154 | 1628 |
|
56207 | 1629 |
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head (polyneg p) = polyneg (head p)" |
52658 | 1630 |
by (induct p arbitrary: n rule: degree.induct) auto |
1631 |
||
33154 | 1632 |
|
60533 | 1633 |
subsection \<open>Correctness of polynomial pseudo division\<close> |
33154 | 1634 |
|
1635 |
lemma polydivide_aux_properties: |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1636 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
52658 | 1637 |
and np: "isnpolyh p n0" |
1638 |
and ns: "isnpolyh s n1" |
|
1639 |
and ap: "head p = a" |
|
56198 | 1640 |
and ndp: "degree p = n" |
1641 |
and pnz: "p \<noteq> 0\<^sub>p" |
|
1642 |
shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and> |
|
56066 | 1643 |
(\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> (polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)" |
33154 | 1644 |
using ns |
52658 | 1645 |
proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct) |
34915 | 1646 |
case less |
33154 | 1647 |
let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)" |
56066 | 1648 |
let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow> k \<le> k' \<and> |
1649 |
(degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths" |
|
33154 | 1650 |
let ?b = "head s" |
34915 | 1651 |
let ?p' = "funpow (degree s - n) shift1 p" |
50282
fe4d4bb9f4c2
more robust syntax that survives collapse of \<^isub> and \<^sub>;
wenzelm
parents:
49962
diff
changeset
|
1652 |
let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p" |
33154 | 1653 |
let ?akk' = "a ^\<^sub>p (k' - k)" |
60533 | 1654 |
note ns = \<open>isnpolyh s n1\<close> |
52803 | 1655 |
from np have np0: "isnpolyh p 0" |
1656 |
using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp |
|
1657 |
have np': "isnpolyh ?p' 0" |
|
1658 |
using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def |
|
1659 |
by simp |
|
1660 |
have headp': "head ?p' = head p" |
|
1661 |
using funpow_shift1_head[OF np pnz] by simp |
|
1662 |
from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" |
|
1663 |
by (simp add: isnpoly_def) |
|
1664 |
from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap |
|
33154 | 1665 |
have nakk':"isnpolyh ?akk' 0" by blast |
56066 | 1666 |
{ |
1667 |
assume sz: "s = 0\<^sub>p" |
|
1668 |
then have ?ths |
|
1669 |
using np polydivide_aux.simps |
|
52658 | 1670 |
apply clarsimp |
1671 |
apply (rule exI[where x="0\<^sub>p"]) |
|
1672 |
apply simp |
|
56066 | 1673 |
done |
1674 |
} |
|
33154 | 1675 |
moreover |
56066 | 1676 |
{ |
1677 |
assume sz: "s \<noteq> 0\<^sub>p" |
|
1678 |
{ |
|
1679 |
assume dn: "degree s < n" |
|
1680 |
then have "?ths" |
|
1681 |
using ns ndp np polydivide_aux.simps |
|
52658 | 1682 |
apply auto |
1683 |
apply (rule exI[where x="0\<^sub>p"]) |
|
1684 |
apply simp |
|
56066 | 1685 |
done |
1686 |
} |
|
52803 | 1687 |
moreover |
56066 | 1688 |
{ |
1689 |
assume dn': "\<not> degree s < n" |
|
1690 |
then have dn: "degree s \<ge> n" |
|
1691 |
by arith |
|
52803 | 1692 |
have degsp': "degree s = degree ?p'" |
56066 | 1693 |
using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] |
1694 |
by simp |
|
1695 |
{ |
|
1696 |
assume ba: "?b = a" |
|
1697 |
then have headsp': "head s = head ?p'" |
|
52803 | 1698 |
using ap headp' by simp |
1699 |
have nr: "isnpolyh (s -\<^sub>p ?p') 0" |
|
1700 |
using polysub_normh[OF ns np'] by simp |
|
34915 | 1701 |
from degree_polysub_samehead[OF ns np' headsp' degsp'] |
56207 | 1702 |
have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" |
1703 |
by simp |
|
52803 | 1704 |
moreover |
56066 | 1705 |
{ |
1706 |
assume deglt:"degree (s -\<^sub>p ?p') < degree s" |
|
41403
7eba049f7310
partial_function (tailrec) replaces function (tailrec);
krauss
parents:
39246
diff
changeset
|
1707 |
from polydivide_aux.simps sz dn' ba |
7eba049f7310
partial_function (tailrec) replaces function (tailrec);
krauss
parents:
39246
diff
changeset
|
1708 |
have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')" |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1709 |
by (simp add: Let_def) |
56066 | 1710 |
{ |
1711 |
assume h1: "polydivide_aux a n p k s = (k', r)" |
|
52803 | 1712 |
from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1] |
1713 |
have kk': "k \<le> k'" |
|
56066 | 1714 |
and nr: "\<exists>nr. isnpolyh r nr" |
52803 | 1715 |
and dr: "degree r = 0 \<or> degree r < degree p" |
56066 | 1716 |
and q1: "\<exists>q nq. isnpolyh q nq \<and> a ^\<^sub>p k' - k *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r" |
52803 | 1717 |
by auto |
1718 |
from q1 obtain q n1 where nq: "isnpolyh q n1" |
|
56066 | 1719 |
and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r" |
1720 |
by blast |
|
1721 |
from nr obtain nr where nr': "isnpolyh r nr" |
|
1722 |
by blast |
|
52803 | 1723 |
from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" |
1724 |
by simp |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1725 |
from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1726 |
have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp |
52803 | 1727 |
from polyadd_normh[OF polymul_normh[OF np |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1728 |
polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr'] |
52803 | 1729 |
have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" |
1730 |
by simp |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1731 |
from asp have "\<forall>bs :: 'a::{field_char_0,field} list. |
56066 | 1732 |
Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)" |
1733 |
by simp |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1734 |
then have "\<forall>bs :: 'a::{field_char_0,field} list. |
56066 | 1735 |
Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) = |
52803 | 1736 |
Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r" |
36349 | 1737 |
by (simp add: field_simps) |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1738 |
then have "\<forall>bs :: 'a::{field_char_0,field} list. |
56066 | 1739 |
Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = |
52803 | 1740 |
Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) + |
1741 |
Ipoly bs p * Ipoly bs q + Ipoly bs r" |
|
1742 |
by (auto simp only: funpow_shift1_1) |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1743 |
then have "\<forall>bs:: 'a::{field_char_0,field} list. |
56066 | 1744 |
Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = |
52803 | 1745 |
Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) + |
1746 |
Ipoly bs q) + Ipoly bs r" |
|
1747 |
by (simp add: field_simps) |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1748 |
then have "\<forall>bs:: 'a::{field_char_0,field} list. |
56066 | 1749 |
Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = |
52803 | 1750 |
Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r)" |
1751 |
by simp |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1752 |
with isnpolyh_unique[OF nakks' nqr'] |
52803 | 1753 |
have "a ^\<^sub>p (k' - k) *\<^sub>p s = |
1754 |
p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r" |
|
1755 |
by blast |
|
56066 | 1756 |
then have ?qths using nq' |
50282
fe4d4bb9f4c2
more robust syntax that survives collapse of \<^isub> and \<^sub>;
wenzelm
parents:
49962
diff
changeset
|
1757 |
apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI) |
52803 | 1758 |
apply (rule_tac x="0" in exI) |
1759 |
apply simp |
|
1760 |
done |
|
56066 | 1761 |
with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> |
1762 |
(\<exists>nr. isnpolyh r nr) \<and> ?qths" |
|
52803 | 1763 |
by blast |
1764 |
} |
|
56066 | 1765 |
then have ?ths by blast |
52803 | 1766 |
} |
1767 |
moreover |
|
56066 | 1768 |
{ |
1769 |
assume spz:"s -\<^sub>p ?p' = 0\<^sub>p" |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1770 |
from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0,field}"] |
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1771 |
have "\<forall>bs:: 'a::{field_char_0,field} list. Ipoly bs s = Ipoly bs ?p'" |
52803 | 1772 |
by simp |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1773 |
then have "\<forall>bs:: 'a::{field_char_0,field} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)" |
52658 | 1774 |
using np nxdn |
1775 |
apply simp |
|
1776 |
apply (simp only: funpow_shift1_1) |
|
1777 |
apply simp |
|
1778 |
done |
|
56066 | 1779 |
then have sp': "s = ?xdn *\<^sub>p p" |
1780 |
using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] |
|
52658 | 1781 |
by blast |
56066 | 1782 |
{ |
56207 | 1783 |
assume h1: "polydivide_aux a n p k s = (k', r)" |
41403
7eba049f7310
partial_function (tailrec) replaces function (tailrec);
krauss
parents:
39246
diff
changeset
|
1784 |
from polydivide_aux.simps sz dn' ba |
7eba049f7310
partial_function (tailrec) replaces function (tailrec);
krauss
parents:
39246
diff
changeset
|
1785 |
have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')" |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1786 |
by (simp add: Let_def) |
52803 | 1787 |
also have "\<dots> = (k,0\<^sub>p)" |
1788 |
using polydivide_aux.simps spz by simp |
|
56066 | 1789 |
finally have "(k', r) = (k, 0\<^sub>p)" |
1790 |
using h1 by simp |
|
34915 | 1791 |
with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]] |
41403
7eba049f7310
partial_function (tailrec) replaces function (tailrec);
krauss
parents:
39246
diff
changeset
|
1792 |
polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1793 |
apply auto |
52803 | 1794 |
apply (rule exI[where x="?xdn"]) |
34915 | 1795 |
apply (auto simp add: polymul_commute[of p]) |
52803 | 1796 |
done |
1797 |
} |
|
1798 |
} |
|
1799 |
ultimately have ?ths by blast |
|
1800 |
} |
|
33154 | 1801 |
moreover |
56066 | 1802 |
{ |
1803 |
assume ba: "?b \<noteq> a" |
|
52803 | 1804 |
from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1805 |
polymul_normh[OF head_isnpolyh[OF ns] np']] |
52803 | 1806 |
have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" |
1807 |
by (simp add: min_def) |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1808 |
have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p" |
52803 | 1809 |
using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns] |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1810 |
polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns] |
52803 | 1811 |
funpow_shift1_nz[OF pnz] |
1812 |
by simp_all |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1813 |
from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz |
34915 | 1814 |
polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"] |
52803 | 1815 |
have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')" |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1816 |
using head_head[OF ns] funpow_shift1_head[OF np pnz] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1817 |
polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1818 |
by (simp add: ap) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1819 |
from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1820 |
head_nz[OF np] pnz sz ap[symmetric] |
34915 | 1821 |
funpow_shift1_nz[OF pnz, where n="degree s - n"] |
52803 | 1822 |
polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns] |
34915 | 1823 |
ndp dn |
52803 | 1824 |
have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')" |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1825 |
by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree) |
56066 | 1826 |
{ |
1827 |
assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s" |
|
52803 | 1828 |
from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] |
1829 |
polymul_normh[OF head_isnpolyh[OF ns]np']] ap |
|
1830 |
have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" |
|
1831 |
by simp |
|
56066 | 1832 |
{ |
1833 |
assume h1:"polydivide_aux a n p k s = (k', r)" |
|
41403
7eba049f7310
partial_function (tailrec) replaces function (tailrec);
krauss
parents:
39246
diff
changeset
|
1834 |
from h1 polydivide_aux.simps sz dn' ba |
7eba049f7310
partial_function (tailrec) replaces function (tailrec);
krauss
parents:
39246
diff
changeset
|
1835 |
have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)" |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1836 |
by (simp add: Let_def) |
34915 | 1837 |
with less(1)[OF dth nasbp', of "Suc k" k' r] |
52803 | 1838 |
obtain q nq nr where kk': "Suc k \<le> k'" |
1839 |
and nr: "isnpolyh r nr" |
|
1840 |
and nq: "isnpolyh q nq" |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1841 |
and dr: "degree r = 0 \<or> degree r < degree p" |
52803 | 1842 |
and qr: "a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = p *\<^sub>p q +\<^sub>p r" |
1843 |
by auto |
|
56066 | 1844 |
from kk' have kk'': "Suc (k' - Suc k) = k' - k" |
1845 |
by arith |
|
52803 | 1846 |
{ |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1847 |
fix bs :: "'a::{field_char_0,field} list" |
52803 | 1848 |
from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric] |
1849 |
have "Ipoly bs (a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)" |
|
1850 |
by simp |
|
56066 | 1851 |
then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s = |
52803 | 1852 |
Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r" |
1853 |
by (simp add: field_simps) |
|
56066 | 1854 |
then have "Ipoly bs a ^ (k' - k) * Ipoly bs s = |
52803 | 1855 |
Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r" |
1856 |
by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"]) |
|
56066 | 1857 |
then have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = |
52803 | 1858 |
Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r" |
1859 |
by (simp add: field_simps) |
|
1860 |
} |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1861 |
then have ieq:"\<forall>bs :: 'a::{field_char_0,field} list. |
56207 | 1862 |
Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = |
1863 |
Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)" |
|
52803 | 1864 |
by auto |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1865 |
let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)" |
56207 | 1866 |
from polyadd_normh[OF nq polymul_normh[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k"] head_isnpolyh[OF ns], simplified ap] nxdn]] |
52803 | 1867 |
have nqw: "isnpolyh ?q 0" |
1868 |
by simp |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1869 |
from ieq isnpolyh_unique[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - k"] ns, simplified ap] polyadd_normh[OF polymul_normh[OF np nqw] nr]] |
52803 | 1870 |
have asth: "(a ^\<^sub>p (k' - k) *\<^sub>p s) = p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r" |
1871 |
by blast |
|
1872 |
from dr kk' nr h1 asth nqw have ?ths |
|
1873 |
apply simp |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1874 |
apply (rule conjI) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1875 |
apply (rule exI[where x="nr"], simp) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1876 |
apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1877 |
apply (rule exI[where x="0"], simp) |
52803 | 1878 |
done |
1879 |
} |
|
56066 | 1880 |
then have ?ths by blast |
52803 | 1881 |
} |
1882 |
moreover |
|
56066 | 1883 |
{ |
1884 |
assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p" |
|
52803 | 1885 |
{ |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1886 |
fix bs :: "'a::{field_char_0,field} list" |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1887 |
from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz |
52803 | 1888 |
have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" |
1889 |
by simp |
|
56066 | 1890 |
then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p" |
52803 | 1891 |
by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"]) |
56066 | 1892 |
then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" |
52803 | 1893 |
by simp |
1894 |
} |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1895 |
then have hth: "\<forall>bs :: 'a::{field_char_0,field} list. |
56066 | 1896 |
Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" .. |
52803 | 1897 |
from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)" |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1898 |
using isnpolyh_unique[where ?'a = "'a::{field_char_0,field}", OF polymul_normh[OF head_isnpolyh[OF np] ns] |
33154 | 1899 |
polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]], |
56066 | 1900 |
simplified ap] |
1901 |
by simp |
|
1902 |
{ |
|
1903 |
assume h1: "polydivide_aux a n p k s = (k', r)" |
|
52803 | 1904 |
from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps |
56066 | 1905 |
have "(k', r) = (Suc k, 0\<^sub>p)" |
1906 |
by (simp add: Let_def) |
|
52803 | 1907 |
with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn] |
1908 |
polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq |
|
1909 |
have ?ths |
|
1910 |
apply (clarsimp simp add: Let_def) |
|
1911 |
apply (rule exI[where x="?b *\<^sub>p ?xdn"]) |
|
1912 |
apply simp |
|
1913 |
apply (rule exI[where x="0"], simp) |
|
1914 |
done |
|
1915 |
} |
|
56066 | 1916 |
then have ?ths by blast |
52803 | 1917 |
} |
52658 | 1918 |
ultimately have ?ths |
56066 | 1919 |
using degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"] |
52658 | 1920 |
head_nz[OF np] pnz sz ap[symmetric] |
56066 | 1921 |
by (auto simp add: degree_eq_degreen0[symmetric]) |
52803 | 1922 |
} |
33154 | 1923 |
ultimately have ?ths by blast |
1924 |
} |
|
52803 | 1925 |
ultimately have ?ths by blast |
1926 |
} |
|
33154 | 1927 |
ultimately show ?ths by blast |
1928 |
qed |
|
1929 |
||
52803 | 1930 |
lemma polydivide_properties: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
1931 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
56066 | 1932 |
and np: "isnpolyh p n0" |
1933 |
and ns: "isnpolyh s n1" |
|
1934 |
and pnz: "p \<noteq> 0\<^sub>p" |
|
1935 |
shows "\<exists>k r. polydivide s p = (k, r) \<and> |
|
52803 | 1936 |
(\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and> |
56066 | 1937 |
(\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)" |
52803 | 1938 |
proof - |
1939 |
have trv: "head p = head p" "degree p = degree p" |
|
1940 |
by simp_all |
|
1941 |
from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)" |
|
1942 |
by auto |
|
1943 |
then obtain k r where kr: "polydivide s p = (k,r)" |
|
1944 |
by blast |
|
56000 | 1945 |
from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr] |
33154 | 1946 |
polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"] |
1947 |
have "(degree r = 0 \<or> degree r < degree p) \<and> |
|
52803 | 1948 |
(\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> head p ^\<^sub>p k - 0 *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)" |
1949 |
by blast |
|
1950 |
with kr show ?thesis |
|
33154 | 1951 |
apply - |
1952 |
apply (rule exI[where x="k"]) |
|
1953 |
apply (rule exI[where x="r"]) |
|
1954 |
apply simp |
|
1955 |
done |
|
1956 |
qed |
|
1957 |
||
52658 | 1958 |
|
60533 | 1959 |
subsection \<open>More about polypoly and pnormal etc\<close> |
33154 | 1960 |
|
56000 | 1961 |
definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p" |
33154 | 1962 |
|
52658 | 1963 |
lemma isnonconstant_pnormal_iff: |
56198 | 1964 |
assumes "isnonconstant p" |
52803 | 1965 |
shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" |
33154 | 1966 |
proof |
52803 | 1967 |
let ?p = "polypoly bs p" |
33154 | 1968 |
assume H: "pnormal ?p" |
56066 | 1969 |
have csz: "coefficients p \<noteq> []" |
56198 | 1970 |
using assms by (cases p) auto |
56066 | 1971 |
from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] pnormal_last_nonzero[OF H] |
1972 |
show "Ipoly bs (head p) \<noteq> 0" |
|
1973 |
by (simp add: polypoly_def) |
|
33154 | 1974 |
next |
1975 |
assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
|
1976 |
let ?p = "polypoly bs p" |
|
56066 | 1977 |
have csz: "coefficients p \<noteq> []" |
56198 | 1978 |
using assms by (cases p) auto |
56066 | 1979 |
then have pz: "?p \<noteq> []" |
1980 |
by (simp add: polypoly_def) |
|
1981 |
then have lg: "length ?p > 0" |
|
1982 |
by simp |
|
52803 | 1983 |
from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] |
56066 | 1984 |
have lz: "last ?p \<noteq> 0" |
1985 |
by (simp add: polypoly_def) |
|
33154 | 1986 |
from pnormal_last_length[OF lg lz] show "pnormal ?p" . |
1987 |
qed |
|
1988 |
||
1989 |
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1" |
|
1990 |
unfolding isnonconstant_def |
|
52658 | 1991 |
apply (cases p) |
1992 |
apply simp_all |
|
58259 | 1993 |
apply (rename_tac nat a, case_tac nat) |
52658 | 1994 |
apply auto |
33154 | 1995 |
done |
52658 | 1996 |
|
1997 |
lemma isnonconstant_nonconstant: |
|
56198 | 1998 |
assumes "isnonconstant p" |
33154 | 1999 |
shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" |
2000 |
proof |
|
2001 |
let ?p = "polypoly bs p" |
|
2002 |
assume nc: "nonconstant ?p" |
|
56198 | 2003 |
from isnonconstant_pnormal_iff[OF assms, of bs] nc |
56066 | 2004 |
show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
2005 |
unfolding nonconstant_def by blast |
|
33154 | 2006 |
next |
2007 |
let ?p = "polypoly bs p" |
|
2008 |
assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
|
56198 | 2009 |
from isnonconstant_pnormal_iff[OF assms, of bs] h |
56066 | 2010 |
have pn: "pnormal ?p" |
2011 |
by blast |
|
56009 | 2012 |
{ |
2013 |
fix x |
|
2014 |
assume H: "?p = [x]" |
|
2015 |
from H have "length (coefficients p) = 1" |
|
2016 |
unfolding polypoly_def by auto |
|
56198 | 2017 |
with isnonconstant_coefficients_length[OF assms] |
2018 |
have False by arith |
|
56009 | 2019 |
} |
2020 |
then show "nonconstant ?p" |
|
2021 |
using pn unfolding nonconstant_def by blast |
|
33154 | 2022 |
qed |
2023 |
||
56066 | 2024 |
lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p" |
52658 | 2025 |
apply (induct p) |
2026 |
apply (simp_all add: pnormal_def) |
|
2027 |
apply (case_tac "p = []") |
|
2028 |
apply simp_all |
|
2029 |
done |
|
33154 | 2030 |
|
52658 | 2031 |
lemma degree_degree: |
56207 | 2032 |
assumes "isnonconstant p" |
33154 | 2033 |
shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
2034 |
proof |
|
52803 | 2035 |
let ?p = "polypoly bs p" |
33154 | 2036 |
assume H: "degree p = Polynomial_List.degree ?p" |
56207 | 2037 |
from isnonconstant_coefficients_length[OF assms] have pz: "?p \<noteq> []" |
33154 | 2038 |
unfolding polypoly_def by auto |
56207 | 2039 |
from H degree_coefficients[of p] isnonconstant_coefficients_length[OF assms] |
56066 | 2040 |
have lg: "length (pnormalize ?p) = length ?p" |
33154 | 2041 |
unfolding Polynomial_List.degree_def polypoly_def by simp |
56066 | 2042 |
then have "pnormal ?p" |
2043 |
using pnormal_length[OF pz] by blast |
|
56207 | 2044 |
with isnonconstant_pnormal_iff[OF assms] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
56066 | 2045 |
by blast |
33154 | 2046 |
next |
56066 | 2047 |
let ?p = "polypoly bs p" |
33154 | 2048 |
assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
56207 | 2049 |
with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p" |
56066 | 2050 |
by blast |
56207 | 2051 |
with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms] |
52803 | 2052 |
show "degree p = Polynomial_List.degree ?p" |
33154 | 2053 |
unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto |
2054 |
qed |
|
2055 |
||
52658 | 2056 |
|
60533 | 2057 |
section \<open>Swaps ; Division by a certain variable\<close> |
52658 | 2058 |
|
56066 | 2059 |
primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" |
2060 |
where |
|
33154 | 2061 |
"swap n m (C x) = C x" |
56198 | 2062 |
| "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)" |
39246 | 2063 |
| "swap n m (Neg t) = Neg (swap n m t)" |
2064 |
| "swap n m (Add s t) = Add (swap n m s) (swap n m t)" |
|
2065 |
| "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)" |
|
2066 |
| "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)" |
|
2067 |
| "swap n m (Pw t k) = Pw (swap n m t) k" |
|
56066 | 2068 |
| "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)" |
33154 | 2069 |
|
52658 | 2070 |
lemma swap: |
56066 | 2071 |
assumes "n < length bs" |
2072 |
and "m < length bs" |
|
33154 | 2073 |
shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t" |
2074 |
proof (induct t) |
|
52658 | 2075 |
case (Bound k) |
56066 | 2076 |
then show ?case |
2077 |
using assms by simp |
|
33154 | 2078 |
next |
52658 | 2079 |
case (CN c k p) |
56066 | 2080 |
then show ?case |
2081 |
using assms by simp |
|
33154 | 2082 |
qed simp_all |
2083 |
||
52658 | 2084 |
lemma swap_swap_id [simp]: "swap n m (swap m n t) = t" |
2085 |
by (induct t) simp_all |
|
2086 |
||
2087 |
lemma swap_commute: "swap n m p = swap m n p" |
|
2088 |
by (induct p) simp_all |
|
33154 | 2089 |
|
2090 |
lemma swap_same_id[simp]: "swap n n t = t" |
|
52658 | 2091 |
by (induct t) simp_all |
33154 | 2092 |
|
2093 |
definition "swapnorm n m t = polynate (swap n m t)" |
|
2094 |
||
52658 | 2095 |
lemma swapnorm: |
2096 |
assumes nbs: "n < length bs" |
|
2097 |
and mbs: "m < length bs" |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
2098 |
shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field})) = |
52658 | 2099 |
Ipoly ((bs[n:= bs!m])[m:= bs!n]) t" |
41807 | 2100 |
using swap[OF assms] swapnorm_def by simp |
33154 | 2101 |
|
52658 | 2102 |
lemma swapnorm_isnpoly [simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58889
diff
changeset
|
2103 |
assumes "SORT_CONSTRAINT('a::{field_char_0,field})" |
33154 | 2104 |
shows "isnpoly (swapnorm n m p)" |
2105 |
unfolding swapnorm_def by simp |
|
2106 |
||
52803 | 2107 |
definition "polydivideby n s p = |
56000 | 2108 |
(let |
2109 |
ss = swapnorm 0 n s; |
|
2110 |
sp = swapnorm 0 n p; |
|
2111 |
h = head sp; |
|
2112 |
(k, r) = polydivide ss sp |
|
2113 |
in (k, swapnorm 0 n h, swapnorm 0 n r))" |
|
33154 | 2114 |
|
56000 | 2115 |
lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" |
52658 | 2116 |
by (induct p) simp_all |
33154 | 2117 |
|
41808 | 2118 |
fun isweaknpoly :: "poly \<Rightarrow> bool" |
2119 |
where |
|
33154 | 2120 |
"isweaknpoly (C c) = True" |
41808 | 2121 |
| "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p" |
2122 |
| "isweaknpoly p = False" |
|
33154 | 2123 |
|
52803 | 2124 |
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p" |
52658 | 2125 |
by (induct p arbitrary: n0) auto |
33154 | 2126 |
|
52803 | 2127 |
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)" |
52658 | 2128 |
by (induct p) auto |
33154 | 2129 |
|
62390 | 2130 |
end |