author krauss Mon, 21 Feb 2011 23:14:36 +0100 changeset 41808 9f436d00248f parent 41806 173e1b50d5c5 child 41809 6799f95479e2
recdef -> function
```--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Mon Feb 21 18:29:47 2011 +0100
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Mon Feb 21 23:14:36 2011 +0100
@@ -50,75 +50,72 @@
| "polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
else CN (polysubst0 t c) n (polysubst0 t p))"

-consts
-  decrpoly:: "poly \<Rightarrow> poly"
-recdef decrpoly "measure polysize"
+fun decrpoly:: "poly \<Rightarrow> poly"
+where
"decrpoly (Bound n) = Bound (n - 1)"
-  "decrpoly (Neg a) = Neg (decrpoly a)"
-  "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
-  "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
-  "decrpoly (Pw p n) = Pw (decrpoly p) n"
-  "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
-  "decrpoly a = a"
+| "decrpoly (Neg a) = Neg (decrpoly a)"
+| "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
+| "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
+| "decrpoly (Pw p n) = Pw (decrpoly p) n"
+| "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
+| "decrpoly a = a"

subsection{* Degrees and heads and coefficients *}

-consts degree:: "poly \<Rightarrow> nat"
-recdef degree "measure size"
+fun degree:: "poly \<Rightarrow> nat"
+where
"degree (CN c 0 p) = 1 + degree p"
-  "degree p = 0"
+| "degree p = 0"

+where
-  (* More general notions of degree and head *)
-consts degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
-recdef degreen "measure size"
+
+(* More general notions of degree and head *)
+fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
+where
"degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
-  "degreen p = (\<lambda>m. 0)"
-
-consts headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
-  "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
-  "headn p = (\<lambda>m. p)"
+ |"degreen p = (\<lambda>m. 0)"

-consts coefficients:: "poly \<Rightarrow> poly list"
-recdef coefficients "measure size"
-  "coefficients (CN c 0 p) = c#(coefficients p)"
-  "coefficients p = [p]"
+fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
+where
+  "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
+| "headn p = (\<lambda>m. p)"

-consts isconstant:: "poly \<Rightarrow> bool"
-recdef isconstant "measure size"
-  "isconstant (CN c 0 p) = False"
-  "isconstant p = True"
+fun coefficients:: "poly \<Rightarrow> poly list"
+where
+  "coefficients (CN c 0 p) = c#(coefficients p)"
+| "coefficients p = [p]"

-  "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
+fun isconstant:: "poly \<Rightarrow> bool"
+where
+  "isconstant (CN c 0 p) = False"
+| "isconstant p = True"

+where
+  "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
+
+where
-  "headconst (C n) = n"
+| "headconst (C n) = n"

subsection{* Operations for normalization *}
consts
-  polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
polysub :: "poly\<times>poly \<Rightarrow> poly"
polymul :: "poly\<times>poly \<Rightarrow> poly"
-  polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
+
abbreviation poly_add :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
where "a +\<^sub>p b \<equiv> polyadd (a,b)"
abbreviation poly_mul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
where "a *\<^sub>p b \<equiv> polymul (a,b)"
abbreviation poly_sub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
where "a -\<^sub>p b \<equiv> polysub (a,b)"
-abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
-  where "a ^\<^sub>p k \<equiv> polypow k a"

recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)"
"polyadd (C c, C c') = C (c+\<^sub>Nc')"
@@ -133,10 +130,11 @@
(hints recdef_simp add: Let_def measure_def split_def inv_image_def recdef_cong del: if_cong)

-recdef polyneg "measure size"
+fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
+where
"polyneg (C c) = C (~\<^sub>N c)"
-  "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
-  "polyneg a = Neg a"
+| "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
+| "polyneg a = Neg a"

defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)"

@@ -152,21 +150,28 @@
then CN (polymul(CN c n p,c')) n' (polymul(CN c n p,p'))
else polyadd(polymul(CN c n p, c'),CN 0\<^sub>p n' (polymul(CN c n p, p'))))"
"polymul (a,b) = Mul a b"
-recdef polypow "measure id"
+
+fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
+where
"polypow 0 = (\<lambda>p. 1\<^sub>p)"
-  "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in
+| "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in
if even n then d else polymul(p,d))"

-consts polynate :: "poly \<Rightarrow> poly"
-recdef polynate "measure polysize"
+abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
+  where "a ^\<^sub>p k \<equiv> polypow k a"
+
+function polynate :: "poly \<Rightarrow> poly"
+where
"polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
-  "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
-  "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
-  "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
-  "polynate (Neg p) = (~\<^sub>p (polynate p))"
-  "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
-  "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
-  "polynate (C c) = C (normNum c)"
+| "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
+| "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
+| "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
+| "polynate (Neg p) = (~\<^sub>p (polynate p))"
+| "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
+| "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
+| "polynate (C c) = C (normNum c)"
+by pat_completeness auto
+termination by (relation "measure polysize") auto

fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
"poly_cmul y (C x) = C (y *\<^sub>N x)"
@@ -235,11 +240,11 @@

subsection {* Normal form and normalization *}

-consts isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
-recdef isnpolyh "measure size"
+fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
+where
"isnpolyh (C c) = (\<lambda>k. isnormNum c)"
-  "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))"
-  "isnpolyh p = (\<lambda>k. False)"
+| "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))"
+| "isnpolyh p = (\<lambda>k. False)"

lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
by (induct p rule: isnpolyh.induct, auto)
@@ -1618,11 +1623,11 @@

lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)

-consts isweaknpoly :: "poly \<Rightarrow> bool"
-recdef isweaknpoly "measure size"
+fun isweaknpoly :: "poly \<Rightarrow> bool"
+where
"isweaknpoly (C c) = True"
-  "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
-  "isweaknpoly p = False"
+| "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
+| "isweaknpoly p = False"

lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
by (induct p arbitrary: n0, auto)```