# HG changeset patch
# User wenzelm
# Date 1373880559 -7200
# Node ID 1e7896c7f781f6649d7b44e0d30e5633e4a4dbb9
# Parent  42c14dba1daa4161aaf16ffef5973f10afbddf6c
tuned specifications and proofs;

diff -r 42c14dba1daa -r 1e7896c7f781 src/HOL/Decision_Procs/Commutative_Ring.thy
--- a/src/HOL/Decision_Procs/Commutative_Ring.thy	Mon Jul 15 10:42:12 2013 +0200
+++ b/src/HOL/Decision_Procs/Commutative_Ring.thy	Mon Jul 15 11:29:19 2013 +0200
@@ -271,7 +271,7 @@
       with 1 have "\<And>P. Ipol ls (pow l P) = Ipol ls P ^ l" by simp
       moreover
       note Suc `even l` even_nat_plus_one_div_two
-      ultimately show ?thesis by (auto simp add: mul_ci power_Suc even_pow)
+      ultimately show ?thesis by (auto simp add: mul_ci even_pow)
     next
       assume "odd l"
       {
@@ -286,7 +286,7 @@
           have two_times: "2 * (w div 2) = w"
             by (simp only: numerals even_nat_div_two_times_two [OF `even w`])
           have "Ipol ls P * Ipol ls P = Ipol ls P ^ Suc (Suc 0)"
-            by (simp add: power_Suc)
+            by simp
           then have "Ipol ls P * Ipol ls P = Ipol ls P ^ 2"
             by (simp add: numerals)
           with Suc show ?thesis
diff -r 42c14dba1daa -r 1e7896c7f781 src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Mon Jul 15 10:42:12 2013 +0200
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Mon Jul 15 11:29:19 2013 +0200
@@ -8,8 +8,6 @@
 imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
 begin
 
-  (* Implementation *)
-
 subsection{* Datatype of polynomial expressions *} 
 
 datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
@@ -18,8 +16,11 @@
 abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
 abbreviation poly_p :: "int \<Rightarrow> poly" ("'((_)')\<^sub>p") where "(i)\<^sub>p \<equiv> C (i)\<^sub>N"
 
+
 subsection{* Boundedness, substitution and all that *}
-primrec polysize:: "poly \<Rightarrow> nat" where
+
+primrec polysize:: "poly \<Rightarrow> nat"
+where
   "polysize (C c) = 1"
 | "polysize (Bound n) = 1"
 | "polysize (Neg p) = 1 + polysize p"
@@ -29,7 +30,8 @@
 | "polysize (Pw p n) = 1 + polysize p"
 | "polysize (CN c n p) = 4 + polysize c + polysize p"
 
-primrec polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *) where
+primrec polybound0:: "poly \<Rightarrow> bool" -- {* a poly is INDEPENDENT of Bound 0 *}
+where
   "polybound0 (C c) = True"
 | "polybound0 (Bound n) = (n>0)"
 | "polybound0 (Neg a) = polybound0 a"
@@ -39,7 +41,8 @@
 | "polybound0 (Pw p n) = (polybound0 p)"
 | "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
 
-primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *) where
+primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" -- {* substitute a poly into a poly for Bound 0 *}
+where
   "polysubst0 t (C c) = (C c)"
 | "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"
 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
@@ -61,6 +64,7 @@
 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
 | "decrpoly a = a"
 
+
 subsection{* Degrees and heads and coefficients *}
 
 fun degree:: "poly \<Rightarrow> nat"
@@ -104,9 +108,9 @@
   "headconst (CN c n p) = headconst p"
 | "headconst (C n) = n"
 
+
 subsection{* Operations for normalization *}
 
-
 declare if_cong[fundef_cong del]
 declare let_cong[fundef_cong del]
 
@@ -131,8 +135,7 @@
 | "polyneg a = Neg a"
 
 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
-where
-  "p -\<^sub>p q = polyadd p (polyneg q)"
+  where "p -\<^sub>p q = polyadd p (polyneg q)"
 
 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
 where
@@ -173,7 +176,8 @@
 by pat_completeness auto
 termination by (relation "measure polysize") auto
 
-fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
+fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
+where
   "poly_cmul y (C x) = C (y *\<^sub>N x)"
 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
 | "poly_cmul y p = C y *\<^sub>p p"
@@ -181,35 +185,39 @@
 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
   "monic p \<equiv> (let h = headconst p in if h = 0\<^sub>N then (p,False) else ((C (Ninv h)) *\<^sub>p p, 0>\<^sub>N h))"
 
+
 subsection{* Pseudo-division *}
 
-definition shift1 :: "poly \<Rightarrow> poly" where
-  "shift1 p \<equiv> CN 0\<^sub>p 0 p"
+definition shift1 :: "poly \<Rightarrow> poly"
+  where "shift1 p \<equiv> CN 0\<^sub>p 0 p"
 
-abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
-  "funpow \<equiv> compow"
+abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
+  where "funpow \<equiv> compow"
 
 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
-  where
+where
   "polydivide_aux a n p k s = 
-  (if s = 0\<^sub>p then (k,s)
-  else (let b = head s; m = degree s in
-  (if m < n then (k,s) else 
-  (let p'= funpow (m - n) shift1 p in 
-  (if a = b then polydivide_aux a n p k (s -\<^sub>p p') 
-  else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
+    (if s = 0\<^sub>p then (k,s)
+    else (let b = head s; m = degree s in
+    (if m < n then (k,s) else 
+    (let p'= funpow (m - n) shift1 p in 
+    (if a = b then polydivide_aux a n p k (s -\<^sub>p p') 
+    else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
 
-definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
-  "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
+definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)"
+  where "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
 
-fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
+fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
+where
   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
 
-fun poly_deriv :: "poly \<Rightarrow> poly" where
+fun poly_deriv :: "poly \<Rightarrow> poly"
+where
   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
 | "poly_deriv p = 0\<^sub>p"
 
+
 subsection{* Semantics of the polynomial representation *}
 
 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
@@ -233,6 +241,7 @@
 
 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
 
+
 subsection {* Normal form and normalization *}
 
 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
@@ -242,10 +251,10 @@
 | "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)
+  by (induct p rule: isnpolyh.induct) auto
 
-definition isnpoly :: "poly \<Rightarrow> bool" where
-  "isnpoly p \<equiv> isnpolyh p 0"
+definition isnpoly :: "poly \<Rightarrow> bool"
+  where "isnpoly p \<equiv> isnpolyh p 0"
 
 text{* polyadd preserves normal forms *}
 
@@ -304,7 +313,8 @@
 qed auto
 
 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
-by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
+  by (induct p q rule: polyadd.induct)
+    (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
 
 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"
   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
@@ -332,14 +342,14 @@
 qed auto
 
 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
-  by (induct p arbitrary: n rule: headn.induct, auto)
+  by (induct p arbitrary: n rule: headn.induct) auto
 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
-  by (induct p arbitrary: n rule: degree.induct, auto)
+  by (induct p arbitrary: n rule: degree.induct) auto
 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
-  by (induct p arbitrary: n rule: degreen.induct, auto)
+  by (induct p arbitrary: n rule: degreen.induct) auto
 
 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
-  by (induct p arbitrary: n rule: degree.induct, auto)
+  by (induct p arbitrary: n rule: degree.induct) auto
 
 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
   using degree_isnpolyh_Suc by auto
@@ -352,9 +362,9 @@
   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
   using np nq m
 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
-  case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
+  case (2 c c' n' p' n0 n1) thus ?case  by (cases n') simp_all
 next
-  case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
+  case (3 c n p c' n0 n1) thus ?case by (cases n) auto
 next
   case (4 c n p c' n' p' n0 n1 m) 
   have "n' = n \<or> n < n' \<or> n' < n" by arith
@@ -377,7 +387,8 @@
     moreover {assume eq: "n = n'" hence ?case using 4 
         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
         apply (auto simp add: Let_def)
-        by blast
+        apply blast
+        done
       }
     ultimately have ?case by blast}
   ultimately show ?case by blast
@@ -385,13 +396,14 @@
 
 lemma polymul_properties:
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
+    and np: "isnpolyh p n0"
+    and nq: "isnpolyh q n1"
+    and m: "m \<le> min n0 n1"
   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)" 
-  and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)" 
-  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)"
+    and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)" 
+    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)"
   using np nq m
-proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
+proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
   case (2 c c' n' p') 
   { case (1 n0 n1) 
     with "2.hyps"(4-6)[of n' n' n']
@@ -430,7 +442,7 @@
         with "4.hyps"(16-17)[OF cnp np', of "n'"]
           "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
         have ?case
-          by (cases "Suc n' = n", simp_all add: min_def)
+          by (cases "Suc n' = n") (simp_all add: min_def)
       } moreover {
         assume "n' = n"
         with "4.hyps"(16-17)[OF cnp np', of n]
@@ -525,35 +537,41 @@
 qed auto
 
 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
-by(induct p q rule: polymul.induct, auto simp add: field_simps)
+  by (induct p q rule: polymul.induct) (auto simp add: field_simps)
 
 lemma polymul_normh: 
-    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
   using polymul_properties(1)  by blast
+
 lemma polymul_eq0_iff: 
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p) "
   using polymul_properties(2)  by blast
-lemma polymul_degreen:  
+
+lemma polymul_degreen:  (* FIXME duplicate? *)
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 else degreen p m + degreen q m)"
+  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
+    degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 else degreen p m + degreen q m)"
   using polymul_properties(3) by blast
+
 lemma polymul_norm:   
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul p q)"
   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
 
 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
-  by (induct p arbitrary: n0 rule: headconst.induct, auto)
+  by (induct p arbitrary: n0 rule: headconst.induct) auto
 
 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
-  by (induct p arbitrary: n0, auto)
+  by (induct p arbitrary: n0) auto
 
-lemma monic_eqI: assumes np: "isnpolyh p n0" 
-  shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
+lemma monic_eqI:
+  assumes np: "isnpolyh p n0" 
+  shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
+    (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
   unfolding monic_def Let_def
-proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
+proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
   let ?h = "headconst p"
   assume pz: "p \<noteq> 0\<^sub>p"
   {assume hz: "INum ?h = (0::'a)"
@@ -567,14 +585,14 @@
 text{* polyneg is a negation and preserves normal forms *}
 
 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
-by (induct p rule: polyneg.induct, auto)
+  by (induct p rule: polyneg.induct) auto
 
 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
-  by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
+  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
-  by (induct p arbitrary: n0 rule: polyneg.induct, auto)
+  by (induct p arbitrary: n0 rule: polyneg.induct) auto
 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
-by (induct p rule: polyneg.induct, auto simp add: polyneg0)
+  by (induct p rule: polyneg.induct) (auto simp add: polyneg0)
 
 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
   using isnpoly_def polyneg_normh by simp
@@ -583,29 +601,31 @@
 text{* polysub is a substraction and preserves normal forms *}
 
 lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"
-by (simp add: polysub_def polyneg polyadd)
+  by (simp add: polysub_def)
 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
-by (simp add: polysub_def polyneg_normh polyadd_normh)
+  by (simp add: polysub_def polyneg_normh polyadd_normh)
 
 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"
   using polyadd_norm polyneg_norm by (simp add: polysub_def) 
-lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+lemma polysub_same_0[simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
-unfolding polysub_def split_def fst_conv snd_conv
-by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
+  unfolding polysub_def split_def fst_conv snd_conv
+  by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
 
 lemma polysub_0: 
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
   unfolding polysub_def split_def fst_conv snd_conv
   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
-  (auto simp: Nsub0[simplified Nsub_def] Let_def)
+    (auto simp: Nsub0[simplified Nsub_def] Let_def)
 
 text{* polypow is a power function and preserves normal forms *}
 
 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
-proof(induct n rule: polypow.induct)
-  case 1 thus ?case by simp
+proof (induct n rule: polypow.induct)
+  case 1
+  thus ?case by simp
 next
   case (2 n)
   let ?q = "polypow ((Suc n) div 2) p"
@@ -613,19 +633,21 @@
   have "odd (Suc n) \<or> even (Suc n)" by simp
   moreover 
   {assume odd: "odd (Suc n)"
-    have th: "(Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat))))) = Suc n div 2 + Suc n div 2 + 1" by arith
+    have th: "(Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat))))) = Suc n div 2 + Suc n div 2 + 1"
+      by arith
     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)
     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
       using "2.hyps" by simp
     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
-      apply (simp only: power_add power_one_right) by simp
+      by (simp only: power_add power_one_right) simp
     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
       by (simp only: th)
     finally have ?case 
     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
   moreover 
   {assume even: "even (Suc n)"
-    have th: "(Suc (Suc (0\<Colon>nat))) * (Suc n div Suc (Suc (0\<Colon>nat))) = Suc n div 2 + Suc n div 2" by arith
+    have th: "(Suc (Suc (0\<Colon>nat))) * (Suc n div Suc (Suc (0\<Colon>nat))) = Suc n div 2 + Suc n div 2"
+      by arith
     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
       using "2.hyps" apply (simp only: power_add) by simp
@@ -634,7 +656,7 @@
 qed
 
 lemma polypow_normh: 
-    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
 proof (induct k arbitrary: n rule: polypow.induct)
   case (2 k n)
@@ -653,23 +675,28 @@
 
 text{* Finally the whole normalization *}
 
-lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
-by (induct p rule:polynate.induct, auto)
+lemma polynate [simp]:
+  "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
+  by (induct p rule:polynate.induct) auto
 
 lemma polynate_norm[simp]: 
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "isnpoly (polynate p)"
-  by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
+  by (induct p rule: polynate.induct)
+     (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
+      simp_all add: isnpoly_def)
 
 text{* shift1 *}
 
 
 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
-by (simp add: shift1_def polymul)
+  by (simp add: shift1_def)
 
 lemma shift1_isnpoly: 
-  assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
-  using pn pnz by (simp add: shift1_def isnpoly_def )
+  assumes pn: "isnpoly p"
+    and pnz: "p \<noteq> 0\<^sub>p"
+  shows "isnpoly (shift1 p) "
+  using pn pnz by (simp add: shift1_def isnpoly_def)
 
 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
   by (simp add: shift1_def)
@@ -678,18 +705,22 @@
   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
 
 lemma funpow_isnpolyh: 
-  assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
+  assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
+    and np: "isnpolyh p n"
   shows "isnpolyh (funpow k f p) n"
-  using f np by (induct k arbitrary: p, auto)
+  using f np by (induct k arbitrary: p) auto
 
-lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
-  by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
+lemma funpow_shift1:
+  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) =
+    Ipoly bs (Mul (Pw (Bound 0) n) p)"
+  by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
 
 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
 
 lemma funpow_shift1_1: 
-  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
+  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) =
+    Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
   by (simp add: funpow_shift1)
 
 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
@@ -697,23 +728,29 @@
 
 lemma behead:
   assumes np: "isnpolyh p n"
-  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
+  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
+    (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
   using np
 proof (induct p arbitrary: n rule: behead.induct)
   case (1 c p n) hence pn: "isnpolyh p n" by simp
   from 1(1)[OF pn] 
   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" . 
-  then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
-    by (simp_all add: th[symmetric] field_simps power_Suc)
+  then show ?case using "1.hyps"
+    apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
+    apply (simp_all add: th[symmetric] field_simps)
+    done
 qed (auto simp add: Let_def)
 
 lemma behead_isnpolyh:
-  assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
-  using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
+  assumes np: "isnpolyh p n"
+  shows "isnpolyh (behead p) n"
+  using np by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
+
 
 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
+
 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
-proof(induct p arbitrary: n rule: poly.induct, auto)
+proof (induct p arbitrary: n rule: poly.induct, auto)
   case (goal1 c n p n')
   hence "n = Suc (n - 1)" by simp
   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
@@ -721,27 +758,29 @@
 qed
 
 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
-by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
+  by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
 
-lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
+lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
+  by (induct p) auto
 
 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
-  apply (induct p arbitrary: n0, auto)
+  apply (induct p arbitrary: n0)
+  apply auto
   apply (atomize)
   apply (erule_tac x = "Suc nat" in allE)
   apply auto
   done
 
 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
- by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
+  by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
 
 lemma polybound0_I:
   assumes nb: "polybound0 a"
   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
-using nb
-by (induct a rule: poly.induct) auto 
-lemma polysubst0_I:
-  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
+  using nb
+  by (induct a rule: poly.induct) auto 
+
+lemma polysubst0_I: "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
   by (induct t) simp_all
 
 lemma polysubst0_I':
@@ -749,16 +788,18 @@
   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
 
-lemma decrpoly: assumes nb: "polybound0 t"
+lemma decrpoly:
+  assumes nb: "polybound0 t"
   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
-  using nb by (induct t rule: decrpoly.induct, simp_all)
+  using nb by (induct t rule: decrpoly.induct) simp_all
 
-lemma polysubst0_polybound0: assumes nb: "polybound0 t"
+lemma polysubst0_polybound0:
+  assumes nb: "polybound0 t"
   shows "polybound0 (polysubst0 t a)"
-using nb by (induct a rule: poly.induct, auto)
+  using nb by (induct a rule: poly.induct) auto
 
 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
-  by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
+  by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
 
 primrec maxindex :: "poly \<Rightarrow> nat" where
   "maxindex (Bound n) = n + 1"
@@ -770,11 +811,11 @@
 | "maxindex (Pw p n) = maxindex p"
 | "maxindex (C x) = 0"
 
-definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
-  "wf_bs bs p = (length bs \<ge> maxindex p)"
+definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
+  where "wf_bs bs p = (length bs \<ge> maxindex p)"
 
 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
-proof(induct p rule: coefficients.induct)
+proof (induct p rule: coefficients.induct)
   case (1 c p) 
   show ?case 
   proof
@@ -791,12 +832,13 @@
 qed simp_all
 
 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
-by (induct p rule: coefficients.induct, auto)
+  by (induct p rule: coefficients.induct) auto
 
 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
-  unfolding wf_bs_def by (induct p, auto simp add: nth_append)
+  unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
 
-lemma take_maxindex_wf: assumes wf: "wf_bs bs p" 
+lemma take_maxindex_wf:
+  assumes wf: "wf_bs bs p" 
   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
 proof-
   let ?ip = "maxindex p"
@@ -808,7 +850,7 @@
 qed
 
 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
-  by (induct p, auto)
+  by (induct p) auto
 
 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
   unfolding wf_bs_def by simp
@@ -819,22 +861,28 @@
 
 
 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
-by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
+  by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
-  by (induct p rule: coefficients.induct, simp_all)
+  by (induct p rule: coefficients.induct) simp_all
 
 
 lemma coefficients_head: "last (coefficients p) = head p"
-  by (induct p rule: coefficients.induct, auto)
+  by (induct p rule: coefficients.induct) auto
 
 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
-  unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
+  unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
 
 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
   apply (rule exI[where x="replicate (n - length xs) z"])
-  by simp
+  apply simp
+  done
+
 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
-by (cases p, auto) (case_tac "nat", simp_all)
+  apply (cases p)
+  apply auto
+  apply (case_tac "nat")
+  apply simp_all
+  done
 
 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
   unfolding wf_bs_def 
@@ -852,11 +900,12 @@
   done
 
 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
-  unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
+  unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
 
 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
 
+
 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
 
 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
@@ -874,8 +923,8 @@
 
 lemma coefficients_isconst:
   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
-  by (induct p arbitrary: n rule: coefficients.induct, 
-    auto simp add: isnpolyh_Suc_const)
+  by (induct p arbitrary: n rule: coefficients.induct) 
+    (auto simp add: isnpolyh_Suc_const)
 
 lemma polypoly_polypoly':
   assumes np: "isnpolyh p n0"
@@ -896,12 +945,14 @@
 qed
 
 lemma polypoly_poly:
-  assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
+  assumes np: "isnpolyh p n0"
+  shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
   using np 
-by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
+  by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
 
 lemma polypoly'_poly: 
-  assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
+  assumes np: "isnpolyh p n0"
+  shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
 
 
@@ -909,29 +960,34 @@
   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
   shows "polypoly bs p = [Ipoly bs p]"
   using np nb unfolding polypoly_def 
-  by (cases p, auto, case_tac nat, auto)
+  apply (cases p)
+  apply auto
+  apply (case_tac nat)
+  apply auto
+  done
 
 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" 
-  by (induct p rule: head.induct, auto)
+  by (induct p rule: head.induct) auto
 
 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
-  by (cases p,auto)
+  by (cases p) auto
 
 lemma head_eq_headn0: "head p = headn p 0"
-  by (induct p rule: head.induct, simp_all)
+  by (induct p rule: head.induct) simp_all
 
 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
   by (simp add: head_eq_headn0)
 
 lemma isnpolyh_zero_iff: 
-  assumes nq: "isnpolyh p n0" and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0, field_inverse_zero, power})"
+  assumes nq: "isnpolyh p n0"
+    and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0, field_inverse_zero, power})"
   shows "p = 0\<^sub>p"
-using nq eq
+  using nq eq
 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
   case less
   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
   {assume nz: "maxindex p = 0"
-    then obtain c where "p = C c" using np by (cases p, auto)
+    then obtain c where "p = C c" using np by (cases p) auto
     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
   moreover
   {assume nz: "maxindex p \<noteq> 0"
@@ -958,7 +1014,7 @@
       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
-      hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext) 
+      hence "poly (polypoly' (?ts @ xs) p) = poly []" by auto
       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
       with coefficients_head[of p, symmetric]
@@ -975,7 +1031,8 @@
 qed
 
 lemma isnpolyh_unique:  
-  assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
+  assumes np:"isnpolyh p n0"
+    and nq: "isnpolyh q n1"
   shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0, field_inverse_zero, power})) \<longleftrightarrow>  p = q"
 proof(auto)
   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
@@ -989,69 +1046,91 @@
 
 text{* consequences of unicity on the algorithms for polynomial normalization *}
 
-lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
+lemma polyadd_commute:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+    and np: "isnpolyh p n0"
+    and nq: "isnpolyh q n1"
+  shows "p +\<^sub>p q = q +\<^sub>p p"
   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
 
 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
 lemma one_normh: "isnpolyh (1)\<^sub>p n" by simp
+
 lemma polyadd_0[simp]: 
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
+    and np: "isnpolyh p n0"
+  shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] 
     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
 
 lemma polymul_1[simp]: 
-    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np: "isnpolyh p n0" shows "p *\<^sub>p (1)\<^sub>p = p" and "(1)\<^sub>p *\<^sub>p p = p"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+    and np: "isnpolyh p n0"
+  shows "p *\<^sub>p (1)\<^sub>p = p" and "(1)\<^sub>p *\<^sub>p p = p"
   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] 
     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
+
 lemma polymul_0[simp]: 
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np: "isnpolyh p n0" shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
+    and np: "isnpolyh p n0"
+  shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] 
     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
 
 lemma polymul_commute: 
-    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+    and np:"isnpolyh p n0"
+    and nq: "isnpolyh q n1"
   shows "p *\<^sub>p q = q *\<^sub>p p"
-using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a\<Colon>{field_char_0, field_inverse_zero, power}"] by simp
+  using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a\<Colon>{field_char_0, field_inverse_zero, power}"]
+  by simp
 
-declare polyneg_polyneg[simp]
+declare polyneg_polyneg [simp]
   
-lemma isnpolyh_polynate_id[simp]: 
+lemma isnpolyh_polynate_id [simp]: 
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np:"isnpolyh p n0" shows "polynate p = p"
-  using isnpolyh_unique[where ?'a= "'a::{field_char_0, field_inverse_zero}", OF polynate_norm[of p, unfolded isnpoly_def] np] polynate[where ?'a = "'a::{field_char_0, field_inverse_zero}"] by simp
+    and np:"isnpolyh p n0"
+  shows "polynate p = p"
+  using isnpolyh_unique[where ?'a= "'a::{field_char_0, field_inverse_zero}", OF polynate_norm[of p, unfolded isnpoly_def] np] polynate[where ?'a = "'a::{field_char_0, field_inverse_zero}"]
+  by simp
 
 lemma polynate_idempotent[simp]: 
-    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "polynate (polynate p) = polynate p"
   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
 
 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
   unfolding poly_nate_def polypoly'_def ..
-lemma poly_nate_poly: shows "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0, field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
+
+lemma poly_nate_poly:
+  "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0, field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
-  unfolding poly_nate_polypoly' by (auto intro: ext)
+  unfolding poly_nate_polypoly' by auto
+
 
 subsection{* heads, degrees and all that *}
+
 lemma degree_eq_degreen0: "degree p = degreen p 0"
-  by (induct p rule: degree.induct, simp_all)
+  by (induct p rule: degree.induct) simp_all
 
-lemma degree_polyneg: assumes n: "isnpolyh p n"
+lemma degree_polyneg:
+  assumes n: "isnpolyh p n"
   shows "degree (polyneg p) = degree p"
-  using n
-  by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
+  apply (induct p arbitrary: n rule: polyneg.induct)
+  using n apply simp_all
+  apply (case_tac na)
+  apply auto
+  done
 
 lemma degree_polyadd:
   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
-using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
+  using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
 
 
-lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
+lemma degree_polysub:
+  assumes np: "isnpolyh p n0"
+    and nq: "isnpolyh q n1"
   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
 proof-
   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
@@ -1060,24 +1139,25 @@
 
 lemma degree_polysub_samehead: 
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q" 
-  and d: "degree p = degree q"
+    and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q" 
+    and d: "degree p = degree q"
   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
-unfolding polysub_def split_def fst_conv snd_conv
-using np nq h d
-proof(induct p q rule:polyadd.induct)
-  case (1 c c') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) 
+  unfolding polysub_def split_def fst_conv snd_conv
+  using np nq h d
+proof (induct p q rule: polyadd.induct)
+  case (1 c c')
+  thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) 
 next
   case (2 c c' n' p') 
   from 2 have "degree (C c) = degree (CN c' n' p')" by simp
-  hence nz:"n' > 0" by (cases n', auto)
-  hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
+  hence nz:"n' > 0" by (cases n') auto
+  hence "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
   with 2 show ?case by simp
 next
   case (3 c n p c') 
   hence "degree (C c') = degree (CN c n p)" by simp
-  hence nz:"n > 0" by (cases n, auto)
-  hence "head (CN c n p) = CN c n p" by (cases n, auto)
+  hence nz:"n > 0" by (cases n) auto
+  hence "head (CN c n p) = CN c n p" by (cases n) auto
   with 3 show ?case by simp
 next
   case (4 c n p c' n' p')
@@ -1096,36 +1176,43 @@
     moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}
     moreover {assume nz: "n > 0"
       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
-      hence ?case using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] polysub_same_0[OF c'nh, simplified polysub_def] using nn' 4 by (simp add: Let_def)}
+      hence ?case
+        using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] polysub_same_0[OF c'nh, simplified polysub_def]
+        using nn' 4 by (simp add: Let_def)}
     ultimately have ?case by blast}
   moreover
   {assume nn': "n < n'" hence n'p: "n' > 0" by simp 
-    hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
-    have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" using 4 nn' by (cases n', simp_all)
-    hence "n > 0" by (cases n, simp_all)
-    hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
+    hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n') simp_all
+    have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
+      using 4 nn' by (cases n', simp_all)
+    hence "n > 0" by (cases n) simp_all
+    hence headcnp: "head (CN c n p) = CN c n p" by (cases n) auto
     from H(3) headcnp headcnp' nn' have ?case by auto}
   moreover
   {assume nn': "n > n'"  hence np: "n > 0" by simp 
-    hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
+    hence headcnp:"head (CN c n p) = CN c n p"  by (cases n) simp_all
     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
-    from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
-    with degcnpeq have "n' > 0" by (cases n', simp_all)
-    hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
+    from np have degcnp: "degree (CN c n p) = 0" by (cases n) simp_all
+    with degcnpeq have "n' > 0" by (cases n') simp_all
+    hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
     from H(3) headcnp headcnp' nn' have ?case by auto}
   ultimately show ?case  by blast
 qed auto
  
 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
-by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
+  by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
 
 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
-proof(induct k arbitrary: n0 p)
-  case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
+proof (induct k arbitrary: n0 p)
+  case 0
+  thus ?case by auto
+next
+  case (Suc k n0 p)
+  hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
     and "head (shift1 p) = head p" by (simp_all add: shift1_head) 
   thus ?case by (simp add: funpow_swap1)
-qed auto  
+qed
 
 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
   by (simp add: shift1_def)
@@ -1133,47 +1220,52 @@
   by (induct k arbitrary: p) (auto simp add: shift1_degree)
 
 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
-  by (induct n arbitrary: p) (simp_all add: funpow.simps)
+  by (induct n arbitrary: p) simp_all
 
 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
-  by (induct p arbitrary: n rule: degree.induct, auto)
+  by (induct p arbitrary: n rule: degree.induct) auto
 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
-  by (induct p arbitrary: n rule: degreen.induct, auto)
+  by (induct p arbitrary: n rule: degreen.induct) auto
 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
-  by (induct p arbitrary: n rule: degree.induct, auto)
+  by (induct p arbitrary: n rule: degree.induct) auto
 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
-  by (induct p rule: head.induct, auto)
+  by (induct p rule: head.induct) auto
 
 lemma polyadd_eq_const_degree: 
-  "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> \<Longrightarrow> degree p = degree q" 
+  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
 
-lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
-  and deg: "degree p \<noteq> degree q"
+lemma polyadd_head:
+  assumes np: "isnpolyh p n0"
+    and nq: "isnpolyh q n1"
+    and deg: "degree p \<noteq> degree q"
   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
-using np nq deg
-apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
-apply (case_tac n', simp, simp)
-apply (case_tac n, simp, simp)
-apply (case_tac n, case_tac n', simp add: Let_def)
-apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
-apply (auto simp add: polyadd_eq_const_degree)
-apply (metis head_nz)
-apply (metis head_nz)
-apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
-by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
+  using np nq deg
+  apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
+  using np
+  apply simp_all
+  apply (case_tac n', simp, simp)
+  apply (case_tac n, simp, simp)
+  apply (case_tac n, case_tac n', simp add: Let_def)
+  apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
+  apply (auto simp add: polyadd_eq_const_degree)
+  apply (metis head_nz)
+  apply (metis head_nz)
+  apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
+  apply (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
+  done
 
 lemma polymul_head_polyeq: 
-   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "\<lbrakk>isnpolyh p n0; isnpolyh q n1 ; p \<noteq> 0\<^sub>p ; q \<noteq> 0\<^sub>p \<rbrakk> \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
   case (2 c c' n' p' n0 n1)
   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"  by (simp_all add: head_isnpolyh)
-  thus ?case using 2 by (cases n', auto) 
+  thus ?case using 2 by (cases n') auto
 next 
   case (3 c n p c' n0 n1) 
   hence "isnpolyh (head (CN c n p)) n0" "isnormNum c'"  by (simp_all add: head_isnpolyh)
-  thus ?case using 3 by (cases n, auto)
+  thus ?case using 3 by (cases n) auto
 next
   case (4 c n p c' n' p' n0 n1)
   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
@@ -1182,13 +1274,22 @@
   have "n < n' \<or> n' < n \<or> n = n'" by arith
   moreover 
   {assume nn': "n < n'" hence ?case 
-      using norm 
-    "4.hyps"(2)[OF norm(1,6)]
-    "4.hyps"(1)[OF norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
+      using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
+      apply simp
+      apply (cases n)
+      apply simp
+      apply (cases n')
+      apply simp_all
+      done }
   moreover {assume nn': "n'< n"
-    hence ?case using norm "4.hyps"(6) [OF norm(5,3)]
-      "4.hyps"(5)[OF norm(5,4)] 
-      by (simp,cases n',simp,cases n,auto)}
+    hence ?case
+      using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] 
+      apply simp
+      apply (cases n')
+      apply simp
+      apply (cases n)
+      apply auto
+      done }
   moreover {assume nn': "n' = n"
     from nn' polymul_normh[OF norm(5,4)] 
     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
@@ -1218,44 +1319,50 @@
 qed simp_all
 
 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
-  by(induct p rule: degree.induct, auto)
+  by (induct p rule: degree.induct) auto
 
 lemma degree_head[simp]: "degree (head p) = 0"
-  by (induct p rule: head.induct, auto)
+  by (induct p rule: head.induct) auto
 
 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
-  by (cases n, simp_all)
+  by (cases n) simp_all
 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
-  by (cases n, simp_all)
+  by (cases n) simp_all
 
-lemma polyadd_different_degree: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow> degree (polyadd p q) = max (degree p) (degree q)"
+lemma polyadd_different_degree:
+  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow>
+    degree (polyadd p q) = max (degree p) (degree q)"
   using polyadd_different_degreen degree_eq_degreen0 by simp
 
 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
-  by (induct p arbitrary: n0 rule: polyneg.induct, auto)
+  by (induct p arbitrary: n0 rule: polyneg.induct) auto
 
 lemma degree_polymul:
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
+    and np: "isnpolyh p n0"
+    and nq: "isnpolyh q n1"
   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
 
 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
-  by (induct p arbitrary: n rule: degree.induct, auto)
+  by (induct p arbitrary: n rule: degree.induct) auto
 
 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
-  by (induct p arbitrary: n rule: degree.induct, auto)
+  by (induct p arbitrary: n rule: degree.induct) auto
+
 
 subsection {* Correctness of polynomial pseudo division *}
 
 lemma polydivide_aux_properties:
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
-  and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
+    and np: "isnpolyh p n0"
+    and ns: "isnpolyh s n1"
+    and ap: "head p = a"
+    and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p) 
           \<and> (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
   using ns
-proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
+proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
   case less
   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) 
@@ -1272,12 +1379,20 @@
   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap 
   have nakk':"isnpolyh ?akk' 0" by blast
-  {assume sz: "s = 0\<^sub>p"
-   hence ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }
+  { assume sz: "s = 0\<^sub>p"
+    hence ?ths using np polydivide_aux.simps
+      apply clarsimp
+      apply (rule exI[where x="0\<^sub>p"])
+      apply simp
+      done }
   moreover
   {assume sz: "s \<noteq> 0\<^sub>p"
     {assume dn: "degree s < n"
-      hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }
+      hence "?ths" using ns ndp np polydivide_aux.simps
+        apply auto
+        apply (rule exI[where x="0\<^sub>p"])
+        apply simp
+        done }
     moreover 
     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
       have degsp': "degree s = degree ?p'" 
@@ -1332,9 +1447,14 @@
         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
           from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0, field_inverse_zero}"]
           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
-          hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)" using np nxdn apply simp
-            by (simp only: funpow_shift1_1) simp
-          hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
+          hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
+            using np nxdn
+            apply simp
+            apply (simp only: funpow_shift1_1)
+            apply simp
+            done
+          hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
+            by blast
           {assume h1: "polydivide_aux a n p k s = (k',r)"
             from polydivide_aux.simps sz dn' ba
             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
@@ -1385,12 +1505,16 @@
             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
               
             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
-            have "Ipoly bs (a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
-            hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
-              by (simp add: field_simps power_Suc)
-            hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
-              by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
-            hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
+            have "Ipoly bs (a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
+              by simp
+            hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
+              Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
+              by (simp add: field_simps)
+            hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
+              Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
+              by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
+            hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
+              Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
               by (simp add: field_simps)}
             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
               Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)" by auto 
@@ -1415,7 +1539,8 @@
             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
         }
-        hence hth: "\<forall> (bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
+        hence hth: "\<forall> (bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) =
+            Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
           from hth
           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)" 
             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns] 
@@ -1426,17 +1551,20 @@
           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
-          have ?ths apply (clarsimp simp add: Let_def)
-            apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
+          have ?ths
+            apply (clarsimp simp add: Let_def)
+            apply (rule exI[where x="?b *\<^sub>p ?xdn"])
+            apply simp
             apply (rule exI[where x="0"], simp)
-            done}
-        hence ?ths by blast}
-        ultimately have ?ths using  degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
-          head_nz[OF np] pnz sz ap[symmetric]
+            done }
+        hence ?ths by blast }
+        ultimately have ?ths
+          using  degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
+            head_nz[OF np] pnz sz ap[symmetric]
           by (simp add: degree_eq_degreen0[symmetric]) blast }
       ultimately have ?ths by blast
     }
-    ultimately have ?ths by blast}
+    ultimately have ?ths by blast }
   ultimately show ?ths by blast
 qed
 
@@ -1462,16 +1590,18 @@
     done
 qed
 
+
 subsection{* More about polypoly and pnormal etc *}
 
 definition "isnonconstant p = (\<not> isconstant p)"
 
-lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p" 
+lemma isnonconstant_pnormal_iff:
+  assumes nc: "isnonconstant p" 
   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" 
 proof
   let ?p = "polypoly bs p"  
   assume H: "pnormal ?p"
-  have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
+  have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
   
   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]  
     pnormal_last_nonzero[OF H]
@@ -1479,7 +1609,7 @@
 next
   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
   let ?p = "polypoly bs p"
-  have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
+  have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
   hence pz: "?p \<noteq> []" by (simp add: polypoly_def) 
   hence lg: "length ?p > 0" by simp
   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] 
@@ -1489,10 +1619,14 @@
 
 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
   unfolding isnonconstant_def
-  apply (cases p, simp_all)
-  apply (case_tac nat, auto)
+  apply (cases p)
+  apply simp_all
+  apply (case_tac nat)
+  apply auto
   done
-lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
+
+lemma isnonconstant_nonconstant:
+  assumes inc: "isnonconstant p"
   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
 proof
   let ?p = "polypoly bs p"
@@ -1511,12 +1645,14 @@
 qed
 
 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
-  unfolding pnormal_def
- apply (induct p)
- apply (simp_all, case_tac "p=[]", simp_all)
- done
+  apply (induct p)
+  apply (simp_all add: pnormal_def)
+  apply (case_tac "p = []")
+  apply simp_all
+  done
 
-lemma degree_degree: assumes inc: "isnonconstant p"
+lemma degree_degree:
+  assumes inc: "isnonconstant p"
   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
 proof
   let  ?p = "polypoly bs p"
@@ -1538,7 +1674,9 @@
     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
 qed
 
+
 section{* Swaps ; Division by a certain variable *}
+
 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
   "swap n m (C x) = C x"
 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
@@ -1550,37 +1688,47 @@
 | "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)"
 
-lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
+lemma swap:
+  assumes nbs: "n < length bs"
+    and mbs: "m < length bs"
   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
 proof (induct t)
-  case (Bound k) thus ?case using nbs mbs by simp 
+  case (Bound k)
+  thus ?case using nbs mbs by simp 
 next
-  case (CN c k p) thus ?case using nbs mbs by simp 
+  case (CN c k p)
+  thus ?case using nbs mbs by simp 
 qed simp_all
-lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
-  by (induct t,simp_all)
 
-lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
+lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
+  by (induct t) simp_all
+
+lemma swap_commute: "swap n m p = swap m n p"
+  by (induct p) simp_all
 
 lemma swap_same_id[simp]: "swap n n t = t"
-  by (induct t, simp_all)
+  by (induct t) simp_all
 
 definition "swapnorm n m t = polynate (swap n m t)"
 
-lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
-  shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
+lemma swapnorm:
+  assumes nbs: "n < length bs"
+    and mbs: "m < length bs"
+  shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) =
+    Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
   using swap[OF assms] swapnorm_def by simp
 
-lemma swapnorm_isnpoly[simp]: 
-    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+lemma swapnorm_isnpoly [simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "isnpoly (swapnorm n m p)"
   unfolding swapnorm_def by simp
 
 definition "polydivideby n s p = 
-    (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
-     in (k,swapnorm 0 n h,swapnorm 0 n r))"
+  (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
+   in (k,swapnorm 0 n h,swapnorm 0 n r))"
 
-lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
+lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)"
+  by (induct p) simp_all
 
 fun isweaknpoly :: "poly \<Rightarrow> bool"
 where
@@ -1589,9 +1737,9 @@
 | "isweaknpoly p = False"
 
 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p" 
-  by (induct p arbitrary: n0, auto)
+  by (induct p arbitrary: n0) auto
 
 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)" 
-  by (induct p, auto)
+  by (induct p) auto
 
 end
\ No newline at end of file