# HG changeset patch
# User wenzelm
# Date 1375218977 -7200
# Node ID bcaa5bbf7e6b928cf7cada4e77283c92d70ad03d
# Parent  0b98561d07907ae2572bc99864bed3fcb3a73485
tuned proofs;

diff -r 0b98561d0790 -r bcaa5bbf7e6b src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Tue Jul 30 22:43:11 2013 +0200
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Tue Jul 30 23:16:17 2013 +0200
@@ -8,7 +8,7 @@
 imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
 begin
 
-subsection{* Datatype of polynomial expressions *} 
+subsection{* Datatype of polynomial expressions *}
 
 datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
@@ -36,7 +36,7 @@
 | "polybound0 (Bound n) = (n>0)"
 | "polybound0 (Neg a) = polybound0 a"
 | "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
-| "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)" 
+| "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)"
 | "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
 | "polybound0 (Pw p n) = (polybound0 p)"
 | "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
@@ -47,13 +47,13 @@
 | "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"
 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
 | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
-| "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)" 
+| "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
 | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
 | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
 | "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))"
 
-fun decrpoly:: "poly \<Rightarrow> poly" 
+fun decrpoly:: "poly \<Rightarrow> poly"
 where
   "decrpoly (Bound n) = Bound (n - 1)"
 | "decrpoly (Neg a) = Neg (decrpoly a)"
@@ -117,12 +117,12 @@
 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
 where
   "polyadd (C c) (C c') = C (c+\<^sub>Nc')"
-|  "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
+| "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
 | "polyadd (CN c n p) (CN c' n' p') =
     (if n < n' then CN (polyadd c (CN c' n' p')) n p
      else if n'<n then CN (polyadd (CN c n p) c') n' p'
-     else (let cc' = polyadd c c' ; 
+     else (let cc' = polyadd c c' ;
                pp' = polyadd p p'
            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
 | "polyadd a b = Add a b"
@@ -140,13 +140,13 @@
 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
 where
   "polymul (C c) (C c') = C (c*\<^sub>Nc')"
-| "polymul (C c) (CN c' n' p') = 
+| "polymul (C c) (CN c' n' p') =
       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
-| "polymul (CN c n p) (C c') = 
+| "polymul (CN c n p) (C c') =
       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
-| "polymul (CN c n p) (CN c' n' p') = 
+| "polymul (CN c n p) (CN c' n' p') =
   (if n<n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
-  else if n' < n 
+  else if n' < n
   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"
@@ -157,7 +157,7 @@
 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)"
 
 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
@@ -196,13 +196,15 @@
 
 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
 where
-  "polydivide_aux a n p k s = 
+  "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')))))))"
+     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"
@@ -234,9 +236,9 @@
   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
 
-lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i" 
+lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
   by (simp add: INum_def)
-lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j" 
+lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
   by (simp  add: INum_def)
 
 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
@@ -258,49 +260,52 @@
 
 text{* polyadd preserves normal forms *}
 
-lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> 
+lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
       \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
-proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
+proof (induct p q arbitrary: n0 n1 rule: polyadd.induct)
   case (2 ab c' n' p' n0 n1)
-  from 2 have  th1: "isnpolyh (C ab) (Suc n')" by simp 
+  from 2 have  th1: "isnpolyh (C ab) (Suc n')" by simp
   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp
-  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
+  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
   thus ?case using 2 th3 by simp
 next
   case (3 c' n' p' ab n1 n0)
-  from 3 have  th1: "isnpolyh (C ab) (Suc n')" by simp 
+  from 3 have  th1: "isnpolyh (C ab) (Suc n')" by simp
   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp
-  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
+  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
   thus ?case using 3 th3 by simp
 next
   case (4 c n p c' n' p' n0 n1)
   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
-  from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all 
+  from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
   from 4 have ngen0: "n \<ge> n0" by simp
-  from 4 have n'gen1: "n' \<ge> n1" by simp 
+  from 4 have n'gen1: "n' \<ge> n1" by simp
   have "n < n' \<or> n' < n \<or> n = n'" by auto
-  moreover {assume eq: "n = n'"
-    with "4.hyps"(3)[OF nc nc'] 
+  moreover {
+    assume eq: "n = n'"
+    with "4.hyps"(3)[OF nc nc']
     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
-    from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
-  moreover {assume lt: "n < n'"
+    from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def) }
+  moreover {
+    assume lt: "n < n'"
     have "min n0 n1 \<le> n0" by simp
-    with 4 lt have th1:"min n0 n1 \<le> n" by auto 
+    with 4 lt have th1:"min n0 n1 \<le> n" by auto
     from 4 have th21: "isnpolyh c (Suc n)" by simp
     from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp
     from lt have th23: "min (Suc n) n' = Suc n" by arith
     from "4.hyps"(1)[OF th21 th22]
     have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp
-    with 4 lt th1 have ?case by simp } 
-  moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
+    with 4 lt th1 have ?case by simp }
+  moreover {
+    assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
     have "min n0 n1 \<le> n1"  by simp
     with 4 gt have th1:"min n0 n1 \<le> n'" by auto
     from 4 have th21: "isnpolyh c' (Suc n')" by simp_all
@@ -308,8 +313,8 @@
     from gt have th23: "min n (Suc n') = Suc n'" by arith
     from "4.hyps"(2)[OF th22 th21]
     have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp
-    with 4 gt th1 have ?case by simp}
-      ultimately show ?case by blast
+    with 4 gt th1 have ?case by simp }
+  ultimately show ?case by blast
 qed auto
 
 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
@@ -321,8 +326,8 @@
 
 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
 
-lemma polyadd_different_degreen: 
-  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 
+lemma polyadd_different_degreen:
+  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
   degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
   case (4 c n p c' n' p' m n0 n1)
@@ -362,11 +367,13 @@
   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) 
+  case (4 c n p c' n' p' n0 n1 m)
   have "n' = n \<or> n < n' \<or> n' < n" by arith
   thus ?case
   proof (elim disjE)
@@ -376,21 +383,21 @@
   qed simp_all
 qed auto
 
-lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> 
+lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk>
   \<Longrightarrow> degreen p m = degreen q m"
 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
-  case (4 c n p c' n' p' m n0 n1 x) 
-  {assume nn': "n' < n" hence ?case using 4 by simp}
-  moreover 
-  {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
-    moreover {assume "n < n'" with 4 have ?case by simp }
-    moreover {assume eq: "n = n'" hence ?case using 4 
+  case (4 c n p c' n' p' m n0 n1 x)
+  { assume nn': "n' < n" hence ?case using 4 by simp }
+  moreover
+  { assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
+    moreover { assume "n < n'" with 4 have ?case by simp }
+    moreover { assume eq: "n = n'" hence ?case using 4
         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
         apply (auto simp add: Let_def)
         apply blast
         done
-      }
-    ultimately have ?case by blast}
+    }
+    ultimately have ?case by blast }
   ultimately show ?case by blast
 qed simp_all
 
@@ -399,37 +406,37 @@
     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)" 
+  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)"
   using np nq m
 proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
-  case (2 c c' n' p') 
-  { case (1 n0 n1) 
+  case (2 c c' n' p')
+  { case (1 n0 n1)
     with "2.hyps"(4-6)[of n' n' n']
       and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
     show ?case by (auto simp add: min_def)
   next
-    case (2 n0 n1) thus ?case by auto 
+    case (2 n0 n1) thus ?case by auto
   next
-    case (3 n0 n1) thus ?case  using "2.hyps" by auto } 
+    case (3 n0 n1) thus ?case  using "2.hyps" by auto }
 next
   case (3 c n p c')
-  { case (1 n0 n1) 
+  { case (1 n0 n1)
     with "3.hyps"(4-6)[of n n n]
       "3.hyps"(1-3)[of "Suc n" "Suc n" n]
     show ?case by (auto simp add: min_def)
   next
     case (2 n0 n1) thus ?case by auto
   next
-    case (3 n0 n1) thus ?case  using "3.hyps" by auto } 
+    case (3 n0 n1) thus ?case  using "3.hyps" by auto }
 next
   case (4 c n p c' n' p')
   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
     {
       case (1 n0 n1)
       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
-        and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)" 
+        and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
         by simp_all
@@ -462,23 +469,24 @@
       let ?d2 = "degreen ?cnp' m"
       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
       have "n'<n \<or> n < n' \<or> n' = n" by auto
-      moreover 
+      moreover
       {assume "n' < n \<or> n < n'"
-        with "4.hyps"(3,6,18) np np' m 
+        with "4.hyps"(3,6,18) np np' m
         have ?eq by auto }
       moreover
-      {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
+      { assume nn': "n' = n"
+        hence nn: "\<not> n' < n \<and> \<not> n < n'" by arith
         from "4.hyps"(16,18)[of n n' n]
           "4.hyps"(13,14)[of n "Suc n'" n]
           np np' nn'
         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
-          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
+          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
-        {assume mn: "m = n" 
+        { assume mn: "m = n"
           from "4.hyps"(17,18)[OF norm(1,4), of n]
             "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
-          have degs:  "degreen (?cnp *\<^sub>p c') n = 
+          have degs:  "degreen (?cnp *\<^sub>p c') n =
             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
           from degs norm
@@ -487,31 +495,31 @@
             by simp
           have nmin: "n \<le> min n n" by (simp add: min_def)
           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
-          have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n = degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 
+          have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n = degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
           from "4.hyps"(16-18)[OF norm(1,4), of n]
             "4.hyps"(13-15)[OF norm(1,2), of n]
             mn norm m nn' deg
-          have ?eq by simp}
+          have ?eq by simp }
         moreover
-        {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
-          from nn' m np have max1: "m \<le> max n n"  by simp 
-          hence min1: "m \<le> min n n" by simp     
+        { assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
+          from nn' m np have max1: "m \<le> max n n"  by simp
+          hence min1: "m \<le> min n n" by simp
           hence min2: "m \<le> min n (Suc n)" by simp
           from "4.hyps"(16-18)[OF norm(1,4) min1]
             "4.hyps"(13-15)[OF norm(1,2) min2]
             degreen_polyadd[OF norm(3,6) max1]
 
-          have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m 
+          have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
             \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
             using mn nn' np np' by simp
           with "4.hyps"(16-18)[OF norm(1,4) min1]
             "4.hyps"(13-15)[OF norm(1,2) min2]
             degreen_0[OF norm(3) mn']
-          have ?eq using nn' mn np np' by clarsimp}
-        ultimately have ?eq by blast}
-      ultimately show ?eq by blast}
+          have ?eq using nn' mn np np' by clarsimp }
+        ultimately have ?eq by blast }
+      ultimately show ?eq by blast }
     { case (2 n0 n1)
-      hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1" 
+      hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
         and m: "m \<le> min n0 n1" by simp_all
       hence mn: "m \<le> n" by simp
       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
@@ -522,32 +530,32 @@
           np np' C(2) mn
         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
-          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
-          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" 
+          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
+          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
           by (simp_all add: min_def)
-            
+
           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
-          have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" 
+          have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
             using norm by simp
         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
         have "False" by simp }
-      thus ?case using "4.hyps" by clarsimp}
+      thus ?case using "4.hyps" by clarsimp }
 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)
 
-lemma polymul_normh: 
+lemma polymul_normh:
   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
+  using polymul_properties(1) by blast
 
-lemma polymul_eq0_iff: 
+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
+  using polymul_properties(2) by blast
 
 lemma polymul_degreen:  (* FIXME duplicate? *)
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
@@ -555,7 +563,7 @@
     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:   
+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
@@ -567,14 +575,14 @@
   by (induct p arbitrary: n0) auto
 
 lemma monic_eqI:
-  assumes np: "isnpolyh p n0" 
+  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])
   let ?h = "headconst p"
   assume pz: "p \<noteq> 0\<^sub>p"
-  {assume hz: "INum ?h = (0::'a)"
+  { assume hz: "INum ?h = (0::'a)"
     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
@@ -602,18 +610,19 @@
 
 lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"
   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)"
+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)
 
 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"
-  using polyadd_norm polyneg_norm by (simp add: polysub_def) 
+  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})"
   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])
 
-lemma polysub_0: 
+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
@@ -631,8 +640,8 @@
   let ?q = "polypow ((Suc n) div 2) p"
   let ?d = "polymul ?q ?q"
   have "odd (Suc n) \<or> even (Suc n)" by simp
-  moreover 
-  {assume odd: "odd (Suc n)"
+  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
     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)
@@ -642,10 +651,10 @@
       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 
+    finally have ?case
     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
-  moreover 
-  {assume even: "even (Suc n)"
+  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
     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
@@ -655,7 +664,7 @@
   ultimately show ?case by blast
 qed
 
-lemma polypow_normh: 
+lemma polypow_normh:
   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)
@@ -666,9 +675,9 @@
   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp
   from dn on show ?case by (simp add: Let_def)
-qed auto 
+qed auto
 
-lemma polypow_norm:   
+lemma polypow_norm:
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
   by (simp add: polypow_normh isnpoly_def)
@@ -679,7 +688,7 @@
   "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]: 
+lemma polynate_norm[simp]:
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "isnpoly (polynate p)"
   by (induct p rule: polynate.induct)
@@ -692,7 +701,7 @@
 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
   by (simp add: shift1_def)
 
-lemma shift1_isnpoly: 
+lemma shift1_isnpoly:
   assumes pn: "isnpoly p"
     and pnz: "p \<noteq> 0\<^sub>p"
   shows "isnpoly (shift1 p) "
@@ -700,11 +709,11 @@
 
 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
   by (simp add: shift1_def)
-lemma funpow_shift1_isnpoly: 
+lemma funpow_shift1_isnpoly:
   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
 
-lemma funpow_isnpolyh: 
+lemma funpow_isnpolyh:
   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
     and np: "isnpolyh p n"
   shows "isnpolyh (funpow k f p) n"
@@ -718,7 +727,7 @@
 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: 
+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)"
   by (simp add: funpow_shift1)
@@ -733,8 +742,8 @@
   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" . 
+  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")
     apply (simp_all add: th[symmetric] field_simps)
@@ -778,7 +787,7 @@
   assumes nb: "polybound0 a"
   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
   using nb
-  by (induct a rule: poly.induct) auto 
+  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
@@ -816,15 +825,15 @@
 
 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)
-  case (1 c p) 
-  show ?case 
+  case (1 c p)
+  show ?case
   proof
     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
     hence "x = c \<or> x \<in> set (coefficients p)" by simp
-    moreover 
+    moreover
     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
-    moreover 
-    {assume H: "x \<in> set (coefficients p)" 
+    moreover
+    {assume H: "x \<in> set (coefficients p)"
       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
       with "1.hyps" H have "wf_bs bs x" by blast }
     ultimately  show "wf_bs bs x" by blast
@@ -838,7 +847,7 @@
   unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
 
 lemma take_maxindex_wf:
-  assumes wf: "wf_bs bs p" 
+  assumes wf: "wf_bs bs p"
   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
 proof-
   let ?ip = "maxindex p"
@@ -885,14 +894,14 @@
   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 
+  unfolding wf_bs_def
   apply (induct p q rule: polyadd.induct)
   apply (auto simp add: Let_def)
   done
 
 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
-  unfolding wf_bs_def 
-  apply (induct p q arbitrary: bs rule: polymul.induct) 
+  unfolding wf_bs_def
+  apply (induct p q arbitrary: bs rule: polymul.induct)
   apply (simp_all add: wf_bs_polyadd)
   apply clarsimp
   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
@@ -918,12 +927,12 @@
   have cp: "isnpolyh (CN c 0 p) n0" by fact
   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
     by (auto simp add: isnpolyh_mono[where n'=0])
-  from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp 
+  from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
 qed auto
 
 lemma coefficients_isconst:
   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
-  by (induct p arbitrary: n rule: coefficients.induct) 
+  by (induct p arbitrary: n rule: coefficients.induct)
     (auto simp add: isnpolyh_Suc_const)
 
 lemma polypoly_polypoly':
@@ -940,17 +949,17 @@
   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
     by auto
-  
-  thus ?thesis unfolding polypoly_def polypoly'_def by simp 
+
+  thus ?thesis unfolding polypoly_def polypoly'_def by simp
 qed
 
 lemma polypoly_poly:
   assumes np: "isnpolyh p n0"
   shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
-  using np 
+  using np
   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
 
-lemma polypoly'_poly: 
+lemma polypoly'_poly:
   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]] .
@@ -959,14 +968,14 @@
 lemma polypoly_poly_polybound0:
   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
   shows "polypoly bs p = [Ipoly bs p]"
-  using np nb unfolding polypoly_def 
+  using np nb unfolding polypoly_def
   apply (cases p)
   apply auto
   apply (case_tac nat)
   apply auto
   done
 
-lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" 
+lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
   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)"
@@ -978,7 +987,7 @@
 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: 
+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})"
   shows "p = 0\<^sub>p"
@@ -994,10 +1003,10 @@
     let ?h = "head p"
     let ?hd = "decrpoly ?h"
     let ?ihd = "maxindex ?hd"
-    from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h" 
+    from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
       by simp_all
     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
-    
+
     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
     have mihn: "maxindex ?h \<le> maxindex p" by auto
     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
@@ -1023,21 +1032,21 @@
       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
-    
+
     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
     hence "?h = 0\<^sub>p" by simp
     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
   ultimately show "p = 0\<^sub>p" by blast
 qed
 
-lemma isnpolyh_unique:  
+lemma isnpolyh_unique:
   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>"
   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
-  hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" 
+  hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
     using wf_bs_polysub[where p=p and q=q] by auto
   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
   show "p = q" by blast
@@ -1056,28 +1065,28 @@
 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
 lemma one_normh: "isnpolyh (1)\<^sub>p n" by simp
 
-lemma polyadd_0[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"
-  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] 
+  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]: 
+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"
-  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] 
+  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]: 
+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"
-  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] 
+  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: 
+lemma polymul_commute:
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
     and np:"isnpolyh p n0"
     and nq: "isnpolyh q n1"
@@ -1086,15 +1095,15 @@
   by 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
 
-lemma polynate_idempotent[simp]: 
+lemma polynate_idempotent[simp]:
   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]] .
@@ -1137,34 +1146,34 @@
   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
 qed
 
-lemma degree_polysub_samehead: 
+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 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]) 
+  thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
 next
-  case (2 c c' n' p') 
+  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
   with 2 show ?case by simp
 next
-  case (3 c n p c') 
+  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
   with 3 show ?case by simp
 next
   case (4 c n p c' n' p')
-  hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1" 
+  hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
-  hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all  
-  hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0" 
+  hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
+  hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
     using H(1-2) degree_polyneg by auto
   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
@@ -1178,10 +1187,10 @@
       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)}
+        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 
+  {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)
@@ -1189,7 +1198,7 @@
     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 
+  {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
     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
@@ -1198,7 +1207,7 @@
     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)
 
@@ -1210,7 +1219,7 @@
   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) 
+    and "head (shift1 p) = head p" by (simp_all add: shift1_head)
   thus ?case by (simp add: funpow_swap1)
 qed
 
@@ -1231,7 +1240,7 @@
 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
   by (induct p rule: head.induct) auto
 
-lemma polyadd_eq_const_degree: 
+lemma polyadd_eq_const_degree:
   "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
 
@@ -1255,15 +1264,15 @@
   apply (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
   done
 
-lemma polymul_head_polyeq: 
+lemma polymul_head_polyeq:
   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
-next 
-  case (3 c n p c' n0 n1) 
+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
 next
@@ -1272,8 +1281,8 @@
     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
     by simp_all
   have "n < n' \<or> n' < n \<or> n = n'" by arith
-  moreover 
-  {assume nn': "n < n'" hence ?case 
+  moreover
+  {assume nn': "n < n'" hence ?case
       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
       apply simp
       apply (cases n)
@@ -1283,7 +1292,7 @@
       done }
   moreover {assume nn': "n'< n"
     hence ?case
-      using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] 
+      using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
       apply simp
       apply (cases n')
       apply simp
@@ -1291,14 +1300,14 @@
       apply auto
       done }
   moreover {assume nn': "n' = n"
-    from nn' polymul_normh[OF norm(5,4)] 
+    from nn' polymul_normh[OF norm(5,4)]
     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
-    from nn' polymul_normh[OF norm(5,3)] norm 
+    from nn' polymul_normh[OF norm(5,3)] norm
     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
-    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 
-    from polyadd_normh[OF ncnpc' ncnpp0'] 
-    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" 
+    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
+    from polyadd_normh[OF ncnpc' ncnpp0']
+    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"
       by (simp add: min_def)
     {assume np: "n > 0"
       with nn' head_isnpolyh_Suc'[OF np nth]
@@ -1314,7 +1323,7 @@
     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
     have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
         "4.hyps"(5)[OF norm(5,4)] nn' nz by simp }
-    ultimately have ?case by (cases n) auto} 
+    ultimately have ?case by (cases n) auto}
   ultimately show ?case by blast
 qed simp_all
 
@@ -1359,25 +1368,29 @@
     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) 
+  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)
   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) 
+  let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
   let ?b = "head s"
   let ?p' = "funpow (degree s - n) shift1 p"
   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
   let ?akk' = "a ^\<^sub>p (k' - k)"
   note ns = `isnpolyh s n1`
-  from np have np0: "isnpolyh p 0" 
-    using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
-  have np': "isnpolyh ?p' 0" using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def by simp
-  have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
-  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 
+  from np have np0: "isnpolyh p 0"
+    using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
+  have np': "isnpolyh ?p' 0"
+    using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
+    by simp
+  have headp': "head ?p' = head p"
+    using funpow_shift1_head[OF np pnz] by simp
+  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
@@ -1386,67 +1399,82 @@
       apply simp
       done }
   moreover
-  {assume sz: "s \<noteq> 0\<^sub>p"
-    {assume dn: "degree s < n"
+  { assume sz: "s \<noteq> 0\<^sub>p"
+    { assume dn: "degree s < n"
       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'" 
+    moreover
+    { assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
+      have degsp': "degree s = degree ?p'"
         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
-      {assume ba: "?b = a"
-        hence headsp': "head s = head ?p'" using ap headp' by simp
-        have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
+      { assume ba: "?b = a"
+        hence headsp': "head s = head ?p'"
+          using ap headp' by simp
+        have nr: "isnpolyh (s -\<^sub>p ?p') 0"
+          using polysub_normh[OF ns np'] by simp
         from degree_polysub_samehead[OF ns np' headsp' degsp']
         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
-        moreover 
-        {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
+        moreover
+        { assume deglt:"degree (s -\<^sub>p ?p') < degree s"
           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')"
             by (simp add: Let_def)
-          {assume h1: "polydivide_aux a n p k s = (k', r)"
-            from less(1)[OF deglt nr, of k k' r]
-              trans[OF eq[symmetric] h1]
-            have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
-              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)" by auto
-            from q1 obtain q n1 where nq: "isnpolyh q n1" 
-              and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
+          { assume h1: "polydivide_aux a n p k s = (k', r)"
+            from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
+            have kk': "k \<le> k'"
+              and nr:"\<exists>nr. isnpolyh r nr"
+              and dr: "degree r = 0 \<or> degree r < degree p"
+              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)"
+              by auto
+            from q1 obtain q n1 where nq: "isnpolyh q n1"
+              and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r" by blast
             from nr obtain nr where nr': "isnpolyh r nr" by blast
-            from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
+            from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
+              by simp
             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
-            from polyadd_normh[OF polymul_normh[OF np 
+            from polyadd_normh[OF polymul_normh[OF np
               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
-            have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp 
-            from asp have "\<forall> (bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = 
+            have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
+              by simp
+            from asp have "\<forall> (bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) =
               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
-            hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) = 
-              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r" 
+            hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
+              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
               by (simp add: field_simps)
-            hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
-              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) 
-              + Ipoly bs p * Ipoly bs q + Ipoly bs r"
-              by (auto simp only: funpow_shift1_1) 
-            hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
-              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) 
-              + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
-            hence "\<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 ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r)" by simp
+            hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
+              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
+              Ipoly bs p * Ipoly bs q + Ipoly bs r"
+              by (auto simp only: funpow_shift1_1)
+            hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
+              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
+              Ipoly bs q) + Ipoly bs r"
+              by (simp add: field_simps)
+            hence "\<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 ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r)"
+              by simp
             with isnpolyh_unique[OF nakks' nqr']
-            have "a ^\<^sub>p (k' - k) *\<^sub>p s = 
-              p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r" by blast
+            have "a ^\<^sub>p (k' - k) *\<^sub>p s =
+              p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
+              by blast
             hence ?qths using nq'
               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
-              apply (rule_tac x="0" in exI) by simp
+              apply (rule_tac x="0" in exI)
+              apply simp
+              done
             with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
-              by blast } hence ?ths by blast }
-        moreover 
-        {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
+              by blast
+          }
+          hence ?ths by blast
+        }
+        moreover
+        { 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
+          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
@@ -1455,134 +1483,162 @@
             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)"
+          { 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')"
               by (simp add: Let_def)
-            also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
+            also have "\<dots> = (k,0\<^sub>p)"
+              using polydivide_aux.simps spz by simp
             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
               apply auto
-              apply (rule exI[where x="?xdn"])        
+              apply (rule exI[where x="?xdn"])
               apply (auto simp add: polymul_commute[of p])
-              done} }
-        ultimately have ?ths by blast }
+              done
+          }
+        }
+        ultimately have ?ths by blast
+      }
       moreover
-      {assume ba: "?b \<noteq> a"
-        from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] 
+      { assume ba: "?b \<noteq> a"
+        from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
           polymul_normh[OF head_isnpolyh[OF ns] np']]
-        have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
+        have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
+          by (simp add: min_def)
         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
-          using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns] 
+          using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
-            funpow_shift1_nz[OF pnz] by simp_all
+            funpow_shift1_nz[OF pnz]
+          by simp_all
         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
-        have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')" 
+        have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
           using head_head[OF ns] funpow_shift1_head[OF np pnz]
             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
           by (simp add: ap)
         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
           head_nz[OF np] pnz sz ap[symmetric]
           funpow_shift1_nz[OF pnz, where n="degree s - n"]
-          polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
+          polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
           ndp dn
-        have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
+        have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
-        {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
-          from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
-          ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
-          {assume h1:"polydivide_aux a n p k s = (k', r)"
+        { assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
+          from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
+            polymul_normh[OF head_isnpolyh[OF ns]np']] ap
+          have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
+            by simp
+          { assume h1:"polydivide_aux a n p k s = (k', r)"
             from h1 polydivide_aux.simps sz dn' ba
             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
               by (simp add: Let_def)
             with less(1)[OF dth nasbp', of "Suc k" k' r]
-            obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq" 
+            obtain q nq nr where kk': "Suc k \<le> k'"
+              and nr: "isnpolyh r nr"
+              and nq: "isnpolyh q nq"
               and dr: "degree r = 0 \<or> degree r < degree p"
-              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" by auto
+              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"
+              by auto
             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
-            {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)
-            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 
+            {
+              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)
+              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
             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
             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]]
-            have nqw: "isnpolyh ?q 0" by simp
+            have nqw: "isnpolyh ?q 0"
+              by simp
             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]]
-            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" by blast
-            from dr kk' nr h1 asth nqw have ?ths apply simp
+            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"
+              by blast
+            from dr kk' nr h1 asth nqw have ?ths
+              apply simp
               apply (rule conjI)
               apply (rule exI[where x="nr"], simp)
               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
               apply (rule exI[where x="0"], simp)
-              done}
-          hence ?ths by blast }
-        moreover 
-        {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
-          {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
+              done
+          }
+          hence ?ths by blast
+        }
+        moreover
+        { assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
+          {
+            fix bs :: "'a::{field_char_0, field_inverse_zero} list"
             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
-          have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
-          hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p" 
-            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) =
+            have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
+              by simp
+            hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
+              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))" ..
-          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] 
+          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]
                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
               simplified ap] by simp
-          {assume h1: "polydivide_aux a n p k s = (k', r)"
-          from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
-          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
-            apply (rule exI[where x="0"], simp)
-            done }
-        hence ?ths by blast }
+          { assume h1: "polydivide_aux a n p k s = (k', r)"
+            from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
+            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
+              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]
-          by (simp add: degree_eq_degreen0[symmetric]) blast }
+          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
 
-lemma polydivide_properties: 
+lemma polydivide_properties:
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
-  shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) 
-  \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
-proof-
-  have trv: "head p = head p" "degree p = degree p" by simp_all
-  from polydivide_def[where s="s" and p="p"] 
-  have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
-  then obtain k r where kr: "polydivide s p = (k,r)" by blast
+    and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
+  shows "\<exists>k r. polydivide s p = (k,r) \<and>
+    (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
+    (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r))"
+proof -
+  have trv: "head p = head p" "degree p = degree p"
+    by simp_all
+  from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
+    by auto
+  then obtain k r where kr: "polydivide s p = (k,r)"
+    by blast
   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
   have "(degree r = 0 \<or> degree r < degree p) \<and>
-   (\<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)" by blast
-  with kr show ?thesis 
+    (\<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)"
+    by blast
+  with kr show ?thesis
     apply -
     apply (rule exI[where x="k"])
     apply (rule exI[where x="r"])
@@ -1596,23 +1652,23 @@
 definition "isnonconstant p = (\<not> isconstant p)"
 
 lemma isnonconstant_pnormal_iff:
-  assumes nc: "isnonconstant p" 
-  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" 
+  assumes nc: "isnonconstant p"
+  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
 proof
-  let ?p = "polypoly bs p"  
+  let ?p = "polypoly bs p"
   assume H: "pnormal ?p"
   have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
-  
-  from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]  
+
+  from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
     pnormal_last_nonzero[OF H]
   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
 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
-  hence pz: "?p \<noteq> []" by (simp add: polypoly_def) 
+  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"] 
+  from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
 qed
@@ -1638,10 +1694,10 @@
   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
   from isnonconstant_pnormal_iff[OF inc, of bs] h
   have pn: "pnormal ?p" by blast
-  {fix x assume H: "?p = [x]" 
+  { fix x assume H: "?p = [x]"
     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
-    with isnonconstant_coefficients_length[OF inc] have False by arith}
-  thus "nonconstant ?p" using pn unfolding nonconstant_def by blast  
+    with isnonconstant_coefficients_length[OF inc] have False by arith }
+  thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
 qed
 
 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
@@ -1655,29 +1711,29 @@
   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"
+  let ?p = "polypoly bs p"
   assume H: "degree p = Polynomial_List.degree ?p"
   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
     unfolding polypoly_def by auto
   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
   have lg:"length (pnormalize ?p) = length ?p"
     unfolding Polynomial_List.degree_def polypoly_def by simp
-  hence "pnormal ?p" using pnormal_length[OF pz] by blast 
-  with isnonconstant_pnormal_iff[OF inc]  
+  hence "pnormal ?p" using pnormal_length[OF pz] by blast
+  with isnonconstant_pnormal_iff[OF inc]
   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
 next
-  let  ?p = "polypoly bs p"  
+  let  ?p = "polypoly bs p"
   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
-  show "degree p = Polynomial_List.degree ?p" 
+  show "degree p = Polynomial_List.degree ?p"
     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
 qed
 
 
-section{* Swaps ; Division by a certain variable *}
+section {* Swaps ; Division by a certain variable *}
 
-primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
+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)"
 | "swap n m (Neg t) = Neg (swap n m t)"
@@ -1685,8 +1741,8 @@
 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
 | "swap n m (Pw t k) = Pw (swap n m t) k"
-| "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)"
+| "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"
@@ -1694,10 +1750,10 @@
   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 
+  thus ?case using nbs mbs by simp
 next
   case (CN c k p)
-  thus ?case using nbs mbs by simp 
+  thus ?case using nbs mbs by simp
 qed simp_all
 
 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
@@ -1723,9 +1779,9 @@
   shows "isnpoly (swapnorm n m p)"
   unfolding swapnorm_def by simp
 
-definition "polydivideby n s p = 
+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))"
+   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
@@ -1736,10 +1792,10 @@
 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
 | "isweaknpoly p = False"
 
-lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p" 
+lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
   by (induct p arbitrary: n0) auto
 
-lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)" 
+lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
   by (induct p) auto
 
 end
\ No newline at end of file