# HG changeset patch
# User wenzelm
# Date 1434823882 -7200
# Node ID 00db0d934a7dc25b7d5907bd1a5740b2d8ae30d4
# Parent  25a3c522cc8f65805a129feec90ef4b1932f9d60
tuned proofs;

diff -r 25a3c522cc8f -r 00db0d934a7d src/HOL/Decision_Procs/Polynomial_List.thy
--- a/src/HOL/Decision_Procs/Polynomial_List.thy	Sat Jun 20 17:29:51 2015 +0200
+++ b/src/HOL/Decision_Procs/Polynomial_List.thy	Sat Jun 20 20:11:22 2015 +0200
@@ -8,71 +8,72 @@
 imports Complex_Main
 begin
 
-text\<open>Application of polynomial as a function.\<close>
+text \<open>Application of polynomial as a function.\<close>
 
 primrec (in semiring_0) poly :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a"
 where
-  poly_Nil:  "poly [] x = 0"
-| poly_Cons: "poly (h#t) x = h + x * poly t x"
+  poly_Nil: "poly [] x = 0"
+| poly_Cons: "poly (h # t) x = h + x * poly t x"
 
 
-subsection\<open>Arithmetic Operations on Polynomials\<close>
+subsection \<open>Arithmetic Operations on Polynomials\<close>
 
-text\<open>addition\<close>
-
+text \<open>Addition\<close>
 primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "+++" 65)
 where
-  padd_Nil:  "[] +++ l2 = l2"
-| padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t else (h + hd l2)#(t +++ tl l2))"
+  padd_Nil: "[] +++ l2 = l2"
+| padd_Cons: "(h # t) +++ l2 = (if l2 = [] then h # t else (h + hd l2) # (t +++ tl l2))"
 
-text\<open>Multiplication by a constant\<close>
+text \<open>Multiplication by a constant\<close>
 primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "%*" 70) where
-  cmult_Nil:  "c %* [] = []"
+  cmult_Nil: "c %* [] = []"
 | cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
 
-text\<open>Multiplication by a polynomial\<close>
+text \<open>Multiplication by a polynomial\<close>
 primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "***" 70)
 where
-  pmult_Nil:  "[] *** l2 = []"
-| pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
-                              else (h %* l2) +++ ((0) # (t *** l2)))"
+  pmult_Nil: "[] *** l2 = []"
+| pmult_Cons: "(h # t) *** l2 = (if t = [] then h %* l2 else (h %* l2) +++ (0 # (t *** l2)))"
 
-text\<open>Repeated multiplication by a polynomial\<close>
-primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a  list \<Rightarrow> 'a list" where
-  mulexp_zero:  "mulexp 0 p q = q"
-| mulexp_Suc:   "mulexp (Suc n) p q = p *** mulexp n p q"
+text \<open>Repeated multiplication by a polynomial\<close>
+primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a  list \<Rightarrow> 'a list"
+where
+  mulexp_zero: "mulexp 0 p q = q"
+| mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q"
 
-text\<open>Exponential\<close>
-primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list"  (infixl "%^" 80) where
-  pexp_0:   "p %^ 0 = [1]"
+text \<open>Exponential\<close>
+primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list"  (infixl "%^" 80)
+where
+  pexp_0: "p %^ 0 = [1]"
 | pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
 
-text\<open>Quotient related value of dividing a polynomial by x + a\<close>
-(* Useful for divisor properties in inductive proofs *)
+text \<open>Quotient related value of dividing a polynomial by x + a.
+  Useful for divisor properties in inductive proofs.\<close>
 primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"
 where
-  pquot_Nil:  "pquot [] a= []"
-| pquot_Cons: "pquot (h#t) a =
-    (if t = [] then [h] else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
+  pquot_Nil: "pquot [] a = []"
+| pquot_Cons: "pquot (h # t) a =
+    (if t = [] then [h] else (inverse a * (h - hd( pquot t a))) # pquot t a)"
 
-text\<open>normalization of polynomials (remove extra 0 coeff)\<close>
-primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where
-  pnormalize_Nil:  "pnormalize [] = []"
-| pnormalize_Cons: "pnormalize (h#p) =
+text \<open>Normalization of polynomials (remove extra 0 coeff).\<close>
+primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list"
+where
+  pnormalize_Nil: "pnormalize [] = []"
+| pnormalize_Cons: "pnormalize (h # p) =
     (if pnormalize p = [] then (if h = 0 then [] else [h]) else h # pnormalize p)"
 
-definition (in semiring_0) "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
-definition (in semiring_0) "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
-text\<open>Other definitions\<close>
+definition (in semiring_0) "pnormal p \<longleftrightarrow> pnormalize p = p \<and> p \<noteq> []"
+definition (in semiring_0) "nonconstant p \<longleftrightarrow> pnormal p \<and> (\<forall>x. p \<noteq> [x])"
+
+text \<open>Other definitions.\<close>
 
 definition (in ring_1) poly_minus :: "'a list \<Rightarrow> 'a list" ("-- _" [80] 80)
   where "-- p = (- 1) %* p"
 
 definition (in semiring_0) divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "divides" 70)
-  where "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
+  where "p1 divides p2 \<longleftrightarrow> (\<exists>q. poly p2 = poly(p1 *** q))"
 
-lemma (in semiring_0) dividesI:
-  "poly p2 = poly (p1 *** q) \<Longrightarrow> p1 divides p2"
+lemma (in semiring_0) dividesI: "poly p2 = poly (p1 *** q) \<Longrightarrow> p1 divides p2"
   by (auto simp add: divides_def)
 
 lemma (in semiring_0) dividesE:
@@ -80,15 +81,15 @@
   obtains q where "poly p2 = poly (p1 *** q)"
   using assms by (auto simp add: divides_def)
 
-    --\<open>order of a polynomial\<close>
-definition (in ring_1) order :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where
-  "order a p = (SOME n. ([-a, 1] %^ n) divides p \<and> ~ (([-a, 1] %^ (Suc n)) divides p))"
+-- \<open>order of a polynomial\<close>
+definition (in ring_1) order :: "'a \<Rightarrow> 'a list \<Rightarrow> nat"
+  where "order a p = (SOME n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ (Suc n)) divides p))"
 
-     --\<open>degree of a polynomial\<close>
+-- \<open>degree of a polynomial\<close>
 definition (in semiring_0) degree :: "'a list \<Rightarrow> nat"
   where "degree p = length (pnormalize p) - 1"
 
-     --\<open>squarefree polynomials --- NB with respect to real roots only.\<close>
+-- \<open>squarefree polynomials --- NB with respect to real roots only\<close>
 definition (in ring_1) rsquarefree :: "'a list \<Rightarrow> bool"
   where "rsquarefree p \<longleftrightarrow> poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)"
 
@@ -108,51 +109,61 @@
 
 end
 
-lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct t) auto
+lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t"
+  by (induct t) auto
 
-lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"
+lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ (0 # t) = a # t"
   by simp
 
-text\<open>Handy general properties\<close>
+
+text \<open>Handy general properties.\<close>
 
 lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
 proof (induct b arbitrary: a)
   case Nil
-  thus ?case by auto
+  then show ?case
+    by auto
 next
   case (Cons b bs a)
-  thus ?case by (cases a) (simp_all add: add.commute)
+  then show ?case
+    by (cases a) (simp_all add: add.commute)
 qed
 
 lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
   apply (induct a)
-  apply (simp, clarify)
-  apply (case_tac b, simp_all add: ac_simps)
+  apply simp
+  apply clarify
+  apply (case_tac b)
+  apply (simp_all add: ac_simps)
   done
 
-lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
+lemma (in semiring_0) poly_cmult_distr: "a %* (p +++ q) = a %* p +++ a %* q"
   apply (induct p arbitrary: q)
   apply simp
-  apply (case_tac q, simp_all add: distrib_left)
+  apply (case_tac q)
+  apply (simp_all add: distrib_left)
   done
 
-lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
+lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = 0 # t"
   apply (induct t)
   apply simp
   apply (auto simp add: padd_commut)
-  apply (case_tac t, auto)
+  apply (case_tac t)
+  apply auto
   done
 
-text\<open>properties of evaluation of polynomials.\<close>
+
+text \<open>Properties of evaluation of polynomials.\<close>
 
 lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
-proof(induct p1 arbitrary: p2)
+proof (induct p1 arbitrary: p2)
   case Nil
-  thus ?case by simp
+  then show ?case
+    by simp
 next
   case (Cons a as p2)
-  thus ?case
-    by (cases p2) (simp_all  add: ac_simps distrib_left)
+  then show ?case
+    by (cases p2) (simp_all add: ac_simps distrib_left)
 qed
 
 lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
@@ -161,7 +172,7 @@
   apply (auto simp add: distrib_left ac_simps)
   done
 
-lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x"
+lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c * poly p x"
   by (induct p) (auto simp add: distrib_left ac_simps)
 
 lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
@@ -172,11 +183,12 @@
 lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
 proof (induct p1 arbitrary: p2)
   case Nil
-  thus ?case by simp
+  then show ?case
+    by simp
 next
   case (Cons a as p2)
-  thus ?case by (cases as)
-    (simp_all add: poly_cmult poly_add distrib_right distrib_left ac_simps)
+  then show ?case
+    by (cases as) (simp_all add: poly_cmult poly_add distrib_right distrib_left ac_simps)
 qed
 
 class idom_char_0 = idom + ring_char_0
@@ -186,7 +198,8 @@
 lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
   by (induct n) (auto simp add: poly_cmult poly_mult)
 
-text\<open>More Polynomial Evaluation Lemmas\<close>
+
+text \<open>More Polynomial Evaluation lemmas.\<close>
 
 lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
   by simp
@@ -197,71 +210,72 @@
 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
   by (induct p) auto
 
-lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
+lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly (p %^ n *** p %^ d) x"
   by (induct n) (auto simp add: poly_mult mult.assoc)
 
-subsection\<open>Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
- @{term "p(x)"}\<close>
+
+subsection \<open>Key Property: if @{term "f a = 0"} then @{term "(x - a)"} divides @{term "p(x)"}.\<close>
 
 lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
-proof(induct t)
+proof (induct t)
   case Nil
-  { fix h have "[h] = [h] +++ [- a, 1] *** []" by simp }
-  thus ?case by blast
+  have "[h] = [h] +++ [- a, 1] *** []" for h by simp
+  then show ?case by blast
 next
   case (Cons  x xs)
-  { fix h
-    from Cons.hyps[rule_format, of x]
-    obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
-    have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
+  have "\<exists>q r. h # x # xs = [r] +++ [-a, 1] *** q" for h
+  proof -
+    from Cons.hyps obtain q r where qr: "x # xs = [r] +++ [- a, 1] *** q"
+      by blast
+    have "h # x # xs = [a * r + h] +++ [-a, 1] *** (r # q)"
       using qr by (cases q) (simp_all add: algebra_simps)
-    hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}
-  thus ?case by blast
+    then show ?thesis by blast
+  qed
+  then show ?case by blast
 qed
 
 lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
   using lemma_poly_linear_rem [where t = t and a = a] by auto
 
-
-lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
-proof -
-  { assume p: "p = []" hence ?thesis by simp }
+lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) \<longleftrightarrow> p = [] \<or> (\<exists>q. p = [-a, 1] *** q)"
+proof (cases p)
+  case Nil
+  then show ?thesis by simp
+next
+  case (Cons x xs)
+  have "poly p a = 0" if "p = [-a, 1] *** q" for q
+    using that by (simp add: poly_add poly_cmult)
   moreover
-  {
-    fix x xs assume p: "p = x#xs"
-    {
-      fix q assume "p = [-a, 1] *** q"
-      hence "poly p a = 0" by (simp add: poly_add poly_cmult)
-    }
-    moreover
-    { assume p0: "poly p a = 0"
-      from poly_linear_rem[of x xs a] obtain q r
-      where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
-      have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp
-      hence "\<exists>q. p = [- a, 1] *** q"
-        using p qr
-        apply -
-        apply (rule exI[where x=q])
-        apply auto
-        apply (cases q)
-        apply auto
-        done
-    }
-    ultimately have ?thesis using p by blast
-  }
-  ultimately show ?thesis by (cases p) auto
+  have "\<exists>q. p = [- a, 1] *** q" if p0: "poly p a = 0"
+  proof -
+    from poly_linear_rem[of x xs a] obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q"
+      by blast
+    have "r = 0"
+      using p0 by (simp only: Cons qr poly_mult poly_add) simp
+    with Cons qr show ?thesis
+      apply -
+      apply (rule exI[where x = q])
+      apply auto
+      apply (cases q)
+      apply auto
+      done
+  qed
+  ultimately show ?thesis using Cons by blast
 qed
 
-lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
+lemma (in semiring_0) lemma_poly_length_mult[simp]:
+  "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
   by (induct p) auto
 
-lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
+lemma (in semiring_0) lemma_poly_length_mult2[simp]:
+  "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
   by (induct p) auto
 
 lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
   by auto
 
-subsection\<open>Polynomial length\<close>
+
+subsection \<open>Polynomial length\<close>
 
 lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
   by (induct p) auto
@@ -279,46 +293,52 @@
 lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
   by (auto simp add: poly_mult)
 
-text\<open>Normalisation Properties\<close>
 
-lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
+text \<open>Normalisation Properties.\<close>
+
+lemma (in semiring_0) poly_normalized_nil: "pnormalize p = [] \<longrightarrow> poly p x = 0"
   by (induct p) auto
 
-text\<open>A nontrivial polynomial of degree n has no more than n roots\<close>
+text \<open>A nontrivial polynomial of degree n has no more than n roots.\<close>
 lemma (in idom) poly_roots_index_lemma:
-   assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n"
+  assumes p: "poly p x \<noteq> poly [] x"
+    and n: "length p = n"
   shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
   using p n
 proof (induct n arbitrary: p x)
   case 0
-  thus ?case by simp
+  then show ?case by simp
 next
   case (Suc n p x)
-  {
-    assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
-    from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto
+  have False if C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
+  proof -
+    from Suc.prems have p0: "poly p x \<noteq> 0" "p \<noteq> []"
+      by auto
     from p0(1)[unfolded poly_linear_divides[of p x]]
-    have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
-    from C obtain a where a: "poly p a = 0" by blast
-    from a[unfolded poly_linear_divides[of p a]] p0(2)
-    obtain q where q: "p = [-a, 1] *** q" by blast
-    have lg: "length q = n" using q Suc.prems(2) by simp
+    have "\<forall>q. p \<noteq> [- x, 1] *** q"
+      by blast
+    from C obtain a where a: "poly p a = 0"
+      by blast
+    from a[unfolded poly_linear_divides[of p a]] p0(2) obtain q where q: "p = [-a, 1] *** q"
+      by blast
+    have lg: "length q = n"
+      using q Suc.prems(2) by simp
     from q p0 have qx: "poly q x \<noteq> poly [] x"
       by (auto simp add: poly_mult poly_add poly_cmult)
-    from Suc.hyps[OF qx lg] obtain i where
-      i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast
+    from Suc.hyps[OF qx lg] obtain i where i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
+      by blast
     let ?i = "\<lambda>m. if m = Suc n then a else i m"
     from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
       by blast
     from y have "y = a \<or> poly q y = 0"
       by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
-    with i[rule_format, of y] y(1) y(2) have False
+    with i[rule_format, of y] y(1) y(2) show ?thesis
       apply auto
       apply (erule_tac x = "m" in allE)
       apply auto
       done
-  }
-  thus ?case by blast
+  qed
+  then show ?case by blast
 qed
 
 
@@ -327,7 +347,7 @@
   by (blast intro: poly_roots_index_lemma)
 
 lemma (in idom) poly_roots_finite_lemma1:
-  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. (n::nat) < N \<and> x = i n)"
+  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n::nat. n < N \<and> x = i n)"
   apply (drule poly_roots_index_length, safe)
   apply (rule_tac x = "Suc (length p)" in exI)
   apply (rule_tac x = i in exI)
@@ -335,61 +355,65 @@
   done
 
 lemma (in idom) idom_finite_lemma:
-  assumes P: "\<forall>x. P x --> (\<exists>n. n < length j \<and> x = j!n)"
+  assumes "\<forall>x. P x \<longrightarrow> (\<exists>n. n < length j \<and> x = j!n)"
   shows "finite {x. P x}"
 proof -
-  let ?M = "{x. P x}"
-  let ?N = "set j"
-  have "?M \<subseteq> ?N" using P by auto
-  thus ?thesis using finite_subset by auto
+  from assms have "{x. P x} \<subseteq> set j" by auto
+  then show ?thesis using finite_subset by auto
 qed
 
 lemma (in idom) poly_roots_finite_lemma2:
   "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> x \<in> set i"
-  apply (drule poly_roots_index_length, safe)
-  apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
+  apply (drule poly_roots_index_length)
+  apply safe
+  apply (rule_tac x = "map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
   apply (auto simp add: image_iff)
-  apply (erule_tac x="x" in allE, clarsimp)
+  apply (erule_tac x="x" in allE)
+  apply clarsimp
   apply (case_tac "n = length p")
   apply (auto simp add: order_le_less)
   done
 
-lemma (in ring_char_0) UNIV_ring_char_0_infinte: "\<not> (finite (UNIV:: 'a set))"
+lemma (in ring_char_0) UNIV_ring_char_0_infinte: "\<not> finite (UNIV :: 'a set)"
 proof
   assume F: "finite (UNIV :: 'a set)"
   have "finite (UNIV :: nat set)"
   proof (rule finite_imageD)
     have "of_nat ` UNIV \<subseteq> UNIV" by simp
-    then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset)
-    show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def)
+    then show "finite (of_nat ` UNIV :: 'a set)"
+      using F by (rule finite_subset)
+    show "inj (of_nat :: nat \<Rightarrow> 'a)"
+      by (simp add: inj_on_def)
   qed
   with infinite_UNIV_nat show False ..
 qed
 
 lemma (in idom_char_0) poly_roots_finite: "poly p \<noteq> poly [] \<longleftrightarrow> finite {x. poly p x = 0}"
+  (is "?lhs \<longleftrightarrow> ?rhs")
 proof
-  assume H: "poly p \<noteq> poly []"
-  show "finite {x. poly p x = (0::'a)}"
-    using H
+  show ?rhs if ?lhs
+    using that
     apply -
-    apply (erule contrapos_np, rule ext)
+    apply (erule contrapos_np)
+    apply (rule ext)
     apply (rule ccontr)
     apply (clarify dest!: poly_roots_finite_lemma2)
     using finite_subset
   proof -
     fix x i
-    assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
-      and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
-    let ?M= "{x. poly p x = (0\<Colon>'a)}"
-    from P have "?M \<subseteq> set i" by auto
-    with finite_subset F show False by auto
+    assume F: "\<not> finite {x. poly p x = 0}"
+      and P: "\<forall>x. poly p x = 0 \<longrightarrow> x \<in> set i"
+    from P have "{x. poly p x = 0} \<subseteq> set i"
+      by auto
+    with finite_subset F show False
+      by auto
   qed
-next
-  assume F: "finite {x. poly p x = (0\<Colon>'a)}"
-  show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
+  show ?lhs if ?rhs
+    using UNIV_ring_char_0_infinte that by auto
 qed
 
-text\<open>Entirety and Cancellation for polynomials\<close>
+
+text \<open>Entirety and Cancellation for polynomials\<close>
 
 lemma (in idom_char_0) poly_entire_lemma2:
   assumes p0: "poly p \<noteq> poly []"
@@ -397,8 +421,10 @@
   shows "poly (p***q) \<noteq> poly []"
 proof -
   let ?S = "\<lambda>p. {x. poly p x = 0}"
-  have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult)
-  with p0 q0 show ?thesis  unfolding poly_roots_finite by auto
+  have "?S (p *** q) = ?S p \<union> ?S q"
+    by (auto simp add: poly_mult)
+  with p0 q0 show ?thesis
+    unfolding poly_roots_finite by auto
 qed
 
 lemma (in idom_char_0) poly_entire:
@@ -410,16 +436,13 @@
   "poly (p *** q) \<noteq> poly [] \<longleftrightarrow> poly p \<noteq> poly [] \<and> poly q \<noteq> poly []"
   by (simp add: poly_entire)
 
-lemma fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
-  by auto
-
 lemma (in comm_ring_1) poly_add_minus_zero_iff:
   "poly (p +++ -- q) = poly [] \<longleftrightarrow> poly p = poly q"
-  by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult)
+  by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq_iff poly_cmult)
 
 lemma (in comm_ring_1) poly_add_minus_mult_eq:
   "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
-  by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult algebra_simps)
+  by (auto simp add: poly_add poly_minus_def fun_eq_iff poly_mult poly_cmult algebra_simps)
 
 subclass (in idom_char_0) comm_ring_1 ..
 
@@ -433,17 +456,16 @@
   finally show ?thesis .
 qed
 
-lemma (in idom) poly_exp_eq_zero[simp]:
-  "poly (p %^ n) = poly [] \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0"
-  apply (simp only: fun_eq add: HOL.all_simps [symmetric])
+lemma (in idom) poly_exp_eq_zero[simp]: "poly (p %^ n) = poly [] \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0"
+  apply (simp only: fun_eq_iff add: HOL.all_simps [symmetric])
   apply (rule arg_cong [where f = All])
   apply (rule ext)
   apply (induct n)
   apply (auto simp add: poly_exp poly_mult)
   done
 
-lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
-  apply (simp add: fun_eq)
+lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a, 1] \<noteq> poly []"
+  apply (simp add: fun_eq_iff)
   apply (rule_tac x = "minus one a" in exI)
   apply (simp add: add.commute [of a])
   done
@@ -451,54 +473,60 @@
 lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) \<noteq> poly []"
   by auto
 
-text\<open>A more constructive notion of polynomials being trivial\<close>
+
+text \<open>A more constructive notion of polynomials being trivial.\<close>
 
 lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] \<Longrightarrow> h = 0 \<and> poly t = poly []"
-  apply (simp add: fun_eq)
+  apply (simp add: fun_eq_iff)
   apply (case_tac "h = zero")
-  apply (drule_tac [2] x = zero in spec, auto)
-  apply (cases "poly t = poly []", simp)
+  apply (drule_tac [2] x = zero in spec)
+  apply auto
+  apply (cases "poly t = poly []")
+  apply simp
 proof -
   fix x
-  assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"
-    and pnz: "poly t \<noteq> poly []"
+  assume H: "\<forall>x. x = 0 \<or> poly t x = 0"
+  assume pnz: "poly t \<noteq> poly []"
   let ?S = "{x. poly t x = 0}"
-  from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
-  hence th: "?S \<supseteq> UNIV - {0}" by auto
-  from poly_roots_finite pnz have th': "finite ?S" by blast
-  from finite_subset[OF th th'] UNIV_ring_char_0_infinte show "poly t x = (0\<Colon>'a)"
+  from H have "\<forall>x. x \<noteq> 0 \<longrightarrow> poly t x = 0"
+    by blast
+  then have th: "?S \<supseteq> UNIV - {0}"
+    by auto
+  from poly_roots_finite pnz have th': "finite ?S"
+    by blast
+  from finite_subset[OF th th'] UNIV_ring_char_0_infinte show "poly t x = 0"
     by simp
 qed
 
-lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
+lemma (in idom_char_0) poly_zero: "poly p = poly [] \<longleftrightarrow> list_all (\<lambda>c. c = 0) p"
   apply (induct p)
   apply simp
   apply (rule iffI)
-  apply (drule poly_zero_lemma', auto)
+  apply (drule poly_zero_lemma')
+  apply auto
   done
 
 lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0"
   unfolding poly_zero[symmetric] by simp
 
 
-
-text\<open>Basics of divisibility.\<close>
+text \<open>Basics of divisibility.\<close>
 
-lemma (in idom) poly_primes:
-  "[a, 1] divides (p *** q) \<longleftrightarrow> [a, 1] divides p \<or> [a, 1] divides q"
-  apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult distrib_right [symmetric])
+lemma (in idom) poly_primes: "[a, 1] divides (p *** q) \<longleftrightarrow> [a, 1] divides p \<or> [a, 1] divides q"
+  apply (auto simp add: divides_def fun_eq_iff poly_mult poly_add poly_cmult distrib_right [symmetric])
   apply (drule_tac x = "uminus a" in spec)
   apply (simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
   apply (cases "p = []")
   apply (rule exI[where x="[]"])
   apply simp
   apply (cases "q = []")
-  apply (erule allE[where x="[]"], simp)
+  apply (erule allE[where x="[]"])
+  apply simp
 
   apply clarsimp
-  apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
+  apply (cases "\<exists>q. p = a %* q +++ (0 # q)")
   apply (clarsimp simp add: poly_add poly_cmult)
-  apply (rule_tac x="qa" in exI)
+  apply (rule_tac x = qa in exI)
   apply (simp add: distrib_right [symmetric])
   apply clarsimp
 
@@ -511,13 +539,14 @@
 lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
   apply (simp add: divides_def)
   apply (rule_tac x = "[one]" in exI)
-  apply (auto simp add: poly_mult fun_eq)
+  apply (auto simp add: poly_mult fun_eq_iff)
   done
 
 lemma (in comm_semiring_1) poly_divides_trans: "p divides q \<Longrightarrow> q divides r \<Longrightarrow> p divides r"
-  apply (simp add: divides_def, safe)
+  apply (simp add: divides_def)
+  apply safe
   apply (rule_tac x = "pmult qa qaa" in exI)
-  apply (auto simp add: poly_mult fun_eq mult.assoc)
+  apply (auto simp add: poly_mult fun_eq_iff mult.assoc)
   done
 
 lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)"
@@ -526,46 +555,43 @@
   apply (rule_tac [2] poly_divides_trans)
   apply (auto simp add: divides_def)
   apply (rule_tac x = p in exI)
-  apply (auto simp add: poly_mult fun_eq ac_simps)
+  apply (auto simp add: poly_mult fun_eq_iff ac_simps)
   done
 
-lemma (in comm_semiring_1) poly_exp_divides:
-  "(p %^ n) divides q \<Longrightarrow> m \<le> n \<Longrightarrow> (p %^ m) divides q"
+lemma (in comm_semiring_1) poly_exp_divides: "(p %^ n) divides q \<Longrightarrow> m \<le> n \<Longrightarrow> (p %^ m) divides q"
   by (blast intro: poly_divides_exp poly_divides_trans)
 
-lemma (in comm_semiring_0) poly_divides_add:
-  "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)"
-  apply (simp add: divides_def, auto)
+lemma (in comm_semiring_0) poly_divides_add: "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)"
+  apply (auto simp add: divides_def)
   apply (rule_tac x = "padd qa qaa" in exI)
-  apply (auto simp add: poly_add fun_eq poly_mult distrib_left)
+  apply (auto simp add: poly_add fun_eq_iff poly_mult distrib_left)
   done
 
-lemma (in comm_ring_1) poly_divides_diff:
-  "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r"
-  apply (simp add: divides_def, auto)
+lemma (in comm_ring_1) poly_divides_diff: "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r"
+  apply (auto simp add: divides_def)
   apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
-  apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
+  apply (auto simp add: poly_add fun_eq_iff poly_mult poly_minus algebra_simps)
   done
 
-lemma (in comm_ring_1) poly_divides_diff2:
-  "p divides r \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides q"
+lemma (in comm_ring_1) poly_divides_diff2: "p divides r \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides q"
   apply (erule poly_divides_diff)
-  apply (auto simp add: poly_add fun_eq poly_mult divides_def ac_simps)
+  apply (auto simp add: poly_add fun_eq_iff poly_mult divides_def ac_simps)
   done
 
 lemma (in semiring_0) poly_divides_zero: "poly p = poly [] \<Longrightarrow> q divides p"
   apply (simp add: divides_def)
-  apply (rule exI[where x="[]"])
-  apply (auto simp add: fun_eq poly_mult)
+  apply (rule exI[where x = "[]"])
+  apply (auto simp add: fun_eq_iff poly_mult)
   done
 
 lemma (in semiring_0) poly_divides_zero2 [simp]: "q divides []"
   apply (simp add: divides_def)
   apply (rule_tac x = "[]" in exI)
-  apply (auto simp add: fun_eq)
+  apply (auto simp add: fun_eq_iff)
   done
 
-text\<open>At last, we can consider the order of a root.\<close>
+
+text \<open>At last, we can consider the order of a root.\<close>
 
 lemma (in idom_char_0) poly_order_exists_lemma:
   assumes lp: "length p = d"
@@ -574,22 +600,24 @@
   using lp p
 proof (induct d arbitrary: p)
   case 0
-  thus ?case by simp
+  then show ?case by simp
 next
   case (Suc n p)
   show ?case
   proof (cases "poly p a = 0")
     case True
-    from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto
-    hence pN: "p \<noteq> []" by auto
+    from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []"
+      by auto
+    then have pN: "p \<noteq> []"
+      by auto
     from True[unfolded poly_linear_divides] pN obtain q where q: "p = [-a, 1] *** q"
       by blast
     from q h True have qh: "length q = n" "poly q \<noteq> poly []"
       apply -
       apply simp
-      apply (simp only: fun_eq)
+      apply (simp only: fun_eq_iff)
       apply (rule ccontr)
-      apply (simp add: fun_eq poly_add poly_cmult)
+      apply (simp add: fun_eq_iff poly_add poly_cmult)
       done
     from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0"
       by blast
@@ -611,13 +639,15 @@
   by (induct n) (auto simp add: poly_mult ac_simps)
 
 lemma (in comm_semiring_1) divides_left_mult:
-  assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r"
+  assumes "(p *** q) divides r"
+  shows "p divides r \<and> q divides r"
 proof-
-  from d obtain t where r:"poly r = poly (p***q *** t)"
+  from assms obtain t where "poly r = poly (p *** q *** t)"
     unfolding divides_def by blast
-  hence "poly r = poly (p *** (q *** t))"
-    "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult ac_simps)
-  thus ?thesis unfolding divides_def by blast
+  then have "poly r = poly (p *** (q *** t))" and "poly r = poly (q *** (p *** t))"
+    by (auto simp add: fun_eq_iff poly_mult ac_simps)
+  then show ?thesis
+    unfolding divides_def by blast
 qed
 
 
@@ -627,12 +657,14 @@
   by (induct n) simp_all
 
 lemma (in idom_char_0) poly_order_exists:
-  assumes "length p = d" and "poly p \<noteq> poly []"
+  assumes "length p = d"
+    and "poly p \<noteq> poly []"
   shows "\<exists>n. [- a, 1] %^ n divides p \<and> \<not> [- a, 1] %^ Suc n divides p"
 proof -
   from assms have "\<exists>n q. p = mulexp n [- a, 1] q \<and> poly q a \<noteq> 0"
     by (rule poly_order_exists_lemma)
-  then obtain n q where p: "p = mulexp n [- a, 1] q" and "poly q a \<noteq> 0" by blast
+  then obtain n q where p: "p = mulexp n [- a, 1] q" and "poly q a \<noteq> 0"
+    by blast
   have "[- a, 1] %^ n divides mulexp n [- a, 1] q"
   proof (rule dividesI)
     show "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ n *** q)"
@@ -645,21 +677,25 @@
       by (rule dividesE)
     moreover have "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** m)"
     proof (induct n)
-      case 0 show ?case
+      case 0
+      show ?case
       proof (rule ccontr)
         assume "\<not> poly (mulexp 0 [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc 0 *** m)"
         then have "poly q a = 0"
           by (simp add: poly_add poly_cmult)
-        with \<open>poly q a \<noteq> 0\<close> show False by simp
+        with \<open>poly q a \<noteq> 0\<close> show False
+          by simp
       qed
     next
-      case (Suc n) show ?case
+      case (Suc n)
+      show ?case
         by (rule pexp_Suc [THEN ssubst], rule ccontr)
           (simp add: poly_mult_left_cancel poly_mult_assoc Suc del: pmult_Cons pexp_Suc)
     qed
     ultimately show False by simp
   qed
-  ultimately show ?thesis by (auto simp add: p)
+  ultimately show ?thesis
+    by (auto simp add: p)
 qed
 
 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
@@ -671,19 +707,19 @@
   apply (cut_tac x = y and y = n in less_linear)
   apply (drule_tac m = n in poly_exp_divides)
   apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
-              simp del: pmult_Cons pexp_Suc)
+    simp del: pmult_Cons pexp_Suc)
   done
 
-text\<open>Order\<close>
+
+text \<open>Order\<close>
 
 lemma some1_equalityD: "n = (SOME n. P n) \<Longrightarrow> \<exists>!n. P n \<Longrightarrow> P n"
   by (blast intro: someI2)
 
 lemma (in idom_char_0) order:
-      "(([-a, 1] %^ n) divides p \<and>
-        ~(([-a, 1] %^ (Suc n)) divides p)) =
-        ((n = order a p) \<and> ~(poly p = poly []))"
-  apply (unfold order_def)
+  "([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p) \<longleftrightarrow>
+    n = order a p \<and> poly p \<noteq> poly []"
+  unfolding order_def
   apply (rule iffI)
   apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
   apply (blast intro!: poly_order [THEN [2] some1_equalityD])
@@ -691,82 +727,88 @@
 
 lemma (in idom_char_0) order2:
   "poly p \<noteq> poly [] \<Longrightarrow>
-    ([-a, 1] %^ (order a p)) divides p \<and> \<not> (([-a, 1] %^ (Suc (order a p))) divides p)"
+    ([-a, 1] %^ (order a p)) divides p \<and> \<not> ([-a, 1] %^ Suc (order a p)) divides p"
   by (simp add: order del: pexp_Suc)
 
 lemma (in idom_char_0) order_unique:
-  "poly p \<noteq> poly [] \<Longrightarrow> ([-a, 1] %^ n) divides p \<Longrightarrow> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
+  "poly p \<noteq> poly [] \<Longrightarrow> ([-a, 1] %^ n) divides p \<Longrightarrow> \<not> ([-a, 1] %^ (Suc n)) divides p \<Longrightarrow>
     n = order a p"
   using order [of a n p] by auto
 
 lemma (in idom_char_0) order_unique_lemma:
-  "poly p \<noteq> poly [] \<and> ([-a, 1] %^ n) divides p \<and> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
+  "poly p \<noteq> poly [] \<and> ([-a, 1] %^ n) divides p \<and> \<not> ([-a, 1] %^ (Suc n)) divides p \<Longrightarrow>
     n = order a p"
   by (blast intro: order_unique)
 
 lemma (in ring_1) order_poly: "poly p = poly q \<Longrightarrow> order a p = order a q"
-  by (auto simp add: fun_eq divides_def poly_mult order_def)
+  by (auto simp add: fun_eq_iff divides_def poly_mult order_def)
 
 lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
-  by (induct "p") auto
+  by (induct p) auto
 
 lemma (in comm_ring_1) lemma_order_root:
-  "0 < n \<and> [- a, 1] %^ n divides p \<and> ~ [- a, 1] %^ (Suc n) divides p \<Longrightarrow> poly p a = 0"
+  "0 < n \<and> [- a, 1] %^ n divides p \<and> \<not> [- a, 1] %^ (Suc n) divides p \<Longrightarrow> poly p a = 0"
   by (induct n arbitrary: a p) (auto simp add: divides_def poly_mult simp del: pmult_Cons)
 
-lemma (in idom_char_0) order_root:
-  "poly p a = 0 \<longleftrightarrow> poly p = poly [] \<or> order a p \<noteq> 0"
+lemma (in idom_char_0) order_root: "poly p a = 0 \<longleftrightarrow> poly p = poly [] \<or> order a p \<noteq> 0"
   apply (cases "poly p = poly []")
   apply auto
-  apply (simp add: poly_linear_divides del: pmult_Cons, safe)
+  apply (simp add: poly_linear_divides del: pmult_Cons)
+  apply safe
   apply (drule_tac [!] a = a in order2)
   apply (rule ccontr)
-  apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
-  using neq0_conv
-  apply (blast intro: lemma_order_root)
+  apply (simp add: divides_def poly_mult fun_eq_iff del: pmult_Cons)
+  apply blast
+  using neq0_conv apply (blast intro: lemma_order_root)
   done
 
 lemma (in idom_char_0) order_divides:
   "([-a, 1] %^ n) divides p \<longleftrightarrow> poly p = poly [] \<or> n \<le> order a p"
   apply (cases "poly p = poly []")
   apply auto
-  apply (simp add: divides_def fun_eq poly_mult)
+  apply (simp add: divides_def fun_eq_iff poly_mult)
   apply (rule_tac x = "[]" in exI)
   apply (auto dest!: order2 [where a=a] intro: poly_exp_divides simp del: pexp_Suc)
   done
 
 lemma (in idom_char_0) order_decomp:
-  "poly p \<noteq> poly [] \<Longrightarrow> \<exists>q. poly p = poly (([-a, 1] %^ (order a p)) *** q) \<and> ~([-a, 1] divides q)"
-  apply (unfold divides_def)
+  "poly p \<noteq> poly [] \<Longrightarrow> \<exists>q. poly p = poly (([-a, 1] %^ order a p) *** q) \<and> \<not> [-a, 1] divides q"
+  unfolding divides_def
   apply (drule order2 [where a = a])
-  apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
-  apply (rule_tac x = q in exI, safe)
+  apply (simp add: divides_def del: pexp_Suc pmult_Cons)
+  apply safe
+  apply (rule_tac x = q in exI)
+  apply safe
   apply (drule_tac x = qa in spec)
-  apply (auto simp add: poly_mult fun_eq poly_exp ac_simps simp del: pmult_Cons)
+  apply (auto simp add: poly_mult fun_eq_iff poly_exp ac_simps simp del: pmult_Cons)
   done
 
-text\<open>Important composition properties of orders.\<close>
+text \<open>Important composition properties of orders.\<close>
 lemma order_mult:
-  "poly (p *** q) \<noteq> poly [] \<Longrightarrow>
-    order a (p *** q) = order a p + order (a::'a::{idom_char_0}) q"
+  fixes a :: "'a::idom_char_0"
+  shows "poly (p *** q) \<noteq> poly [] \<Longrightarrow> order a (p *** q) = order a p + order a q"
   apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
   apply (auto simp add: poly_entire simp del: pmult_Cons)
   apply (drule_tac a = a in order2)+
   apply safe
-  apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
+  apply (simp add: divides_def fun_eq_iff poly_exp_add poly_mult del: pmult_Cons, safe)
   apply (rule_tac x = "qa *** qaa" in exI)
   apply (simp add: poly_mult ac_simps del: pmult_Cons)
   apply (drule_tac a = a in order_decomp)+
   apply safe
-  apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
+  apply (subgoal_tac "[-a, 1] divides (qa *** qaa) ")
   apply (simp add: poly_primes del: pmult_Cons)
   apply (auto simp add: divides_def simp del: pmult_Cons)
   apply (rule_tac x = qb in exI)
-  apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
-  apply (drule poly_mult_left_cancel [THEN iffD1], force)
-  apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) = poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ")
-  apply (drule poly_mult_left_cancel [THEN iffD1], force)
-  apply (simp add: fun_eq poly_exp_add poly_mult ac_simps del: pmult_Cons)
+  apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) =
+    poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
+  apply (drule poly_mult_left_cancel [THEN iffD1])
+  apply force
+  apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) =
+    poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ")
+  apply (drule poly_mult_left_cancel [THEN iffD1])
+  apply force
+  apply (simp add: fun_eq_iff poly_exp_add poly_mult ac_simps del: pmult_Cons)
   done
 
 lemma (in idom_char_0) order_mult:
@@ -777,7 +819,8 @@
   apply (auto simp add: poly_entire simp del: pmult_Cons)
   apply (drule_tac a = a in order2)+
   apply safe
-  apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
+  apply (simp add: divides_def fun_eq_iff poly_exp_add poly_mult del: pmult_Cons)
+  apply safe
   apply (rule_tac x = "pmult qa qaa" in exI)
   apply (simp add: poly_mult ac_simps del: pmult_Cons)
   apply (drule_tac a = a in order_decomp)+
@@ -794,42 +837,43 @@
     poly (pmult (pexp [uminus a, one] (order a q))
       (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb)))")
   apply (drule poly_mult_left_cancel [THEN iffD1], force)
-  apply (simp add: fun_eq poly_exp_add poly_mult ac_simps del: pmult_Cons)
+  apply (simp add: fun_eq_iff poly_exp_add poly_mult ac_simps del: pmult_Cons)
   done
 
 lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] \<Longrightarrow> poly p a = 0 \<longleftrightarrow> order a p \<noteq> 0"
   by (rule order_root [THEN ssubst]) auto
 
-lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto
+lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p"
+  by auto
 
 lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
-  by (simp add: fun_eq)
+  by (simp add: fun_eq_iff)
 
 lemma (in idom_char_0) rsquarefree_decomp:
-  "rsquarefree p \<Longrightarrow> poly p a = 0 \<Longrightarrow>
-    \<exists>q. poly p = poly ([-a, 1] *** q) \<and> poly q a \<noteq> 0"
-  apply (simp add: rsquarefree_def, safe)
+  "rsquarefree p \<Longrightarrow> poly p a = 0 \<Longrightarrow> \<exists>q. poly p = poly ([-a, 1] *** q) \<and> poly q a \<noteq> 0"
+  apply (simp add: rsquarefree_def)
+  apply safe
   apply (frule_tac a = a in order_decomp)
   apply (drule_tac x = a in spec)
   apply (drule_tac a = a in order_root2 [symmetric])
   apply (auto simp del: pmult_Cons)
   apply (rule_tac x = q in exI, safe)
-  apply (simp add: poly_mult fun_eq)
+  apply (simp add: poly_mult fun_eq_iff)
   apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
   apply (simp add: divides_def del: pmult_Cons, safe)
   apply (drule_tac x = "[]" in spec)
-  apply (auto simp add: fun_eq)
+  apply (auto simp add: fun_eq_iff)
   done
 
 
-text\<open>Normalization of a polynomial.\<close>
+text \<open>Normalization of a polynomial.\<close>
 
 lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
-  by (induct p) (auto simp add: fun_eq)
+  by (induct p) (auto simp add: fun_eq_iff)
 
-text\<open>The degree of a polynomial.\<close>
+text \<open>The degree of a polynomial.\<close>
 
-lemma (in semiring_0) lemma_degree_zero: "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
+lemma (in semiring_0) lemma_degree_zero: "list_all (\<lambda>c. c = 0) p \<longleftrightarrow> pnormalize p = []"
   by (induct p) auto
 
 lemma (in idom_char_0) degree_zero:
@@ -838,18 +882,17 @@
   using assms
   by (cases "pnormalize p = []") (auto simp add: degree_def poly_zero lemma_degree_zero)
 
-lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0"
+lemma (in semiring_0) pnormalize_sing: "pnormalize [x] = [x] \<longleftrightarrow> x \<noteq> 0"
   by simp
 
-lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])"
+lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> pnormalize [x, y] = [x, y]"
   by simp
 
-lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
+lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c # p)"
   unfolding pnormal_def by simp
 
-lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
-  unfolding pnormal_def by(auto split: split_if_asm)
-
+lemma (in semiring_0) pnormal_tail: "p \<noteq> [] \<Longrightarrow> pnormal (c # p) \<Longrightarrow> pnormal p"
+  unfolding pnormal_def by (auto split: split_if_asm)
 
 lemma (in semiring_0) pnormal_last_nonzero: "pnormal p \<Longrightarrow> last p \<noteq> 0"
   by (induct p) (simp_all add: pnormal_def split: split_if_asm)
@@ -860,7 +903,6 @@
 lemma (in semiring_0) pnormal_last_length: "0 < length p \<Longrightarrow> last p \<noteq> 0 \<Longrightarrow> pnormal p"
   by (induct p) (auto simp: pnormal_def  split: split_if_asm)
 
-
 lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> 0 < length p \<and> last p \<noteq> 0"
   using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
 
@@ -868,42 +910,55 @@
   "poly (c # cs) = poly (d # ds) \<longleftrightarrow> c = d \<and> poly cs = poly ds"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
-  assume eq: ?lhs
-  hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
-    by (simp only: poly_minus poly_add algebra_simps) (simp add: algebra_simps)
-  hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add: fun_eq_iff)
-  hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))"
-    unfolding poly_zero by (simp add: poly_minus_def algebra_simps)
-  hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
-    unfolding poly_zero[symmetric] by simp
-  then show ?rhs by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
-next
-  assume ?rhs
-  then show ?lhs by(simp add:fun_eq_iff)
+  show ?rhs if ?lhs
+  proof -
+    from that have "poly ((c # cs) +++ -- (d # ds)) x = 0" for x
+      by (simp only: poly_minus poly_add algebra_simps) (simp add: algebra_simps)
+    then have "poly ((c # cs) +++ -- (d # ds)) = poly []"
+      by (simp add: fun_eq_iff)
+    then have "c = d" and "list_all (\<lambda>x. x = 0) ((cs +++ -- ds))"
+      unfolding poly_zero by (simp_all add: poly_minus_def algebra_simps)
+    from this(2) have "poly (cs +++ -- ds) x = 0" for x
+      unfolding poly_zero[symmetric] by simp
+    with \<open>c = d\<close> show ?thesis
+      by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
+  qed
+  show ?lhs if ?rhs
+    using that by (simp add:fun_eq_iff)
 qed
 
 lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
 proof (induct q arbitrary: p)
   case Nil
-  thus ?case by (simp only: poly_zero lemma_degree_zero) simp
+  then show ?case
+    by (simp only: poly_zero lemma_degree_zero) simp
 next
   case (Cons c cs p)
-  thus ?case
+  then show ?case
   proof (induct p)
     case Nil
-    hence "poly [] = poly (c#cs)" by blast
-    then have "poly (c#cs) = poly [] " by simp
-    thus ?case by (simp only: poly_zero lemma_degree_zero) simp
+    then have "poly [] = poly (c # cs)"
+      by blast
+    then have "poly (c#cs) = poly []"
+      by simp
+    then show ?case
+      by (simp only: poly_zero lemma_degree_zero) simp
   next
     case (Cons d ds)
-    hence eq: "poly (d # ds) = poly (c # cs)" by blast
-    hence eq': "\<And>x. poly (d # ds) x = poly (c # cs) x" by simp
-    hence "poly (d # ds) 0 = poly (c # cs) 0" by blast
-    hence dc: "d = c" by auto
+    then have eq: "poly (d # ds) = poly (c # cs)"
+      by blast
+    then have eq': "\<And>x. poly (d # ds) x = poly (c # cs) x"
+      by simp
+    then have "poly (d # ds) 0 = poly (c # cs) 0"
+      by blast
+    then have dc: "d = c"
+      by auto
     with eq have "poly ds = poly cs"
       unfolding  poly_Cons_eq by simp
-    with Cons.prems have "pnormalize ds = pnormalize cs" by blast
-    with dc show ?case by simp
+    with Cons.prems have "pnormalize ds = pnormalize cs"
+      by blast
+    with dc show ?case
+      by simp
   qed
 qed
 
@@ -912,11 +967,11 @@
   shows "degree p = degree q"
   using pnormalize_unique[OF pq] unfolding degree_def by simp
 
-lemma (in semiring_0) pnormalize_length:
-  "length (pnormalize p) \<le> length p" by (induct p) auto
+lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p"
+  by (induct p) auto
 
 lemma (in semiring_0) last_linear_mul_lemma:
-  "last ((a %* p) +++ (x#(b %* p))) = (if p = [] then x else b * last p)"
+  "last ((a %* p) +++ (x # (b %* p))) = (if p = [] then x else b * last p)"
   apply (induct p arbitrary: a x b)
   apply auto
   apply (rename_tac a p aa x b)
@@ -928,13 +983,14 @@
 
 lemma (in semiring_1) last_linear_mul:
   assumes p: "p \<noteq> []"
-  shows "last ([a,1] *** p) = last p"
+  shows "last ([a, 1] *** p) = last p"
 proof -
-  from p obtain c cs where cs: "p = c#cs" by (cases p) auto
-  from cs have eq: "[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
+  from p obtain c cs where cs: "p = c # cs"
+    by (cases p) auto
+  from cs have eq: "[a, 1] *** p = (a %* (c # cs)) +++ (0 # (1 %* (c # cs)))"
     by (simp add: poly_cmult_distr)
-  show ?thesis using cs
-    unfolding eq last_linear_mul_lemma by simp
+  show ?thesis
+    using cs unfolding eq last_linear_mul_lemma by simp
 qed
 
 lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
@@ -947,27 +1003,27 @@
   using pnormalize_eq[of p] unfolding degree_def by simp
 
 lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)"
-  by (rule ext) simp
+  by auto
 
 lemma (in idom_char_0) linear_mul_degree:
   assumes p: "poly p \<noteq> poly []"
-  shows "degree ([a,1] *** p) = degree p + 1"
+  shows "degree ([a, 1] *** p) = degree p + 1"
 proof -
   from p have pnz: "pnormalize p \<noteq> []"
     unfolding poly_zero lemma_degree_zero .
 
   from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
   have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp
+
   from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
     pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
-
   have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
     by simp
 
   have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
     by (rule ext) (simp add: poly_mult poly_add poly_cmult)
-  from degree_unique[OF eqs] th
-  show ?thesis by (simp add: degree_unique[OF poly_normalize])
+  from degree_unique[OF eqs] th show ?thesis
+    by (simp add: degree_unique[OF poly_normalize])
 qed
 
 lemma (in idom_char_0) linear_pow_mul_degree:
@@ -981,40 +1037,39 @@
       using degree_unique[OF True] by (simp add: degree_def)
   next
     case False
-    then show ?thesis by (auto simp add: poly_Nil_ext)
+    then show ?thesis
+      by (auto simp add: poly_Nil_ext)
   qed
 next
   case (Suc n a p)
-  have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"
+  have eq: "poly ([a, 1] %^(Suc n) *** p) = poly ([a, 1] %^ n *** ([a, 1] *** p))"
     apply (rule ext)
     apply (simp add: poly_mult poly_add poly_cmult)
-    apply (simp add: ac_simps ac_simps distrib_left)
+    apply (simp add: ac_simps distrib_left)
     done
   note deq = degree_unique[OF eq]
   show ?case
   proof (cases "poly p = poly []")
     case True
-    with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
-      apply -
-      apply (rule ext)
-      apply (simp add: poly_mult poly_cmult poly_add)
-      done
+    with eq have eq': "poly ([a, 1] %^(Suc n) *** p) = poly []"
+      by (auto simp add: poly_mult poly_cmult poly_add)
     from degree_unique[OF eq'] True show ?thesis
       by (simp add: degree_def)
   next
     case False
     then have ap: "poly ([a,1] *** p) \<noteq> poly []"
       using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
-    have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
-      by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
-    from ap have ap': "(poly ([a,1] *** p) = poly []) = False"
+    have eq: "poly ([a, 1] %^(Suc n) *** p) = poly ([a, 1]%^n *** ([a, 1] *** p))"
+      by (auto simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
+    from ap have ap': "poly ([a, 1] *** p) = poly [] \<longleftrightarrow> False"
       by blast
-    have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
+    have th0: "degree ([a, 1]%^n *** ([a, 1] *** p)) = degree ([a, 1] *** p) + n"
       apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
       apply simp
       done
     from degree_unique[OF eq] ap False th0 linear_mul_degree[OF False, of a]
-    show ?thesis by (auto simp del: poly.simps)
+    show ?thesis
+      by (auto simp del: poly.simps)
   qed
 qed
 
@@ -1023,24 +1078,25 @@
   shows "order a p \<le> degree p"
 proof -
   from order2[OF p0, unfolded divides_def]
-  obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast
-  {
-    assume "poly q = poly []"
-    with q p0 have False by (simp add: poly_mult poly_entire)
-  }
+  obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)"
+    by blast
+  with q p0 have "poly q \<noteq> poly []"
+    by (simp add: poly_mult poly_entire)
   with degree_unique[OF q, unfolded linear_pow_mul_degree] show ?thesis
     by auto
 qed
 
-text\<open>Tidier versions of finiteness of roots.\<close>
 
+text \<open>Tidier versions of finiteness of roots.\<close>
 lemma (in idom_char_0) poly_roots_finite_set:
   "poly p \<noteq> poly [] \<Longrightarrow> finite {x. poly p x = 0}"
   unfolding poly_roots_finite .
 
-text\<open>bound for polynomial.\<close>
 
-lemma poly_mono: "abs(x) \<le> k \<Longrightarrow> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
+text \<open>Bound for polynomial.\<close>
+lemma poly_mono:
+  fixes x :: "'a::linordered_idom"
+  shows "abs x \<le> k \<Longrightarrow> abs (poly p x) \<le> poly (map abs p) k"
   apply (induct p)
   apply auto
   apply (rename_tac a p)
@@ -1049,6 +1105,7 @@
   apply (auto intro!: mult_mono simp add: abs_mult)
   done
 
-lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp
+lemma (in semiring_0) poly_Sing: "poly [c] x = c"
+  by simp
 
 end