tuned proofs;
authorwenzelm
Sun, 25 Aug 2013 23:20:25 +0200
changeset 53196 942a1b48bb31
parent 53195 e4b18828a817
child 53197 6c5e7143e1f6
tuned proofs;
src/HOL/Library/Formal_Power_Series.thy
src/HOL/Library/Fraction_Field.thy
src/HOL/Library/Univ_Poly.thy
--- a/src/HOL/Library/Formal_Power_Series.thy	Sun Aug 25 21:25:17 2013 +0200
+++ b/src/HOL/Library/Formal_Power_Series.thy	Sun Aug 25 23:20:25 2013 +0200
@@ -469,7 +469,7 @@
 
 lemma fps_nonzero_least_unique:
   assumes a0: "a \<noteq> 0"
-  shows "\<exists>! n. leastP (\<lambda>n. a$n \<noteq> 0) n"
+  shows "\<exists>!n. leastP (\<lambda>n. a$n \<noteq> 0) n"
 proof -
   from fps_nonzero_nth_minimal [of a] a0
   obtain n where "a$n \<noteq> 0" "\<forall>m < n. a$m = 0" by blast
@@ -490,9 +490,9 @@
 qed
 
 lemma fps_eq_least_unique:
-  assumes ab: "(a::('a::ab_group_add) fps) \<noteq> b"
+  assumes "(a::('a::ab_group_add) fps) \<noteq> b"
   shows "\<exists>! n. leastP (\<lambda>n. a$n \<noteq> b$n) n"
-  using fps_nonzero_least_unique[of "a - b"] ab
+  using fps_nonzero_least_unique[of "a - b"] assms
   by auto
 
 instantiation fps :: (comm_ring_1) metric_space
@@ -1316,7 +1316,8 @@
       case (Suc k)
       note th = Suc.hyps[symmetric]
       have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. Suc k})$n =
-        (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) $ n"
+        (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. k} -
+          fps_const (a (Suc k)) * X^ Suc k) $ n"
         by (simp add: field_simps)
       also have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)$n"
         using th unfolding fps_sub_nth by simp
@@ -1594,33 +1595,36 @@
 lemma fps_setprod_nth:
   fixes m :: nat
     and a :: "nat \<Rightarrow> ('a::comm_ring_1) fps"
-  shows "(setprod a {0 .. m})$n = setsum (\<lambda>v. setprod (\<lambda>j. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))"
+  shows "(setprod a {0 .. m})$n =
+    setsum (\<lambda>v. setprod (\<lambda>j. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))"
   (is "?P m n")
 proof (induct m arbitrary: n rule: nat_less_induct)
   fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"
-  {
-    assume m0: "m = 0"
-    hence "?P m n" apply simp
-      unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] by simp
-  }
-  moreover
-  {
-    fix k assume k: "m = Suc k"
-    have km: "k < m" using k by arith
+  show "?P m n"
+  proof (cases m)
+    case 0
+    then show ?thesis
+      apply simp
+      unfolding natlist_trivial_1[where n = n, unfolded One_nat_def]
+      apply simp
+      done
+  next
+    case (Suc k)
+    then have km: "k < m" by arith
     have u0: "{0 .. k} \<union> {m} = {0..m}"
-      using k apply (simp add: set_eq_iff)
+      using Suc apply (simp add: set_eq_iff)
       apply presburger
       done
     have f0: "finite {0 .. k}" "finite {m}" by auto
-    have d0: "{0 .. k} \<inter> {m} = {}" using k by auto
+    have d0: "{0 .. k} \<inter> {m} = {}" using Suc by auto
     have "(setprod a {0 .. m}) $ n = (setprod a {0 .. k} * a m) $ n"
       unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0] by simp
     also have "\<dots> = (\<Sum>i = 0..n. (\<Sum>v\<in>natpermute i (k + 1). \<Prod>j\<in>{0..k}. a j $ v ! j) * a m $ (n - i))"
       unfolding fps_mult_nth H[rule_format, OF km] ..
     also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j $ v ! j)"
-      apply (simp add: k)
+      apply (simp add: Suc)
       unfolding natpermute_split[of m "m + 1", simplified, of n,
-        unfolded natlist_trivial_1[unfolded One_nat_def] k]
+        unfolded natlist_trivial_1[unfolded One_nat_def] Suc]
       apply (subst setsum_UN_disjoint)
       apply simp
       apply simp
@@ -1636,12 +1640,11 @@
       apply (rule_tac f="\<lambda>xs. xs @[n - x]" in  setsum_reindex_cong)
       apply (simp add: inj_on_def)
       apply auto
-      unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded k]
+      unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded Suc]
       apply (clarsimp simp add: natpermute_def nth_append)
       done
-    finally have "?P m n" .
-  }
-  ultimately show "?P m n " by (cases m) auto
+    finally show ?thesis .
+  qed
 qed
 
 text{* The special form for powers *}
@@ -1656,7 +1659,8 @@
 
 lemma fps_power_nth:
   fixes m :: nat and a :: "('a::comm_ring_1) fps"
-  shows "(a ^m)$n = (if m=0 then 1$n else setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m - 1}) (natpermute n m))"
+  shows "(a ^m)$n =
+    (if m=0 then 1$n else setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m - 1}) (natpermute n m))"
   by (cases m) (simp_all add: fps_power_nth_Suc del: power_Suc)
 
 lemma fps_nth_power_0:
@@ -1679,45 +1683,47 @@
   assumes a0: "a$0 = (0::'a::{idom})"
     and a1: "a$1 \<noteq> 0"
   shows "(b oo a = c oo a) \<longleftrightarrow> b = c" (is "?lhs \<longleftrightarrow>?rhs")
-proof-
-  { assume ?rhs then have "?lhs" by simp }
-  moreover
-  { assume h: ?lhs
-    { fix n have "b$n = c$n"
-      proof (induct n rule: nat_less_induct)
-        fix n
-        assume H: "\<forall>m<n. b$m = c$m"
-        {
-          assume n0: "n=0"
-          from h have "(b oo a)$n = (c oo a)$n" by simp
-          hence "b$n = c$n" using n0 by (simp add: fps_compose_nth)
-        }
-        moreover
-        {
-          fix n1 assume n1: "n = Suc n1"
-          have f: "finite {0 .. n1}" "finite {n}" by simp_all
-          have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto
-          have d: "{0 .. n1} \<inter> {n} = {}" using n1 by auto
-          have seq: "(\<Sum>i = 0..n1. b $ i * a ^ i $ n) = (\<Sum>i = 0..n1. c $ i * a ^ i $ n)"
-            apply (rule setsum_cong2)
-            using H n1 apply auto
-            done
-          have th0: "(b oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n"
-            unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq] seq
-            using startsby_zero_power_nth_same[OF a0]
-            by simp
-          have th1: "(c oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n"
-            unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq]
-            using startsby_zero_power_nth_same[OF a0]
-            by simp
-          from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
-          have "b$n = c$n" by auto
-        }
-        ultimately show "b$n = c$n" by (cases n) auto
-      qed}
-    then have ?rhs by (simp add: fps_eq_iff)
-  }
-  ultimately show ?thesis by blast
+proof
+  assume ?rhs
+  then show "?lhs" by simp
+next
+  assume h: ?lhs
+  {
+    fix n
+    have "b$n = c$n"
+    proof (induct n rule: nat_less_induct)
+      fix n
+      assume H: "\<forall>m<n. b$m = c$m"
+      {
+        assume n0: "n=0"
+        from h have "(b oo a)$n = (c oo a)$n" by simp
+        hence "b$n = c$n" using n0 by (simp add: fps_compose_nth)
+      }
+      moreover
+      {
+        fix n1 assume n1: "n = Suc n1"
+        have f: "finite {0 .. n1}" "finite {n}" by simp_all
+        have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto
+        have d: "{0 .. n1} \<inter> {n} = {}" using n1 by auto
+        have seq: "(\<Sum>i = 0..n1. b $ i * a ^ i $ n) = (\<Sum>i = 0..n1. c $ i * a ^ i $ n)"
+          apply (rule setsum_cong2)
+          using H n1
+          apply auto
+          done
+        have th0: "(b oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n"
+          unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq] seq
+          using startsby_zero_power_nth_same[OF a0]
+          by simp
+        have th1: "(c oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n"
+          unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq]
+          using startsby_zero_power_nth_same[OF a0]
+          by simp
+        from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
+        have "b$n = c$n" by auto
+      }
+      ultimately show "b$n = c$n" by (cases n) auto
+    qed}
+  then show ?rhs by (simp add: fps_eq_iff)
 qed
 
 
@@ -1789,27 +1795,22 @@
 lemma fps_radical_power_nth[simp]:
   assumes r: "(r k (a$0)) ^ k = a$0"
   shows "fps_radical r k a ^ k $ 0 = (if k = 0 then 1 else a$0)"
-proof-
-  {
-    assume "k = 0"
-    hence ?thesis by simp
-  }
-  moreover
-  {
-    fix h
-    assume h: "k = Suc h"
-    have eq1: "fps_radical r k a ^ k $ 0 = (\<Prod>j\<in>{0..h}. fps_radical r k a $ (replicate k 0) ! j)"
-      unfolding fps_power_nth h by simp
-    also have "\<dots> = (\<Prod>j\<in>{0..h}. r k (a$0))"
-      apply (rule setprod_cong)
-      apply simp
-      using h
-      apply (subgoal_tac "replicate k (0::nat) ! x = 0")
-      apply (auto intro: nth_replicate simp del: replicate.simps)
-      done
-    also have "\<dots> = a$0" using r by (simp add: h setprod_constant)
-    finally have ?thesis using h by simp}
-  ultimately show ?thesis by (cases k) auto
+proof (cases k)
+  case 0
+  then show ?thesis by simp
+next
+  case (Suc h)
+  have eq1: "fps_radical r k a ^ k $ 0 = (\<Prod>j\<in>{0..h}. fps_radical r k a $ (replicate k 0) ! j)"
+    unfolding fps_power_nth Suc by simp
+  also have "\<dots> = (\<Prod>j\<in>{0..h}. r k (a$0))"
+    apply (rule setprod_cong)
+    apply simp
+    using Suc
+    apply (subgoal_tac "replicate k (0::nat) ! x = 0")
+    apply (auto intro: nth_replicate simp del: replicate.simps)
+    done
+  also have "\<dots> = a$0" using r Suc by (simp add: setprod_constant)
+  finally show ?thesis using Suc by simp
 qed
 
 lemma natpermute_max_card:
@@ -1886,7 +1887,8 @@
             have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) =
                 (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))"
               apply (rule setprod_cong, simp)
-              using i r0 apply (simp del: replicate.simps)
+              using i r0
+              apply (simp del: replicate.simps)
               done
             also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k"
               using i r0 by (simp add: setprod_gen_delta)
@@ -1990,7 +1992,7 @@
     and a0: "r (Suc k) (b$0 ::'a::field_char_0) = a$0"
     and b0: "b$0 \<noteq> 0"
   shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b"
-proof-
+proof -
   let ?r = "fps_radical r (Suc k) b"
   have r00: "r (Suc k) (b$0) \<noteq> 0" using b0 r0 by auto
   {
@@ -2131,7 +2133,8 @@
   fixes a:: "'a::field_char_0 fps"
   assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0"
     and a0: "a$0 \<noteq> 0"
-  shows "fps_deriv (fps_radical r (Suc k) a) = fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
+  shows "fps_deriv (fps_radical r (Suc k) a) =
+    fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
 proof -
   let ?r = "fps_radical r (Suc k) a"
   let ?w = "(fps_const (of_nat (Suc k)) * ?r ^ k)"
@@ -2278,7 +2281,8 @@
     and ra0: "r k (a $ 0) ^ k = a $ 0"
     and r1: "(r k 1)^k = 1"
     and a0: "a$0 \<noteq> 0"
-  shows "r k (inverse (a $ 0)) = r k 1 / (r k (a $ 0)) \<longleftrightarrow> fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a"
+  shows "r k (inverse (a $ 0)) = r k 1 / (r k (a $ 0)) \<longleftrightarrow>
+    fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a"
   using radical_divide[where k=k and r=r and a=1 and b=a, OF k ] ra0 r1 a0
   by (simp add: divide_inverse fps_divide_def)
 
@@ -2344,10 +2348,7 @@
   have "((1+X)*a) $n = setsum (\<lambda>i. (1+X)$i * a$(n-i)) {0..n}"
     by (simp add: fps_mult_nth)
   also have "\<dots> = setsum (\<lambda>i. (1+X)$i * a$(n-i)) {0.. 1}"
-    unfolding Suc
-    apply (rule setsum_mono_zero_right)
-    apply auto
-    done
+    unfolding Suc by (rule setsum_mono_zero_right) auto
   also have "\<dots> = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))"
     by (simp add: Suc)
   finally show ?thesis .
@@ -2747,7 +2748,7 @@
   case 0
   then show ?thesis by simp
 next
-  case(Suc h)
+  case (Suc h)
   {
     fix n
     {
@@ -3019,13 +3020,15 @@
     apply (rule setsum_cong2)
     apply simp
     apply (frule binomial_fact[where ?'a = 'a, symmetric])
-    by (simp add: field_simps of_nat_mult)
+    apply (simp add: field_simps of_nat_mult)
+    done
 qed
 
 text{* And the nat-form -- also available from Binomial.thy *}
 lemma binomial_theorem: "(a+b) ^ n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
   using gbinomial_theorem[of "of_nat a" "of_nat b" n]
-  unfolding of_nat_add[symmetric] of_nat_power[symmetric] of_nat_mult[symmetric] of_nat_setsum[symmetric]
+  unfolding of_nat_add[symmetric] of_nat_power[symmetric] of_nat_mult[symmetric]
+    of_nat_setsum[symmetric]
   by simp
 
 
@@ -3089,6 +3092,7 @@
     unfolding fps_deriv_eq_iff by simp
 qed
 
+
 subsubsection{* Binomial series *}
 
 definition "fps_binomial a = Abs_fps (\<lambda>n. a gchoose n)"
@@ -3123,47 +3127,59 @@
         by (cases n, simp_all add: field_simps del: of_nat_Suc)
     }
     note th0 = this
-    {fix n have "a$n = (c gchoose n) * a$0"
-      proof(induct n)
-        case 0 thus ?case by simp
+    {
+      fix n
+      have "a$n = (c gchoose n) * a$0"
+      proof (induct n)
+        case 0
+        thus ?case by simp
       next
         case (Suc m)
         thus ?case unfolding th0
           apply (simp add: field_simps del: of_nat_Suc)
           unfolding mult_assoc[symmetric] gbinomial_mult_1
-          by (simp add: field_simps)
-      qed}
+          apply (simp add: field_simps)
+          done
+      qed
+    }
     note th1 = this
     have ?rhs
       apply (simp add: fps_eq_iff)
       apply (subst th1)
-      by (simp add: field_simps)}
+      apply (simp add: field_simps)
+      done
+  }
   moreover
-  {assume h: ?rhs
-  have th00:"\<And>x y. x * (a$0 * y) = a$0 * (x*y)" by (simp add: mult_commute)
+  {
+    assume h: ?rhs
+    have th00: "\<And>x y. x * (a$0 * y) = a$0 * (x*y)"
+      by (simp add: mult_commute)
     have "?l = ?r"
       apply (subst h)
       apply (subst (2) h)
       apply (clarsimp simp add: fps_eq_iff field_simps)
       unfolding mult_assoc[symmetric] th00 gbinomial_mult_1
-      by (simp add: field_simps gbinomial_mult_1)}
+      apply (simp add: field_simps gbinomial_mult_1)
+      done
+  }
   ultimately show ?thesis by blast
 qed
 
 lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + X)"
-proof-
+proof -
   let ?a = "fps_binomial c"
   have th0: "?a = fps_const (?a$0) * ?a" by (simp)
   from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis .
 qed
 
 lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r")
-proof-
+proof -
   let ?P = "?r - ?l"
   let ?b = "fps_binomial"
   let ?db = "\<lambda>x. fps_deriv (?b x)"
   have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)"  by simp
-  also have "\<dots> = inverse (1 + X) * (fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))"
+  also have "\<dots> = inverse (1 + X) *
+      (fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))"
     unfolding fps_binomial_deriv
     by (simp add: fps_divide_def field_simps)
   also have "\<dots> = (fps_const (c + d)/ (1 + X)) * ?P"
@@ -3182,7 +3198,8 @@
 proof-
   have th: "?r$0 \<noteq> 0" by simp
   have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + X)"
-    by (simp add: fps_inverse_deriv[OF th] fps_divide_def power2_eq_square mult_commute fps_const_neg[symmetric] del: fps_const_neg minus_one)
+    by (simp add: fps_inverse_deriv[OF th] fps_divide_def
+      power2_eq_square mult_commute fps_const_neg[symmetric] del: fps_const_neg minus_one)
   have eq: "inverse ?r $ 0 = 1"
     by (simp add: fps_inverse_def)
   from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + X)" "- 1"] th'] eq
@@ -3190,8 +3207,9 @@
 qed
 
 text{* Vandermonde's Identity as a consequence *}
-lemma gbinomial_Vandermonde: "setsum (\<lambda>k. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
-proof-
+lemma gbinomial_Vandermonde:
+  "setsum (\<lambda>k. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
+proof -
   let ?ba = "fps_binomial a"
   let ?bb = "fps_binomial b"
   let ?bab = "fps_binomial (a + b)"
@@ -3199,9 +3217,11 @@
   then show ?thesis by (simp add: fps_mult_nth)
 qed
 
-lemma binomial_Vandermonde: "setsum (\<lambda>k. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
+lemma binomial_Vandermonde:
+  "setsum (\<lambda>k. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
   using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n]
-  apply (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric] of_nat_setsum[symmetric] of_nat_add[symmetric])
+  apply (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric]
+    of_nat_setsum[symmetric] of_nat_add[symmetric])
   apply simp
   done
 
@@ -3216,36 +3236,49 @@
 lemma Vandermonde_pochhammer_lemma:
   fixes a :: "'a::field_char_0"
   assumes b: "\<forall> j\<in>{0 ..<n}. b \<noteq> of_nat j"
-  shows "setsum (%k. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) / (of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} = pochhammer (- (a+ b)) n / pochhammer (- b) n" (is "?l = ?r")
-proof-
+  shows "setsum (%k. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) /
+      (of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} =
+    pochhammer (- (a+ b)) n / pochhammer (- b) n"
+  (is "?l = ?r")
+proof -
   let ?m1 = "%m. (- 1 :: 'a) ^ m"
   let ?f = "%m. of_nat (fact m)"
   let ?p = "%(x::'a). pochhammer (- x)"
   from b have bn0: "?p b n \<noteq> 0" unfolding pochhammer_eq_0_iff by simp
-  {fix k assume kn: "k \<in> {0..n}"
-    {assume c:"pochhammer (b - of_nat n + 1) n = 0"
+  {
+    fix k
+    assume kn: "k \<in> {0..n}"
+    {
+      assume c:"pochhammer (b - of_nat n + 1) n = 0"
       then obtain j where j: "j < n" "b - of_nat n + 1 = - of_nat j"
         unfolding pochhammer_eq_0_iff by blast
       from j have "b = of_nat n - of_nat j - of_nat 1"
         by (simp add: algebra_simps)
       then have "b = of_nat (n - j - 1)"
         using j kn by (simp add: of_nat_diff)
-      with b have False using j by auto}
+      with b have False using j by auto
+    }
     then have nz: "pochhammer (1 + b - of_nat n) n \<noteq> 0"
       by (auto simp add: algebra_simps)
 
     from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k \<noteq> 0"
       by (rule pochhammer_neq_0_mono)
-    {assume k0: "k = 0 \<or> n =0"
-      then have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
+    {
+      assume k0: "k = 0 \<or> n =0"
+      then have "b gchoose (n - k) =
+        (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
         using kn
-        by (cases "k=0", simp_all add: gbinomial_pochhammer)}
+        by (cases "k = 0") (simp_all add: gbinomial_pochhammer)
+    }
     moreover
-    { assume n0: "n \<noteq> 0" and k0: "k \<noteq> 0"
-      then obtain m where m: "n = Suc m" by (cases n, auto)
-      from k0 obtain h where h: "k = Suc h" by (cases k, auto)
-      { assume kn: "k = n"
-        then have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
+    {
+      assume n0: "n \<noteq> 0" and k0: "k \<noteq> 0"
+      then obtain m where m: "n = Suc m" by (cases n) auto
+      from k0 obtain h where h: "k = Suc h" by (cases k) auto
+      {
+        assume kn: "k = n"
+        then have "b gchoose (n - k) =
+          (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
           using kn pochhammer_minus'[where k=k and n=n and b=b]
           apply (simp add:  pochhammer_same)
           using bn0
@@ -3253,9 +3286,9 @@
           done
       }
       moreover
-      { assume nk: "k \<noteq> n"
-        have m1nk: "?m1 n = setprod (%i. - 1) {0..m}"
-          "?m1 k = setprod (%i. - 1) {0..h}"
+      {
+        assume nk: "k \<noteq> n"
+        have m1nk: "?m1 n = setprod (%i. - 1) {0..m}" "?m1 k = setprod (%i. - 1) {0..h}"
           by (simp_all add: setprod_constant m h)
         from kn nk have kn': "k < n" by simp
         have bnz0: "pochhammer (b - of_nat n + 1) k \<noteq> 0"
@@ -3263,8 +3296,10 @@
           unfolding pochhammer_eq_0_iff
           apply auto
           apply (erule_tac x= "n - ka - 1" in allE)
-          by (auto simp add: algebra_simps of_nat_diff)
-        have eq1: "setprod (%k. (1::'a) + of_nat m - of_nat k) {0 .. h} = setprod of_nat {Suc (m - h) .. Suc m}"
+          apply (auto simp add: algebra_simps of_nat_diff)
+          done
+        have eq1: "setprod (%k. (1::'a) + of_nat m - of_nat k) {0 .. h} =
+          setprod of_nat {Suc (m - h) .. Suc m}"
           apply (rule strong_setprod_reindex_cong[where f="%k. Suc m - k "])
           using kn' h m
           apply (auto simp add: inj_on_def image_def)
@@ -3274,7 +3309,6 @@
 
         have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
           unfolding m1nk
-
           unfolding m h pochhammer_Suc_setprod
           apply (simp add: field_simps del: fact_Suc minus_one)
           unfolding fact_altdef_nat id_def
@@ -3301,9 +3335,11 @@
           using kn
           apply (auto simp add: inj_on_def m h image_def)
           apply (rule_tac x= "m - x" in bexI)
-          by (auto simp add: of_nat_diff)
-
-        have "?m1 n * ?p b n = pochhammer (b - of_nat n + 1) k * setprod (%i. b - of_nat i) {0.. n - k - 1}"
+          apply (auto simp add: of_nat_diff)
+          done
+
+        have "?m1 n * ?p b n =
+          pochhammer (b - of_nat n + 1) k * setprod (%i. b - of_nat i) {0.. n - k - 1}"
           unfolding th20 th21
           unfolding h m
           apply (subst setprod_Un_disjoint[symmetric])
@@ -3312,19 +3348,28 @@
           apply (rule setprod_cong)
           apply auto
           done
-        then have th2: "(?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k = setprod (%i. b - of_nat i) {0.. n - k - 1}"
+        then have th2: "(?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k =
+          setprod (%i. b - of_nat i) {0.. n - k - 1}"
           using nz' by (simp add: field_simps)
-        have "(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k) = ((?m1 k * ?p (of_nat n) k) / ?f n) * ((?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k)"
+        have "(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k) =
+          ((?m1 k * ?p (of_nat n) k) / ?f n) * ((?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k)"
           using bnz0
           by (simp add: field_simps)
         also have "\<dots> = b gchoose (n - k)"
           unfolding th1 th2
           using kn' by (simp add: gbinomial_def)
-        finally have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" by simp}
-      ultimately have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
+        finally have "b gchoose (n - k) =
+          (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
+          by simp
+      }
+      ultimately
+      have "b gchoose (n - k) =
+        (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
         by (cases "k = n") auto
     }
-    ultimately have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" "pochhammer (1 + b - of_nat n) k \<noteq> 0 "
+    ultimately have "b gchoose (n - k) =
+        (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
+      "pochhammer (1 + b - of_nat n) k \<noteq> 0 "
       apply (cases "n = 0")
       using nz'
       apply auto
@@ -3340,14 +3385,15 @@
     unfolding gbinomial_Vandermonde[symmetric]
     apply (simp add: th00)
     unfolding gbinomial_pochhammer
-    using bn0 apply (simp add: setsum_left_distrib setsum_right_distrib field_simps)
+    using bn0
+    apply (simp add: setsum_left_distrib setsum_right_distrib field_simps)
     apply (rule setsum_cong2)
     apply (drule th00(2))
-    by (simp add: field_simps power_add[symmetric])
+    apply (simp add: field_simps power_add[symmetric])
+    done
   finally show ?thesis by simp
 qed
 
-
 lemma Vandermonde_pochhammer:
   fixes a :: "'a::field_char_0"
   assumes c: "ALL i : {0..< n}. c \<noteq> - of_nat i"
@@ -3481,9 +3527,14 @@
 lemma nat_induct2: "P 0 \<Longrightarrow> P 1 \<Longrightarrow> (\<And>n. P n \<Longrightarrow> P (n + 2)) \<Longrightarrow> P (n::nat)"
   unfolding One_nat_def numeral_2_eq_2
   apply (induct n rule: nat_less_induct)
-  apply (case_tac n, simp)
-  apply (rename_tac m, case_tac m, simp)
-  apply (rename_tac k, case_tac k, simp_all)
+  apply (case_tac n)
+  apply simp
+  apply (rename_tac m)
+  apply (case_tac m)
+  apply simp
+  apply (rename_tac k)
+  apply (case_tac k)
+  apply simp_all
   done
 
 lemma nat_add_1_add_1: "(n::nat) + 1 + 1 = n + 2"
@@ -3633,7 +3684,8 @@
 subsection {* Hypergeometric series *}
 
 definition "F as bs (c::'a::{field_char_0, field_inverse_zero}) =
-  Abs_fps (%n. (foldl (%r a. r* pochhammer a n) 1 as * c^n)/ (foldl (%r b. r * pochhammer b n) 1 bs * of_nat (fact n)))"
+  Abs_fps (%n. (foldl (%r a. r* pochhammer a n) 1 as * c^n) /
+    (foldl (%r b. r * pochhammer b n) 1 bs * of_nat (fact n)))"
 
 lemma F_nth[simp]: "F as bs c $ n =
   (foldl (\<lambda>r a. r* pochhammer a n) 1 as * c^n) /
@@ -3644,10 +3696,12 @@
   "foldl (%r x. r * f x) (v::'a::comm_ring_1) as * x = foldl (%r x. r * f x) (v * x) as "
   by (induct as arbitrary: x v) (auto simp add: algebra_simps)
 
-lemma foldr_mult_foldl: "foldr (%x r. r * f x) as v = foldl (%r x. r * f x) (v :: 'a::comm_ring_1) as"
+lemma foldr_mult_foldl:
+  "foldr (%x r. r * f x) as v = foldl (%r x. r * f x) (v :: 'a::comm_ring_1) as"
   by (induct as arbitrary: v) (auto simp add: foldl_mult_start)
 
-lemma F_nth_alt: "F as bs c $ n = foldr (\<lambda>a r. r * pochhammer a n) as (c ^ n) /
+lemma F_nth_alt:
+  "F as bs c $ n = foldr (\<lambda>a r. r * pochhammer a n) as (c ^ n) /
     foldr (\<lambda>b r. r * pochhammer b n) bs (of_nat (fact n))"
   by (simp add: foldl_mult_start foldr_mult_foldl)
 
@@ -3659,7 +3713,8 @@
   let ?a = "(Abs_fps (\<lambda>n. 1)) oo (fps_const c * X)"
   have th0: "(fps_const c * X) $ 0 = 0" by simp
   show ?thesis unfolding gp[OF th0, symmetric]
-    by (auto simp add: fps_eq_iff pochhammer_fact[symmetric] fps_compose_nth power_mult_distrib cond_value_iff setsum_delta' cong del: if_weak_cong)
+    by (auto simp add: fps_eq_iff pochhammer_fact[symmetric]
+      fps_compose_nth power_mult_distrib cond_value_iff setsum_delta' cong del: if_weak_cong)
 qed
 
 lemma F_B[simp]: "F [-a] [] (- 1) = fps_binomial a"
@@ -3673,12 +3728,15 @@
   apply auto
   done
 
-lemma foldl_prod_prod: "foldl (%(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (%r x. r * g x) w as =
+lemma foldl_prod_prod:
+  "foldl (%(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (%r x. r * g x) w as =
     foldl (%r x. r * f x * g x) (v*w) as"
   by (induct as arbitrary: v w) (auto simp add: algebra_simps)
 
 
-lemma F_rec: "F as bs c $ Suc n = ((foldl (%r a. r* (a + of_nat n)) c as)/ (foldl (%r b. r * (b + of_nat n)) (of_nat (Suc n)) bs )) * F as bs c $ n"
+lemma F_rec:
+  "F as bs c $ Suc n = ((foldl (%r a. r* (a + of_nat n)) c as) /
+    (foldl (%r b. r * (b + of_nat n)) (of_nat (Suc n)) bs )) * F as bs c $ n"
   apply (simp del: of_nat_Suc of_nat_add fact_Suc)
   apply (simp add: foldl_mult_start del: fact_Suc of_nat_Suc)
   unfolding foldl_prod_prod[unfolded foldl_mult_start] pochhammer_Suc
--- a/src/HOL/Library/Fraction_Field.thy	Sun Aug 25 21:25:17 2013 +0200
+++ b/src/HOL/Library/Fraction_Field.thy	Sun Aug 25 23:20:25 2013 +0200
@@ -75,23 +75,21 @@
 
 code_datatype Fract
 
-lemma Fract_cases [case_names Fract, cases type: fract]:
-  assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C"
-  shows C
-  using assms by (cases q) (clarsimp simp add: Fract_def fract_def quotient_def)
+lemma Fract_cases [cases type: fract]:
+  obtains (Fract) a b where "q = Fract a b" "b \<noteq> 0"
+  by (cases q) (clarsimp simp add: Fract_def fract_def quotient_def)
 
 lemma Fract_induct [case_names Fract, induct type: fract]:
-  assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)"
-  shows "P q"
-  using assms by (cases q) simp
+  shows "(\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)) \<Longrightarrow> P q"
+  by (cases q) simp
 
 lemma eq_fract:
   shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b"
-  and "\<And>a. Fract a 0 = Fract 0 1"
-  and "\<And>a c. Fract 0 a = Fract 0 c"
+    and "\<And>a. Fract a 0 = Fract 0 1"
+    and "\<And>a c. Fract 0 a = Fract 0 c"
   by (simp_all add: Fract_def)
 
-instantiation fract :: (idom) "{comm_ring_1, power}"
+instantiation fract :: (idom) "{comm_ring_1,power}"
 begin
 
 definition Zero_fract_def [code_unfold]: "0 = Fract 0 1"
@@ -103,13 +101,16 @@
     fractrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"
 
 lemma add_fract [simp]:
-  assumes "b \<noteq> (0::'a::idom)" and "d \<noteq> 0"
+  assumes "b \<noteq> (0::'a::idom)"
+    and "d \<noteq> 0"
   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
 proof -
   have "(\<lambda>x y. fractrel``{(fst x * snd y + fst y * snd x, snd x * snd y :: 'a)})
     respects2 fractrel"
-  apply (rule equiv_fractrel [THEN congruent2_commuteI]) apply (auto simp add: algebra_simps)
-  unfolding mult_assoc[symmetric] .
+    apply (rule equiv_fractrel [THEN congruent2_commuteI])
+    apply (auto simp add: algebra_simps)
+    unfolding mult_assoc[symmetric]
+    done
   with assms show ?thesis by (simp add: Fract_def add_fract_def UN_fractrel2)
 qed
 
@@ -140,8 +141,9 @@
 lemma mult_fract [simp]: "Fract (a::'a::idom) b * Fract c d = Fract (a * c) (b * d)"
 proof -
   have "(\<lambda>x y. fractrel `` {(fst x * fst y, snd x * snd y :: 'a)}) respects2 fractrel"
-    apply (rule equiv_fractrel [THEN congruent2_commuteI]) apply (auto simp add: algebra_simps)
-    unfolding mult_assoc[symmetric] .
+    apply (rule equiv_fractrel [THEN congruent2_commuteI])
+    apply (auto simp add: algebra_simps)
+    done
   then show ?thesis by (simp add: Fract_def mult_fract_def UN_fractrel2)
 qed
 
@@ -155,34 +157,27 @@
 
 instance
 proof
-  fix q r s :: "'a fract" show "(q * r) * s = q * (r * s)" 
+  fix q r s :: "'a fract"
+  show "(q * r) * s = q * (r * s)" 
     by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
-next
-  fix q r :: "'a fract" show "q * r = r * q"
+  show "q * r = r * q"
     by (cases q, cases r) (simp add: eq_fract algebra_simps)
-next
-  fix q :: "'a fract" show "1 * q = q"
+  show "1 * q = q"
     by (cases q) (simp add: One_fract_def eq_fract)
-next
-  fix q r s :: "'a fract" show "(q + r) + s = q + (r + s)"
+  show "(q + r) + s = q + (r + s)"
     by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
-next
-  fix q r :: "'a fract" show "q + r = r + q"
+  show "q + r = r + q"
     by (cases q, cases r) (simp add: eq_fract algebra_simps)
-next
-  fix q :: "'a fract" show "0 + q = q"
+  show "0 + q = q"
     by (cases q) (simp add: Zero_fract_def eq_fract)
-next
-  fix q :: "'a fract" show "- q + q = 0"
+  show "- q + q = 0"
     by (cases q) (simp add: Zero_fract_def eq_fract)
-next
-  fix q r :: "'a fract" show "q - r = q + - r"
+  show "q - r = q + - r"
     by (cases q, cases r) (simp add: eq_fract)
-next
-  fix q r s :: "'a fract" show "(q + r) * s = q * s + r * s"
+  show "(q + r) * s = q * s + r * s"
     by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
-next
-  show "(0::'a fract) \<noteq> 1" by (simp add: Zero_fract_def One_fract_def eq_fract)
+  show "(0::'a fract) \<noteq> 1"
+    by (simp add: Zero_fract_def One_fract_def eq_fract)
 qed
 
 end
@@ -205,28 +200,28 @@
   "1 = Fract 1 1"
   by (simp_all add: fract_collapse)
 
-lemma Fract_cases_nonzero [case_names Fract 0]:
-  assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"
-  assumes 0: "q = 0 \<Longrightarrow> C"
-  shows C
+lemma Fract_cases_nonzero:
+  obtains (Fract) a b where "q = Fract a b" "b \<noteq> 0" "a \<noteq> 0"
+    | (0) "q = 0"
 proof (cases "q = 0")
-  case True then show C using 0 by auto
+  case True
+  then show thesis using 0 by auto
 next
   case False
   then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto
   moreover with False have "0 \<noteq> Fract a b" by simp
   with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_fract_def eq_fract)
-  with Fract `q = Fract a b` `b \<noteq> 0` show C by auto
+  with Fract `q = Fract a b` `b \<noteq> 0` show thesis by auto
 qed
   
 
-
 subsubsection {* The field of rational numbers *}
 
 context idom
 begin
+
 subclass ring_no_zero_divisors ..
-thm mult_eq_0_iff
+
 end
 
 instantiation fract :: (idom) field_inverse_zero
@@ -236,12 +231,11 @@
   "inverse q = Abs_fract (\<Union>x \<in> Rep_fract q.
      fractrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"
 
-
 lemma inverse_fract [simp]: "inverse (Fract a b) = Fract (b::'a::idom) a"
 proof -
-  have stupid: "\<And>x. (0::'a) = x \<longleftrightarrow> x = 0" by auto
+  have *: "\<And>x. (0::'a) = x \<longleftrightarrow> x = 0" by auto
   have "(\<lambda>x. fractrel `` {if fst x = 0 then (0, 1) else (snd x, fst x :: 'a)}) respects fractrel"
-    by (auto simp add: congruent_def stupid algebra_simps)
+    by (auto simp add: congruent_def * algebra_simps)
   then show ?thesis by (simp add: Fract_def inverse_fract_def UN_fractrel)
 qed
 
@@ -315,20 +309,20 @@
 begin
 
 definition le_fract_def:
-   "q \<le> r \<longleftrightarrow> the_elem (\<Union>x \<in> Rep_fract q. \<Union>y \<in> Rep_fract r.
-      {(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)})"
+  "q \<le> r \<longleftrightarrow> the_elem (\<Union>x \<in> Rep_fract q. \<Union>y \<in> Rep_fract r.
+    {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"
 
 definition less_fract_def: "z < (w::'a fract) \<longleftrightarrow> z \<le> w \<and> \<not> w \<le> z"
 
 lemma le_fract [simp]:
   assumes "b \<noteq> 0" and "d \<noteq> 0"
   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
-by (simp add: Fract_def le_fract_def le_congruent2 UN_fractrel2 assms)
+  by (simp add: Fract_def le_fract_def le_congruent2 UN_fractrel2 assms)
 
 lemma less_fract [simp]:
   assumes "b \<noteq> 0" and "d \<noteq> 0"
   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
-by (simp add: less_fract_def less_le_not_le mult_ac assms)
+  by (simp add: less_fract_def less_le_not_le mult_ac assms)
 
 instance
 proof
@@ -427,10 +421,11 @@
 instance fract :: (linordered_idom) linordered_field_inverse_zero
 proof
   fix q r s :: "'a fract"
-  show "q \<le> r ==> s + q \<le> s + r"
+  assume "q \<le> r"
+  then show "s + q \<le> s + r"
   proof (induct q, induct r, induct s)
     fix a b c d e f :: 'a
-    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
+    assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
     assume le: "Fract a b \<le> Fract c d"
     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
     proof -
@@ -443,7 +438,10 @@
       with neq show ?thesis by (simp add: field_simps)
     qed
   qed
-  show "q < r ==> 0 < s ==> s * q < s * r"
+next
+  fix q r s :: "'a fract"
+  assume "q < r" and "0 < s"
+  then show "s * q < s * r"
   proof (induct q, induct r, induct s)
     fix a b c d e f :: 'a
     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
@@ -483,36 +481,28 @@
   thus "P q" by (force simp add: linorder_neq_iff step step')
 qed
 
-lemma zero_less_Fract_iff:
-  "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
+lemma zero_less_Fract_iff: "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
   by (auto simp add: Zero_fract_def zero_less_mult_iff)
 
-lemma Fract_less_zero_iff:
-  "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
+lemma Fract_less_zero_iff: "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0"
   by (auto simp add: Zero_fract_def mult_less_0_iff)
 
-lemma zero_le_Fract_iff:
-  "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
+lemma zero_le_Fract_iff: "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
   by (auto simp add: Zero_fract_def zero_le_mult_iff)
 
-lemma Fract_le_zero_iff:
-  "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
+lemma Fract_le_zero_iff: "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0"
   by (auto simp add: Zero_fract_def mult_le_0_iff)
 
-lemma one_less_Fract_iff:
-  "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
+lemma one_less_Fract_iff: "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"
   by (auto simp add: One_fract_def mult_less_cancel_right_disj)
 
-lemma Fract_less_one_iff:
-  "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
+lemma Fract_less_one_iff: "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"
   by (auto simp add: One_fract_def mult_less_cancel_right_disj)
 
-lemma one_le_Fract_iff:
-  "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
+lemma one_le_Fract_iff: "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"
   by (auto simp add: One_fract_def mult_le_cancel_right)
 
-lemma Fract_le_one_iff:
-  "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
+lemma Fract_le_one_iff: "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"
   by (auto simp add: One_fract_def mult_le_cancel_right)
 
 end
--- a/src/HOL/Library/Univ_Poly.thy	Sun Aug 25 21:25:17 2013 +0200
+++ b/src/HOL/Library/Univ_Poly.thy	Sun Aug 25 23:20:25 2013 +0200
@@ -10,7 +10,8 @@
 
 text{* Application of polynomial as a function. *}
 
-primrec (in semiring_0) poly :: "'a list => 'a  => 'a" where
+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"
 
@@ -22,173 +23,171 @@
 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_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t else (h + hd l2)#(t +++ tl l2))"
 
 text{*Multiplication by a constant*}
 primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "%*" 70) where
-   cmult_Nil:  "c %* [] = []"
-|  cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
+  cmult_Nil:  "c %* [] = []"
+| cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
 
 text{*Multiplication by a polynomial*}
 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
+  pmult_Nil:  "[] *** l2 = []"
+| pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
                               else (h %* l2) +++ ((0) # (t *** l2)))"
 
 text{*Repeated multiplication by a polynomial*}
 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"
+  mulexp_zero:  "mulexp 0 p q = q"
+| mulexp_Suc:   "mulexp (Suc n) p q = p *** mulexp n p q"
 
 text{*Exponential*}
 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)"
+  pexp_0:   "p %^ 0 = [1]"
+| pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
 
 text{*Quotient related value of dividing a polynomial by x + a*}
 (* Useful for divisor properties in inductive proofs *)
-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))"
+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))"
 
 text{*normalization of polynomials (remove extra 0 coeff)*}
 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)))"
+| 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{*Other definitions*}
 
-definition (in ring_1)
-  poly_minus :: "'a list => 'a list" ("-- _" [80] 80) where
-  "-- p = (- 1) %* p"
+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))"
+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))"
 
     --{*order of a polynomial*}
-definition (in ring_1) order :: "'a => 'a list => nat" where
-  "order a p = (SOME n. ([-a, 1] %^ n) divides p &
-                      ~ (([-a, 1] %^ (Suc n)) divides p))"
+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))"
 
      --{*degree of a polynomial*}
-definition (in semiring_0) degree :: "'a list => nat" where
-  "degree p = length (pnormalize p) - 1"
+definition (in semiring_0) degree :: "'a list \<Rightarrow> nat"
+  where "degree p = length (pnormalize p) - 1"
 
      --{*squarefree polynomials --- NB with respect to real roots only.*}
-definition (in ring_1)
-  rsquarefree :: "'a list => bool" where
-  "rsquarefree p = (poly p \<noteq> poly [] &
-                     (\<forall>a. (order a p = 0) | (order a p = 1)))"
+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)"
 
 context semiring_0
 begin
 
 lemma padd_Nil2[simp]: "p +++ [] = p"
-by (induct p) auto
+  by (induct p) auto
 
 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
-by auto
+  by auto
 
 lemma pminus_Nil: "-- [] = []"
-by (simp add: poly_minus_def)
+  by (simp add: poly_minus_def)
 
 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
+
 end
 
 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)"
-by simp
+  by simp
 
 text{*Handy general properties*}
 
 lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
-proof(induct b arbitrary: a)
-  case Nil thus ?case by auto
+proof (induct b arbitrary: a)
+  case Nil
+  thus ?case by auto
 next
-  case (Cons b bs a) thus ?case by (cases a) (simp_all add: add_commute)
+  case (Cons b bs a)
+  thus ?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: add_ac)
-done
+  apply (induct a)
+  apply (simp, clarify)
+  apply (case_tac b, simp_all add: add_ac)
+  done
 
 lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
-apply (induct p arbitrary: q, simp)
-apply (case_tac q, simp_all add: distrib_left)
-done
+  apply (induct p arbitrary: q)
+  apply simp
+  apply (case_tac q, simp_all add: distrib_left)
+  done
 
 lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
-apply (induct "t", simp)
-apply (auto simp add: padd_commut)
-apply (case_tac t, auto)
-done
+  apply (induct t)
+  apply simp
+  apply (auto simp add: padd_commut)
+  apply (case_tac t, auto)
+  done
 
 text{*properties of evaluation of polynomials.*}
 
 lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
 proof(induct p1 arbitrary: p2)
-  case Nil thus ?case by simp
+  case Nil
+  thus ?case by simp
 next
-  case (Cons a as p2) thus ?case
+  case (Cons a as p2)
+  thus ?case
     by (cases p2) (simp_all  add: add_ac distrib_left)
 qed
 
 lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
-apply (induct "p")
-apply (case_tac [2] "x=zero")
-apply (auto simp add: distrib_left mult_ac)
-done
+  apply (induct p)
+  apply (case_tac [2] "x = zero")
+  apply (auto simp add: distrib_left mult_ac)
+  done
 
 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 mult_ac)
+  by (induct p) (auto simp add: distrib_left mult_ac)
 
 lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
-apply (simp add: poly_minus_def)
-apply (auto simp add: poly_cmult)
-done
+  apply (simp add: poly_minus_def)
+  apply (auto simp add: poly_cmult)
+  done
 
 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
+proof (induct p1 arbitrary: p2)
+  case Nil
+  thus ?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 mult_ac)
+  thus ?case by (cases as)
+    (simp_all add: poly_cmult poly_add distrib_right distrib_left mult_ac)
 qed
 
 class idom_char_0 = idom + ring_char_0
 
 lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
-apply (induct "n")
-apply (auto simp add: poly_cmult poly_mult)
-done
+  by (induct n) (auto simp add: poly_cmult poly_mult)
 
 text{*More Polynomial Evaluation Lemmas*}
 
-lemma  (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
-by simp
+lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
+  by simp
 
 lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
   by (simp add: poly_mult mult_assoc)
 
 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
-by (induct "p", auto)
+  by (induct p) auto
 
 lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
-apply (induct "n")
-apply (auto simp add: poly_mult mult_assoc)
-done
+  by (induct n) (auto simp add: poly_mult mult_assoc)
 
 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
  @{term "p(x)"} *}
@@ -196,11 +195,11 @@
 lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
 proof(induct t)
   case Nil
-  {fix h have "[h] = [h] +++ [- a, 1] *** []" by simp}
+  { fix h have "[h] = [h] +++ [- a, 1] *** []" by simp }
   thus ?case by blast
 next
   case (Cons  x xs)
-  {fix h
+  { 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)"
@@ -210,12 +209,12 @@
 qed
 
 lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
-by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto)
+  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}
+proof -
+  { assume p: "p = []" hence ?thesis by simp }
   moreover
   {
     fix x xs assume p: "p = x#xs"
@@ -224,59 +223,68 @@
       hence "poly p a = 0" by (simp add: poly_add poly_cmult)
     }
     moreover
-    {assume p0: "poly p a = 0"
+    { 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)
+      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
 qed
 
 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)
+  by (induct p) auto
 
 lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
-by (induct "p", auto)
+  by (induct p) auto
 
 lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
-by auto
+  by auto
 
 subsection{*Polynomial length*}
 
 lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
-by (induct "p", auto)
+  by (induct p) auto
 
 lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
-apply (induct p1 arbitrary: p2, simp_all)
-apply arith
-done
+  by (induct p1 arbitrary: p2) (simp_all, arith)
 
 lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
-by (simp add: poly_add_length)
+  by (simp add: poly_add_length)
 
 lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
- "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
-by (auto simp add: poly_mult)
+  "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
+  by (auto simp add: poly_mult)
 
 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)
+  by (auto simp add: poly_mult)
 
 text{*Normalisation Properties*}
 
 lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
-by (induct "p", auto)
+  by (induct p) auto
 
 text{*A nontrivial polynomial of degree n has no more than n roots*}
 lemma (in idom) poly_roots_index_lemma:
    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
+proof (induct n arbitrary: p x)
+  case 0
+  thus ?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)"
+  {
+    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
     from p0(1)[unfolded poly_linear_divides[of p x]]
     have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
@@ -293,48 +301,49 @@
       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 apply auto
-      apply (erule_tac x="m" in allE)
+    with i[rule_format, of y] y(1) y(2) have False
       apply auto
-      done}
+      apply (erule_tac x = "m" in allE)
+      apply auto
+      done
+  }
   thus ?case by blast
 qed
 
 
-lemma (in idom) poly_roots_index_length: "poly p x \<noteq> poly [] x ==>
-      \<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"
-by (blast intro: poly_roots_index_lemma)
+lemma (in idom) poly_roots_index_length:
+  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. n \<le> length p \<and> x = i n)"
+  by (blast intro: poly_roots_index_lemma)
 
-lemma (in idom) poly_roots_finite_lemma1: "poly p x \<noteq> poly [] x ==>
-      \<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & 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)
-apply (simp add: less_Suc_eq_le)
-done
-
+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)"
+  apply (drule poly_roots_index_length, safe)
+  apply (rule_tac x = "Suc (length p)" in exI)
+  apply (rule_tac x = i in exI)
+  apply (simp add: less_Suc_eq_le)
+  done
 
 lemma (in idom) idom_finite_lemma:
-  assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)"
+  assumes P: "\<forall>x. P x --> (\<exists>n. n < length j \<and> x = j!n)"
   shows "finite {x. P x}"
-proof-
+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
 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 (auto simp add: image_iff)
+  apply (erule_tac x="x" in allE, clarsimp)
+  apply (case_tac "n = length p")
+  apply (auto simp add: order_le_less)
+  done
 
-lemma (in idom) poly_roots_finite_lemma2: "poly p x \<noteq> poly [] x ==>
-      \<exists>i. \<forall>x. (poly p x = 0) --> 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 (auto simp add: image_iff)
-apply (erule_tac x="x" in allE, clarsimp)
-by (case_tac "n=length p", auto simp add: order_le_less)
-
-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)"
@@ -346,8 +355,7 @@
   with infinite_UNIV_nat show False ..
 qed
 
-lemma (in idom_char_0) poly_roots_finite: "(poly p \<noteq> poly []) =
-  finite {x. poly p x = 0}"
+lemma (in idom_char_0) poly_roots_finite: "poly p \<noteq> poly [] \<longleftrightarrow> finite {x. poly p x = 0}"
 proof
   assume H: "poly p \<noteq> poly []"
   show "finite {x. poly p x = (0::'a)}"
@@ -357,7 +365,7 @@
     apply (rule ccontr)
     apply (clarify dest!: poly_roots_finite_lemma2)
     using finite_subset
-  proof-
+  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"
@@ -373,9 +381,10 @@
 text{*Entirety and Cancellation for polynomials*}
 
 lemma (in idom_char_0) poly_entire_lemma2:
-  assumes p0: "poly p \<noteq> poly []" and q0: "poly q \<noteq> poly []"
+  assumes p0: "poly p \<noteq> poly []"
+    and q0: "poly q \<noteq> poly []"
   shows "poly (p***q) \<noteq> poly []"
-proof-
+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
@@ -383,74 +392,82 @@
 
 lemma (in idom_char_0) poly_entire:
   "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
-using poly_entire_lemma2[of p q]
-by (auto simp add: fun_eq_iff poly_mult)
+  using poly_entire_lemma2[of p q]
+  by (auto simp add: fun_eq_iff poly_mult)
 
-lemma (in idom_char_0) poly_entire_neg: "(poly (p *** q) \<noteq> poly []) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
-by (simp add: poly_entire)
+lemma (in idom_char_0) poly_entire_neg:
+  "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) = (\<forall>x. f x = g x)"
-by auto
+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 []) = (poly p = poly q)"
-by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult)
+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)
 
-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 distrib_left)
+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 distrib_left)
 
 subclass (in idom_char_0) comm_ring_1 ..
-lemma (in idom_char_0) poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)"
-proof-
-  have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []" by (simp only: poly_add_minus_zero_iff)
-  also have "\<dots> \<longleftrightarrow> poly p = poly [] | poly q = poly r"
+
+lemma (in idom_char_0) poly_mult_left_cancel:
+  "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
+proof -
+  have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []"
+    by (simp only: poly_add_minus_zero_iff)
+  also have "\<dots> \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
     by (auto intro: simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
   finally show ?thesis .
 qed
 
 lemma (in idom) poly_exp_eq_zero[simp]:
-     "(poly (p %^ n) = poly []) = (poly p = poly [] & n \<noteq> 0)"
-apply (simp only: fun_eq 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
+  "poly (p %^ n) = poly [] \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0"
+  apply (simp only: fun_eq 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)
-apply (rule_tac x = "minus one a" in exI)
-apply (unfold diff_minus)
-apply (subst add_commute)
-apply (subst add_assoc)
-apply simp
-done
+  apply (simp add: fun_eq)
+  apply (rule_tac x = "minus one a" in exI)
+  apply (unfold diff_minus)
+  apply (subst add_commute)
+  apply (subst add_assoc)
+  apply simp
+  done
 
-lemma (in idom) poly_exp_prime_eq_zero: "(poly ([a, 1] %^ n) \<noteq> poly [])"
-by auto
+lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) \<noteq> poly []"
+  by auto
 
 text{*A more constructive notion of polynomials being trivial*}
 
-lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] ==> h = 0 & poly t = poly []"
-apply(simp add: fun_eq)
-apply (case_tac "h = zero")
-apply (drule_tac [2] x = zero in spec, auto)
-apply (cases "poly t = poly []", simp)
-proof-
+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 (case_tac "h = zero")
+  apply (drule_tac [2] x = zero in spec, auto)
+  apply (cases "poly t = poly []", 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\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"
+    and 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)" by simp
-  qed
+  from finite_subset[OF th th'] UNIV_ring_char_0_infinte show "poly t x = (0\<Colon>'a)"
+    by simp
+qed
 
 lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
-apply (induct "p", simp)
-apply (rule iffI)
-apply (drule poly_zero_lemma', auto)
-done
+  apply (induct p)
+  apply simp
+  apply (rule iffI)
+  apply (drule poly_zero_lemma', 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
@@ -459,115 +476,126 @@
 
 text{*Basics of divisibility.*}
 
-lemma (in idom) poly_primes: "([a, 1] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"
-apply (auto simp add: divides_def fun_eq 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)
+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])
+  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 clarsimp
-apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
-apply (clarsimp simp add: poly_add poly_cmult)
-apply (rule_tac x="qa" in exI)
-apply (simp add: distrib_right [symmetric])
-apply clarsimp
+  apply clarsimp
+  apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
+  apply (clarsimp simp add: poly_add poly_cmult)
+  apply (rule_tac x="qa" in exI)
+  apply (simp add: distrib_right [symmetric])
+  apply clarsimp
 
-apply (auto simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
-apply (rule_tac x = "pmult qa q" in exI)
-apply (rule_tac [2] x = "pmult p qa" in exI)
-apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
-done
+  apply (auto simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
+  apply (rule_tac x = "pmult qa q" in exI)
+  apply (rule_tac [2] x = "pmult p qa" in exI)
+  apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
+  done
 
 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)
-done
+  apply (simp add: divides_def)
+  apply (rule_tac x = "[one]" in exI)
+  apply (auto simp add: poly_mult fun_eq)
+  done
 
-lemma (in comm_semiring_1) poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r"
-apply (simp add: divides_def, safe)
-apply (rule_tac x = "pmult qa qaa" in exI)
-apply (auto simp add: poly_mult fun_eq mult_assoc)
-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 (rule_tac x = "pmult qa qaa" in exI)
+  apply (auto simp add: poly_mult fun_eq mult_assoc)
+  done
 
-lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)"
-apply (auto simp add: le_iff_add)
-apply (induct_tac k)
-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 mult_ac)
-done
+lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)"
+  apply (auto simp add: le_iff_add)
+  apply (induct_tac k)
+  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 mult_ac)
+  done
 
-lemma (in comm_semiring_1) poly_exp_divides: "[| (p %^ n) divides q;  m\<le>n |] ==> (p %^ m) divides q"
-by (blast intro: poly_divides_exp poly_divides_trans)
+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; p divides r |] ==> p divides (q +++ r)"
-apply (simp add: divides_def, auto)
-apply (rule_tac x = "padd qa qaa" in exI)
-apply (auto simp add: poly_add fun_eq poly_mult distrib_left)
-done
+  "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)"
+  apply (simp add: divides_def, auto)
+  apply (rule_tac x = "padd qa qaa" in exI)
+  apply (auto simp add: poly_add fun_eq poly_mult distrib_left)
+  done
 
 lemma (in comm_ring_1) poly_divides_diff:
-   "[| p divides q; p divides (q +++ r) |] ==> p divides r"
-apply (simp add: divides_def, auto)
-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)
-done
+  "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r"
+  apply (simp add: divides_def, auto)
+  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)
+  done
 
-lemma (in comm_ring_1) poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
-apply (erule poly_divides_diff)
-apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
-done
+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 add_ac)
+  done
 
-lemma (in semiring_0) poly_divides_zero: "poly p = poly [] ==> q divides p"
-apply (simp add: divides_def)
-apply (rule exI[where x="[]"])
-apply (auto simp add: fun_eq poly_mult)
-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)
+  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)
-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)
+  done
 
 text{*At last, we can consider the order of a root.*}
 
-lemma (in idom_char_0)  poly_order_exists_lemma:
-  assumes lp: "length p = d" and p: "poly p \<noteq> poly []"
+lemma (in idom_char_0) poly_order_exists_lemma:
+  assumes lp: "length p = d"
+    and p: "poly p \<noteq> poly []"
   shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
-using lp p
-proof(induct d arbitrary: p)
-  case 0 thus ?case by simp
+  using lp p
+proof (induct d arbitrary: p)
+  case 0
+  thus ?case by simp
 next
   case (Suc n p)
-  {assume p0: "poly p a = 0"
+  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 p0[unfolded poly_linear_divides] pN  obtain q where
-      q: "p = [-a, 1] *** q" by blast
-    from q h p0 have qh: "length q = n" "poly q \<noteq> poly []"
+    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 (rule ccontr)
       apply (simp add: fun_eq 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
+    from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0"
+      by blast
     from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
-    hence ?case by blast}
-  moreover
-  {assume p0: "poly p a \<noteq> 0"
-    hence ?case using Suc.prems apply simp by (rule exI[where x="0::nat"], simp)}
-  ultimately show ?case by blast
+    then show ?thesis by blast
+  next
+    case False
+    then show ?thesis
+      using Suc.prems
+      apply simp
+      apply (rule exI[where x="0::nat"])
+      apply simp
+      done
+  qed
 qed
 
 
@@ -585,263 +613,240 @@
 qed
 
 
-
 (* FIXME: Tidy up *)
 
-lemma (in semiring_1)
-  zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
+lemma (in semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
   by (induct n) simp_all
 
 lemma (in idom_char_0) poly_order_exists:
-  assumes lp: "length p = d" and p0: "poly p \<noteq> poly []"
-  shows "\<exists>n. ([-a, 1] %^ n) divides p & ~(([-a, 1] %^ (Suc n)) divides p)"
-proof-
-let ?poly = poly
-let ?mulexp = mulexp
-let ?pexp = pexp
-from lp p0
-show ?thesis
-apply -
-apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
-apply (rule_tac x = n in exI, safe)
-apply (unfold divides_def)
-apply (rule_tac x = q in exI)
-apply (induct_tac "n", simp)
-apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult distrib_left mult_ac)
-apply safe
-apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \<noteq> ?poly (pmult (?pexp [uminus a, one] (Suc n)) qa)")
-apply simp
-apply (induct_tac "n")
-apply (simp del: pmult_Cons pexp_Suc)
-apply (erule_tac Q = "?poly q a = zero" in contrapos_np)
-apply (simp add: poly_add poly_cmult)
-apply (rule pexp_Suc [THEN ssubst])
-apply (rule ccontr)
-apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
-done
-qed
-
+  assumes "length p = d" and "poly p \<noteq> poly []"
+  shows "\<exists>n. ([-a, 1] %^ n) divides p \<and> ~(([-a, 1] %^ (Suc n)) divides p)"
+  using assms
+  apply -
+  apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
+  apply (rule_tac x = n in exI, safe)
+  apply (unfold divides_def)
+  apply (rule_tac x = q in exI)
+  apply (induct_tac n, simp)
+  apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult distrib_left mult_ac)
+  apply safe
+  apply (subgoal_tac "poly (mulexp n [uminus a, one] q) \<noteq>
+    poly (pmult (pexp [uminus a, one] (Suc n)) qa)")
+  apply simp
+  apply (induct_tac n)
+  apply (simp del: pmult_Cons pexp_Suc)
+  apply (erule_tac Q = "poly q a = zero" in contrapos_np)
+  apply (simp add: poly_add poly_cmult)
+  apply (rule pexp_Suc [THEN ssubst])
+  apply (rule ccontr)
+  apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
+  done
 
 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
-by (simp add: divides_def, auto)
+  by (auto simp add: divides_def)
 
-lemma (in idom_char_0) poly_order: "poly p \<noteq> poly []
-      ==> EX! n. ([-a, 1] %^ n) divides p &
-                 ~(([-a, 1] %^ (Suc n)) divides p)"
-apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
-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)
-done
+lemma (in idom_char_0) poly_order:
+  "poly p \<noteq> poly [] \<Longrightarrow> \<exists>!n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p)"
+  apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
+  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)
+  done
 
 text{*Order*}
 
-lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n"
-by (blast intro: someI2)
+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 &
+      "(([-a, 1] %^ n) divides p \<and>
         ~(([-a, 1] %^ (Suc n)) divides p)) =
-        ((n = order a p) & ~(poly p = poly []))"
-apply (unfold 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])
-done
+        ((n = order a p) \<and> ~(poly p = poly []))"
+  apply (unfold 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])
+  done
 
-lemma (in idom_char_0) order2: "[| poly p \<noteq> poly [] |]
-      ==> ([-a, 1] %^ (order a p)) divides p &
-              ~(([-a, 1] %^ (Suc(order a p))) divides p)"
-by (simp add: order del: pexp_Suc)
+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)"
+  by (simp add: order del: pexp_Suc)
 
-lemma (in idom_char_0) order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p;
-         ~(([-a, 1] %^ (Suc n)) divides p)
-      |] ==> (n = order a p)"
-by (insert order [of a n p], auto)
+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>
+    n = order a p"
+  using order [of a n p] by auto
 
-lemma (in idom_char_0) order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &
-         ~(([-a, 1] %^ (Suc n)) divides p))
-      ==> (n = order a p)"
-by (blast intro: order_unique)
+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>
+    n = order a p"
+  by (blast intro: order_unique)
 
-lemma (in ring_1) order_poly: "poly p = poly q ==> order a p = order a q"
+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)
 
 lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
   by (induct "p") auto
 
 lemma (in comm_ring_1) lemma_order_root:
-     " 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p
-             \<Longrightarrow> poly p a = 0"
-apply (induct n arbitrary: a p, blast)
-apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
-done
+  "0 < n \<and> [- a, 1] %^ n divides p \<and> ~ [- 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) = ((poly p = poly []) | order a p \<noteq> 0)"
-proof-
-  let ?poly = poly
-  show ?thesis
-apply (case_tac "?poly p = ?poly []", auto)
-apply (simp add: poly_linear_divides del: pmult_Cons, 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)
-done
-qed
+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 (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)
+  done
 
-lemma (in idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"
-proof-
-  let ?poly = poly
-  show ?thesis
-apply (case_tac "?poly p = ?poly []", auto)
-apply (simp add: divides_def fun_eq poly_mult)
-apply (rule_tac x = "[]" in exI)
-apply (auto dest!: order2 [where a=a]
-            intro: poly_exp_divides simp del: pexp_Suc)
-done
-qed
+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 (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 []
-      ==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &
-                ~([-a, 1] divides q)"
-apply (unfold 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 (drule_tac x = qa in spec)
-apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
-done
+  "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)
+  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 (drule_tac x = qa in spec)
+  apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
+  done
 
 text{*Important composition properties of orders.*}
-lemma order_mult: "poly (p *** q) \<noteq> poly []
-      ==> order a (p *** q) = order a p + order (a::'a::{idom_char_0}) 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 (rule_tac x = "qa *** qaa" in exI)
-apply (simp add: poly_mult mult_ac del: pmult_Cons)
-apply (drule_tac a = a in order_decomp)+
-apply safe
-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 mult_ac del: pmult_Cons)
-done
+lemma order_mult:
+  "poly (p *** q) \<noteq> poly [] \<Longrightarrow>
+    order a (p *** q) = order a p + order (a::'a::{idom_char_0}) 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 (rule_tac x = "qa *** qaa" in exI)
+  apply (simp add: poly_mult mult_ac del: pmult_Cons)
+  apply (drule_tac a = a in order_decomp)+
+  apply safe
+  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 mult_ac del: pmult_Cons)
+  done
 
 lemma (in idom_char_0) order_mult:
-  assumes pq0: "poly (p *** q) \<noteq> poly []"
+  assumes "poly (p *** q) \<noteq> poly []"
   shows "order a (p *** q) = order a p + order a q"
-proof-
-  let ?order = order
-  let ?divides = "op divides"
-  let ?poly = poly
-from pq0
-show ?thesis
-apply (cut_tac a = a and p = "pmult 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 (rule_tac x = "pmult qa qaa" in exI)
-apply (simp add: poly_mult mult_ac del: pmult_Cons)
-apply (drule_tac a = a in order_decomp)+
-apply safe
-apply (subgoal_tac "?divides [uminus a,one ] (pmult 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 (pmult (pexp [uminus a, one] (?order a p)) (pmult qa qaa)) = ?poly (pmult (pexp [uminus a, one] (?order a p)) (pmult [uminus a, one] qb))")
-apply (drule poly_mult_left_cancel [THEN iffD1], force)
-apply (subgoal_tac "?poly (pmult (pexp [uminus a, one ] (order a q)) (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) = ?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 mult_ac del: pmult_Cons)
-done
-qed
+  using assms
+  apply (cut_tac a = a and p = "pmult 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 (rule_tac x = "pmult qa qaa" in exI)
+  apply (simp add: poly_mult mult_ac del: pmult_Cons)
+  apply (drule_tac a = a in order_decomp)+
+  apply safe
+  apply (subgoal_tac "[uminus a, one] divides pmult 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 (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa)) =
+    poly (pmult (pexp [uminus a, one] (?order a p)) (pmult [uminus a, one] qb))")
+  apply (drule poly_mult_left_cancel [THEN iffD1], force)
+  apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a q))
+      (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) =
+    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 mult_ac del: pmult_Cons)
+  done
 
-lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order a p \<noteq> 0)"
-by (rule order_root [THEN ssubst], auto)
+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_0) poly_Nil_zero: "poly [] = poly [0]"
-by (simp add: fun_eq)
+  by (simp add: fun_eq)
 
 lemma (in idom_char_0) rsquarefree_decomp:
-     "[| rsquarefree p; poly p a = 0 |]
-      ==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0"
-apply (simp add: rsquarefree_def, 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 (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)
-done
+  "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)
+  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 (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)
+  done
 
 
 text{*Normalization of a polynomial.*}
 
 lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
-apply (induct "p")
-apply (auto simp add: fun_eq)
-done
+  by (induct p) (auto simp add: fun_eq)
 
 text{*The degree of a polynomial.*}
 
-lemma (in semiring_0) lemma_degree_zero:
-     "list_all (%c. c = 0) p \<longleftrightarrow>  pnormalize p = []"
-by (induct "p", auto)
+lemma (in semiring_0) lemma_degree_zero: "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
+  by (induct p) auto
 
 lemma (in idom_char_0) degree_zero:
-  assumes pN: "poly p = poly []" shows"degree p = 0"
-proof-
-  let ?pn = pnormalize
-  from pN
-  show ?thesis
-    apply (simp add: degree_def)
-    apply (case_tac "?pn p = []")
-    apply (auto simp add: poly_zero lemma_degree_zero )
-    done
-qed
+  assumes "poly p = poly []"
+  shows "degree p = 0"
+  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"
-by simp
-lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp
+  by simp
+
+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)"
   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_last_nonzero: "pnormal p ==> last p \<noteq> 0"
-by(induct p) (simp_all add: pnormal_def 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)
 
 lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
   unfolding pnormal_def length_greater_0_conv by blast
 
-lemma (in semiring_0) pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p"
-by (induct p) (auto simp: pnormal_def  split: split_if_asm)
+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)"
+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
 
-lemma (in idom_char_0) poly_Cons_eq: "poly (c#cs) = poly (d#ds) \<longleftrightarrow> c=d \<and> poly cs = poly ds" (is "?lhs \<longleftrightarrow> ?rhs")
+lemma (in idom_char_0) poly_Cons_eq:
+  "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"
@@ -851,18 +856,20 @@
     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
-  thus ?rhs  by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
+  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)
+  assume ?rhs
+  then show ?lhs 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
+proof (induct q arbitrary: p)
+  case Nil
+  thus ?case by (simp only: poly_zero lemma_degree_zero) simp
 next
   case (Cons c cs p)
   thus ?case
-  proof(induct p)
+  proof (induct p)
     case Nil
     hence "poly [] = poly (c#cs)" by blast
     then have "poly (c#cs) = poly [] " by simp
@@ -880,44 +887,51 @@
   qed
 qed
 
-lemma (in idom_char_0) degree_unique: assumes pq: "poly p = poly q"
+lemma (in idom_char_0) degree_unique:
+  assumes pq: "poly p = poly q"
   shows "degree p = degree q"
-using pnormalize_unique[OF pq] unfolding degree_def by simp
+  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 (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []")
+  apply simp
+  apply (induct_tac p)
+  apply auto
+  done
 
-apply (induct p arbitrary: a x b, auto)
-apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []", simp)
-apply (induct_tac p, auto)
-done
-
-lemma (in semiring_1) last_linear_mul: assumes p:"p\<noteq>[]" 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)))"
+lemma (in semiring_1) last_linear_mul:
+  assumes p: "p \<noteq> []"
+  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)))"
     by (simp add: poly_cmult_distr)
   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"
-by (induct p) (auto split: split_if_asm)
+  by (induct p) (auto split: split_if_asm)
 
 lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
-  by (induct p, auto)
+  by (induct p) auto
 
 lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
   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
+lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)"
+  by (rule ext) simp
 
 lemma (in idom_char_0) linear_mul_degree:
   assumes p: "poly p \<noteq> poly []"
   shows "degree ([a,1] *** p) = degree p + 1"
-proof-
+proof -
   from p have pnz: "pnormalize p \<noteq> []"
     unfolding poly_zero lemma_degree_zero .
 
@@ -926,7 +940,6 @@
   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
 
@@ -938,64 +951,81 @@
 
 lemma (in idom_char_0) linear_pow_mul_degree:
   "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
-proof(induct n arbitrary: a p)
+proof (induct n arbitrary: a p)
   case (0 a p)
-  {assume p: "poly p = poly []"
-    hence ?case using degree_unique[OF p] by (simp add: degree_def)}
-  moreover
-  {assume p: "poly p \<noteq> poly []" hence ?case by (auto simp add: poly_Nil_ext) }
-  ultimately show ?case by blast
+  show ?case
+  proof (cases "poly p = poly []")
+    case True
+    then show ?thesis
+      using degree_unique[OF True] by (simp add: degree_def)
+  next
+    case False
+    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))"
-    apply (rule ext, simp add: poly_mult poly_add poly_cmult)
-    by (simp add: mult_ac add_ac distrib_left)
+    apply (rule ext)
+    apply (simp add: poly_mult poly_add poly_cmult)
+    apply (simp add: mult_ac add_ac distrib_left)
+    done
   note deq = degree_unique[OF eq]
-  {assume p: "poly p = poly []"
+  show ?case
+  proof (cases "poly p = poly []")
+    case True
     with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
-      by - (rule ext,simp add: poly_mult poly_cmult poly_add)
-    from degree_unique[OF eq'] p have ?case by (simp add: degree_def)}
-  moreover
-  {assume p: "poly p \<noteq> poly []"
-    from p have ap: "poly ([a,1] *** p) \<noteq> poly []"
+      apply -
+      apply (rule ext)
+      apply (simp add: poly_mult poly_cmult poly_add)
+      done
+    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" by blast
-   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')
-     by simp
-
-   from degree_unique[OF eq] ap p th0 linear_mul_degree[OF p, of a]
-   have ?case by (auto simp del: poly.simps)}
-  ultimately show ?case by blast
+      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"
+      by blast
+    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)
+  qed
 qed
 
 lemma (in idom_char_0) order_degree:
   assumes p0: "poly p \<noteq> poly []"
   shows "order a p \<le> degree p"
-proof-
+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)}
-  with degree_unique[OF q, unfolded linear_pow_mul_degree]
-  show ?thesis by auto
+  {
+    assume "poly q = poly []"
+    with q p0 have False by (simp add: poly_mult poly_entire)
+  }
+  with degree_unique[OF q, unfolded linear_pow_mul_degree] show ?thesis
+    by auto
 qed
 
 text{*Tidier versions of finiteness of roots.*}
 
-lemma (in idom_char_0) poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x. poly p x = 0}"
-unfolding poly_roots_finite .
+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{*bound for polynomial.*}
 
-lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
-apply (induct "p", auto)
-apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
-apply (rule abs_triangle_ineq)
-apply (auto intro!: mult_mono simp add: abs_mult)
-done
+lemma poly_mono: "abs(x) \<le> k \<Longrightarrow> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
+  apply (induct p)
+  apply auto
+  apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
+  apply (rule abs_triangle_ineq)
+  apply (auto intro!: mult_mono simp add: abs_mult)
+  done
 
 lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp