src/HOL/Library/Fundamental_Theorem_Algebra.thy

        if 0 \<le> Re z then Complex (sqrt(Re z)) 0
        else Complex 0 (sqrt(- Re z))
      else Complex (sqrt((cmod z + Re z) /2)) ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
 
 lemma csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
-proof-
+proof -
   obtain x y where xy: "z = Complex x y" by (cases z)
-  {assume y0: "y = 0"
-    {assume x0: "x \<ge> 0"
-      then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
-        by (simp add: csqrt_def power2_eq_square)}
+  {
+    assume y0: "y = 0"
+    {
+      assume x0: "x \<ge> 0"
+      then have ?thesis
+        using y0 xy real_sqrt_pow2[OF x0]
+        by (simp add: csqrt_def power2_eq_square)
+    }
     moreover
-    {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
-      then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
-        by (simp add: csqrt_def power2_eq_square) }
-    ultimately have ?thesis by blast}
+    {
+      assume "\<not> x \<ge> 0"
+      then have x0: "- x \<ge> 0" by arith
+      then have ?thesis
+        using y0 xy real_sqrt_pow2[OF x0]
+        by (simp add: csqrt_def power2_eq_square)
+    }
+    ultimately have ?thesis by blast
+  }
   moreover
-  {assume y0: "y\<noteq>0"
-    {fix x y
+  {
+    assume y0: "y \<noteq> 0"
+    {
+      fix x y
       let ?z = "Complex x y"
-      from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
-      hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
-      hence "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0" by (simp_all add: power2_eq_square) }
+      from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z"
+        by auto
+      then have "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0"
+        by arith+
+      then have "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0"
+        by (simp_all add: power2_eq_square)
+    }
     note th = this
     have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
       by (simp add: power2_eq_square)
     from th[of x y]
     have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
       "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
       unfolding sq4 by simp_all
-    then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) - sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x"
+    then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) -
+        sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x"
       unfolding power2_eq_square by simp
-    have "sqrt 4 = sqrt (2\<^sup>2)" by simp
-    hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
-    have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
+    have "sqrt 4 = sqrt (2\<^sup>2)"
+      by simp
+    then have sqrt4: "sqrt 4 = 2"
+      by (simp only: real_sqrt_abs)
+    have th2: "2 * (y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
       unfolding power2_eq_square
       by (simp add: algebra_simps real_sqrt_divide sqrt4)
-     from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
-       apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y] real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square] real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square] real_sqrt_mult[symmetric])
-      using th1 th2  ..}
+    from y0 xy have ?thesis
+      apply (simp add: csqrt_def power2_eq_square)
+      apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y]
+        real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square]
+        real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square]
+        real_sqrt_mult[symmetric])
+      using th1 th2  ..
+  }
   ultimately show ?thesis by blast
 qed
 
 lemma csqrt_Complex: "x \<ge> 0 \<Longrightarrow> csqrt (Complex x 0) = Complex (sqrt x) 0"
   by (simp add: csqrt_def)

     using real_sqrt_sum_squares_ge1 [of "x" y]
           real_sqrt_sum_squares_ge1 [of "-x" y]
           real_sqrt_sum_squares_eq_cancel [of x y]
     apply (auto simp: csqrt_def intro!: Rings.ordered_ring_class.split_mult_pos_le)
     apply (metis add_commute diff_add_cancel le_add_same_cancel1 real_sqrt_sum_squares_ge1)
-    by (metis add_commute less_eq_real_def power_minus_Bit0 real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
+    apply (metis add_commute less_eq_real_def power_minus_Bit0
+            real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
+    done
 qed
 
 lemma Re_csqrt: "0 \<le> Re(csqrt z)"
   by (metis csqrt_principal le_less)
 
-lemma csqrt_square: "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> csqrt (z^2) = z"
-  using csqrt [of "z^2"] csqrt_principal [of "z^2"]
+lemma csqrt_square: "0 < Re z \<or> Re z = 0 \<and> 0 \<le> Im z \<Longrightarrow> csqrt (z\<^sup>2) = z"
+  using csqrt [of "z\<^sup>2"] csqrt_principal [of "z\<^sup>2"]
   by (cases z) (auto simp: power2_eq_iff)
 
 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
   by auto (metis csqrt power_eq_0_iff)
 
 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
   by auto (metis csqrt power2_eq_1_iff)
 
-subsection{* More lemmas about module of complex numbers *}
+
+subsection {* More lemmas about module of complex numbers *}
 
 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
   by (rule of_real_power [symmetric])
 
 text{* The triangle inequality for cmod *}
 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
 
-subsection{* Basic lemmas about polynomials *}
+
+subsection {* Basic lemmas about polynomials *}
 
 lemma poly_bound_exists:
-  fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly" 
-  shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z <= r \<longrightarrow> norm (poly p z) \<le> m)"
-proof(induct p)
-  case 0 thus ?case by (rule exI[where x=1], simp)
+  fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
+  shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
+proof (induct p)
+  case 0
+  then show ?case by (rule exI[where x=1]) simp
 next
   case (pCons c cs)
   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
     by blast
   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
-  {fix z :: 'a 
+  {
+    fix z :: 'a
     assume H: "norm z \<le> r"
-    from m H have th: "norm (poly cs z) \<le> m" by blast
-    from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
+    from m H have th: "norm (poly cs z) \<le> m"
+      by blast
+    from H have rp: "r \<ge> 0" using norm_ge_zero[of z]
+      by arith
     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z* poly cs z)"
       using norm_triangle_ineq[of c "z* poly cs z"] by simp
-    also have "\<dots> \<le> norm c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
+    also have "\<dots> \<le> norm c + r * m"
+      using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
       by (simp add: norm_mult)
-    also have "\<dots> \<le> ?k" by simp
-    finally have "norm (poly (pCons c cs) z) \<le> ?k" .}
+    also have "\<dots> \<le> ?k"
+      by simp
+    finally have "norm (poly (pCons c cs) z) \<le> ?k" .
+  }
   with kp show ?case by blast
 qed
 
 
 text{* Offsetting the variable in a polynomial gives another of same degree *}
 
 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
-where
-  "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
+  where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
 
 lemma offset_poly_0: "offset_poly 0 h = 0"
   by (simp add: offset_poly_def)
 
 lemma offset_poly_pCons:
   "offset_poly (pCons a p) h =
     smult h (offset_poly p h) + pCons a (offset_poly p h)"
   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
 
 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
-by (simp add: offset_poly_pCons offset_poly_0)
+  by (simp add: offset_poly_pCons offset_poly_0)
 
 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
-apply (induct p)
-apply (simp add: offset_poly_0)
-apply (simp add: offset_poly_pCons algebra_simps)
-done
+  apply (induct p)
+  apply (simp add: offset_poly_0)
+  apply (simp add: offset_poly_pCons algebra_simps)
+  done
 
 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
-by (induct p arbitrary: a, simp, force)
+  by (induct p arbitrary: a) (simp, force)
 
 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
-apply (safe intro!: offset_poly_0)
-apply (induct p, simp)
-apply (simp add: offset_poly_pCons)
-apply (frule offset_poly_eq_0_lemma, simp)
-done
+  apply (safe intro!: offset_poly_0)
+  apply (induct p, simp)
+  apply (simp add: offset_poly_pCons)
+  apply (frule offset_poly_eq_0_lemma, simp)
+  done
 
 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
-apply (induct p)
-apply (simp add: offset_poly_0)
-apply (case_tac "p = 0")
-apply (simp add: offset_poly_0 offset_poly_pCons)
-apply (simp add: offset_poly_pCons)
-apply (subst degree_add_eq_right)
-apply (rule le_less_trans [OF degree_smult_le])
-apply (simp add: offset_poly_eq_0_iff)
-apply (simp add: offset_poly_eq_0_iff)
-done
-
-definition
-  "psize p = (if p = 0 then 0 else Suc (degree p))"
+  apply (induct p)
+  apply (simp add: offset_poly_0)
+  apply (case_tac "p = 0")
+  apply (simp add: offset_poly_0 offset_poly_pCons)
+  apply (simp add: offset_poly_pCons)
+  apply (subst degree_add_eq_right)
+  apply (rule le_less_trans [OF degree_smult_le])
+  apply (simp add: offset_poly_eq_0_iff)
+  apply (simp add: offset_poly_eq_0_iff)
+  done
+
+definition "psize p = (if p = 0 then 0 else Suc (degree p))"
 
 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
   unfolding psize_def by simp
 
-lemma poly_offset: 
-  fixes p:: "('a::comm_ring_1) poly" 
-  shows "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
+lemma poly_offset:
+  fixes p :: "'a::comm_ring_1 poly"
+  shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
 proof (intro exI conjI)
   show "psize (offset_poly p a) = psize p"
     unfolding psize_def
     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
     by (simp add: poly_offset_poly)
 qed
 
 text{* An alternative useful formulation of completeness of the reals *}
-lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
-  shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
+lemma real_sup_exists:
+  assumes ex: "\<exists>x. P x"
+    and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
+  shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
 proof
   from bz have "bdd_above (Collect P)"
     by (force intro: less_imp_le)
   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
     using ex bz by (subst less_cSup_iff) auto

 
 subsection {* Fundamental theorem of algebra *}
 lemma  unimodular_reduce_norm:
   assumes md: "cmod z = 1"
   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
-proof-
-  obtain x y where z: "z = Complex x y " by (cases z, auto)
-  from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
-  {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
-    from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
+proof -
+  obtain x y where z: "z = Complex x y "
+    by (cases z) auto
+  from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
+    by (simp add: cmod_def)
+  {
+    assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
+    from C z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1"
       by (simp_all add: cmod_def power2_eq_square algebra_simps)
-    hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
-    hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
+    then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1"
+      by simp_all
+    then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2"
       by - (rule power_mono, simp, simp)+
-    hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
+    then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"
       by (simp_all add: power_mult_distrib)
-    from add_mono[OF th0] xy have False by simp }
-  thus ?thesis unfolding linorder_not_le[symmetric] by blast
+    from add_mono[OF th0] xy have False by simp
+  }
+  then show ?thesis
+    unfolding linorder_not_le[symmetric] by blast
 qed
 
 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
 lemma reduce_poly_simple:
- assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
+  assumes b: "b \<noteq> 0"
+    and n: "n \<noteq> 0"
   shows "\<exists>z. cmod (1 + b * z^n) < 1"
-using n
-proof(induct n rule: nat_less_induct)
+  using n
+proof (induct n rule: nat_less_induct)
   fix n
-  assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
+  assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)"
+  assume n: "n \<noteq> 0"
   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
-  {assume e: "even n"
-    hence "\<exists>m. n = 2*m" by presburger
-    then obtain m where m: "n = 2*m" by blast
-    from n m have "m\<noteq>0" "m < n" by presburger+
-    with IH[rule_format, of m] obtain z where z: "?P z m" by blast
+  {
+    assume e: "even n"
+    then have "\<exists>m. n = 2 * m"
+      by presburger
+    then obtain m where m: "n = 2 * m"
+      by blast
+    from n m have "m \<noteq> 0" "m < n"
+      by presburger+
+    with IH[rule_format, of m] obtain z where z: "?P z m"
+      by blast
     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
-    hence "\<exists>z. ?P z n" ..}
+    then have "\<exists>z. ?P z n" ..
+  }
   moreover
-  {assume o: "odd n"
+  {
+    assume o: "odd n"
     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
       using b by (simp add: norm_divide)
-    from o have "\<exists>m. n = Suc (2*m)" by presburger+
-    then obtain m where m: "n = Suc (2*m)" by blast
+    from o have "\<exists>m. n = Suc (2 * m)"
+      by presburger+
+    then obtain m where m: "n = Suc (2*m)"
+      by blast
     from unimodular_reduce_norm[OF th0] o
     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp)
       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")

       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
       apply (auto simp add: m power_mult)
       apply (rule_tac x="ii" in exI)
       apply (auto simp add: m power_mult)
       done
-    then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
+    then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
+      by blast
     let ?w = "v / complex_of_real (root n (cmod b))"
     from odd_real_root_pow[OF o, of "cmod b"]
     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
       by (simp add: power_divide complex_of_real_power)
-    have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
-    hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
+    have th2:"cmod (complex_of_real (cmod b) / b) = 1"
+      using b by (simp add: norm_divide)
+    then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
+      by simp
     have th4: "cmod (complex_of_real (cmod b) / b) *
-   cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
-   < cmod (complex_of_real (cmod b) / b) * 1"
+        cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
+        cmod (complex_of_real (cmod b) / b) * 1"
       apply (simp only: norm_mult[symmetric] distrib_left)
-      using b v by (simp add: th2)
-
+      using b v
+      apply (simp add: th2)
+      done
     from mult_less_imp_less_left[OF th4 th3]
     have "?P ?w n" unfolding th1 .
-    hence "\<exists>z. ?P z n" .. }
+    then have "\<exists>z. ?P z n" ..
+  }
   ultimately show "\<exists>z. ?P z n" by blast
 qed
 
 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
 
-lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
+lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
   unfolding cmod_def by simp
 
 lemma bolzano_weierstrass_complex_disc:
   assumes r: "\<forall>n. cmod (s n) \<le> r"
   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
 proof-
-  from seq_monosub[of "Re o s"]
+  from seq_monosub[of "Re \<circ> s"]
   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
     unfolding o_def by blast
-  from seq_monosub[of "Im o s o f"]
-  obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
-  let ?h = "f o g"
-  from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
-  have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
+  from seq_monosub[of "Im \<circ> s \<circ> f"]
+  obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))"
+    unfolding o_def by blast
+  let ?h = "f \<circ> g"
+  from r[rule_format, of 0] have rp: "r \<ge> 0"
+    using norm_ge_zero[of "s 0"] by arith
+  have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
   proof
     fix n
-    from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
+    from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
+    show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
   qed
-  have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
+  have conv1: "convergent (\<lambda>n. Re (s (f n)))"
     apply (rule Bseq_monoseq_convergent)
     apply (simp add: Bseq_def)
     apply (metis gt_ex le_less_linear less_trans order.trans th)
-    using f(2) .
-  have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
+    apply (rule f(2))
+    done
+  have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
   proof
     fix n
-    from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
+    from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
+    show "\<bar>Im (s n)\<bar> \<le> r + 1"
+      by arith
   qed
 
   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
     apply (rule Bseq_monoseq_convergent)
     apply (simp add: Bseq_def)
     apply (metis gt_ex le_less_linear less_trans order.trans th)
-    using g(2) .
+    apply (rule g(2))
+    done
 
   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
     by blast
-  hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
+  then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
     unfolding LIMSEQ_iff real_norm_def .
 
   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
     by blast
-  hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
+  then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
     unfolding LIMSEQ_iff real_norm_def .
   let ?w = "Complex x y"
-  from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
-  {fix e assume ep: "e > (0::real)"
-    hence e2: "e/2 > 0" by simp
+  from f(1) g(1) have hs: "subseq ?h"
+    unfolding subseq_def by auto
+  {
+    fix e :: real
+    assume ep: "e > 0"
+    then have e2: "e/2 > 0" by simp
     from x[rule_format, OF e2] y[rule_format, OF e2]
-    obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2" and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" by blast
-    {fix n assume nN12: "n \<ge> N1 + N2"
-      hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
+    obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
+      and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" by blast
+    {
+      fix n
+      assume nN12: "n \<ge> N1 + N2"
+      then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
+        using seq_suble[OF g(1), of n] by arith+
       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
       have "cmod (s (?h n) - ?w) < e"
-        using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
-    hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
-  with hs show ?thesis  by blast
+        using metric_bound_lemma[of "s (f (g n))" ?w] by simp
+    }
+    then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast
+  }
+  with hs show ?thesis by blast
 qed
 
 text{* Polynomial is continuous. *}
 
 lemma poly_cont:
-  fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly" 
+  fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
   assumes ep: "e > 0"
   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
-proof-
+proof -
   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
   proof
     show "degree (offset_poly p z) = degree p"
       by (rule degree_offset_poly)
     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
       by (rule poly_offset_poly)
   qed
-  {fix w
-    note q(2)[of "w - z", simplified]}
-  note th = this
+  have th: "\<And>w. poly q (w - z) = poly p w"
+    using q(2)[of "w - z" for w] by simp
   show ?thesis unfolding th[symmetric]
-  proof(induct q)
-    case 0 thus ?case  using ep by auto
+  proof (induct q)
+    case 0
+    then show ?case
+      using ep by auto
   next
     case (pCons c cs)
     from poly_bound_exists[of 1 "cs"]
-    obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m" by blast
-    from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
-    have one0: "1 > (0::real)"  by arith
+    obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
+      by blast
+    from ep m(1) have em0: "e/m > 0"
+      by (simp add: field_simps)
+    have one0: "1 > (0::real)"
+      by arith
     from real_lbound_gt_zero[OF one0 em0]
-    obtain d where d: "d >0" "d < 1" "d < e / m" by blast
-    from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
+    obtain d where d: "d > 0" "d < 1" "d < e / m"
+      by blast
+    from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
       by (simp_all add: field_simps)
     show ?case
-      proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
-        fix d w
-        assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "norm (w-z) < d"
-        hence d1: "norm (w-z) \<le> 1" "d \<ge> 0" by simp_all
-        from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
-        from H have th: "norm (w-z) \<le> d" by simp
-        from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
-        show "norm (w - z) * norm (poly cs (w - z)) < e" by simp
-      qed
+    proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
+      fix d w
+      assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
+      then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
+        by simp_all
+      from H(3) m(1) have dme: "d*m < e"
+        by (simp add: field_simps)
+      from H have th: "norm (w - z) \<le> d"
+        by simp
+      from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
+      show "norm (w - z) * norm (poly cs (w - z)) < e"
+        by simp
     qed
+  qed
 qed
 
 text{* Hence a polynomial attains minimum on a closed disc
   in the complex plane. *}
-lemma  poly_minimum_modulus_disc:
-  "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
-proof-
-  {assume "\<not> r \<ge> 0" hence ?thesis
-    by (metis norm_ge_zero order.trans)}
+lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
+proof -
+  {
+    assume "\<not> r \<ge> 0"
+    then have ?thesis
+      by (metis norm_ge_zero order.trans)
+  }
   moreover
-  {assume rp: "r \<ge> 0"
-    from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
-    hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
-    {fix x z
-      assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
-      hence "- x < 0 " by arith
-      with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
-    then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
+  {
+    assume rp: "r \<ge> 0"
+    from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
+      by simp
+    then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
+      by blast
+    {
+      fix x z
+      assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"
+      then have "- x < 0 "
+        by arith
+      with H(2) norm_ge_zero[of "poly p z"] have False
+        by simp
+    }
+    then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
+      by blast
     from real_sup_exists[OF mth1 mth2] obtain s where
-      s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow>(y < s)" by blast
-    let ?m = "-s"
-    {fix y
-      from s[rule_format, of "-y"] have
-    "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
-        unfolding minus_less_iff[of y ] equation_minus_iff by blast }
+      s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow> y < s" by blast
+    let ?m = "- s"
+    {
+      fix y
+      from s[rule_format, of "-y"]
+      have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
+        unfolding minus_less_iff[of y ] equation_minus_iff by blast
+    }
     note s1 = this[unfolded minus_minus]
     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
       by auto
-    {fix n::nat
+    {
+      fix n :: nat
       from s1[rule_format, of "?m + 1/real (Suc n)"]
       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
-        by simp}
-    hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
+        by simp
+    }
+    then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
     from choice[OF th] obtain g where
-      g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
+        g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
       by blast
     from bolzano_weierstrass_complex_disc[OF g(1)]
     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
       by blast
-    {fix w
+    {
+      fix w
       assume wr: "cmod w \<le> r"
       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
-      {assume e: "?e > 0"
-        hence e2: "?e/2 > 0" by simp
+      {
+        assume e: "?e > 0"
+        then have e2: "?e/2 > 0" by simp
         from poly_cont[OF e2, of z p] obtain d where
-          d: "d>0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2" by blast
-        {fix w assume w: "cmod (w - z) < d"
+            d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
+          by blast
+        {
+          fix w
+          assume w: "cmod (w - z) < d"
           have "cmod(poly p w - poly p z) < ?e / 2"
-            using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
+            using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
+        }
         note th1 = this
 
-        from fz(2)[rule_format, OF d(1)] obtain N1 where
-          N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
-        from reals_Archimedean2[of "2/?e"] obtain N2::nat where
-          N2: "2/?e < real N2" by blast
-        have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
+        from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
+          by blast
+        from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
+          by blast
+        have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
           using N1[rule_format, of "N1 + N2"] th1 by simp
-        {fix a b e2 m :: real
-        have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
-          ==> False" by arith}
-      note th0 = this
-      have ath:
-        "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
-      from s1m[OF g(1)[rule_format]]
-      have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
-      from seq_suble[OF fz(1), of "N1+N2"]
-      have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
-      have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
-        using N2 by auto
-      from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
-      from g(2)[rule_format, of "f (N1 + N2)"]
-      have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
-      from order_less_le_trans[OF th01 th00]
-      have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
-      from N2 have "2/?e < real (Suc (N1 + N2))" by arith
-      with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
-      have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
-      with ath[OF th31 th32]
-      have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
-      have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
-        by arith
-      have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
-\<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
-        by (simp add: norm_triangle_ineq3)
-      from ath2[OF th22, of ?m]
-      have thc2: "2*(?e/2) \<le> \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)" by simp
-      from th0[OF th2 thc1 thc2] have False .}
-      hence "?e = 0" by auto
-      then have "cmod (poly p z) = ?m" by simp
-      with s1m[OF wr]
-      have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
-    hence ?thesis by blast}
+        {
+          fix a b e2 m :: real
+          have "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
+            by arith
+        }
+        note th0 = this
+        have ath: "\<And>m x e::real. m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e"
+          by arith
+        from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
+        from seq_suble[OF fz(1), of "N1+N2"]
+        have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
+          by simp
+        have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
+          using N2 by auto
+        from frac_le[OF th000 th00]
+        have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
+          by simp
+        from g(2)[rule_format, of "f (N1 + N2)"]
+        have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
+        from order_less_le_trans[OF th01 th00]
+        have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
+        from N2 have "2/?e < real (Suc (N1 + N2))"
+          by arith
+        with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
+        have "?e/2 > 1/ real (Suc (N1 + N2))"
+          by (simp add: inverse_eq_divide)
+        with ath[OF th31 th32]
+        have thc1: "\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
+          by arith
+        have ath2: "\<And>a b c m::real. \<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c"
+          by arith
+        have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
+            cmod (poly p (g (f (N1 + N2))) - poly p z)"
+          by (simp add: norm_triangle_ineq3)
+        from ath2[OF th22, of ?m]
+        have thc2: "2 * (?e/2) \<le>
+            \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
+          by simp
+        from th0[OF th2 thc1 thc2] have False .
+      }
+      then have "?e = 0"
+        by auto
+      then have "cmod (poly p z) = ?m"
+        by simp
+      with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
+        by simp
+    }
+    then have ?thesis by blast
+  }
   ultimately show ?thesis by blast
 qed
 
 lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
   unfolding power2_eq_square
   apply (simp add: rcis_mult)
   apply (simp add: power2_eq_square[symmetric])
   done
 
 lemma cispi: "cis pi = -1"
-  unfolding cis_def
-  by simp
+  by (simp add: cis_def)
 
 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
   unfolding power2_eq_square
   apply (simp add: rcis_mult add_divide_distrib)
   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
   done
 
 text {* Nonzero polynomial in z goes to infinity as z does. *}
 
 lemma poly_infinity:
-  fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly" 
+  fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
   assumes ex: "p \<noteq> 0"
   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
-using ex
-proof(induct p arbitrary: a d)
+  using ex
+proof (induct p arbitrary: a d)
   case (pCons c cs a d)
-  {assume H: "cs \<noteq> 0"
-    with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)" by blast
+  {
+    assume H: "cs \<noteq> 0"
+    with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
+      by blast
     let ?r = "1 + \<bar>r\<bar>"
-    {fix z::'a assume h: "1 + \<bar>r\<bar> \<le> norm z"
+    {
+      fix z::'a
+      assume h: "1 + \<bar>r\<bar> \<le> norm z"
       have r0: "r \<le> norm z" using h by arith
-      from r[rule_format, OF r0]
-      have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)" by arith
-      from h have z1: "norm z \<ge> 1" by arith
+      from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
+        by arith
+      from h have z1: "norm z \<ge> 1"
+        by arith
       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
         unfolding norm_mult by (simp add: algebra_simps)
       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
       have th2: "norm(z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
         by (simp add: algebra_simps)
-      from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"  by arith}
-    hence ?case by blast}
+      from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)" by arith
+    }
+    then have ?case by blast
+  }
   moreover
-  {assume cs0: "\<not> (cs \<noteq> 0)"
-    with pCons.prems have c0: "c \<noteq> 0" by simp
-    from cs0 have cs0': "cs = 0" by simp
-    {fix z::'a
+  {
+    assume cs0: "\<not> (cs \<noteq> 0)"
+    with pCons.prems have c0: "c \<noteq> 0"
+      by simp
+    from cs0 have cs0': "cs = 0"
+      by simp
+    {
+      fix z::'a
       assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
-      from c0 have "norm c > 0" by simp
+      from c0 have "norm c > 0"
+        by simp
       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
         by (simp add: field_simps norm_mult)
-      have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
-      from norm_diff_ineq[of "z * c" a ]
-      have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
+      have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
+        by arith
+      from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
         by (simp add: algebra_simps)
       from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
-        using cs0' by simp}
-    then have ?case  by blast}
+        using cs0' by simp
+    }
+    then have ?case  by blast
+  }
   ultimately show ?case by blast
 qed simp
 
 text {* Hence polynomial's modulus attains its minimum somewhere. *}
-lemma poly_minimum_modulus:
-  "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
-proof(induct p)
+lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
+proof (induct p)
+  case 0
+  then show ?case by simp
+next
   case (pCons c cs)
-  {assume cs0: "cs \<noteq> 0"
-    from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
-    obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)" by blast
-    have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
+  show ?case
+  proof (cases "cs = 0")
+    case False
+    from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
+    obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
+      by blast
+    have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
+      by arith
     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
-    obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)" by blast
-    {fix z assume z: "r \<le> cmod z"
-      from v[of 0] r[OF z]
-      have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
-        by simp }
+    obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)"
+      by blast
+    {
+      fix z
+      assume z: "r \<le> cmod z"
+      from v[of 0] r[OF z] have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
+        by simp
+    }
     note v0 = this
-    from v0 v ath[of r] have ?case by blast}
-  moreover
-  {assume cs0: "\<not> (cs \<noteq> 0)"
-    hence th:"cs = 0" by simp
-    from th pCons.hyps have ?case by simp}
-  ultimately show ?case by blast
-qed simp
+    from v0 v ath[of r] show ?thesis
+      by blast
+  next
+    case True
+    with pCons.hyps show ?thesis by simp
+  qed
+qed
 
 text{* Constant function (non-syntactic characterization). *}
 definition "constant f = (\<forall>x y. f x = f y)"
 
-lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
-  unfolding constant_def psize_def
-  apply (induct p, auto)
-  done
+lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
+  by (induct p) (auto simp: constant_def psize_def)
 
 lemma poly_replicate_append:
   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
   by (simp add: poly_monom)
 
 text {* Decomposition of polynomial, skipping zero coefficients
   after the first.  *}
 
 lemma poly_decompose_lemma:
- assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
-  shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
-                 (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
-unfolding psize_def
-using nz
-proof(induct p)
-  case 0 thus ?case by simp
+  assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
+  shows "\<exists>k a q. a \<noteq> 0 \<and> Suc (psize q + k) = psize p \<and>
+    (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
+  unfolding psize_def
+  using nz
+proof (induct p)
+  case 0
+  then show ?case by simp
 next
   case (pCons c cs)
-  {assume c0: "c = 0"
-    from pCons.hyps pCons.prems c0 have ?case
+  show ?case
+  proof (cases "c = 0")
+    case True
+    from pCons.hyps pCons.prems True show ?thesis
       apply (auto)
       apply (rule_tac x="k+1" in exI)
       apply (rule_tac x="a" in exI, clarsimp)
       apply (rule_tac x="q" in exI)
-      by (auto)}
-  moreover
-  {assume c0: "c\<noteq>0"
-    have ?case 
+      apply auto
+      done
+  next
+    case False
+    show ?thesis
       apply (rule exI[where x=0])
-      apply (rule exI[where x=c], auto simp add: c0)
-      done}
-  ultimately show ?case by blast
+      apply (rule exI[where x=c], auto simp add: False)
+      done
+  qed
 qed
 
 lemma poly_decompose:
   assumes nc: "\<not> constant (poly p)"
-  shows "\<exists>k a q. a \<noteq> (0::'a::{idom}) \<and> k \<noteq> 0 \<and>
+  shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
                psize q + k + 1 = psize p \<and>
               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
   using nc
 proof (induct p)
   case 0

     {
       fix x y
       from C have "poly (pCons c cs) x = poly (pCons c cs) y"
         by (cases "x = 0") auto
     }
-    with pCons.prems have False by (auto simp add: constant_def)
+    with pCons.prems have False
+      by (auto simp add: constant_def)
   }
   then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
   from poly_decompose_lemma[OF th]
   show ?case
     apply clarsimp

   let ?ths = "\<exists>z. ?p z = 0"
 
   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
   from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
     by blast
-  {assume pc: "?p c = 0" hence ?ths by blast}
-  moreover
-  {assume pc0: "?p c \<noteq> 0"
-    from poly_offset[of p c] obtain q where
-      q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
-    {assume h: "constant (poly q)"
+
+  show ?ths
+  proof (cases "?p c = 0")
+    case True
+    then show ?thesis by blast
+  next
+    case False
+    note pc0 = this
+    from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
+      by blast
+    {
+      assume h: "constant (poly q)"
       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
-      {fix x y
+      {
+        fix x y
         from th have "?p x = poly q (x - c)" by auto
         also have "\<dots> = poly q (y - c)"
           using h unfolding constant_def by blast
         also have "\<dots> = ?p y" using th by auto
-        finally have "?p x = ?p y" .}
-      with less(2) have False unfolding constant_def by blast }
-    hence qnc: "\<not> constant (poly q)" by blast
-    from q(2) have pqc0: "?p c = poly q 0" by simp
-    from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
+        finally have "?p x = ?p y" .
+      }
+      with less(2) have False
+        unfolding constant_def by blast
+    }
+    then have qnc: "\<not> constant (poly q)"
+      by blast
+    from q(2) have pqc0: "?p c = poly q 0"
+      by simp
+    from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
+      by simp
     let ?a0 = "poly q 0"
-    from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
-    from a00
-    have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
+    from pc0 pqc0 have a00: "?a0 \<noteq> 0"
+      by simp
+    from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
       by simp
     let ?r = "smult (inverse ?a0) q"
     have lgqr: "psize q = psize ?r"
-      using a00 unfolding psize_def degree_def
+      using a00
+      unfolding psize_def degree_def
       by (simp add: poly_eq_iff)
-    {assume h: "\<And>x y. poly ?r x = poly ?r y"
-      {fix x y
-        from qr[rule_format, of x]
-        have "poly q x = poly ?r x * ?a0" by auto
-        also have "\<dots> = poly ?r y * ?a0" using h by simp
-        also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
-        finally have "poly q x = poly q y" .}
-      with qnc have False unfolding constant_def by blast}
-    hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
-    from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
-    {fix w
+    {
+      assume h: "\<And>x y. poly ?r x = poly ?r y"
+      {
+        fix x y
+        from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
+          by auto
+        also have "\<dots> = poly ?r y * ?a0"
+          using h by simp
+        also have "\<dots> = poly q y"
+          using qr[rule_format, of y] by simp
+        finally have "poly q x = poly q y" .
+      }
+      with qnc have False unfolding constant_def by blast
+    }
+    then have rnc: "\<not> constant (poly ?r)"
+      unfolding constant_def by blast
+    from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
+      by auto
+    {
+      fix w
       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
         using a00 unfolding norm_divide by (simp add: field_simps)
-      finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
+      finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .
+    }
     note mrmq_eq = this
     from poly_decompose[OF rnc] obtain k a s where
-      kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
-      "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
-    {assume "psize p = k + 1"
-      with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
-      {fix w
+      kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
+        "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
+    {
+      assume "psize p = k + 1"
+      with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
+        by auto
+      {
+        fix w
         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
-          using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
+          using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
+      }
       note hth = this [symmetric]
-        from reduce_poly_simple[OF kas(1,2)]
-      have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
+      from reduce_poly_simple[OF kas(1,2)] have "\<exists>w. cmod (poly ?r w) < 1"
+        unfolding hth by blast
+    }
     moreover
-    {assume kn: "psize p \<noteq> k+1"
-      from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
+    {
+      assume kn: "psize p \<noteq> k + 1"
+      from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
+        by simp
       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
         unfolding constant_def poly_pCons poly_monom
         using kas(1) apply simp
-        by (rule exI[where x=0], rule exI[where x=1], simp)
-      from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
+        apply (rule exI[where x=0])
+        apply (rule exI[where x=1])
+        apply simp
+        done
+      from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
         by (simp add: psize_def degree_monom_eq)
       from less(1) [OF k1n [simplified th02] th01]
       obtain w where w: "1 + w^k * a = 0"
         unfolding poly_pCons poly_monom
-        using kas(2) by (cases k, auto simp add: algebra_simps)
+        using kas(2) by (cases k) (auto simp add: algebra_simps)
       from poly_bound_exists[of "cmod w" s] obtain m where
         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
-      have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
-      from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
-      then have wm1: "w^k * a = - 1" by simp
+      have w0: "w \<noteq> 0" using kas(2) w
+        by (auto simp add: power_0_left)
+      from w have "(1 + w ^ k * a) - 1 = 0 - 1"
+        by simp
+      then have wm1: "w^k * a = - 1"
+        by simp
       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
         using norm_ge_zero[of w] w0 m(1)
-          by (simp add: inverse_eq_divide zero_less_mult_iff)
+        by (simp add: inverse_eq_divide zero_less_mult_iff)
       with real_lbound_gt_zero[OF zero_less_one] obtain t where
         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
       let ?ct = "complex_of_real t"
       let ?w = "?ct * w"
-      have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" using kas(1) by (simp add: algebra_simps power_mult_distrib)
+      have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
+        using kas(1) by (simp add: algebra_simps power_mult_distrib)
       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
-        unfolding wm1 by (simp)
-      finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
+        unfolding wm1 by simp
+      finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
+        cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
         by metis
       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
-      have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)" unfolding norm_of_real by simp
-      have ath: "\<And>x (t::real). 0\<le> x \<Longrightarrow> x < t \<Longrightarrow> t\<le>1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1" by arith
-      have "t * cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
-      then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
-      from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
+      have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"
+        unfolding norm_of_real by simp
+      have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"
+        by arith
+      have "t * cmod w \<le> 1 * cmod w"
+        apply (rule mult_mono)
+        using t(1,2)
+        apply auto
+        done
+      then have tw: "cmod ?w \<le> cmod w"
+        using t(1) by (simp add: norm_mult)
+      from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
         by (simp add: inverse_eq_divide field_simps)
-      with zero_less_power[OF t(1), of k]
-      have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
+      with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
         by (metis comm_mult_strict_left_mono)
-      have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
+      have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
+        using w0 t(1)
         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
         using t(1,2) m(2)[rule_format, OF tw] w0
         by auto
-      with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
+      with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
+        by simp
       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
         by auto
       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
-      from th11 th12
-      have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
+      from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
+        by arith
       then have "cmod (poly ?r ?w) < 1"
         unfolding kas(4)[rule_format, of ?w] r01 by simp
-      then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
-    ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
-    from cr0_contr cq0 q(2)
-    have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
-  ultimately show ?ths by blast
+      then have "\<exists>w. cmod (poly ?r w) < 1"
+        by blast
+    }
+    ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1"
+      by blast
+    from cr0_contr cq0 q(2) show ?thesis
+      unfolding mrmq_eq not_less[symmetric] by auto
+  qed
 qed
 
 text {* Alternative version with a syntactic notion of constant polynomial. *}
 
 lemma fundamental_theorem_of_algebra_alt:
-  assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
+  assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
   shows "\<exists>z. poly p z = (0::complex)"
-using nc
-proof(induct p)
+  using nc
+proof (induct p)
+  case 0
+  then show ?case by simp
+next
   case (pCons c cs)
-  {assume "c=0" hence ?case by auto}
-  moreover
-  {assume c0: "c\<noteq>0"
-    {assume nc: "constant (poly (pCons c cs))"
+  show ?case
+  proof (cases "c = 0")
+    case True
+    then show ?thesis by auto
+  next
+    case False
+    {
+      assume nc: "constant (poly (pCons c cs))"
       from nc[unfolded constant_def, rule_format, of 0]
       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
-      hence "cs = 0"
-        proof(induct cs)
-          case (pCons d ds)
-          {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
-          moreover
-          {assume d0: "d\<noteq>0"
-            from poly_bound_exists[of 1 ds] obtain m where
-              m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
-            have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
-            from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
-              x: "x > 0" "x < cmod d / m" "x < 1" by blast
-            let ?x = "complex_of_real x"
-            from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
-            from pCons.prems[rule_format, OF cx(1)]
-            have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
-            from m(2)[rule_format, OF cx(2)] x(1)
-            have th0: "cmod (?x*poly ds ?x) \<le> x*m"
-              by (simp add: norm_mult)
-            from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
-            with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
-            with cth  have ?case by blast}
-          ultimately show ?case by blast
-        qed simp}
-      then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
-        by blast
-      from fundamental_theorem_of_algebra[OF nc] have ?case .}
-  ultimately show ?case by blast
-qed simp
+      then have "cs = 0"
+      proof (induct cs)
+        case 0
+        then show ?case by simp
+      next
+        case (pCons d ds)
+        show ?case
+        proof (cases "d = 0")
+          case True
+          then show ?thesis using pCons.prems pCons.hyps by simp
+        next
+          case False
+          from poly_bound_exists[of 1 ds] obtain m where
+            m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
+          have dm: "cmod d / m > 0" using False m(1) by (simp add: field_simps)
+          from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
+            x: "x > 0" "x < cmod d / m" "x < 1" by blast
+          let ?x = "complex_of_real x"
+          from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
+          from pCons.prems[rule_format, OF cx(1)]
+          have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
+          from m(2)[rule_format, OF cx(2)] x(1)
+          have th0: "cmod (?x*poly ds ?x) \<le> x*m"
+            by (simp add: norm_mult)
+          from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
+          with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
+          with cth show ?thesis by blast
+        qed
+      qed
+    }
+    then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems False
+      by blast
+    from fundamental_theorem_of_algebra[OF nc] show ?thesis .
+  qed
+qed
 
 
 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
 
 lemma nullstellensatz_lemma:

   fix p q :: "complex poly"
   assume IH: "\<forall>m<n. \<forall>p q.
                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
-    and dpn: "degree p = n" and n0: "n \<noteq> 0"
+    and dpn: "degree p = n"
+    and n0: "n \<noteq> 0"
   from dpn n0 have pne: "p \<noteq> 0" by auto
   let ?ths = "p dvd (q ^ n)"
-  {fix a assume a: "poly p a = 0"
-    {assume oa: "order a p \<noteq> 0"
+  {
+    fix a
+    assume a: "poly p a = 0"
+    {
+      assume oa: "order a p \<noteq> 0"
       let ?op = "order a p"
-      from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
-        "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
+      from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
+        using order by blast+
       note oop = order_degree[OF pne, unfolded dpn]
-      {assume q0: "q = 0"
-        hence ?ths using n0
-          by (simp add: power_0_left)}
+      {
+        assume q0: "q = 0"
+        then have ?ths using n0
+          by (simp add: power_0_left)
+      }
       moreover
-      {assume q0: "q \<noteq> 0"
+      {
+        assume q0: "q \<noteq> 0"
         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
-        from ap(1) obtain s where
-          s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
-        have sne: "s \<noteq> 0"
-          using s pne by auto
-        {assume ds0: "degree s = 0"
+        from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
+          by (rule dvdE)
+        have sne: "s \<noteq> 0" using s pne by auto
+        {
+          assume ds0: "degree s = 0"
           from ds0 obtain k where kpn: "s = [:k:]"
             by (cases s) (auto split: if_splits)
           from sne kpn have k: "k \<noteq> 0" by simp
           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
           have "q ^ n = p * ?w"
             apply (subst r, subst s, subst kpn)
-            using k oop [of a] 
+            using k oop [of a]
             apply (subst power_mult_distrib, simp)
             apply (subst power_add [symmetric], simp)
             done
-          hence ?ths unfolding dvd_def by blast}
+          then have ?ths unfolding dvd_def by blast
+        }
         moreover
-        {assume ds0: "degree s \<noteq> 0"
+        {
+          assume ds0: "degree s \<noteq> 0"
           from ds0 sne dpn s oa
-            have dsn: "degree s < n" apply auto
+            have dsn: "degree s < n"
+              apply auto
               apply (erule ssubst)
               apply (simp add: degree_mult_eq degree_linear_power)
               done
-            {fix x assume h: "poly s x = 0"
-              {assume xa: "x = a"
-                from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
-                  u: "s = [:- a, 1:] * u" by (rule dvdE)
+            {
+              fix x assume h: "poly s x = 0"
+              {
+                assume xa: "x = a"
+                from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
+                  by (rule dvdE)
                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
                   by (subst s, subst u, simp only: power_Suc mult_ac)
-                with ap(2)[unfolded dvd_def] have False by blast}
+                with ap(2)[unfolded dvd_def] have False by blast
+              }
               note xa = this
-              from h have "poly p x = 0" by (subst s, simp)
+              from h have "poly p x = 0" by (subst s) simp
               with pq0 have "poly q x = 0" by blast
               with r xa have "poly r x = 0"
-                by auto}
+                by auto
+            }
             note impth = this
             from IH[rule_format, OF dsn, of s r] impth ds0
             have "s dvd (r ^ (degree s))" by blast
             then obtain u where u: "r ^ (degree s) = s * u" ..
-            hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
+            then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
               by (simp only: poly_mult[symmetric] poly_power[symmetric])
             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
             from oop[of a] dsn have "q ^ n = p * ?w"
               apply -
               apply (subst s, subst r)

               apply (subst mult_assoc [where a=u])
               apply (subst mult_assoc [where b=u, symmetric])
               apply (subst u [symmetric])
               apply (simp add: mult_ac power_add [symmetric])
               done
-            hence ?ths unfolding dvd_def by blast}
-      ultimately have ?ths by blast }
-      ultimately have ?ths by blast}
-    then have ?ths using a order_root pne by blast}
+            then have ?ths unfolding dvd_def by blast
+        }
+        ultimately have ?ths by blast
+      }
+      ultimately have ?ths by blast
+    }
+    then have ?ths using a order_root pne by blast
+  }
   moreover
-  {assume exa: "\<not> (\<exists>a. poly p a = 0)"
-    from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
-      ccs: "c\<noteq>0" "p = pCons c 0" by blast
-
-    then have pp: "\<And>x. poly p x =  c" by simp
+  {
+    assume exa: "\<not> (\<exists>a. poly p a = 0)"
+    from fundamental_theorem_of_algebra_alt[of p] exa
+    obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
+      by blast
+    then have pp: "\<And>x. poly p x = c"
+      by simp
     let ?w = "[:1/c:] * (q ^ n)"
-    from ccs have "(q ^ n) = (p * ?w)" by simp
-    hence ?ths unfolding dvd_def by blast}
+    from ccs have "(q ^ n) = (p * ?w)"
+      by simp
+    then have ?ths
+      unfolding dvd_def by blast
+  }
   ultimately show ?ths by blast
 qed
 
 lemma nullstellensatz_univariate:
   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
 proof -
-  {assume pe: "p = 0"
-    hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
+  {
+    assume pe: "p = 0"
+    then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
       by (auto simp add: poly_all_0_iff_0)
-    {assume "p dvd (q ^ (degree p))"
+    {
+      assume "p dvd (q ^ (degree p))"
       then obtain r where r: "q ^ (degree p) = p * r" ..
-      from r pe have False by simp}
-    with eq pe have ?thesis by blast}
+      from r pe have False by simp
+    }
+    with eq pe have ?thesis by blast
+  }
   moreover
-  {assume pe: "p \<noteq> 0"
-    {assume dp: "degree p = 0"
-      then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
+  {
+    assume pe: "p \<noteq> 0"
+    {
+      assume dp: "degree p = 0"
+      then obtain k where k: "p = [:k:]" "k \<noteq> 0" using pe
         by (cases p) (simp split: if_splits)
-      hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
+      then have th1: "\<forall>x. poly p x \<noteq> 0"
+        by simp
       from k dp have "q ^ (degree p) = p * [:1/k:]"
         by (simp add: one_poly_def)
-      hence th2: "p dvd (q ^ (degree p))" ..
-      from th1 th2 pe have ?thesis by blast}
+      then have th2: "p dvd (q ^ (degree p))" ..
+      from th1 th2 pe have ?thesis by blast
+    }
     moreover
-    {assume dp: "degree p \<noteq> 0"
-      then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
-      {assume "p dvd (q ^ (Suc n))"
+    {
+      assume dp: "degree p \<noteq> 0"
+      then obtain n where n: "degree p = Suc n "
+        by (cases "degree p") auto
+      {
+        assume "p dvd (q ^ (Suc n))"
         then obtain u where u: "q ^ (Suc n) = p * u" ..
-        {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
-          hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
-          hence False using u h(1) by (simp only: poly_mult) simp}}
-        with n nullstellensatz_lemma[of p q "degree p"] dp
-        have ?thesis by auto}
-    ultimately have ?thesis by blast}
+        {
+          fix x
+          assume h: "poly p x = 0" "poly q x \<noteq> 0"
+          then have "poly (q ^ (Suc n)) x \<noteq> 0"
+            by simp
+          then have False using u h(1)
+            by (simp only: poly_mult) simp
+        }
+      }
+      with n nullstellensatz_lemma[of p q "degree p"] dp
+      have ?thesis by auto
+    }
+    ultimately have ?thesis by blast
+  }
   ultimately show ?thesis by blast
 qed
 
 text{* Useful lemma *}
 

   by (metis dvd_imp_degree_le pq)
 
 (* Arithmetic operations on multivariate polynomials.                        *)
 
 lemma mpoly_base_conv:
-  fixes x :: "'a::comm_ring_1" 
+  fixes x :: "'a::comm_ring_1"
   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
   by simp_all
 
 lemma mpoly_norm_conv:
-  fixes x :: "'a::comm_ring_1" 
+  fixes x :: "'a::comm_ring_1"
   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
   by simp_all
 
 lemma mpoly_sub_conv:
-  fixes x :: "'a::comm_ring_1" 
+  fixes x :: "'a::comm_ring_1"
   shows "poly p x - poly q x = poly p x + -1 * poly q x"
   by simp
 
-lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = 0" by simp
+lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
+  by simp
 
 lemma poly_cancel_eq_conv:
-  fixes x :: "'a::field" 
-  shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)" 
+  fixes x :: "'a::field"
+  shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)"
   by auto
 
 lemma poly_divides_pad_rule:
-  fixes p:: "('a::comm_ring_1) poly" 
+  fixes p:: "('a::comm_ring_1) poly"
   assumes pq: "p dvd q"
-shows "p dvd (pCons 0 q)"
-proof-
+  shows "p dvd (pCons 0 q)"
+proof -
   have "pCons 0 q = q * [:0,1:]" by simp
   then have "q dvd (pCons 0 q)" ..
   with pq show ?thesis by (rule dvd_trans)
 qed
 
 lemma poly_divides_conv0:
-  fixes p:: "'a::field poly" 
+  fixes p:: "'a::field poly"
   assumes lgpq: "degree q < degree p"
     and lq: "p \<noteq> 0"
   shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume r: ?rhs
   then have "q = p * 0" by simp
   then show ?lhs ..
 next
   assume l: ?lhs
-  {
-    assume q0: "q = 0"
-    then have ?rhs by simp
-  }
-  moreover
-  {
+  show ?rhs
+  proof (cases "q = 0")
+    case True
+    then show ?thesis by simp
+  next
     assume q0: "q \<noteq> 0"
     from l q0 have "degree p \<le> degree q"
       by (rule dvd_imp_degree_le)
-    with lgpq have ?rhs by simp
-  }
-  ultimately show ?rhs by blast
+    with lgpq show ?thesis by simp
+  qed
 qed
 
 lemma poly_divides_conv1:
-  fixes p :: "('a::field) poly" 
+  fixes p :: "'a::field poly"
   assumes a0: "a \<noteq> 0"
     and pp': "p dvd p'"
     and qrp': "smult a q - p' = r"
   shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
 proof

   then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
     by simp
 qed
 
 lemma poly_const_conv:
-  fixes x :: "'a::comm_ring_1" 
+  fixes x :: "'a::comm_ring_1"
   shows "poly [:c:] x = y \<longleftrightarrow> c = y"
   by simp
 
 end