# HG changeset patch
# User wenzelm
# Date 1468353236 -7200
# Node ID 2100fbbdc3f1d66e43a319e0882dad08521601d1
# Parent  d7610beb98bc978edb8b8721d7755d77f6b28fe4
misc tuning and modernization;

diff -r d7610beb98bc -r 2100fbbdc3f1 src/HOL/Binomial.thy
--- a/src/HOL/Binomial.thy	Tue Jul 12 20:03:18 2016 +0200
+++ b/src/HOL/Binomial.thy	Tue Jul 12 21:53:56 2016 +0200
@@ -1,15 +1,15 @@
-(*  Title       : Binomial.thy
-    Author      : Jacques D. Fleuriot
-    Copyright   : 1998  University of Cambridge
-    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
-    Various additions by Jeremy Avigad.
-    Additional binomial identities by Chaitanya Mangla and Manuel Eberl
+(*  Title:      HOL/Binomial.thy
+    Author:     Jacques D. Fleuriot
+    Author:     Lawrence C Paulson
+    Author:     Jeremy Avigad
+    Author:     Chaitanya Mangla
+    Author:     Manuel Eberl
 *)
 
 section \<open>Combinatorial Functions: Factorial Function, Rising Factorials, Binomial Coefficients and Binomial Theorem\<close>
 
 theory Binomial
-imports Main
+  imports Main
 begin
 
 subsection \<open>Factorial\<close>
@@ -18,17 +18,18 @@
 begin
 
 definition fact :: "nat \<Rightarrow> 'a"
-where
-  fact_setprod: "fact n = of_nat (\<Prod>{1..n})"
+  where fact_setprod: "fact n = of_nat (\<Prod>{1..n})"
 
-lemma fact_setprod_Suc:
-  "fact n = of_nat (setprod Suc {0..<n})"
-  by (cases n) (simp_all add: fact_setprod setprod.atLeast_Suc_atMost_Suc_shift atLeastLessThanSuc_atLeastAtMost)
+lemma fact_setprod_Suc: "fact n = of_nat (setprod Suc {0..<n})"
+  by (cases n)
+    (simp_all add: fact_setprod setprod.atLeast_Suc_atMost_Suc_shift
+      atLeastLessThanSuc_atLeastAtMost)
 
-lemma fact_setprod_rev:
-  "fact n = of_nat (\<Prod>i = 0..<n. n - i)"
+lemma fact_setprod_rev: "fact n = of_nat (\<Prod>i = 0..<n. n - i)"
   using setprod.atLeast_atMost_rev [of "\<lambda>i. i" 1 n]
-  by (cases n) (simp_all add: fact_setprod_Suc setprod.atLeast_Suc_atMost_Suc_shift atLeastLessThanSuc_atLeastAtMost)
+  by (cases n)
+    (simp_all add: fact_setprod_Suc setprod.atLeast_Suc_atMost_Suc_shift
+      atLeastLessThanSuc_atLeastAtMost)
 
 lemma fact_0 [simp]: "fact 0 = 1"
   by (simp add: fact_setprod)
@@ -42,24 +43,19 @@
 lemma fact_Suc [simp]: "fact (Suc n) = of_nat (Suc n) * fact n"
   by (simp add: fact_setprod atLeastAtMostSuc_conv algebra_simps)
 
-lemma fact_2 [simp]:
-  "fact 2 = 2"
+lemma fact_2 [simp]: "fact 2 = 2"
   by (simp add: numeral_2_eq_2)
 
-lemma fact_split:
-  assumes "k \<le> n"
-  shows "fact n = of_nat (setprod Suc {n - k..<n}) * fact (n - k)"
-  using assms by (simp add: fact_setprod_Suc setprod.union_disjoint [symmetric] ivl_disj_un
-    ac_simps of_nat_mult [symmetric])
+lemma fact_split: "k \<le> n \<Longrightarrow> fact n = of_nat (setprod Suc {n - k..<n}) * fact (n - k)"
+  by (simp add: fact_setprod_Suc setprod.union_disjoint [symmetric]
+    ivl_disj_un ac_simps of_nat_mult [symmetric])
 
 end
 
-lemma of_nat_fact [simp]:
-  "of_nat (fact n) = fact n"
+lemma of_nat_fact [simp]: "of_nat (fact n) = fact n"
   by (simp add: fact_setprod)
 
-lemma of_int_fact [simp]:
-  "of_int (fact n) = fact n"
+lemma of_int_fact [simp]: "of_int (fact n) = fact n"
   by (simp only: fact_setprod of_int_of_nat_eq)
 
 lemma fact_reduce: "n > 0 \<Longrightarrow> fact n = of_nat n * fact (n - 1)"
@@ -68,81 +64,87 @@
 lemma fact_nonzero [simp]: "fact n \<noteq> (0::'a::{semiring_char_0,semiring_no_zero_divisors})"
   apply (induct n)
   apply auto
-  using of_nat_eq_0_iff by fastforce
+  using of_nat_eq_0_iff
+  apply fastforce
+  done
 
 lemma fact_mono_nat: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: nat)"
   by (induct n) (auto simp: le_Suc_eq)
 
-lemma fact_in_Nats: "fact n \<in> \<nat>" by (induction n) auto
+lemma fact_in_Nats: "fact n \<in> \<nat>"
+  by (induct n) auto
 
-lemma fact_in_Ints: "fact n \<in> \<int>" by (induction n) auto
+lemma fact_in_Ints: "fact n \<in> \<int>"
+  by (induct n) auto
 
 context
   assumes "SORT_CONSTRAINT('a::linordered_semidom)"
 begin
 
-  lemma fact_mono: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: 'a)"
-    by (metis of_nat_fact of_nat_le_iff fact_mono_nat)
+lemma fact_mono: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: 'a)"
+  by (metis of_nat_fact of_nat_le_iff fact_mono_nat)
 
-  lemma fact_ge_1 [simp]: "fact n \<ge> (1 :: 'a)"
-    by (metis le0 fact_0 fact_mono)
+lemma fact_ge_1 [simp]: "fact n \<ge> (1 :: 'a)"
+  by (metis le0 fact_0 fact_mono)
 
-  lemma fact_gt_zero [simp]: "fact n > (0 :: 'a)"
-    using fact_ge_1 less_le_trans zero_less_one by blast
+lemma fact_gt_zero [simp]: "fact n > (0 :: 'a)"
+  using fact_ge_1 less_le_trans zero_less_one by blast
 
-  lemma fact_ge_zero [simp]: "fact n \<ge> (0 :: 'a)"
-    by (simp add: less_imp_le)
+lemma fact_ge_zero [simp]: "fact n \<ge> (0 :: 'a)"
+  by (simp add: less_imp_le)
 
-  lemma fact_not_neg [simp]: "~ (fact n < (0 :: 'a))"
-    by (simp add: not_less_iff_gr_or_eq)
+lemma fact_not_neg [simp]: "\<not> fact n < (0 :: 'a)"
+  by (simp add: not_less_iff_gr_or_eq)
 
-  lemma fact_le_power:
-      "fact n \<le> (of_nat (n^n) ::'a)"
-  proof (induct n)
-    case (Suc n)
-    then have *: "fact n \<le> (of_nat (Suc n ^ n) ::'a)"
-      by (rule order_trans) (simp add: power_mono del: of_nat_power)
-    have "fact (Suc n) = (of_nat (Suc n) * fact n ::'a)"
-      by (simp add: algebra_simps)
-    also have "... \<le> (of_nat (Suc n) * of_nat (Suc n ^ n) ::'a)"
-      by (simp add: "*" ordered_comm_semiring_class.comm_mult_left_mono del: of_nat_power)
-    also have "... \<le> (of_nat (Suc n ^ Suc n) ::'a)"
-      by (metis of_nat_mult order_refl power_Suc)
-    finally show ?case .
-  qed simp
+lemma fact_le_power: "fact n \<le> (of_nat (n^n) :: 'a)"
+proof (induct n)
+  case 0
+  then show ?case by simp
+next
+  case (Suc n)
+  then have *: "fact n \<le> (of_nat (Suc n ^ n) ::'a)"
+    by (rule order_trans) (simp add: power_mono del: of_nat_power)
+  have "fact (Suc n) = (of_nat (Suc n) * fact n ::'a)"
+    by (simp add: algebra_simps)
+  also have "\<dots> \<le> of_nat (Suc n) * of_nat (Suc n ^ n)"
+    by (simp add: * ordered_comm_semiring_class.comm_mult_left_mono del: of_nat_power)
+  also have "\<dots> \<le> of_nat (Suc n ^ Suc n)"
+    by (metis of_nat_mult order_refl power_Suc)
+  finally show ?case .
+qed
 
 end
 
-text\<open>Note that @{term "fact 0 = fact 1"}\<close>
-lemma fact_less_mono_nat: "\<lbrakk>0 < m; m < n\<rbrakk> \<Longrightarrow> fact m < (fact n :: nat)"
+text \<open>Note that @{term "fact 0 = fact 1"}\<close>
+lemma fact_less_mono_nat: "0 < m \<Longrightarrow> m < n \<Longrightarrow> fact m < (fact n :: nat)"
   by (induct n) (auto simp: less_Suc_eq)
 
-lemma fact_less_mono:
-  "\<lbrakk>0 < m; m < n\<rbrakk> \<Longrightarrow> fact m < (fact n :: 'a::linordered_semidom)"
+lemma fact_less_mono: "0 < m \<Longrightarrow> m < n \<Longrightarrow> fact m < (fact n :: 'a::linordered_semidom)"
   by (metis of_nat_fact of_nat_less_iff fact_less_mono_nat)
 
 lemma fact_ge_Suc_0_nat [simp]: "fact n \<ge> Suc 0"
   by (metis One_nat_def fact_ge_1)
 
-lemma dvd_fact:
-  shows "1 \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> m dvd fact n"
+lemma dvd_fact: "1 \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> m dvd fact n"
   by (induct n) (auto simp: dvdI le_Suc_eq)
 
 lemma fact_ge_self: "fact n \<ge> n"
   by (cases "n = 0") (simp_all add: dvd_imp_le dvd_fact)
 
-lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd (fact m :: 'a :: {semiring_div,linordered_semidom})"
+lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd (fact m :: 'a::{semiring_div,linordered_semidom})"
   by (induct m) (auto simp: le_Suc_eq)
 
-lemma fact_mod: "m \<le> n \<Longrightarrow> fact n mod (fact m :: 'a :: {semiring_div,linordered_semidom}) = 0"
+lemma fact_mod: "m \<le> n \<Longrightarrow> fact n mod (fact m :: 'a::{semiring_div,linordered_semidom}) = 0"
   by (auto simp add: fact_dvd)
 
 lemma fact_div_fact:
   assumes "m \<ge> n"
-  shows "(fact m) div (fact n) = \<Prod>{n + 1..m}"
+  shows "fact m div fact n = \<Prod>{n + 1..m}"
 proof -
-  obtain d where "d = m - n" by auto
-  from assms this have "m = n + d" by auto
+  obtain d where "d = m - n"
+    by auto
+  with assms have "m = n + d"
+    by auto
   have "fact (n + d) div (fact n) = \<Prod>{n + 1..n + d}"
   proof (induct d)
     case 0
@@ -151,44 +153,41 @@
     case (Suc d')
     have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n"
       by simp
-    also from Suc.hyps have "... = Suc (n + d') * \<Prod>{n + 1..n + d'}"
+    also from Suc.hyps have "\<dots> = Suc (n + d') * \<Prod>{n + 1..n + d'}"
       unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod)
-    also have "... = \<Prod>{n + 1..n + Suc d'}"
+    also have "\<dots> = \<Prod>{n + 1..n + Suc d'}"
       by (simp add: atLeastAtMostSuc_conv)
     finally show ?case .
   qed
-  from this \<open>m = n + d\<close> show ?thesis by simp
+  with \<open>m = n + d\<close> show ?thesis by simp
 qed
 
-lemma fact_num_eq_if:
-    "fact m = (if m=0 then 1 else of_nat m * fact (m - 1))"
-by (cases m) auto
+lemma fact_num_eq_if: "fact m = (if m = 0 then 1 else of_nat m * fact (m - 1))"
+  by (cases m) auto
 
 lemma fact_div_fact_le_pow:
-  assumes "r \<le> n" shows "fact n div fact (n - r) \<le> n ^ r"
+  assumes "r \<le> n"
+  shows "fact n div fact (n - r) \<le> n ^ r"
 proof -
-  have "\<And>r. r \<le> n \<Longrightarrow> \<Prod>{n - r..n} = (n - r) * \<Prod>{Suc (n - r)..n}"
+  have "r \<le> n \<Longrightarrow> \<Prod>{n - r..n} = (n - r) * \<Prod>{Suc (n - r)..n}" for r
     by (subst setprod.insert[symmetric]) (auto simp: atLeastAtMost_insertL)
   with assms show ?thesis
     by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono)
 qed
 
-lemma fact_numeral:  \<comment>\<open>Evaluation for specific numerals\<close>
-  "fact (numeral k) = (numeral k) * (fact (pred_numeral k))"
+lemma fact_numeral: "fact (numeral k) = numeral k * fact (pred_numeral k)"
+  \<comment> \<open>Evaluation for specific numerals\<close>
   by (metis fact_Suc numeral_eq_Suc of_nat_numeral)
 
 
-text \<open>This development is based on the work of Andy Gordon and
-  Florian Kammueller.\<close>
-
-
 subsection \<open>Binomial coefficients\<close>
 
+text \<open>This development is based on the work of Andy Gordon and Florian Kammueller.\<close>
+
 text \<open>Combinatorial definition\<close>
 
-definition binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65)
-where
-  "n choose k = card {K\<in>Pow {0..<n}. card K = k}"
+definition binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat"  (infixl "choose" 65)
+  where "n choose k = card {K\<in>Pow {0..<n}. card K = k}"
 
 theorem n_subsets:
   assumes "finite A"
@@ -207,7 +206,7 @@
   ultimately have "inj_on (image f) {K. K \<subseteq> {0..<card A} \<and> card K = k}"
     by (rule inj_on_subset)
   then have "card {K. K \<subseteq> {0..<card A} \<and> card K = k} =
-    card (image f ` {K. K \<subseteq> {0..<card A} \<and> card K = k})" (is "_ = card ?C")
+      card (image f ` {K. K \<subseteq> {0..<card A} \<and> card K = k})" (is "_ = card ?C")
     by (simp add: card_image)
   also have "?C = {K. K \<subseteq> f ` {0..<card A} \<and> card K = k}"
     by (auto elim!: subset_imageE)
@@ -216,11 +215,10 @@
   finally show ?thesis
     by (simp add: binomial_def)
 qed
-    
+
 text \<open>Recursive characterization\<close>
 
-lemma binomial_n_0 [simp, code]:
-  "n choose 0 = 1"
+lemma binomial_n_0 [simp, code]: "n choose 0 = 1"
 proof -
   have "{K \<in> Pow {0..<n}. card K = 0} = {{}}"
     by (auto dest: finite_subset)
@@ -228,17 +226,15 @@
     by (simp add: binomial_def)
 qed
 
-lemma binomial_0_Suc [simp, code]:
-  "0 choose Suc k = 0"
+lemma binomial_0_Suc [simp, code]: "0 choose Suc k = 0"
   by (simp add: binomial_def)
 
-lemma binomial_Suc_Suc [simp, code]:
-  "Suc n choose Suc k = (n choose k) + (n choose Suc k)"
+lemma binomial_Suc_Suc [simp, code]: "Suc n choose Suc k = (n choose k) + (n choose Suc k)"
 proof -
   let ?P = "\<lambda>n k. {K. K \<subseteq> {0..<n} \<and> card K = k}"
   let ?Q = "?P (Suc n) (Suc k)"
   have inj: "inj_on (insert n) (?P n k)"
-    by rule (auto, (metis atLeastLessThan_iff insert_iff less_irrefl subsetCE)+)
+    by rule (auto; metis atLeastLessThan_iff insert_iff less_irrefl subsetCE)
   have disjoint: "insert n ` ?P n k \<inter> ?P n (Suc k) = {}"
     by auto
   have "?Q = {K\<in>?Q. n \<in> K} \<union> {K\<in>?Q. n \<notin> K}"
@@ -257,27 +253,25 @@
     qed
     show "K \<in> ?A \<longleftrightarrow> K \<in> ?B"
       by (subst in_image_insert_iff)
-        (auto simp add: card_insert subset_eq_atLeast0_lessThan_finite Diff_subset_conv K_finite Suc_card_K)
-  qed    
+        (auto simp add: card_insert subset_eq_atLeast0_lessThan_finite
+          Diff_subset_conv K_finite Suc_card_K)
+  qed
   also have "{K\<in>?Q. n \<notin> K} = ?P n (Suc k)"
     by (auto simp add: atLeast0_lessThan_Suc)
   finally show ?thesis using inj disjoint
     by (simp add: binomial_def card_Un_disjoint card_image)
 qed
 
-lemma binomial_eq_0:
-  "n < k \<Longrightarrow> n choose k = 0"
+lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
   by (auto simp add: binomial_def dest: subset_eq_atLeast0_lessThan_card)
 
 lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0"
   by (induct n k rule: diff_induct) simp_all
 
-lemma binomial_eq_0_iff [simp]:
-  "n choose k = 0 \<longleftrightarrow> n < k"
+lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < k"
   by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial)
 
-lemma zero_less_binomial_iff [simp]:
-  "n choose k > 0 \<longleftrightarrow> k \<le> n"
+lemma zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> n"
   by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial)
 
 lemma binomial_n_n [simp]: "n choose n = 1"
@@ -290,120 +284,123 @@
   by (induct n) simp_all
 
 lemma choose_reduce_nat:
-  "0 < (n::nat) \<Longrightarrow> 0 < k \<Longrightarrow>
-    (n choose k) = ((n - 1) choose (k - 1)) + ((n - 1) choose k)"
+  "0 < n \<Longrightarrow> 0 < k \<Longrightarrow>
+    n choose k = ((n - 1) choose (k - 1)) + ((n - 1) choose k)"
   using binomial_Suc_Suc [of "n - 1" "k - 1"] by simp
 
-lemma Suc_times_binomial_eq:
-  "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
+lemma Suc_times_binomial_eq: "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
   apply (induct n arbitrary: k)
-  apply simp apply arith
+   apply simp
+   apply arith
   apply (case_tac k)
    apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0)
   done
 
 lemma binomial_le_pow2: "n choose k \<le> 2^n"
   apply (induct n arbitrary: k)
-  apply (case_tac k) apply simp_all
+   apply (case_tac k)
+    apply simp_all
   apply (case_tac k)
-  apply auto
+   apply auto
   apply (simp add: add_le_mono mult_2)
   done
 
-text\<open>The absorption property\<close>
-lemma Suc_times_binomial:
-  "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)"
+text \<open>The absorption property.\<close>
+lemma Suc_times_binomial: "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)"
   using Suc_times_binomial_eq by auto
 
-text\<open>This is the well-known version of absorption, but it's harder to use because of the
-  need to reason about division.\<close>
-lemma binomial_Suc_Suc_eq_times:
-    "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
+text \<open>This is the well-known version of absorption, but it's harder to use
+  because of the need to reason about division.\<close>
+lemma binomial_Suc_Suc_eq_times: "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
   by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)
 
-text\<open>Another version of absorption, with -1 instead of Suc.\<close>
-lemma times_binomial_minus1_eq:
-  "0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 1))"
+text \<open>Another version of absorption, with \<open>-1\<close> instead of \<open>Suc\<close>.\<close>
+lemma times_binomial_minus1_eq: "0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 1))"
   using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"]
   by (auto split add: nat_diff_split)
 
 
 subsection \<open>The binomial theorem (courtesy of Tobias Nipkow):\<close>
 
-text\<open>Avigad's version, generalized to any commutative ring\<close>
-theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n =
-  (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n")
+text \<open>Avigad's version, generalized to any commutative ring\<close>
+theorem binomial_ring: "(a + b :: 'a::{comm_ring_1,power})^n =
+  (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))"
 proof (induct n)
-  case 0 then show "?P 0" by simp
+  case 0
+  then show ?case by simp
 next
   case (Suc n)
-  have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}"
+  have decomp: "{0..n+1} = {0} \<union> {n + 1} \<union> {1..n}"
     by auto
-  have decomp2: "{0..n} = {0} Un {1..n}"
+  have decomp2: "{0..n} = {0} \<union> {1..n}"
     by auto
-  have "(a+b)^(n+1) =
-      (a+b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
+  have "(a + b)^(n+1) = (a + b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n - k))"
     using Suc.hyps by simp
-  also have "\<dots> = a*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
-                   b*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
+  also have "\<dots> = a * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
+      b * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
     by (rule distrib_right)
   also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
-                  (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))"
+      (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n - k + 1))"
     by (auto simp add: setsum_right_distrib ac_simps)
-  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) +
-                  (\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-k))"
-    by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps
-        del:setsum_cl_ivl_Suc)
-  also have "\<dots> = a^(n+1) + b^(n+1) +
-                  (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) +
-                  (\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))"
+  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n + 1 - k)) +
+      (\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k))"
+    by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps del: setsum_cl_ivl_Suc)
+  also have "\<dots> = a^(n + 1) + b^(n + 1) +
+      (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k)) +
+      (\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n + 1 - k))"
     by (simp add: decomp2)
-  also have
-      "\<dots> = a^(n+1) + b^(n+1) +
-            (\<Sum>k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-k))"
+  also have "\<dots> = a^(n + 1) + b^(n + 1) +
+      (\<Sum>k=1..n. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))"
     by (auto simp add: field_simps setsum.distrib [symmetric] choose_reduce_nat)
-  also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))"
+  also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))"
     using decomp by (simp add: field_simps)
-  finally show "?P (Suc n)" by simp
+  finally show ?case
+    by simp
 qed
 
-text\<open>Original version for the naturals\<close>
-corollary binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))"
-    using binomial_ring [of "int a" "int b" n]
+text \<open>Original version for the naturals.\<close>
+corollary binomial: "(a + b :: nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n - k))"
+  using binomial_ring [of "int a" "int b" n]
   by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric]
-           of_nat_setsum [symmetric]
-           of_nat_eq_iff of_nat_id)
+      of_nat_setsum [symmetric] of_nat_eq_iff of_nat_id)
 
 lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
 proof (induct n arbitrary: k rule: nat_less_induct)
-  fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
-                      fact m" and kn: "k \<le> n"
+  fix n k
+  assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) = fact m"
+  assume kn: "k \<le> n"
   let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
-  { assume "n=0" then have ?ths using kn by simp }
-  moreover
-  { assume "k=0" then have ?ths using kn by simp }
-  moreover
-  { assume nk: "n=k" then have ?ths by simp }
-  moreover
-  { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m"
-    from n have mn: "m < n" by arith
-    from hm have hm': "h \<le> m" by arith
-    from hm h n kn have km: "k \<le> m" by arith
-    have "m - h = Suc (m - Suc h)" using  h km hm by arith
-    with km h have th0: "fact (m - h) = (m - h) * fact (m - k)"
+  consider "n = 0 \<or> k = 0 \<or> n = k" | m h where "n = Suc m" "k = Suc h" "h < m"
+    using kn by atomize_elim presburger
+  then show "fact k * fact (n - k) * (n choose k) = fact n"
+  proof cases
+    case 1
+    with kn show ?thesis by auto
+  next
+    case 2
+    note n = \<open>n = Suc m\<close>
+    note k = \<open>k = Suc h\<close>
+    note hm = \<open>h < m\<close>
+    have mn: "m < n"
+      using n by arith
+    have hm': "h \<le> m"
+      using hm by arith
+    have km: "k \<le> m"
+      using hm k n kn by arith
+    have "m - h = Suc (m - Suc h)"
+      using  k km hm by arith
+    with km k have "fact (m - h) = (m - h) * fact (m - k)"
       by simp
-    from n h th0
-    have "fact k * fact (n - k) * (n choose k) =
+    with n k have "fact k * fact (n - k) * (n choose k) =
         k * (fact h * fact (m - h) * (m choose h)) +
         (m - h) * (fact k * fact (m - k) * (m choose k))"
       by (simp add: field_simps)
     also have "\<dots> = (k + (m - h)) * fact m"
       using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
       by (simp add: field_simps)
-    finally have ?ths using h n km by simp }
-  moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (\<exists>m h. n = Suc m \<and> k = Suc h \<and> h < m)"
-    using kn by presburger
-  ultimately show ?ths by blast
+    finally show ?thesis
+      using k n km by simp
+  qed
 qed
 
 lemma binomial_fact':
@@ -414,19 +411,18 @@
 
 lemma binomial_fact:
   assumes kn: "k \<le> n"
-  shows "(of_nat (n choose k) :: 'a::field_char_0) =
-         (fact n) / (fact k * fact(n - k))"
+  shows "(of_nat (n choose k) :: 'a::field_char_0) = fact n / (fact k * fact (n - k))"
   using binomial_fact_lemma[OF kn]
   apply (simp add: field_simps)
-  by (metis mult.commute of_nat_fact of_nat_mult)
+  apply (metis mult.commute of_nat_fact of_nat_mult)
+  done
 
 lemma fact_binomial:
   assumes "k \<le> n"
   shows "fact k * of_nat (n choose k) = (fact n / fact (n - k) :: 'a::field_char_0)"
   unfolding binomial_fact [OF assms] by (simp add: field_simps)
 
-lemma choose_two:
-  "n choose 2 = n * (n - 1) div 2"
+lemma choose_two: "n choose 2 = n * (n - 1) div 2"
 proof (cases "n \<ge> 2")
   case False
   then have "n = 0 \<or> n = 1"
@@ -444,8 +440,7 @@
 qed
 
 lemma choose_row_sum: "(\<Sum>k=0..n. n choose k) = 2^n"
-  using binomial [of 1 "1" n]
-  by (simp add: numeral_2_eq_2)
+  using binomial [of 1 "1" n] by (simp add: numeral_2_eq_2)
 
 lemma sum_choose_lower: "(\<Sum>k=0..n. (r+k) choose k) = Suc (r+n) choose n"
   by (induct n) auto
@@ -454,12 +449,13 @@
   by (induct n) auto
 
 lemma choose_alternating_sum:
-  "n > 0 \<Longrightarrow> (\<Sum>i\<le>n. (-1)^i * of_nat (n choose i)) = (0 :: 'a :: comm_ring_1)"
-  using binomial_ring[of "-1 :: 'a" 1 n] by (simp add: atLeast0AtMost mult_of_nat_commute zero_power)
+  "n > 0 \<Longrightarrow> (\<Sum>i\<le>n. (-1)^i * of_nat (n choose i)) = (0 :: 'a::comm_ring_1)"
+  using binomial_ring[of "-1 :: 'a" 1 n]
+  by (simp add: atLeast0AtMost mult_of_nat_commute zero_power)
 
 lemma choose_even_sum:
   assumes "n > 0"
-  shows   "2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0) = (2 ^ n :: 'a :: comm_ring_1)"
+  shows "2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)"
 proof -
   have "2 ^ n = (\<Sum>i\<le>n. of_nat (n choose i)) + (\<Sum>i\<le>n. (-1) ^ i * of_nat (n choose i) :: 'a)"
     using choose_row_sum[of n]
@@ -473,7 +469,7 @@
 
 lemma choose_odd_sum:
   assumes "n > 0"
-  shows   "2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0) = (2 ^ n :: 'a :: comm_ring_1)"
+  shows "2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)"
 proof -
   have "2 ^ n = (\<Sum>i\<le>n. of_nat (n choose i)) - (\<Sum>i\<le>n. (-1) ^ i * of_nat (n choose i) :: 'a)"
     using choose_row_sum[of n]
@@ -490,20 +486,21 @@
 
 text\<open>NW diagonal sum property\<close>
 lemma sum_choose_diagonal:
-  assumes "m\<le>n" shows "(\<Sum>k=0..m. (n-k) choose (m-k)) = Suc n choose m"
+  assumes "m \<le> n"
+  shows "(\<Sum>k=0..m. (n - k) choose (m - k)) = Suc n choose m"
 proof -
-  have "(\<Sum>k=0..m. (n-k) choose (m-k)) = (\<Sum>k=0..m. (n-m+k) choose k)"
+  have "(\<Sum>k=0..m. (n-k) choose (m - k)) = (\<Sum>k=0..m. (n - m + k) choose k)"
     using setsum.atLeast_atMost_rev [of "\<lambda>k. (n - k) choose (m - k)" 0 m] assms
       by simp
-  also have "... = Suc (n-m+m) choose m"
+  also have "\<dots> = Suc (n - m + m) choose m"
     by (rule sum_choose_lower)
-  also have "... = Suc n choose m" using assms
-    by simp
+  also have "\<dots> = Suc n choose m"
+    using assms by simp
   finally show ?thesis .
 qed
 
 
-subsection \<open>Pochhammer's symbol : generalized rising factorial\<close>
+subsection \<open>Pochhammer's symbol: generalized rising factorial\<close>
 
 text \<open>See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"}\<close>
 
@@ -511,34 +508,30 @@
 begin
 
 definition pochhammer :: "'a \<Rightarrow> nat \<Rightarrow> 'a"
-where
-  pochhammer_setprod: "pochhammer a n = setprod (\<lambda>i. a + of_nat i) {0..<n}"
+  where pochhammer_setprod: "pochhammer a n = setprod (\<lambda>i. a + of_nat i) {0..<n}"
 
-lemma pochhammer_setprod_rev:
-  "pochhammer a n = setprod (\<lambda>i. a + of_nat (n - i)) {1..n}"
+lemma pochhammer_setprod_rev: "pochhammer a n = setprod (\<lambda>i. a + of_nat (n - i)) {1..n}"
   using setprod.atLeast_lessThan_rev_at_least_Suc_atMost [of "\<lambda>i. a + of_nat i" 0 n]
   by (simp add: pochhammer_setprod)
 
-lemma pochhammer_Suc_setprod:
-  "pochhammer a (Suc n) = setprod (\<lambda>i. a + of_nat i) {0..n}"
+lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>i. a + of_nat i) {0..n}"
   by (simp add: pochhammer_setprod atLeastLessThanSuc_atLeastAtMost)
 
-lemma pochhammer_Suc_setprod_rev:
-  "pochhammer a (Suc n) = setprod (\<lambda>i. a + of_nat (n - i)) {0..n}"
+lemma pochhammer_Suc_setprod_rev: "pochhammer a (Suc n) = setprod (\<lambda>i. a + of_nat (n - i)) {0..n}"
   by (simp add: pochhammer_setprod_rev setprod.atLeast_Suc_atMost_Suc_shift)
 
 lemma pochhammer_0 [simp]: "pochhammer a 0 = 1"
   by (simp add: pochhammer_setprod)
- 
+
 lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
   by (simp add: pochhammer_setprod lessThan_Suc)
- 
+
 lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
   by (simp add: pochhammer_setprod lessThan_Suc)
- 
+
 lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
   by (simp add: pochhammer_setprod atLeast0_lessThan_Suc ac_simps)
- 
+
 end
 
 lemma pochhammer_of_nat: "pochhammer (of_nat x) n = of_nat (pochhammer x n)"
@@ -556,14 +549,12 @@
 lemma pochhammer_fact: "fact n = pochhammer 1 n"
   by (simp add: pochhammer_setprod fact_setprod_Suc)
 
-lemma pochhammer_of_nat_eq_0_lemma:
-  assumes "k > n"
-  shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
-  using assms by (auto simp add: pochhammer_setprod)
+lemma pochhammer_of_nat_eq_0_lemma: "k > n \<Longrightarrow> pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
+  by (auto simp add: pochhammer_setprod)
 
 lemma pochhammer_of_nat_eq_0_lemma':
   assumes kn: "k \<le> n"
-  shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \<noteq> 0"
+  shows "pochhammer (- (of_nat n :: 'a::{idom,ring_char_0})) k \<noteq> 0"
 proof (cases k)
   case 0
   then show ?thesis by simp
@@ -571,12 +562,13 @@
   case (Suc h)
   then show ?thesis
     apply (simp add: pochhammer_Suc_setprod)
-    using Suc kn apply (auto simp add: algebra_simps)
+    using Suc kn
+    apply (auto simp add: algebra_simps)
     done
 qed
 
 lemma pochhammer_of_nat_eq_0_iff:
-  shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \<longleftrightarrow> k > n"
+  "pochhammer (- (of_nat n :: 'a::{idom,ring_char_0})) k = 0 \<longleftrightarrow> k > n"
   (is "?l = ?r")
   using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
     pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
@@ -609,131 +601,142 @@
 qed
 
 lemma pochhammer_minus':
-    "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
-  unfolding pochhammer_minus[where b=b]
-  unfolding mult.assoc[symmetric]
-  unfolding power_add[symmetric]
-  by simp
+  "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
+  apply (simp only: pochhammer_minus [where b = b])
+  apply (simp only: mult.assoc [symmetric])
+  apply (simp only: power_add [symmetric])
+  apply simp
+  done
 
 lemma pochhammer_same: "pochhammer (- of_nat n) n =
-    ((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * (fact n)"
+    ((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * fact n"
   unfolding pochhammer_minus
   by (simp add: of_nat_diff pochhammer_fact)
 
-lemma pochhammer_product':
-  "pochhammer z (n + m) = pochhammer z n * pochhammer (z + of_nat n) m"
-proof (induction n arbitrary: z)
+lemma pochhammer_product': "pochhammer z (n + m) = pochhammer z n * pochhammer (z + of_nat n) m"
+proof (induct n arbitrary: z)
+  case 0
+  then show ?case by simp
+next
   case (Suc n z)
   have "pochhammer z (Suc n) * pochhammer (z + of_nat (Suc n)) m =
-            z * (pochhammer (z + 1) n * pochhammer (z + 1 + of_nat n) m)"
+      z * (pochhammer (z + 1) n * pochhammer (z + 1 + of_nat n) m)"
     by (simp add: pochhammer_rec ac_simps)
   also note Suc[symmetric]
   also have "z * pochhammer (z + 1) (n + m) = pochhammer z (Suc (n + m))"
     by (subst pochhammer_rec) simp
-  finally show ?case by simp
-qed simp
+  finally show ?case
+    by simp
+qed
 
 lemma pochhammer_product:
   "m \<le> n \<Longrightarrow> pochhammer z n = pochhammer z m * pochhammer (z + of_nat m) (n - m)"
   using pochhammer_product'[of z m "n - m"] by simp
 
 lemma pochhammer_times_pochhammer_half:
-  fixes z :: "'a :: field_char_0"
+  fixes z :: "'a::field_char_0"
   shows "pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n) = (\<Prod>k=0..2*n+1. z + of_nat k / 2)"
-proof (induction n)
+proof (induct n)
+  case 0
+  then show ?case
+    by (simp add: atLeast0_atMost_Suc)
+next
   case (Suc n)
   define n' where "n' = Suc n"
   have "pochhammer z (Suc n') * pochhammer (z + 1 / 2) (Suc n') =
-          (pochhammer z n' * pochhammer (z + 1 / 2) n') *
-          ((z + of_nat n') * (z + 1/2 + of_nat n'))" (is "_ = _ * ?A")
-     by (simp_all add: pochhammer_rec' mult_ac)
+      (pochhammer z n' * pochhammer (z + 1 / 2) n') * ((z + of_nat n') * (z + 1/2 + of_nat n'))"
+    (is "_ = _ * ?A")
+    by (simp_all add: pochhammer_rec' mult_ac)
   also have "?A = (z + of_nat (Suc (2 * n + 1)) / 2) * (z + of_nat (Suc (Suc (2 * n + 1))) / 2)"
-    (is "_ = ?A") by (simp add: field_simps n'_def)
+    (is "_ = ?B")
+    by (simp add: field_simps n'_def)
   also note Suc[folded n'_def]
-  also have "(\<Prod>k=0..2 * n + 1. z + of_nat k / 2) * ?A = (\<Prod>k=0..2 * Suc n + 1. z + of_nat k / 2)"
+  also have "(\<Prod>k=0..2 * n + 1. z + of_nat k / 2) * ?B = (\<Prod>k=0..2 * Suc n + 1. z + of_nat k / 2)"
     by (simp add: atLeast0_atMost_Suc)
-  finally show ?case by (simp add: n'_def)
-qed (simp add: atLeast0_atMost_Suc)
+  finally show ?case
+    by (simp add: n'_def)
+qed
 
 lemma pochhammer_double:
-  fixes z :: "'a :: field_char_0"
+  fixes z :: "'a::field_char_0"
   shows "pochhammer (2 * z) (2 * n) = of_nat (2^(2*n)) * pochhammer z n * pochhammer (z+1/2) n"
-proof (induction n)
+proof (induct n)
+  case 0
+  then show ?case by simp
+next
   case (Suc n)
   have "pochhammer (2 * z) (2 * (Suc n)) = pochhammer (2 * z) (2 * n) *
-          (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1)"
+      (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1)"
     by (simp add: pochhammer_rec' ac_simps)
   also note Suc
   also have "of_nat (2 ^ (2 * n)) * pochhammer z n * pochhammer (z + 1/2) n *
-                 (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1) =
-             of_nat (2 ^ (2 * (Suc n))) * pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n)"
+        (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1) =
+      of_nat (2 ^ (2 * (Suc n))) * pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n)"
     by (simp add: field_simps pochhammer_rec')
   finally show ?case .
-qed simp
+qed
 
 lemma fact_double:
-  "fact (2 * n) = (2 ^ (2 * n) * pochhammer (1 / 2) n * fact n :: 'a :: field_char_0)"
+  "fact (2 * n) = (2 ^ (2 * n) * pochhammer (1 / 2) n * fact n :: 'a::field_char_0)"
   using pochhammer_double[of "1/2::'a" n] by (simp add: pochhammer_fact)
 
-lemma pochhammer_absorb_comp:
-  "((r :: 'a :: comm_ring_1) - of_nat k) * pochhammer (- r) k = r * pochhammer (-r + 1) k"
+lemma pochhammer_absorb_comp: "(r - of_nat k) * pochhammer (- r) k = r * pochhammer (-r + 1) k"
   (is "?lhs = ?rhs")
+  for r :: "'a::comm_ring_1"
 proof -
-  have "?lhs = -pochhammer (-r) (Suc k)" by (subst pochhammer_rec') (simp add: algebra_simps)
-  also have "\<dots> = ?rhs" by (subst pochhammer_rec) simp
+  have "?lhs = - pochhammer (- r) (Suc k)"
+    by (subst pochhammer_rec') (simp add: algebra_simps)
+  also have "\<dots> = ?rhs"
+    by (subst pochhammer_rec) simp
   finally show ?thesis .
 qed
 
 
 subsection \<open>Generalized binomial coefficients\<close>
 
-definition gbinomial :: "'a :: {semidom_divide, semiring_char_0} \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
-where
-  gbinomial_setprod_rev: "a gchoose n = setprod (\<lambda>i. a - of_nat i) {0..<n} div fact n"
+definition gbinomial :: "'a::{semidom_divide,semiring_char_0} \<Rightarrow> nat \<Rightarrow> 'a"  (infixl "gchoose" 65)
+  where gbinomial_setprod_rev: "a gchoose n = setprod (\<lambda>i. a - of_nat i) {0..<n} div fact n"
 
 lemma gbinomial_0 [simp]:
   "a gchoose 0 = 1"
   "0 gchoose (Suc n) = 0"
   by (simp_all add: gbinomial_setprod_rev setprod.atLeast0_lessThan_Suc_shift)
 
-lemma gbinomial_Suc:
-  "a gchoose (Suc k) = setprod (\<lambda>i. a - of_nat i) {0..k} div fact (Suc k)"
+lemma gbinomial_Suc: "a gchoose (Suc k) = setprod (\<lambda>i. a - of_nat i) {0..k} div fact (Suc k)"
   by (simp add: gbinomial_setprod_rev atLeastLessThanSuc_atLeastAtMost)
 
-lemma gbinomial_mult_fact:
-  fixes a :: "'a::field_char_0"
-  shows
-    "fact n * (a gchoose n) = (\<Prod>i = 0..<n. a - of_nat i)"
+lemma gbinomial_mult_fact: "fact n * (a gchoose n) = (\<Prod>i = 0..<n. a - of_nat i)"
+  for a :: "'a::field_char_0"
   by (simp_all add: gbinomial_setprod_rev field_simps)
 
-lemma gbinomial_mult_fact':
-  fixes a :: "'a::field_char_0"
-  shows
-    "(a gchoose n) * fact n = (\<Prod>i = 0..<n. a - of_nat i)"
+lemma gbinomial_mult_fact': "(a gchoose n) * fact n = (\<Prod>i = 0..<n. a - of_nat i)"
+  for a :: "'a::field_char_0"
   using gbinomial_mult_fact [of n a] by (simp add: ac_simps)
 
-lemma gbinomial_pochhammer:
-  fixes a :: "'a::field_char_0"
-  shows "a gchoose n = (- 1) ^ n * pochhammer (- a) n / fact n"
+lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / fact n"
+  for a :: "'a::field_char_0"
   by (cases n)
-    (simp_all add: pochhammer_minus, simp_all add: gbinomial_setprod_rev pochhammer_setprod_rev
-      power_mult_distrib [symmetric] atLeastLessThanSuc_atLeastAtMost setprod.atLeast_Suc_atMost_Suc_shift of_nat_diff)
+    (simp_all add: pochhammer_minus,
+     simp_all add: gbinomial_setprod_rev pochhammer_setprod_rev
+       power_mult_distrib [symmetric] atLeastLessThanSuc_atLeastAtMost
+       setprod.atLeast_Suc_atMost_Suc_shift of_nat_diff)
 
-lemma gbinomial_pochhammer':
-  fixes s :: "'a::field_char_0"
-  shows "s gchoose n = pochhammer (s - of_nat n + 1) n / fact n"
+lemma gbinomial_pochhammer': "s gchoose n = pochhammer (s - of_nat n + 1) n / fact n"
+  for s :: "'a::field_char_0"
 proof -
   have "s gchoose n = ((-1)^n * (-1)^n) * pochhammer (s - of_nat n + 1) n / fact n"
     by (simp add: gbinomial_pochhammer pochhammer_minus mult_ac)
-  also have "(-1 :: 'a)^n * (-1)^n = 1" by (subst power_add [symmetric]) simp
-  finally show ?thesis by simp
+  also have "(-1 :: 'a)^n * (-1)^n = 1"
+    by (subst power_add [symmetric]) simp
+  finally show ?thesis
+    by simp
 qed
 
-lemma gbinomial_binomial:
-  "n gchoose k = n choose k"
+lemma gbinomial_binomial: "n gchoose k = n choose k"
 proof (cases "k \<le> n")
   case False
-  then have "n < k" by (simp add: not_le)
+  then have "n < k"
+    by (simp add: not_le)
   then have "0 \<in> (op - n) ` {0..<k}"
     by auto
   then have "setprod (op - n) {0..<k} = 0"
@@ -747,47 +750,44 @@
   then have "\<Prod>(op - n ` {0..<k}) = setprod (op - n) {0..<k}"
     by (auto dest: setprod.reindex)
   also have "op - n ` {0..<k} = {Suc (n - k)..n}"
-    using True by (auto simp add: image_def Bex_def) arith
+    using True by (auto simp add: image_def Bex_def) presburger  (* FIXME slow *)
   finally have *: "setprod (\<lambda>q. n - q) {0..<k} = \<Prod>{Suc (n - k)..n}" ..
-  from True have "(n choose k) = fact n div (fact k * fact (n - k))"
+  from True have "n choose k = fact n div (fact k * fact (n - k))"
     by (rule binomial_fact')
   with * show ?thesis
     by (simp add: gbinomial_setprod_rev mult.commute [of "fact k"] div_mult2_eq fact_div_fact)
 qed
 
-lemma of_nat_gbinomial:
-  "of_nat (n gchoose k) = (of_nat n gchoose k :: 'a::field_char_0)"
+lemma of_nat_gbinomial: "of_nat (n gchoose k) = (of_nat n gchoose k :: 'a::field_char_0)"
 proof (cases "k \<le> n")
-  case False then show ?thesis
+  case False
+  then show ?thesis
     by (simp add: not_le gbinomial_binomial binomial_eq_0 gbinomial_setprod_rev)
 next
-  case True 
-  moreover define m where "m = n - k"
-  ultimately have n: "n = m + k"
+  case True
+  define m where "m = n - k"
+  with True have n: "n = m + k"
     by arith
   from n have "fact n = ((\<Prod>i = 0..<m + k. of_nat (m + k - i) ):: 'a)"
     by (simp add: fact_setprod_rev)
   also have "\<dots> = ((\<Prod>i\<in>{0..<k} \<union> {k..<m + k}. of_nat (m + k - i)) :: 'a)"
     by (simp add: ivl_disj_un)
-  finally have 
-    "fact n = (fact m * (\<Prod>i = 0..<k. of_nat m + of_nat k - of_nat i) :: 'a)"
+  finally have "fact n = (fact m * (\<Prod>i = 0..<k. of_nat m + of_nat k - of_nat i) :: 'a)"
     using setprod_shift_bounds_nat_ivl [of "\<lambda>i. of_nat (m + k - i) :: 'a" 0 k m]
     by (simp add: fact_setprod_rev [of m] setprod.union_disjoint of_nat_diff)
-  then have "fact n / fact (n - k) =
-    ((\<Prod>i = 0..<k. of_nat n - of_nat i) :: 'a)"
+  then have "fact n / fact (n - k) = ((\<Prod>i = 0..<k. of_nat n - of_nat i) :: 'a)"
     by (simp add: n)
   with True have "fact k * of_nat (n gchoose k) = (fact k * (of_nat n gchoose k) :: 'a)"
-    by (simp only: gbinomial_mult_fact [of k "of_nat n"]
-      gbinomial_binomial [of n k]
-      fact_binomial)
-  then show ?thesis by simp
+    by (simp only: gbinomial_mult_fact [of k "of_nat n"] gbinomial_binomial [of n k] fact_binomial)
+  then show ?thesis
+    by simp
 qed
 
-lemma binomial_gbinomial:
-  "of_nat (n choose k) = (of_nat n gchoose k :: 'a::field_char_0)"
+lemma binomial_gbinomial: "of_nat (n choose k) = (of_nat n gchoose k :: 'a::field_char_0)"
   by (simp add: gbinomial_binomial [symmetric] of_nat_gbinomial)
 
-setup \<open>Sign.add_const_constraint (@{const_name gbinomial}, SOME @{typ "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a"})\<close>
+setup
+  \<open>Sign.add_const_constraint (@{const_name gbinomial}, SOME @{typ "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a"})\<close>
 
 lemma gbinomial_1[simp]: "a gchoose 1 = a"
   by (simp add: gbinomial_setprod_rev lessThan_Suc)
@@ -796,29 +796,27 @@
   by (simp add: gbinomial_setprod_rev lessThan_Suc)
 
 lemma gbinomial_mult_1:
-  fixes a :: "'a :: field_char_0"
-  shows "a * (a gchoose n) =
-    of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"  (is "?l = ?r")
+  fixes a :: "'a::field_char_0"
+  shows "a * (a gchoose n) = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
+  (is "?l = ?r")
 proof -
   have "?r = ((- 1) ^n * pochhammer (- a) n / (fact n)) * (of_nat n - (- a + of_nat n))"
-    unfolding gbinomial_pochhammer
-      pochhammer_Suc right_diff_distrib power_Suc
+    apply (simp only: gbinomial_pochhammer pochhammer_Suc right_diff_distrib power_Suc)
     apply (simp del: of_nat_Suc fact_Suc)
     apply (auto simp add: field_simps simp del: of_nat_Suc)
     done
-  also have "\<dots> = ?l" unfolding gbinomial_pochhammer
-    by (simp add: field_simps)
+  also have "\<dots> = ?l"
+    by (simp add: field_simps gbinomial_pochhammer)
   finally show ?thesis ..
 qed
 
 lemma gbinomial_mult_1':
-  fixes a :: "'a :: field_char_0"
-  shows "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
+  "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
+  for a :: "'a::field_char_0"
   by (simp add: mult.commute gbinomial_mult_1)
 
-lemma gbinomial_Suc_Suc:
-  fixes a :: "'a :: field_char_0"
-  shows "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
+lemma gbinomial_Suc_Suc: "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
+  for a :: "'a::field_char_0"
 proof (cases k)
   case 0
   then show ?thesis by simp
@@ -830,24 +828,25 @@
       apply (auto simp add: image_Suc_atMost)
     done
   have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) =
-        (a gchoose Suc h) * (fact (Suc (Suc h))) +
-        (a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))"
+      (a gchoose Suc h) * (fact (Suc (Suc h))) +
+      (a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))"
     by (simp add: Suc field_simps del: fact_Suc)
-  also have "... = (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) +
-                   (\<Prod>i=0..Suc h. a - of_nat i)"
+  also have "\<dots> =
+    (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) + (\<Prod>i=0..Suc h. a - of_nat i)"
     apply (simp del: fact_Suc add: gbinomial_mult_fact field_simps mult.left_commute [of _ "2"])
-    apply (simp del: fact_Suc add: fact_Suc [of "Suc h"] field_simps gbinomial_mult_fact mult.left_commute [of _ "2"] atLeastLessThanSuc_atLeastAtMost)
+    apply (simp del: fact_Suc add: fact_Suc [of "Suc h"] field_simps gbinomial_mult_fact
+      mult.left_commute [of _ "2"] atLeastLessThanSuc_atLeastAtMost)
     done
-  also have "... = (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) +
-                   (\<Prod>i=0..Suc h. a - of_nat i)"
+  also have "\<dots> =
+    (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) + (\<Prod>i=0..Suc h. a - of_nat i)"
     by (simp only: fact_Suc mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult)
-  also have "... =  of_nat (Suc (Suc h)) * (\<Prod>i=0..h. a - of_nat i) +
-                    (\<Prod>i=0..Suc h. a - of_nat i)"
+  also have "\<dots> =
+    of_nat (Suc (Suc h)) * (\<Prod>i=0..h. a - of_nat i) + (\<Prod>i=0..Suc h. a - of_nat i)"
     unfolding gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost by auto
-  also have "... = (\<Prod>i=0..Suc h. a - of_nat i) +
-                   (of_nat h * (\<Prod>i=0..h. a - of_nat i) + 2 * (\<Prod>i=0..h. a - of_nat i))"
+  also have "\<dots> =
+    (\<Prod>i=0..Suc h. a - of_nat i) + (of_nat h * (\<Prod>i=0..h. a - of_nat i) + 2 * (\<Prod>i=0..h. a - of_nat i))"
     by (simp add: field_simps)
-  also have "... =
+  also have "\<dots> =
     ((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0..Suc h}. a - of_nat i)"
     unfolding gbinomial_mult_fact'
     by (simp add: comm_semiring_class.distrib field_simps Suc atLeastLessThanSuc_atLeastAtMost)
@@ -858,132 +857,147 @@
     using eq0
     by (simp add: Suc setprod.atLeast0_atMost_Suc_shift)
   also have "\<dots> = (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
-    unfolding gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost ..
+    by (simp only: gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost)
   finally show ?thesis
     using fact_nonzero [of "Suc k"] by auto
 qed
 
-lemma gbinomial_reduce_nat:
-  fixes a :: "'a :: field_char_0"
-  shows "0 < k \<Longrightarrow> a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)"
+lemma gbinomial_reduce_nat: "0 < k \<Longrightarrow> a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)"
+  for a :: "'a::field_char_0"
   by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc)
 
 lemma gchoose_row_sum_weighted:
-  fixes r :: "'a::field_char_0"
-  shows "(\<Sum>k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))"
-proof (induct m)
-  case 0 show ?case by simp
-next
-  case (Suc m)
-  from Suc show ?case
-    by (simp add: field_simps distrib gbinomial_mult_1)
-qed
+  "(\<Sum>k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))"
+  for r :: "'a::field_char_0"
+  by (induct m) (simp_all add: field_simps distrib gbinomial_mult_1)
 
 lemma binomial_symmetric:
   assumes kn: "k \<le> n"
   shows "n choose k = n choose (n - k)"
-proof-
-  from kn have kn': "n - k \<le> n" by arith
+proof -
+  have kn': "n - k \<le> n"
+    using kn by arith
   from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
-  have "fact k * fact (n - k) * (n choose k) =
-    fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp
-  then show ?thesis using kn by simp
+  have "fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))"
+    by simp
+  then show ?thesis
+    using kn by simp
 qed
 
 lemma choose_rising_sum:
   "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))"
   "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose m)"
 proof -
-  show "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))" by (induction m) simp_all
-  also have "... = ((n + m + 1) choose m)" by (subst binomial_symmetric) simp_all
-  finally show "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose m)" .
+  show "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))"
+    by (induct m) simp_all
+  also have "\<dots> = (n + m + 1) choose m"
+    by (subst binomial_symmetric) simp_all
+  finally show "(\<Sum>j\<le>m. ((n + j) choose n)) = (n + m + 1) choose m" .
 qed
 
-lemma choose_linear_sum:
-  "(\<Sum>i\<le>n. i * (n choose i)) = n * 2 ^ (n - 1)"
+lemma choose_linear_sum: "(\<Sum>i\<le>n. i * (n choose i)) = n * 2 ^ (n - 1)"
 proof (cases n)
+  case 0
+  then show ?thesis by simp
+next
   case (Suc m)
-  have "(\<Sum>i\<le>n. i * (n choose i)) = (\<Sum>i\<le>Suc m. i * (Suc m choose i))" by (simp add: Suc)
-  also have "... = Suc m * 2 ^ m"
+  have "(\<Sum>i\<le>n. i * (n choose i)) = (\<Sum>i\<le>Suc m. i * (Suc m choose i))"
+    by (simp add: Suc)
+  also have "\<dots> = Suc m * 2 ^ m"
     by (simp only: setsum_atMost_Suc_shift Suc_times_binomial setsum_right_distrib[symmetric])
        (simp add: choose_row_sum')
-  finally show ?thesis using Suc by simp
-qed simp_all
+  finally show ?thesis
+    using Suc by simp
+qed
 
 lemma choose_alternating_linear_sum:
   assumes "n \<noteq> 1"
-  shows   "(\<Sum>i\<le>n. (-1)^i * of_nat i * of_nat (n choose i) :: 'a :: comm_ring_1) = 0"
+  shows "(\<Sum>i\<le>n. (-1)^i * of_nat i * of_nat (n choose i) :: 'a::comm_ring_1) = 0"
 proof (cases n)
+  case 0
+  then show ?thesis by simp
+next
   case (Suc m)
-  with assms have "m > 0" by simp
+  with assms have "m > 0"
+    by simp
   have "(\<Sum>i\<le>n. (-1) ^ i * of_nat i * of_nat (n choose i) :: 'a) =
-            (\<Sum>i\<le>Suc m. (-1) ^ i * of_nat i * of_nat (Suc m choose i))" by (simp add: Suc)
-  also have "... = (\<Sum>i\<le>m. (-1) ^ (Suc i) * of_nat (Suc i * (Suc m choose Suc i)))"
+      (\<Sum>i\<le>Suc m. (-1) ^ i * of_nat i * of_nat (Suc m choose i))"
+    by (simp add: Suc)
+  also have "\<dots> = (\<Sum>i\<le>m. (-1) ^ (Suc i) * of_nat (Suc i * (Suc m choose Suc i)))"
     by (simp only: setsum_atMost_Suc_shift setsum_right_distrib[symmetric] mult_ac of_nat_mult) simp
-  also have "... = -of_nat (Suc m) * (\<Sum>i\<le>m. (-1) ^ i * of_nat ((m choose i)))"
+  also have "\<dots> = - of_nat (Suc m) * (\<Sum>i\<le>m. (-1) ^ i * of_nat (m choose i))"
     by (subst setsum_right_distrib, rule setsum.cong[OF refl], subst Suc_times_binomial)
        (simp add: algebra_simps)
   also have "(\<Sum>i\<le>m. (-1 :: 'a) ^ i * of_nat ((m choose i))) = 0"
     using choose_alternating_sum[OF \<open>m > 0\<close>] by simp
-  finally show ?thesis by simp
-qed simp
+  finally show ?thesis
+    by simp
+qed
 
-lemma vandermonde:
-  "(\<Sum>k\<le>r. (m choose k) * (n choose (r - k))) = (m + n) choose r"
-proof (induction n arbitrary: r)
+lemma vandermonde: "(\<Sum>k\<le>r. (m choose k) * (n choose (r - k))) = (m + n) choose r"
+proof (induct n arbitrary: r)
   case 0
   have "(\<Sum>k\<le>r. (m choose k) * (0 choose (r - k))) = (\<Sum>k\<le>r. if k = r then (m choose k) else 0)"
     by (intro setsum.cong) simp_all
-  also have "... = m choose r" by (simp add: setsum.delta)
-  finally show ?case by simp
+  also have "\<dots> = m choose r"
+    by (simp add: setsum.delta)
+  finally show ?case
+    by simp
 next
   case (Suc n r)
-  show ?case by (cases r) (simp_all add: Suc [symmetric] algebra_simps setsum.distrib Suc_diff_le)
+  show ?case
+    by (cases r) (simp_all add: Suc [symmetric] algebra_simps setsum.distrib Suc_diff_le)
 qed
 
-lemma choose_square_sum:
-  "(\<Sum>k\<le>n. (n choose k)^2) = ((2*n) choose n)"
-  using vandermonde[of n n n] by (simp add: power2_eq_square mult_2 binomial_symmetric [symmetric])
+lemma choose_square_sum: "(\<Sum>k\<le>n. (n choose k)^2) = ((2*n) choose n)"
+  using vandermonde[of n n n]
+  by (simp add: power2_eq_square mult_2 binomial_symmetric [symmetric])
 
 lemma pochhammer_binomial_sum:
-  fixes a b :: "'a :: comm_ring_1"
+  fixes a b :: "'a::comm_ring_1"
   shows "pochhammer (a + b) n = (\<Sum>k\<le>n. of_nat (n choose k) * pochhammer a k * pochhammer b (n - k))"
 proof (induction n arbitrary: a b)
+  case 0
+  then show ?case by simp
+next
   case (Suc n a b)
   have "(\<Sum>k\<le>Suc n. of_nat (Suc n choose k) * pochhammer a k * pochhammer b (Suc n - k)) =
-            (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
-            ((\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
-            pochhammer b (Suc n))"
+      (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
+      ((\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
+      pochhammer b (Suc n))"
     by (subst setsum_atMost_Suc_shift) (simp add: ring_distribs setsum.distrib)
   also have "(\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) =
-               a * pochhammer ((a + 1) + b) n"
+      a * pochhammer ((a + 1) + b) n"
     by (subst Suc) (simp add: setsum_right_distrib pochhammer_rec mult_ac)
-  also have "(\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) + pochhammer b (Suc n) =
-               (\<Sum>i=0..Suc n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
-    by (subst setsum_head_Suc, simp, subst setsum_shift_bounds_cl_Suc_ivl) (simp add: atLeast0AtMost)
-  also have "... = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
+  also have "(\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
+        pochhammer b (Suc n) =
+      (\<Sum>i=0..Suc n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
+    apply (subst setsum_head_Suc)
+    apply simp
+    apply (subst setsum_shift_bounds_cl_Suc_ivl)
+    apply (simp add: atLeast0AtMost)
+    done
+  also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
     using Suc by (intro setsum.mono_neutral_right) (auto simp: not_le binomial_eq_0)
-  also have "... = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc (n - i)))"
+  also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc (n - i)))"
     by (intro setsum.cong) (simp_all add: Suc_diff_le)
-  also have "... = b * pochhammer (a + (b + 1)) n"
+  also have "\<dots> = b * pochhammer (a + (b + 1)) n"
     by (subst Suc) (simp add: setsum_right_distrib mult_ac pochhammer_rec)
   also have "a * pochhammer ((a + 1) + b) n + b * pochhammer (a + (b + 1)) n =
-               pochhammer (a + b) (Suc n)" by (simp add: pochhammer_rec algebra_simps)
+      pochhammer (a + b) (Suc n)"
+    by (simp add: pochhammer_rec algebra_simps)
   finally show ?case ..
-qed simp_all
-
+qed
 
-text\<open>Contributed by Manuel Eberl, generalised by LCP.
-  Alternative definition of the binomial coefficient as @{term "\<Prod>i<k. (n - i) / (k - i)"}\<close>
-lemma gbinomial_altdef_of_nat:
-  fixes k :: nat
-    and x :: "'a :: field_char_0"
-  shows "x gchoose k = (\<Prod>i = 0..<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
+text \<open>Contributed by Manuel Eberl, generalised by LCP.
+  Alternative definition of the binomial coefficient as @{term "\<Prod>i<k. (n - i) / (k - i)"}.\<close>
+lemma gbinomial_altdef_of_nat: "x gchoose k = (\<Prod>i = 0..<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
+  for k :: nat and x :: "'a::field_char_0"
   by (simp add: setprod_dividef gbinomial_setprod_rev fact_setprod_rev)
 
 lemma gbinomial_ge_n_over_k_pow_k:
   fixes k :: nat
-    and x :: "'a :: linordered_field"
+    and x :: "'a::linordered_field"
   assumes "of_nat k \<le> x"
   shows "(x / of_nat k :: 'a) ^ k \<le> x gchoose k"
 proof -
@@ -991,22 +1005,26 @@
     using assms of_nat_0_le_iff order_trans by blast
   have "(x / of_nat k :: 'a) ^ k = (\<Prod>i = 0..<k. x / of_nat k :: 'a)"
     by (simp add: setprod_constant)
-  also have "\<dots> \<le> x gchoose k"
+  also have "\<dots> \<le> x gchoose k" (* FIXME *)
     unfolding gbinomial_altdef_of_nat
-  proof (safe intro!: setprod_mono, simp_all) -- \<open>FIXME\<close>
-    fix i :: nat
-    assume ik: "i < k"
-    from assms have "x * of_nat i \<ge> of_nat (i * k)"
-      by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult)
-    then have "x * of_nat k - x * of_nat i \<le> x * of_nat k - of_nat (i * k)" by arith
-    then have "x * of_nat (k - i) \<le> (x - of_nat i) * of_nat k"
-      using ik
-      by (simp add: algebra_simps zero_less_mult_iff of_nat_diff)
-    then have "x * of_nat (k - i) \<le> (x - of_nat i) * (of_nat k :: 'a)"
-      unfolding of_nat_mult[symmetric] of_nat_le_iff .
-    with assms show "x / of_nat k \<le> (x - of_nat i) / (of_nat (k - i) :: 'a)"
-      using \<open>i < k\<close> by (simp add: field_simps)
-  qed (simp add: x zero_le_divide_iff)
+    apply (safe intro!: setprod_mono)
+    apply simp_all
+    prefer 2
+    subgoal premises for i
+    proof -
+      from assms have "x * of_nat i \<ge> of_nat (i * k)"
+        by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult)
+      then have "x * of_nat k - x * of_nat i \<le> x * of_nat k - of_nat (i * k)"
+        by arith
+      then have "x * of_nat (k - i) \<le> (x - of_nat i) * of_nat k"
+        using \<open>i < k\<close> by (simp add: algebra_simps zero_less_mult_iff of_nat_diff)
+      then have "x * of_nat (k - i) \<le> (x - of_nat i) * (of_nat k :: 'a)"
+        by (simp only: of_nat_mult[symmetric] of_nat_le_iff)
+      with assms show ?thesis
+        using \<open>i < k\<close> by (simp add: field_simps)
+    qed
+    apply (simp add: x zero_le_divide_iff)
+    done
   finally show ?thesis .
 qed
 
@@ -1016,33 +1034,36 @@
 lemma gbinomial_minus: "((-a) gchoose b) = (-1) ^ b * ((a + of_nat b - 1) gchoose b)"
   by (subst gbinomial_negated_upper) (simp add: add_ac)
 
-lemma Suc_times_gbinomial:
-  "of_nat (Suc b) * ((a + 1) gchoose (Suc b)) = (a + 1) * (a gchoose b)"
+lemma Suc_times_gbinomial: "of_nat (Suc b) * ((a + 1) gchoose (Suc b)) = (a + 1) * (a gchoose b)"
 proof (cases b)
+  case 0
+  then show ?thesis by simp
+next
   case (Suc b)
-  hence "((a + 1) gchoose (Suc (Suc b))) =
-             (\<Prod>i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)"
+  then have "((a + 1) gchoose (Suc (Suc b))) = (\<Prod>i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)"
     by (simp add: field_simps gbinomial_setprod_rev atLeastLessThanSuc_atLeastAtMost)
   also have "(\<Prod>i = 0..Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
     by (simp add: setprod.atLeast0_atMost_Suc_shift)
-  also have "... / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
+  also have "\<dots> / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
     by (simp_all add: gbinomial_setprod_rev atLeastLessThanSuc_atLeastAtMost)
   finally show ?thesis by (simp add: Suc field_simps del: of_nat_Suc)
-qed simp
+qed
 
-lemma gbinomial_factors:
-  "((a + 1) gchoose (Suc b)) = (a + 1) / of_nat (Suc b) * (a gchoose b)"
+lemma gbinomial_factors: "((a + 1) gchoose (Suc b)) = (a + 1) / of_nat (Suc b) * (a gchoose b)"
 proof (cases b)
+  case 0
+  then show ?thesis by simp
+next
   case (Suc b)
-  hence "((a + 1) gchoose (Suc (Suc b))) =
-             (\<Prod>i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)"
+  then have "((a + 1) gchoose (Suc (Suc b))) = (\<Prod>i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)"
     by (simp add: field_simps gbinomial_setprod_rev atLeastLessThanSuc_atLeastAtMost)
   also have "(\<Prod>i = 0 .. Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
     by (simp add: setprod.atLeast0_atMost_Suc_shift)
-  also have "... / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
+  also have "\<dots> / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
     by (simp_all add: gbinomial_setprod_rev atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
-  finally show ?thesis by (simp add: Suc)
-qed simp
+  finally show ?thesis
+    by (simp add: Suc)
+qed
 
 lemma gbinomial_rec: "((r + 1) gchoose (Suc k)) = (r gchoose k) * ((r + 1) / of_nat (Suc k))"
   using gbinomial_mult_1[of r k]
@@ -1052,41 +1073,45 @@
   using binomial_symmetric[of k n] by (simp add: binomial_gbinomial [symmetric])
 
 
-text \<open>The absorption identity (equation 5.5 \cite[p.~157]{GKP}):\[
+text \<open>The absorption identity (equation 5.5 \cite[p.~157]{GKP}):
+\[
 {r \choose k} = \frac{r}{k}{r - 1 \choose k - 1},\quad \textnormal{integer } k \neq 0.
 \]\<close>
-lemma gbinomial_absorption':
-  "k > 0 \<Longrightarrow> (r gchoose k) = (r / of_nat(k)) * (r - 1 gchoose (k - 1))"
+lemma gbinomial_absorption': "k > 0 \<Longrightarrow> r gchoose k = (r / of_nat k) * (r - 1 gchoose (k - 1))"
   using gbinomial_rec[of "r - 1" "k - 1"]
   by (simp_all add: gbinomial_rec field_simps del: of_nat_Suc)
 
 text \<open>The absorption identity is written in the following form to avoid
 division by $k$ (the lower index) and therefore remove the $k \neq 0$
-restriction\cite[p.~157]{GKP}:\[
+restriction\cite[p.~157]{GKP}:
+\[
 k{r \choose k} = r{r - 1 \choose k - 1}, \quad \textnormal{integer } k.
 \]\<close>
-lemma gbinomial_absorption:
-  "of_nat (Suc k) * (r gchoose Suc k) = r * ((r - 1) gchoose k)"
+lemma gbinomial_absorption: "of_nat (Suc k) * (r gchoose Suc k) = r * ((r - 1) gchoose k)"
   using gbinomial_absorption'[of "Suc k" r] by (simp add: field_simps del: of_nat_Suc)
 
 text \<open>The absorption identity for natural number binomial coefficients:\<close>
-lemma binomial_absorption:
-  "Suc k * (n choose Suc k) = n * ((n - 1) choose k)"
+lemma binomial_absorption: "Suc k * (n choose Suc k) = n * ((n - 1) choose k)"
   by (cases n) (simp_all add: binomial_eq_0 Suc_times_binomial del: binomial_Suc_Suc mult_Suc)
 
 text \<open>The absorption companion identity for natural number coefficients,
-following the proof by GKP \cite[p.~157]{GKP}:\<close>
-lemma binomial_absorb_comp:
-  "(n - k) * (n choose k) = n * ((n - 1) choose k)" (is "?lhs = ?rhs")
+  following the proof by GKP \cite[p.~157]{GKP}:\<close>
+lemma binomial_absorb_comp: "(n - k) * (n choose k) = n * ((n - 1) choose k)"
+  (is "?lhs = ?rhs")
 proof (cases "n \<le> k")
+  case True
+  then show ?thesis by auto
+next
   case False
   then have "?rhs = Suc ((n - 1) - k) * (n choose Suc ((n - 1) - k))"
     using binomial_symmetric[of k "n - 1"] binomial_absorption[of "(n - 1) - k" n]
     by simp
-  also from False have "Suc ((n - 1) - k) = n - k" by simp
-  also from False have "n choose \<dots> = n choose k" by (intro binomial_symmetric [symmetric]) simp_all
+  also have "Suc ((n - 1) - k) = n - k"
+    using False by simp
+  also have "n choose \<dots> = n choose k"
+    using False by (intro binomial_symmetric [symmetric]) simp_all
   finally show ?thesis ..
-qed auto
+qed
 
 text \<open>The generalised absorption companion identity:\<close>
 lemma gbinomial_absorb_comp: "(r - of_nat k) * (r gchoose k) = r * ((r - 1) gchoose k)"
@@ -1100,35 +1125,45 @@
   "0 < n \<Longrightarrow> n choose (Suc k) = ((n - 1) choose (Suc k)) + ((n - 1) choose k)"
   by (subst choose_reduce_nat) simp_all
 
-
 text \<open>
   Equation 5.9 of the reference material \cite[p.~159]{GKP} is a useful
-  summation formula, operating on both indices:\[
-  \sum\limits_{k \leq n}{r + k \choose k} = {r + n + 1 \choose n},
+  summation formula, operating on both indices:
+  \[
+   \sum\limits_{k \leq n}{r + k \choose k} = {r + n + 1 \choose n},
    \quad \textnormal{integer } n.
   \]
 \<close>
-lemma gbinomial_parallel_sum:
-"(\<Sum>k\<le>n. (r + of_nat k) gchoose k) = (r + of_nat n + 1) gchoose n"
-proof (induction n)
+lemma gbinomial_parallel_sum: "(\<Sum>k\<le>n. (r + of_nat k) gchoose k) = (r + of_nat n + 1) gchoose n"
+proof (induct n)
+  case 0
+  then show ?case by simp
+next
   case (Suc m)
-  thus ?case using gbinomial_Suc_Suc[of "(r + of_nat m + 1)" m] by (simp add: add_ac)
-qed auto
+  then show ?case
+    using gbinomial_Suc_Suc[of "(r + of_nat m + 1)" m]
+    by (simp add: add_ac)
+qed
+
 
 subsubsection \<open>Summation on the upper index\<close>
+
 text \<open>
   Another summation formula is equation 5.10 of the reference material \cite[p.~160]{GKP},
   aptly named \emph{summation on the upper index}:\[\sum_{0 \leq k \leq n} {k \choose m} =
   {n + 1 \choose m + 1}, \quad \textnormal{integers } m, n \geq 0.\]
 \<close>
 lemma gbinomial_sum_up_index:
-  "(\<Sum>k = 0..n. (of_nat k gchoose m) :: 'a :: field_char_0) = (of_nat n + 1) gchoose (m + 1)"
-proof (induction n)
+  "(\<Sum>k = 0..n. (of_nat k gchoose m) :: 'a::field_char_0) = (of_nat n + 1) gchoose (m + 1)"
+proof (induct n)
   case 0
-  show ?case using gbinomial_Suc_Suc[of 0 m] by (cases m) auto
+  show ?case
+    using gbinomial_Suc_Suc[of 0 m]
+    by (cases m) auto
 next
   case (Suc n)
-  thus ?case using gbinomial_Suc_Suc[of "of_nat (Suc n) :: 'a" m] by (simp add: add_ac)
+  then show ?case
+    using gbinomial_Suc_Suc[of "of_nat (Suc n) :: 'a" m]
+    by (simp add: add_ac)
 qed
 
 lemma gbinomial_index_swap:
@@ -1137,92 +1172,114 @@
 proof -
   have "?lhs = (of_nat (m + n) gchoose m)"
     by (subst gbinomial_negated_upper) (simp add: power_mult_distrib [symmetric])
-  also have "\<dots> = (of_nat (m + n) gchoose n)" by (subst gbinomial_of_nat_symmetric) simp_all
-  also have "\<dots> = ?rhs" by (subst gbinomial_negated_upper) simp
+  also have "\<dots> = (of_nat (m + n) gchoose n)"
+    by (subst gbinomial_of_nat_symmetric) simp_all
+  also have "\<dots> = ?rhs"
+    by (subst gbinomial_negated_upper) simp
   finally show ?thesis .
 qed
 
-lemma gbinomial_sum_lower_neg:
-  "(\<Sum>k\<le>m. (r gchoose k) * (- 1) ^ k) = (- 1) ^ m * (r - 1 gchoose m)" (is "?lhs = ?rhs")
+lemma gbinomial_sum_lower_neg: "(\<Sum>k\<le>m. (r gchoose k) * (- 1) ^ k) = (- 1) ^ m * (r - 1 gchoose m)"
+  (is "?lhs = ?rhs")
 proof -
   have "?lhs = (\<Sum>k\<le>m. -(r + 1) + of_nat k gchoose k)"
     by (intro setsum.cong[OF refl]) (subst gbinomial_negated_upper, simp add: power_mult_distrib)
-  also have "\<dots>  = -r + of_nat m gchoose m" by (subst gbinomial_parallel_sum) simp
-  also have "\<dots> = ?rhs" by (subst gbinomial_negated_upper) (simp add: power_mult_distrib)
+  also have "\<dots>  = - r + of_nat m gchoose m"
+    by (subst gbinomial_parallel_sum) simp
+  also have "\<dots> = ?rhs"
+    by (subst gbinomial_negated_upper) (simp add: power_mult_distrib)
   finally show ?thesis .
 qed
 
 lemma gbinomial_partial_row_sum:
-"(\<Sum>k\<le>m. (r gchoose k) * ((r / 2) - of_nat k)) = ((of_nat m + 1)/2) * (r gchoose (m + 1))"
-proof (induction m)
+  "(\<Sum>k\<le>m. (r gchoose k) * ((r / 2) - of_nat k)) = ((of_nat m + 1)/2) * (r gchoose (m + 1))"
+proof (induct m)
+  case 0
+  then show ?case by simp
+next
   case (Suc mm)
-  hence "(\<Sum>k\<le>Suc mm. (r gchoose k) * (r / 2 - of_nat k)) =
-             (r - of_nat (Suc mm)) * (r gchoose Suc mm) / 2" by (simp add: field_simps)
-  also have "\<dots> = r * (r - 1 gchoose Suc mm) / 2" by (subst gbinomial_absorb_comp) (rule refl)
+  then have "(\<Sum>k\<le>Suc mm. (r gchoose k) * (r / 2 - of_nat k)) =
+      (r - of_nat (Suc mm)) * (r gchoose Suc mm) / 2"
+    by (simp add: field_simps)
+  also have "\<dots> = r * (r - 1 gchoose Suc mm) / 2"
+    by (subst gbinomial_absorb_comp) (rule refl)
   also have "\<dots> = (of_nat (Suc mm) + 1) / 2 * (r gchoose (Suc mm + 1))"
     by (subst gbinomial_absorption [symmetric]) simp
   finally show ?case .
-qed simp_all
+qed
 
 lemma setsum_bounds_lt_plus1: "(\<Sum>k<mm. f (Suc k)) = (\<Sum>k=1..mm. f k)"
-  by (induction mm) simp_all
+  by (induct mm) simp_all
 
 lemma gbinomial_partial_sum_poly:
   "(\<Sum>k\<le>m. (of_nat m + r gchoose k) * x^k * y^(m-k)) =
-       (\<Sum>k\<le>m. (-r gchoose k) * (-x)^k * (x + y)^(m-k))" (is "?lhs m = ?rhs m")
+    (\<Sum>k\<le>m. (-r gchoose k) * (-x)^k * (x + y)^(m-k))"
+  (is "?lhs m = ?rhs m")
 proof (induction m)
+  case 0
+  then show ?case by simp
+next
   case (Suc mm)
-  define G where "G i k = (of_nat i + r gchoose k) * x^k * y^(i-k)" for i k
+  define G where "G i k = (of_nat i + r gchoose k) * x^k * y^(i - k)" for i k
   define S where "S = ?lhs"
-  have SG_def: "S = (\<lambda>i. (\<Sum>k\<le>i. (G i k)))" unfolding S_def G_def ..
+  have SG_def: "S = (\<lambda>i. (\<Sum>k\<le>i. (G i k)))"
+    unfolding S_def G_def ..
 
   have "S (Suc mm) = G (Suc mm) 0 + (\<Sum>k=Suc 0..Suc mm. G (Suc mm) k)"
     using SG_def by (simp add: setsum_head_Suc atLeast0AtMost [symmetric])
   also have "(\<Sum>k=Suc 0..Suc mm. G (Suc mm) k) = (\<Sum>k=0..mm. G (Suc mm) (Suc k))"
     by (subst setsum_shift_bounds_cl_Suc_ivl) simp
-  also have "\<dots> = (\<Sum>k=0..mm. ((of_nat mm + r gchoose (Suc k))
-                    + (of_nat mm + r gchoose k)) * x^(Suc k) * y^(mm - k))"
+  also have "\<dots> = (\<Sum>k=0..mm. ((of_nat mm + r gchoose (Suc k)) +
+      (of_nat mm + r gchoose k)) * x^(Suc k) * y^(mm - k))"
     unfolding G_def by (subst gbinomial_addition_formula) simp
-  also have "\<dots> = (\<Sum>k=0..mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k))
-                  + (\<Sum>k=0..mm. (of_nat mm + r gchoose k) * x^(Suc k) * y^(mm - k))"
+  also have "\<dots> = (\<Sum>k=0..mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) +
+      (\<Sum>k=0..mm. (of_nat mm + r gchoose k) * x^(Suc k) * y^(mm - k))"
     by (subst setsum.distrib [symmetric]) (simp add: algebra_simps)
   also have "(\<Sum>k=0..mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) =
-               (\<Sum>k<Suc mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k))"
+      (\<Sum>k<Suc mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k))"
     by (simp only: atLeast0AtMost lessThan_Suc_atMost)
-  also have "\<dots> = (\<Sum>k<mm. (of_nat mm + r gchoose Suc k) * x^(Suc k) * y^(mm-k))
-                      + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)" (is "_ = ?A + ?B")
+  also have "\<dots> = (\<Sum>k<mm. (of_nat mm + r gchoose Suc k) * x^(Suc k) * y^(mm-k)) +
+      (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
+    (is "_ = ?A + ?B")
     by (subst setsum_lessThan_Suc) simp
   also have "?A = (\<Sum>k=1..mm. (of_nat mm + r gchoose k) * x^k * y^(mm - k + 1))"
   proof (subst setsum_bounds_lt_plus1 [symmetric], intro setsum.cong[OF refl], clarify)
-    fix k assume "k < mm"
-    hence "mm - k = mm - Suc k + 1" by linarith
-    thus "(of_nat mm + r gchoose Suc k) * x ^ Suc k * y ^ (mm - k) =
-            (of_nat mm + r gchoose Suc k) * x ^ Suc k * y ^ (mm - Suc k + 1)" by (simp only:)
+    fix k
+    assume "k < mm"
+    then have "mm - k = mm - Suc k + 1"
+      by linarith
+    then show "(of_nat mm + r gchoose Suc k) * x ^ Suc k * y ^ (mm - k) =
+        (of_nat mm + r gchoose Suc k) * x ^ Suc k * y ^ (mm - Suc k + 1)"
+      by (simp only:)
   qed
   also have "\<dots> + ?B = y * (\<Sum>k=1..mm. (G mm k)) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
     unfolding G_def by (subst setsum_right_distrib) (simp add: algebra_simps)
   also have "(\<Sum>k=0..mm. (of_nat mm + r gchoose k) * x^(Suc k) * y^(mm - k)) = x * (S mm)"
-      unfolding S_def by (subst setsum_right_distrib) (simp add: atLeast0AtMost algebra_simps)
-  also have "(G (Suc mm) 0) = y * (G mm 0)" by (simp add: G_def)
-  finally have "S (Suc mm) = y * ((G mm 0) + (\<Sum>k=1..mm. (G mm k)))
-                + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm) + x * (S mm)"
+    unfolding S_def by (subst setsum_right_distrib) (simp add: atLeast0AtMost algebra_simps)
+  also have "(G (Suc mm) 0) = y * (G mm 0)"
+    by (simp add: G_def)
+  finally have "S (Suc mm) =
+      y * (G mm 0 + (\<Sum>k=1..mm. (G mm k))) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm) + x * (S mm)"
     by (simp add: ring_distribs)
-  also have "(G mm 0) + (\<Sum>k=1..mm. (G mm k)) = S mm"
+  also have "G mm 0 + (\<Sum>k=1..mm. (G mm k)) = S mm"
     by (simp add: setsum_head_Suc[symmetric] SG_def atLeast0AtMost)
   finally have "S (Suc mm) = (x + y) * (S mm) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
     by (simp add: algebra_simps)
-  also have "(of_nat mm + r gchoose (Suc mm)) = (-1) ^ (Suc mm) * (-r gchoose (Suc mm))"
+  also have "(of_nat mm + r gchoose (Suc mm)) = (-1) ^ (Suc mm) * (- r gchoose (Suc mm))"
     by (subst gbinomial_negated_upper) simp
   also have "(-1) ^ Suc mm * (- r gchoose Suc mm) * x ^ Suc mm =
-                 (-r gchoose (Suc mm)) * (-x) ^ Suc mm" by (simp add: power_minus[of x])
-  also have "(x + y) * S mm + \<dots> = (x + y) * ?rhs mm + (-r gchoose (Suc mm)) * (-x)^Suc mm"
+      (- r gchoose (Suc mm)) * (-x) ^ Suc mm"
+    by (simp add: power_minus[of x])
+  also have "(x + y) * S mm + \<dots> = (x + y) * ?rhs mm + (- r gchoose (Suc mm)) * (- x)^Suc mm"
     unfolding S_def by (subst Suc.IH) simp
   also have "(x + y) * ?rhs mm = (\<Sum>n\<le>mm. ((- r gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n)))"
     by (subst setsum_right_distrib, rule setsum.cong) (simp_all add: Suc_diff_le)
   also have "\<dots> + (-r gchoose (Suc mm)) * (-x)^Suc mm =
-                 (\<Sum>n\<le>Suc mm. (- r gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n))" by simp
-  finally show ?case unfolding S_def .
-qed simp_all
+      (\<Sum>n\<le>Suc mm. (- r gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n))"
+    by simp
+  finally show ?case
+    by (simp only: S_def)
+qed
 
 lemma gbinomial_partial_sum_poly_xpos:
   "(\<Sum>k\<le>m. (of_nat m + r gchoose k) * x^k * y^(m-k)) =
@@ -1237,85 +1294,89 @@
 proof -
   have "2 * 2^(2*m) = (\<Sum>k = 0..(2 * m + 1). (2 * m + 1 choose k))"
     using choose_row_sum[where n="2 * m + 1"] by simp
-  also have "(\<Sum>k = 0..(2 * m + 1). (2 * m + 1 choose k)) = (\<Sum>k = 0..m. (2 * m + 1 choose k))
-                + (\<Sum>k = m+1..2*m+1. (2 * m + 1 choose k))"
-    using setsum_ub_add_nat[of 0 m "\<lambda>k. 2 * m + 1 choose k" "m+1"] by (simp add: mult_2)
+  also have "(\<Sum>k = 0..(2 * m + 1). (2 * m + 1 choose k)) =
+      (\<Sum>k = 0..m. (2 * m + 1 choose k)) +
+      (\<Sum>k = m+1..2*m+1. (2 * m + 1 choose k))"
+    using setsum_ub_add_nat[of 0 m "\<lambda>k. 2 * m + 1 choose k" "m+1"]
+    by (simp add: mult_2)
   also have "(\<Sum>k = m+1..2*m+1. (2 * m + 1 choose k)) =
-                 (\<Sum>k = 0..m. (2 * m + 1 choose (k + (m + 1))))"
+      (\<Sum>k = 0..m. (2 * m + 1 choose (k + (m + 1))))"
     by (subst setsum_shift_bounds_cl_nat_ivl [symmetric]) (simp add: mult_2)
   also have "\<dots> = (\<Sum>k = 0..m. (2 * m + 1 choose (m - k)))"
     by (intro setsum.cong[OF refl], subst binomial_symmetric) simp_all
   also have "\<dots> = (\<Sum>k = 0..m. (2 * m + 1 choose k))"
     using setsum.atLeast_atMost_rev [of "\<lambda>k. 2 * m + 1 choose (m - k)" 0 m]
     by simp
-  also have "\<dots> + \<dots> = 2 * \<dots>" by simp
-  finally show ?thesis by (subst (asm) mult_cancel1) (simp add: atLeast0AtMost)
+  also have "\<dots> + \<dots> = 2 * \<dots>"
+    by simp
+  finally show ?thesis
+    by (subst (asm) mult_cancel1) (simp add: atLeast0AtMost)
 qed
 
-lemma gbinomial_r_part_sum:
-  "(\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)" (is "?lhs = ?rhs")
+lemma gbinomial_r_part_sum: "(\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)"
+  (is "?lhs = ?rhs")
 proof -
   have "?lhs = of_nat (\<Sum>k\<le>m. (2 * m + 1) choose k)"
     by (simp add: binomial_gbinomial add_ac)
-  also have "\<dots> = of_nat (2 ^ (2 * m))" by (subst binomial_r_part_sum) (rule refl)
+  also have "\<dots> = of_nat (2 ^ (2 * m))"
+    by (subst binomial_r_part_sum) (rule refl)
   finally show ?thesis by simp
 qed
 
 lemma gbinomial_sum_nat_pow2:
-   "(\<Sum>k\<le>m. (of_nat (m + k) gchoose k :: 'a :: field_char_0) / 2 ^ k) = 2 ^ m" (is "?lhs = ?rhs")
+  "(\<Sum>k\<le>m. (of_nat (m + k) gchoose k :: 'a::field_char_0) / 2 ^ k) = 2 ^ m"
+  (is "?lhs = ?rhs")
 proof -
-  have "2 ^ m * 2 ^ m = (2 ^ (2*m) :: 'a)" by (induction m) simp_all
-  also have "\<dots> = (\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k))" using gbinomial_r_part_sum ..
+  have "2 ^ m * 2 ^ m = (2 ^ (2*m) :: 'a)"
+    by (induct m) simp_all
+  also have "\<dots> = (\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k))"
+    using gbinomial_r_part_sum ..
   also have "\<dots> = (\<Sum>k\<le>m. (of_nat (m + k) gchoose k) * 2 ^ (m - k))"
     using gbinomial_partial_sum_poly_xpos[where x="1" and y="1" and r="of_nat m + 1" and m="m"]
     by (simp add: add_ac)
   also have "\<dots> = 2 ^ m * (\<Sum>k\<le>m. (of_nat (m + k) gchoose k) / 2 ^ k)"
     by (subst setsum_right_distrib) (simp add: algebra_simps power_diff)
-  finally show ?thesis by (subst (asm) mult_left_cancel) simp_all
+  finally show ?thesis
+    by (subst (asm) mult_left_cancel) simp_all
 qed
 
 lemma gbinomial_trinomial_revision:
   assumes "k \<le> m"
-  shows   "(r gchoose m) * (of_nat m gchoose k) = (r gchoose k) * (r - of_nat k gchoose (m - k))"
+  shows "(r gchoose m) * (of_nat m gchoose k) = (r gchoose k) * (r - of_nat k gchoose (m - k))"
 proof -
-  have "(r gchoose m) * (of_nat m gchoose k) =
-            (r gchoose m) * fact m / (fact k * fact (m - k))"
+  have "(r gchoose m) * (of_nat m gchoose k) = (r gchoose m) * fact m / (fact k * fact (m - k))"
     using assms by (simp add: binomial_gbinomial [symmetric] binomial_fact)
-  also have "\<dots> = (r gchoose k) * (r - of_nat k gchoose (m - k))" using assms
-    by (simp add: gbinomial_pochhammer power_diff pochhammer_product)
+  also have "\<dots> = (r gchoose k) * (r - of_nat k gchoose (m - k))"
+    using assms by (simp add: gbinomial_pochhammer power_diff pochhammer_product)
   finally show ?thesis .
 qed
 
-
-text\<open>Versions of the theorems above for the natural-number version of "choose"\<close>
+text \<open>Versions of the theorems above for the natural-number version of "choose"\<close>
 lemma binomial_altdef_of_nat:
-  fixes n k :: nat
-    and x :: "'a :: field_char_0"
-  assumes "k \<le> n"
-  shows "of_nat (n choose k) = (\<Prod>i = 0..<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
-  using assms by (simp add: gbinomial_altdef_of_nat binomial_gbinomial of_nat_diff)
+  "k \<le> n \<Longrightarrow> of_nat (n choose k) = (\<Prod>i = 0..<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
+  for n k :: nat and x :: "'a::field_char_0"
+  by (simp add: gbinomial_altdef_of_nat binomial_gbinomial of_nat_diff)
 
-lemma binomial_ge_n_over_k_pow_k:
-  fixes k n :: nat
-    and x :: "'a :: linordered_field"
-  assumes "k \<le> n"
-  shows "(of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
-by (simp add: assms gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff)
+lemma binomial_ge_n_over_k_pow_k: "k \<le> n \<Longrightarrow> (of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
+  for k n :: nat and x :: "'a::linordered_field"
+  by (simp add: gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff)
 
 lemma binomial_le_pow:
   assumes "r \<le> n"
   shows "n choose r \<le> n ^ r"
 proof -
   have "n choose r \<le> fact n div fact (n - r)"
-    using \<open>r \<le> n\<close> by (subst binomial_fact_lemma[symmetric]) auto
-  with fact_div_fact_le_pow [OF assms] show ?thesis by auto
+    using assms by (subst binomial_fact_lemma[symmetric]) auto
+  with fact_div_fact_le_pow [OF assms] show ?thesis
+    by auto
 qed
 
-lemma binomial_altdef_nat: "(k::nat) \<le> n \<Longrightarrow>
-    n choose k = fact n div (fact k * fact (n - k))"
- by (subst binomial_fact_lemma [symmetric]) auto
+lemma binomial_altdef_nat: "k \<le> n \<Longrightarrow> n choose k = fact n div (fact k * fact (n - k))"
+  for k n :: nat
+  by (subst binomial_fact_lemma [symmetric]) auto
 
-lemma choose_dvd: "k \<le> n \<Longrightarrow> fact k * fact (n - k) dvd (fact n :: 'a :: {semiring_div,linordered_semidom})"
+lemma choose_dvd:
+  "k \<le> n \<Longrightarrow> fact k * fact (n - k) dvd (fact n :: 'a::{semiring_div,linordered_semidom})"
   unfolding dvd_def
   apply (rule exI [where x="of_nat (n choose k)"])
   using binomial_fact_lemma [of k n, THEN arg_cong [where f = of_nat and 'b='a]]
@@ -1323,26 +1384,28 @@
   done
 
 lemma fact_fact_dvd_fact:
-    "fact k * fact n dvd (fact (k+n) :: 'a :: {semiring_div,linordered_semidom})"
-by (metis add.commute add_diff_cancel_left' choose_dvd le_add2)
+  "fact k * fact n dvd (fact (k + n) :: 'a::{semiring_div,linordered_semidom})"
+  by (metis add.commute add_diff_cancel_left' choose_dvd le_add2)
 
 lemma choose_mult_lemma:
-     "((m+r+k) choose (m+k)) * ((m+k) choose k) = ((m+r+k) choose k) * ((m+r) choose m)"
+  "((m + r + k) choose (m + k)) * ((m + k) choose k) = ((m + r + k) choose k) * ((m + r) choose m)"
+  (is "?lhs = _")
 proof -
-  have "((m+r+k) choose (m+k)) * ((m+k) choose k) =
-        fact (m+r + k) div (fact (m + k) * fact (m+r - m)) * (fact (m + k) div (fact k * fact m))"
+  have "?lhs =
+      fact (m + r + k) div (fact (m + k) * fact (m + r - m)) * (fact (m + k) div (fact k * fact m))"
     by (simp add: binomial_altdef_nat)
-  also have "... = fact (m+r+k) div (fact r * (fact k * fact m))"
+  also have "\<dots> = fact (m + r + k) div (fact r * (fact k * fact m))"
     apply (subst div_mult_div_if_dvd)
     apply (auto simp: algebra_simps fact_fact_dvd_fact)
     apply (metis add.assoc add.commute fact_fact_dvd_fact)
     done
-  also have "... = (fact (m+r+k) * fact (m+r)) div (fact r * (fact k * fact m) * fact (m+r))"
+  also have "\<dots> = (fact (m + r + k) * fact (m + r)) div (fact r * (fact k * fact m) * fact (m + r))"
     apply (subst div_mult_div_if_dvd [symmetric])
     apply (auto simp add: algebra_simps)
     apply (metis fact_fact_dvd_fact dvd_trans nat_mult_dvd_cancel_disj)
     done
-  also have "... = (fact (m+r+k) div (fact k * fact (m+r)) * (fact (m+r) div (fact r * fact m)))"
+  also have "\<dots> =
+      (fact (m + r + k) div (fact k * fact (m + r)) * (fact (m + r) div (fact r * fact m)))"
     apply (subst div_mult_div_if_dvd)
     apply (auto simp: fact_fact_dvd_fact algebra_simps)
     done
@@ -1350,23 +1413,21 @@
     by (simp add: binomial_altdef_nat mult.commute)
 qed
 
-text\<open>The "Subset of a Subset" identity\<close>
+text \<open>The "Subset of a Subset" identity.\<close>
 lemma choose_mult:
-  assumes "k\<le>m" "m\<le>n"
-    shows "(n choose m) * (m choose k) = (n choose k) * ((n-k) choose (m-k))"
-using assms choose_mult_lemma [of "m-k" "n-m" k]
-by simp
+  "k \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> (n choose m) * (m choose k) = (n choose k) * ((n - k) choose (m - k))"
+  using choose_mult_lemma [of "m-k" "n-m" k] by simp
 
 
 subsection \<open>More on Binomial Coefficients\<close>
 
-lemma choose_one: "(n::nat) choose 1 = n"
+lemma choose_one: "n choose 1 = n" for n :: nat
   by simp
 
 (*FIXME: messy and apparently unused*)
-lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \<longrightarrow>
-    (ALL n. P (Suc n) 0) \<longrightarrow> (ALL n. (ALL k < n. P n k \<longrightarrow> P n (Suc k) \<longrightarrow>
-    P (Suc n) (Suc k))) \<longrightarrow> (ALL k <= n. P n k)"
+lemma binomial_induct [rule_format]: "(\<forall>n::nat. P n n) \<longrightarrow>
+    (\<forall>n. P (Suc n) 0) \<longrightarrow> (\<forall>n. (\<forall>k < n. P n k \<longrightarrow> P n (Suc k) \<longrightarrow>
+    P (Suc n) (Suc k))) \<longrightarrow> (\<forall>k \<le> n. P n k)"
   apply (induct n)
   apply auto
   apply (case_tac "k = 0")
@@ -1376,118 +1437,133 @@
   done
 
 lemma card_UNION:
-  assumes "finite A" and "\<forall>k \<in> A. finite k"
+  assumes "finite A"
+    and "\<forall>k \<in> A. finite k"
   shows "card (\<Union>A) = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * int (card (\<Inter>I)))"
   (is "?lhs = ?rhs")
 proof -
-  have "?rhs = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * (\<Sum>_\<in>\<Inter>I. 1))" by simp
-  also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))" (is "_ = nat ?rhs")
-    by(subst setsum_right_distrib) simp
+  have "?rhs = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * (\<Sum>_\<in>\<Inter>I. 1))"
+    by simp
+  also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))"
+    (is "_ = nat ?rhs")
+    by (subst setsum_right_distrib) simp
   also have "?rhs = (\<Sum>(I, _)\<in>Sigma {I. I \<subseteq> A \<and> I \<noteq> {}} Inter. (- 1) ^ (card I + 1))"
-    using assms by(subst setsum.Sigma)(auto)
+    using assms by (subst setsum.Sigma) auto
   also have "\<dots> = (\<Sum>(x, I)\<in>(SIGMA x:UNIV. {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}). (- 1) ^ (card I + 1))"
     by (rule setsum.reindex_cong [where l = "\<lambda>(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta)
   also have "\<dots> = (\<Sum>(x, I)\<in>(SIGMA x:\<Union>A. {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}). (- 1) ^ (card I + 1))"
-    using assms by(auto intro!: setsum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\<Union>A"])
+    using assms
+    by (auto intro!: setsum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\<Union>A"])
   also have "\<dots> = (\<Sum>x\<in>\<Union>A. (\<Sum>I|I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I. (- 1) ^ (card I + 1)))"
-    using assms by(subst setsum.Sigma) auto
+    using assms by (subst setsum.Sigma) auto
   also have "\<dots> = (\<Sum>_\<in>\<Union>A. 1)" (is "setsum ?lhs _ = _")
-  proof(rule setsum.cong[OF refl])
+  proof (rule setsum.cong[OF refl])
     fix x
     assume x: "x \<in> \<Union>A"
     define K where "K = {X \<in> A. x \<in> X}"
-    with \<open>finite A\<close> have K: "finite K" by auto
+    with \<open>finite A\<close> have K: "finite K"
+      by auto
     let ?I = "\<lambda>i. {I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I}"
     have "inj_on snd (SIGMA i:{1..card A}. ?I i)"
-      using assms by(auto intro!: inj_onI)
+      using assms by (auto intro!: inj_onI)
     moreover have [symmetric]: "snd ` (SIGMA i:{1..card A}. ?I i) = {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}"
-      using assms by(auto intro!: rev_image_eqI[where x="(card a, a)" for a]
+      using assms
+      by (auto intro!: rev_image_eqI[where x="(card a, a)" for a]
         simp add: card_gt_0_iff[folded Suc_le_eq]
         dest: finite_subset intro: card_mono)
     ultimately have "?lhs x = (\<Sum>(i, I)\<in>(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))"
       by (rule setsum.reindex_cong [where l = snd]) fastforce
     also have "\<dots> = (\<Sum>i=1..card A. (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. (- 1) ^ (i + 1)))"
-      using assms by(subst setsum.Sigma) auto
+      using assms by (subst setsum.Sigma) auto
     also have "\<dots> = (\<Sum>i=1..card A. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1))"
-      by(subst setsum_right_distrib) simp
-    also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))" (is "_ = ?rhs")
-    proof(rule setsum.mono_neutral_cong_right[rule_format])
-      show "{1..card K} \<subseteq> {1..card A}" using \<open>finite A\<close>
-        by(auto simp add: K_def intro: card_mono)
+      by (subst setsum_right_distrib) simp
+    also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))"
+      (is "_ = ?rhs")
+    proof (rule setsum.mono_neutral_cong_right[rule_format])
+      show "finite {1..card A}"
+        by simp
+      show "{1..card K} \<subseteq> {1..card A}"
+        using \<open>finite A\<close> by (auto simp add: K_def intro: card_mono)
     next
       fix i
       assume "i \<in> {1..card A} - {1..card K}"
-      hence i: "i \<le> card A" "card K < i" by auto
+      then have i: "i \<le> card A" "card K < i"
+        by auto
       have "{I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I} = {I. I \<subseteq> K \<and> card I = i}"
-        by(auto simp add: K_def)
-      also have "\<dots> = {}" using \<open>finite A\<close> i
-        by(auto simp add: K_def dest: card_mono[rotated 1])
+        by (auto simp add: K_def)
+      also have "\<dots> = {}"
+        using \<open>finite A\<close> i by (auto simp add: K_def dest: card_mono[rotated 1])
       finally show "(- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1 :: int) = 0"
-        by(simp only:) simp
+        by (simp only:) simp
     next
       fix i
       have "(\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1) = (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)"
         (is "?lhs = ?rhs")
-        by(rule setsum.cong)(auto simp add: K_def)
-      thus "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" by simp
-    qed simp
-    also have "{I. I \<subseteq> K \<and> card I = 0} = {{}}" using assms
-      by(auto simp add: card_eq_0_iff K_def dest: finite_subset)
-    hence "?rhs = (\<Sum>i = 0..card K. (- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)) + 1"
-      by(subst (2) setsum_head_Suc)(simp_all )
+        by (rule setsum.cong) (auto simp add: K_def)
+      then show "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs"
+        by simp
+    qed
+    also have "{I. I \<subseteq> K \<and> card I = 0} = {{}}"
+      using assms by (auto simp add: card_eq_0_iff K_def dest: finite_subset)
+    then have "?rhs = (\<Sum>i = 0..card K. (- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)) + 1"
+      by (subst (2) setsum_head_Suc) simp_all
     also have "\<dots> = (\<Sum>i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1"
-      using K by(subst n_subsets[symmetric]) simp_all
+      using K by (subst n_subsets[symmetric]) simp_all
     also have "\<dots> = - (\<Sum>i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1"
-      by(subst setsum_right_distrib[symmetric]) simp
+      by (subst setsum_right_distrib[symmetric]) simp
     also have "\<dots> =  - ((-1 + 1) ^ card K) + 1"
-      by(subst binomial_ring)(simp add: ac_simps)
-    also have "\<dots> = 1" using x K by(auto simp add: K_def card_gt_0_iff)
+      by (subst binomial_ring) (simp add: ac_simps)
+    also have "\<dots> = 1"
+      using x K by (auto simp add: K_def card_gt_0_iff)
     finally show "?lhs x = 1" .
   qed
-  also have "nat \<dots> = card (\<Union>A)" by simp
+  also have "nat \<dots> = card (\<Union>A)"
+    by simp
   finally show ?thesis ..
 qed
 
-text\<open>The number of nat lists of length \<open>m\<close> summing to \<open>N\<close> is
-@{term "(N + m - 1) choose N"}:\<close>
-
+text \<open>The number of nat lists of length \<open>m\<close> summing to \<open>N\<close> is @{term "(N + m - 1) choose N"}:\<close>
 lemma card_length_listsum_rec:
-  assumes "m\<ge>1"
+  assumes "m \<ge> 1"
   shows "card {l::nat list. length l = m \<and> listsum l = N} =
-    (card {l. length l = (m - 1) \<and> listsum l = N} +
-    card {l. length l = m \<and> listsum l + 1 =  N})"
-    (is "card ?C = (card ?A + card ?B)")
+      card {l. length l = (m - 1) \<and> listsum l = N} +
+      card {l. length l = m \<and> listsum l + 1 = N}"
+    (is "card ?C = card ?A + card ?B")
 proof -
-  let ?A'="{l. length l = m \<and> listsum l = N \<and> hd l = 0}"
-  let ?B'="{l. length l = m \<and> listsum l = N \<and> hd l \<noteq> 0}"
-  let ?f ="\<lambda> l. 0#l"
-  let ?g ="\<lambda> l. (hd l + 1) # tl l"
-  have 1: "\<And>xs x. xs \<noteq> [] \<Longrightarrow> x = hd xs \<Longrightarrow> x # tl xs = xs" by simp
-  have 2: "\<And>xs. (xs::nat list) \<noteq> [] \<Longrightarrow> listsum(tl xs) = listsum xs - hd xs"
-    by(auto simp add: neq_Nil_conv)
+  let ?A' = "{l. length l = m \<and> listsum l = N \<and> hd l = 0}"
+  let ?B' = "{l. length l = m \<and> listsum l = N \<and> hd l \<noteq> 0}"
+  let ?f = "\<lambda>l. 0 # l"
+  let ?g = "\<lambda>l. (hd l + 1) # tl l"
+  have 1: "xs \<noteq> [] \<Longrightarrow> x = hd xs \<Longrightarrow> x # tl xs = xs" for x xs
+    by simp
+  have 2: "xs \<noteq> [] \<Longrightarrow> listsum(tl xs) = listsum xs - hd xs" for xs :: "nat list"
+    by (auto simp add: neq_Nil_conv)
   have f: "bij_betw ?f ?A ?A'"
-    apply(rule bij_betw_byWitness[where f' = tl])
+    apply (rule bij_betw_byWitness[where f' = tl])
     using assms
-    by (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv)
-  have 3: "\<And>xs:: nat list. xs \<noteq> [] \<Longrightarrow> hd xs + (listsum xs - hd xs) = listsum xs"
+    apply (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv)
+    done
+  have 3: "xs \<noteq> [] \<Longrightarrow> hd xs + (listsum xs - hd xs) = listsum xs" for xs :: "nat list"
     by (metis 1 listsum_simps(2) 2)
   have g: "bij_betw ?g ?B ?B'"
-    apply(rule bij_betw_byWitness[where f' = "\<lambda> l. (hd l - 1) # tl l"])
+    apply (rule bij_betw_byWitness[where f' = "\<lambda>l. (hd l - 1) # tl l"])
     using assms
     by (auto simp: 2 length_0_conv[symmetric] intro!: 3
-      simp del: length_greater_0_conv length_0_conv)
-  { fix M N :: nat have "finite {xs. size xs = M \<and> set xs \<subseteq> {0..<N}}"
-    using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto }
-    note fin = this
+        simp del: length_greater_0_conv length_0_conv)
+  have fin: "finite {xs. size xs = M \<and> set xs \<subseteq> {0..<N}}" for M N :: nat
+    using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto
   have fin_A: "finite ?A" using fin[of _ "N+1"]
-    by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 \<and> set xs \<subseteq> {0..<N+1}}"],
-      auto simp: member_le_listsum_nat less_Suc_eq_le)
+    by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 \<and> set xs \<subseteq> {0..<N+1}}"])
+      (auto simp: member_le_listsum_nat less_Suc_eq_le)
   have fin_B: "finite ?B"
-    by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m \<and> set xs \<subseteq> {0..<N}}"],
-      auto simp: member_le_listsum_nat less_Suc_eq_le fin)
-  have uni: "?C = ?A' \<union> ?B'" by auto
-  have disj: "?A' \<inter> ?B' = {}" by auto
-  have "card ?C = card(?A' \<union> ?B')" using uni by simp
+    by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m \<and> set xs \<subseteq> {0..<N}}"])
+      (auto simp: member_le_listsum_nat less_Suc_eq_le fin)
+  have uni: "?C = ?A' \<union> ?B'"
+    by auto
+  have disj: "?A' \<inter> ?B' = {}"
+    by auto
+  have "card ?C = card(?A' \<union> ?B')"
+    using uni by simp
   also have "\<dots> = card ?A + card ?B"
     using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g]
       bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B
@@ -1495,59 +1571,65 @@
   finally show ?thesis .
 qed
 
-lemma card_length_listsum: \<comment>"By Holden Lee, tidied by Tobias Nipkow"
-  "card {l::nat list. size l = m \<and> listsum l = N} = (N + m - 1) choose N"
+lemma card_length_listsum: "card {l::nat list. size l = m \<and> listsum l = N} = (N + m - 1) choose N"
+  \<comment> "by Holden Lee, tidied by Tobias Nipkow"
 proof (cases m)
-  case 0 then show ?thesis
-    by (cases N) (auto simp: cong: conj_cong)
+  case 0
+  then show ?thesis
+    by (cases N) (auto cong: conj_cong)
 next
   case (Suc m')
-    have m: "m\<ge>1" by (simp add: Suc)
-    then show ?thesis
-    proof (induct "N + m - 1" arbitrary: N m)
-      case 0   \<comment> "In the base case, the only solution is [0]."
-      have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 0)} = {[0]}"
-        by (auto simp: length_Suc_conv)
-      have "m=1 \<and> N=0" using 0 by linarith
-      then show ?case by simp
+  have m: "m \<ge> 1"
+    by (simp add: Suc)
+  then show ?thesis
+  proof (induct "N + m - 1" arbitrary: N m)
+    case 0  \<comment> "In the base case, the only solution is [0]."
+    have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 0)} = {[0]}"
+      by (auto simp: length_Suc_conv)
+    have "m = 1 \<and> N = 0"
+      using 0 by linarith
+    then show ?case
+      by simp
+  next
+    case (Suc k)
+    have c1: "card {l::nat list. size l = (m - 1) \<and> listsum l =  N} = (N + (m - 1) - 1) choose N"
+    proof (cases "m = 1")
+      case True
+      with Suc.hyps have "N \<ge> 1"
+        by auto
+      with True show ?thesis
+        by (simp add: binomial_eq_0)
     next
-      case (Suc k)
-
-      have c1: "card {l::nat list. size l = (m - 1) \<and> listsum l =  N} =
-        (N + (m - 1) - 1) choose N"
-      proof cases
-        assume "m = 1"
-        with Suc.hyps have "N\<ge>1" by auto
-        with \<open>m = 1\<close> show ?thesis by (simp add: binomial_eq_0)
-      next
-        assume "m \<noteq> 1" thus ?thesis using Suc by fastforce
-      qed
-
-      from Suc have c2: "card {l::nat list. size l = m \<and> listsum l + 1 = N} =
-        (if N>0 then ((N - 1) + m - 1) choose (N - 1) else 0)"
-      proof -
-        have aux: "\<And>m n. n > 0 \<Longrightarrow> Suc m = n \<longleftrightarrow> m = n - 1" by arith
-        from Suc have "N>0 \<Longrightarrow>
-          card {l::nat list. size l = m \<and> listsum l + 1 = N} =
-          ((N - 1) + m - 1) choose (N - 1)" by (simp add: aux)
-        thus ?thesis by auto
-      qed
-
-      from Suc.prems have "(card {l::nat list. size l = (m - 1) \<and> listsum l = N} +
+      case False
+      then show ?thesis
+        using Suc by fastforce
+    qed
+    from Suc have c2: "card {l::nat list. size l = m \<and> listsum l + 1 = N} =
+      (if N > 0 then ((N - 1) + m - 1) choose (N - 1) else 0)"
+    proof -
+      have *: "n > 0 \<Longrightarrow> Suc m = n \<longleftrightarrow> m = n - 1" for m n
+        by arith
+      from Suc have "N > 0 \<Longrightarrow>
+        card {l::nat list. size l = m \<and> listsum l + 1 = N} =
+          ((N - 1) + m - 1) choose (N - 1)"
+        by (simp add: *)
+      then show ?thesis
+        by auto
+    qed
+    from Suc.prems have "(card {l::nat list. size l = (m - 1) \<and> listsum l = N} +
           card {l::nat list. size l = m \<and> listsum l + 1 = N}) = (N + m - 1) choose N"
-        by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def)
-      thus ?case using card_length_listsum_rec[OF Suc.prems] by auto
-    qed
+      by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def)
+    then show ?case
+      using card_length_listsum_rec[OF Suc.prems] by auto
+  qed
 qed
 
-
-lemma Suc_times_binomial_add: \<comment> \<open>by Lukas Bulwahn\<close>
-  "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)"
+lemma Suc_times_binomial_add: "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)"
+  \<comment> \<open>by Lukas Bulwahn\<close>
 proof -
   have dvd: "Suc a * (fact a * fact b) dvd fact (Suc (a + b))" for a b
     using fact_fact_dvd_fact[of "Suc a" "b", where 'a=nat]
     by (simp only: fact_Suc add_Suc[symmetric] of_nat_id mult.assoc)
-
   have "Suc a * (fact (Suc (a + b)) div (Suc a * fact a * fact b)) =
       Suc a * fact (Suc (a + b)) div (Suc a * (fact a * fact b))"
     by (subst div_mult_swap[symmetric]; simp only: mult.assoc dvd)
@@ -1557,14 +1639,14 @@
     using dvd[of b a] by (subst div_mult_swap[symmetric]; simp only: ac_simps dvd)
   finally show ?thesis
     by (subst (1 2) binomial_altdef_nat)
-       (simp_all only: ac_simps diff_Suc_Suc Suc_diff_le diff_add_inverse fact_Suc of_nat_id)
+      (simp_all only: ac_simps diff_Suc_Suc Suc_diff_le diff_add_inverse fact_Suc of_nat_id)
 qed
 
 
 subsection \<open>Misc\<close>
 
 lemma fact_code [code]:
-  "fact n = (of_nat (fold_atLeastAtMost_nat (op *) 2 n 1) :: 'a :: semiring_char_0)"
+  "fact n = (of_nat (fold_atLeastAtMost_nat (op *) 2 n 1) :: 'a::semiring_char_0)"
 proof -
   have "fact n = (of_nat (\<Prod>{1..n}) :: 'a)"
     by (simp add: fact_setprod)
@@ -1576,15 +1658,22 @@
 qed
 
 lemma pochhammer_code [code]:
-  "pochhammer a n = (if n = 0 then 1 else
-     fold_atLeastAtMost_nat (\<lambda>n acc. (a + of_nat n) * acc) 0 (n - 1) 1)"
-  by (cases n) (simp_all add: pochhammer_setprod setprod_atLeastAtMost_code [symmetric] atLeastLessThanSuc_atLeastAtMost)
+  "pochhammer a n =
+    (if n = 0 then 1
+     else fold_atLeastAtMost_nat (\<lambda>n acc. (a + of_nat n) * acc) 0 (n - 1) 1)"
+  by (cases n)
+    (simp_all add: pochhammer_setprod setprod_atLeastAtMost_code [symmetric]
+      atLeastLessThanSuc_atLeastAtMost)
 
 lemma gbinomial_code [code]:
-  "a gchoose n = (if n = 0 then 1 else
-     fold_atLeastAtMost_nat (\<lambda>n acc. (a - of_nat n) * acc) 0 (n - 1) 1 / fact n)"
-  by (cases n) (simp_all add: gbinomial_setprod_rev setprod_atLeastAtMost_code [symmetric] atLeastLessThanSuc_atLeastAtMost)
+  "a gchoose n =
+    (if n = 0 then 1
+     else fold_atLeastAtMost_nat (\<lambda>n acc. (a - of_nat n) * acc) 0 (n - 1) 1 / fact n)"
+  by (cases n)
+    (simp_all add: gbinomial_setprod_rev setprod_atLeastAtMost_code [symmetric]
+      atLeastLessThanSuc_atLeastAtMost)
 
+(* FIXME *)
 (*TODO: This code equation breaks Scala code generation in HOL-Codegenerator_Test. We have to figure out why and how to prevent that. *)
 
 (*
@@ -1596,11 +1685,11 @@
 proof -
   {
     assume "k \<le> n"
-    hence "{1..n} = {1..n-k} \<union> {n-k+1..n}" by auto
-    hence "(fact n :: nat) = fact (n-k) * \<Prod>{n-k+1..n}"
+    then have "{1..n} = {1..n-k} \<union> {n-k+1..n}" by auto
+    then have "(fact n :: nat) = fact (n-k) * \<Prod>{n-k+1..n}"
       by (simp add: setprod.union_disjoint fact_altdef_nat)
   }
-  thus ?thesis by (auto simp: binomial_altdef_nat mult_ac setprod_atLeastAtMost_code)
+  then show ?thesis by (auto simp: binomial_altdef_nat mult_ac setprod_atLeastAtMost_code)
 qed
 *)