author wenzelm Thu, 16 Aug 2012 17:16:20 +0200 changeset 48830 72efe3e0a46b parent 48827 8791d106e30b child 48831 2cc51d1cabd7
tuned proofs;
```--- a/src/HOL/Library/Binomial.thy	Thu Aug 16 15:40:26 2012 +0200
+++ b/src/HOL/Library/Binomial.thy	Thu Aug 16 17:16:20 2012 +0200
@@ -14,55 +14,53 @@

primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65) where
binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)"
-  | binomial_Suc: "(Suc n choose k) =
+| binomial_Suc: "(Suc n choose k) =
(if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"

lemma binomial_n_0 [simp]: "(n choose 0) = 1"
-by (cases n) simp_all
+  by (cases n) simp_all

lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
-by simp
+  by simp

lemma binomial_Suc_Suc [simp]:
"(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
-by simp
+  by simp

lemma binomial_eq_0: "!!k. n < k ==> (n choose k) = 0"
-by (induct n) auto
+  by (induct n) auto

declare binomial_0 [simp del] binomial_Suc [simp del]

lemma binomial_n_n [simp]: "(n choose n) = 1"
-by (induct n) (simp_all add: binomial_eq_0)
+  by (induct n) (simp_all add: binomial_eq_0)

lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
-by (induct n) simp_all
+  by (induct n) simp_all

lemma binomial_1 [simp]: "(n choose Suc 0) = n"
-by (induct n) simp_all
+  by (induct n) simp_all

lemma zero_less_binomial: "k \<le> n ==> (n choose k) > 0"
-by (induct n k rule: diff_induct) simp_all
+  by (induct n k rule: diff_induct) simp_all

lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
-apply (safe intro!: binomial_eq_0)
-apply (erule contrapos_pp)
-done
+  apply (safe intro!: binomial_eq_0)
+  apply (erule contrapos_pp)
+  done

lemma zero_less_binomial_iff: "(n choose k > 0) = (k\<le>n)"
-        del:neq0_conv)
+  by (simp add: linorder_not_less binomial_eq_0_iff neq0_conv[symmetric] del: neq0_conv)

(*Might be more useful if re-oriented*)
lemma Suc_times_binomial_eq:
"!!k. k \<le> n ==> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
-apply (induct n)
-apply (case_tac k)
-    binomial_eq_0)
-done
+  apply (induct n)
+   apply (case_tac k)
+  done

text{*This is the well-known version, but it's harder to use because of the
@@ -74,7 +72,7 @@
lemma times_binomial_minus1_eq:
"[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
-  apply (simp split add: nat_diff_split, auto)
+   apply (simp split add: nat_diff_split, auto)
done

@@ -85,20 +83,19 @@
Kamm\"uller, tidied by LCP.
*}

-lemma card_s_0_eq_empty:
-    "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
+lemma card_s_0_eq_empty: "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"

lemma choose_deconstruct: "finite M ==> x \<notin> M
==> {s. s <= insert x M & card(s) = Suc k}
= {s. s <= M & card(s) = Suc k} Un
{s. EX t. t <= M & card(t) = k & s = insert x t}"
apply safe
-   apply (auto intro: finite_subset [THEN card_insert_disjoint])
+     apply (auto intro: finite_subset [THEN card_insert_disjoint])
apply (drule_tac x = "xa - {x}" in spec)
apply (subgoal_tac "x \<notin> xa", auto)
apply (erule rev_mp, subst card_Diff_singleton)
-  apply (auto intro: finite_subset)
+    apply (auto intro: finite_subset)
done
(*
lemma "finite(UN y. {x. P x y})"
@@ -111,10 +108,10 @@

lemma finite_bex_subset[simp]:
"finite B \<Longrightarrow> (!!A. A<=B \<Longrightarrow> finite{x. P x A}) \<Longrightarrow> finite{x. EX A<=B. P x A}"
-apply(subgoal_tac "{x. EX A<=B. P x A} = (UN A:Pow B. {x. P x A})")
- apply simp
-apply blast
-done
+  apply (subgoal_tac "{x. EX A<=B. P x A} = (UN A:Pow B. {x. P x A})")
+   apply simp
+  apply blast
+  done

text{*There are as many subsets of @{term A} having cardinality @{term k}
as there are sets obtained from the former by inserting a fixed element
@@ -123,10 +120,10 @@
"[|finite A; x \<notin> A|] ==>
card {B. EX C. C <= A & card(C) = k & B = insert x C} =
card {B. B <= A & card(B) = k}"
-apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
-     apply (auto elim!: equalityE simp add: inj_on_def)
-apply (subst Diff_insert0, auto)
-done
+  apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
+       apply (auto elim!: equalityE simp add: inj_on_def)
+  apply (subst Diff_insert0, auto)
+  done

text {*
Main theorem: combinatorial statement about number of subsets of a set.
@@ -191,11 +188,11 @@

definition "pochhammer (a::'a::comm_semiring_1) n = (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"

-lemma pochhammer_0[simp]: "pochhammer a 0 = 1"
+lemma pochhammer_0[simp]: "pochhammer a 0 = 1"

lemma pochhammer_1[simp]: "pochhammer a 1 = a" by (simp add: pochhammer_def)
-lemma pochhammer_Suc0[simp]: "pochhammer a (Suc 0) = a"
+lemma pochhammer_Suc0[simp]: "pochhammer a (Suc 0) = a"

lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"
@@ -216,18 +213,18 @@

lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
proof-
-  {assume "n=0" then have ?thesis by simp}
+  { assume "n=0" then have ?thesis by simp }
moreover
-  {fix m assume m: "n = Suc m"
-    have ?thesis unfolding m pochhammer_Suc_setprod setprod_nat_ivl_Suc ..}
-  ultimately show ?thesis by (cases n, auto)
-qed
+  { fix m assume m: "n = Suc m"
+    have ?thesis unfolding m pochhammer_Suc_setprod setprod_nat_ivl_Suc .. }
+  ultimately show ?thesis by (cases n) auto
+qed

lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
proof-
-  {assume "n=0" then have ?thesis by (simp add: pochhammer_Suc_setprod)}
+  { assume "n=0" then have ?thesis by (simp add: pochhammer_Suc_setprod) }
moreover
-  {assume n0: "n \<noteq> 0"
+  { assume n0: "n \<noteq> 0"
have th0: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
have eq: "insert 0 {1 .. n} = {0..n}" by auto
have th1: "(\<Prod>n\<in>{1\<Colon>nat..n}. a + of_nat n) =
@@ -236,74 +233,80 @@
using n0 by (auto simp add: fun_eq_iff field_simps)
have ?thesis apply (simp add: pochhammer_def)
unfolding setprod_insert[OF th0, unfolded eq]
-    using th1 by (simp add: field_simps)}
-ultimately show ?thesis by blast
+    using th1 by (simp add: field_simps) }
+  ultimately show ?thesis by blast
qed

lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n"
unfolding fact_altdef_nat
-
-  apply (cases n, simp_all add: of_nat_setprod pochhammer_Suc_setprod)
+  apply (cases n)
+   apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod)
apply (rule setprod_reindex_cong[where f=Suc])
-  by (auto simp add: fun_eq_iff)
+    apply (auto simp add: fun_eq_iff)
+  done

-lemma pochhammer_of_nat_eq_0_lemma: assumes kn: "k > n"
+lemma pochhammer_of_nat_eq_0_lemma:
+  assumes kn: "k > n"
shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
proof-
-  from kn obtain h where h: "k = Suc h" by (cases k, auto)
-  {assume n0: "n=0" then have ?thesis using kn
-      by (cases k) (simp_all add: pochhammer_rec)}
+  from kn obtain h where h: "k = Suc h" by (cases k) auto
+  { assume n0: "n=0" then have ?thesis using kn
+      by (cases k) (simp_all add: pochhammer_rec) }
moreover
-  {assume n0: "n \<noteq> 0"
-    then have ?thesis apply (simp add: h pochhammer_Suc_setprod)
-  apply (rule_tac x="n" in bexI)
-  using h kn by auto}
-ultimately show ?thesis by blast
+  { assume n0: "n \<noteq> 0"
+    then have ?thesis
+      apply (simp add: h pochhammer_Suc_setprod)
+      apply (rule_tac x="n" in bexI)
+      using h kn
+      apply auto
+      done }
+  ultimately show ?thesis by blast
qed

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"
proof-
-  {assume "k=0" then have ?thesis by simp}
+  { assume "k=0" then have ?thesis by simp }
moreover
-  {fix h assume h: "k = Suc h"
+  { fix h assume h: "k = Suc h"
then have ?thesis apply (simp add: pochhammer_Suc_setprod)
-      using h kn by (auto simp add: algebra_simps)}
-  ultimately show ?thesis by (cases k, auto)
+      using h kn by (auto simp add: algebra_simps) }
+  ultimately show ?thesis by (cases k) auto
qed

-lemma pochhammer_of_nat_eq_0_iff:
+lemma pochhammer_of_nat_eq_0_iff:
shows "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]
+  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]

-lemma pochhammer_eq_0_iff:
+lemma pochhammer_eq_0_iff:
"pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (EX k < n . a = - of_nat k) "
+  apply (cases n)
apply (rule_tac x=x in exI)
apply auto
done

-lemma pochhammer_eq_0_mono:
+lemma pochhammer_eq_0_mono:
"pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
-  unfolding pochhammer_eq_0_iff by auto
+  unfolding pochhammer_eq_0_iff by auto

-lemma pochhammer_neq_0_mono:
+lemma pochhammer_neq_0_mono:
"pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
-  unfolding pochhammer_eq_0_iff by auto
+  unfolding pochhammer_eq_0_iff by auto

lemma pochhammer_minus:
-  assumes kn: "k \<le> n"
+  assumes kn: "k \<le> n"
shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
proof-
-  {assume k0: "k = 0" then have ?thesis by simp}
-  moreover
-  {fix h assume h: "k = Suc h"
+  { assume k0: "k = 0" then have ?thesis by simp }
+  moreover
+  { fix h assume h: "k = Suc h"
have eq: "((- 1) ^ Suc h :: 'a) = setprod (%i. - 1) {0 .. h}"
using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]
by auto
@@ -312,12 +315,13 @@
apply (rule strong_setprod_reindex_cong[where f = "%i. h - i"])
apply (auto simp add: inj_on_def image_def h )
apply (rule_tac x="h - x" in bexI)
-      by (auto simp add: fun_eq_iff h of_nat_diff)}
-  ultimately show ?thesis by (cases k, auto)
+      apply (auto simp add: fun_eq_iff h of_nat_diff)
+      done }
+  ultimately show ?thesis by (cases k) auto
qed

lemma pochhammer_minus':
-  assumes kn: "k \<le> n"
+  assumes kn: "k \<le> n"
shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
unfolding pochhammer_minus[OF kn, where b=b]
unfolding mult_assoc[symmetric]
@@ -332,103 +336,112 @@
subsection{* Generalized binomial coefficients *}

definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
-  where "a gchoose n = (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))"
+  where "a gchoose n =
+    (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))"

lemma gbinomial_0[simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
-apply (subgoal_tac "(\<Prod>i\<Colon>nat\<in>{0\<Colon>nat..n}. - of_nat i) = (0::'b)")
- apply (simp del:setprod_zero_iff)
-apply simp
-done
+  apply (subgoal_tac "(\<Prod>i\<Colon>nat\<in>{0\<Colon>nat..n}. - of_nat i) = (0::'b)")
+   apply (simp del:setprod_zero_iff)
+  apply simp
+  done

lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)"
-proof-
-  {assume "n=0" then have ?thesis by simp}
+proof -
+  { assume "n=0" then have ?thesis by simp }
moreover
-  {assume n0: "n\<noteq>0"
+  { assume n0: "n\<noteq>0"
from n0 setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]
have eq: "(- (1\<Colon>'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"
by auto
-    from n0 have ?thesis
-      by (simp add: pochhammer_def gbinomial_def field_simps eq setprod_timesf[symmetric] del: minus_one) (* FIXME: del: minus_one *)}
+    from n0 have ?thesis
+      by (simp add: pochhammer_def gbinomial_def field_simps
+        eq setprod_timesf[symmetric] del: minus_one) (* FIXME: del: minus_one *) }
ultimately show ?thesis by blast
qed

-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)
+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"
-    let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
-  {assume "n=0" then have ?ths using kn by simp}
+  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}
+  { assume "k=0" then have ?ths using kn by simp }
moreover
-  {assume nk: "n=k" then have ?ths by simp}
+  { 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"
+  { 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
+    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)"
by simp
-    from n h th0
-    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))"
+    from n h th0
+    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))"
also have "\<dots> = (k + (m - h)) * fact m"
using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
-    finally have ?ths using h n km by simp}
-  moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (EX m h. n=Suc m \<and> k = Suc h \<and> h < m)" using kn by presburger
+    finally have ?ths using h n km by simp }
+  moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (EX m h. n=Suc m \<and> k = Suc h \<and> h < m)"
+    using kn by presburger
ultimately show ?ths by blast
qed
-
-lemma binomial_fact:
-  assumes kn: "k \<le> n"
-  shows "(of_nat (n choose k) :: 'a::field_char_0) = of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"
+
+lemma binomial_fact:
+  assumes kn: "k \<le> n"
+  shows "(of_nat (n choose k) :: 'a::field_char_0) =
+    of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"
using binomial_fact_lemma[OF kn]
by (simp add: field_simps of_nat_mult [symmetric])

lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k"
-proof-
-  {assume kn: "k > n"
-    from kn binomial_eq_0[OF kn] have ?thesis
-      by (simp add: gbinomial_pochhammer field_simps
-        pochhammer_of_nat_eq_0_iff)}
+proof -
+  { assume kn: "k > n"
+    from kn binomial_eq_0[OF kn] have ?thesis
+      by (simp add: gbinomial_pochhammer field_simps  pochhammer_of_nat_eq_0_iff) }
moreover
-  {assume "k=0" then have ?thesis by simp}
+  { assume "k=0" then have ?thesis by simp }
moreover
-  {assume kn: "k \<le> n" and k0: "k\<noteq> 0"
-    from k0 obtain h where h: "k = Suc h" by (cases k, auto)
+  { assume kn: "k \<le> n" and k0: "k\<noteq> 0"
+    from k0 obtain h where h: "k = Suc h" by (cases k) auto
from h
have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
by (subst setprod_constant, auto)
have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
apply (rule strong_setprod_reindex_cong[where f="op - n"])
-      using h kn
-      apply (simp_all add: inj_on_def image_iff Bex_def set_eq_iff)
-      apply clarsimp
-      apply (presburger)
-      apply presburger
-    have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"
-"{1..n - Suc h} \<inter> {n - h .. n} = {}" and eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}" using h kn by auto
+        using h kn
+        apply (simp_all add: inj_on_def image_iff Bex_def set_eq_iff)
+        apply clarsimp
+        apply presburger
+       apply presburger
+      done
+    have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"
+        "{1..n - Suc h} \<inter> {n - h .. n} = {}" and
+        eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}"
+      using h kn by auto
from eq[symmetric]
have ?thesis using kn
-      apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
+      apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
gbinomial_pochhammer field_simps pochhammer_Suc_setprod del: minus_one)
-      apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc del: minus_one)
+      apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h
+        of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc del: minus_one)
unfolding setprod_Un_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
-      unfolding mult_assoc[symmetric]
+      unfolding mult_assoc[symmetric]
unfolding setprod_timesf[symmetric]
apply simp
apply (rule strong_setprod_reindex_cong[where f= "op - n"])
-      apply (auto simp add: inj_on_def image_iff Bex_def)
-      apply presburger
+        apply (auto simp add: inj_on_def image_iff Bex_def)
+       apply presburger
apply (subgoal_tac "(of_nat (n - x) :: 'a) = of_nat n - of_nat x")
+       apply simp
+      apply (rule of_nat_diff)
apply simp
-      by (rule of_nat_diff, simp)
+      done
}
moreover
have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith
@@ -441,72 +454,86 @@
lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"

-lemma gbinomial_mult_1: "a * (a gchoose n) = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" (is "?l = ?r")
-proof-
+lemma gbinomial_mult_1:
+  "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 / of_nat (fact n)) * (of_nat n - (- a + of_nat n))"
unfolding gbinomial_pochhammer
-    pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc
+      pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc
by (simp add:  field_simps del: of_nat_Suc)
also have "\<dots> = ?l" unfolding gbinomial_pochhammer
finally show ?thesis ..
qed

-lemma gbinomial_mult_1': "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
+lemma gbinomial_mult_1':
+    "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"

-lemma gbinomial_Suc: "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))"
+lemma gbinomial_Suc:
+    "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))"
-
+
lemma gbinomial_mult_fact:
-  "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
-  unfolding gbinomial_Suc
-  by (simp_all add: field_simps del: fact_Suc)
+  "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) =
+    (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
+  by (simp_all add: gbinomial_Suc field_simps del: fact_Suc)

lemma gbinomial_mult_fact':
-  "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
+  "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) =
+    (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
using gbinomial_mult_fact[of k a]
-  apply (subst mult_commute) .
+  apply (subst mult_commute)
+  apply assumption
+  done

-lemma gbinomial_Suc_Suc: "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
-proof-
-  {assume "k = 0" then have ?thesis by simp}
+
+lemma gbinomial_Suc_Suc:
+  "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
+proof -
+  { assume "k = 0" then have ?thesis by simp }
moreover
-  {fix h assume h: "k = Suc h"
-   have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
-     apply (rule strong_setprod_reindex_cong[where f = Suc])
-     using h by auto
+  { fix h assume h: "k = Suc h"
+    have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
+      apply (rule strong_setprod_reindex_cong[where f = Suc])
+        using h
+        apply auto
+      done

-    have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) = ((a gchoose Suc h) * of_nat (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0\<Colon>nat..Suc h}. a - of_nat i)"
-      unfolding h
-      apply (simp add: field_simps del: fact_Suc)
+    have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) =
+      ((a gchoose Suc h) * of_nat (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0\<Colon>nat..Suc h}. a - of_nat i)"
+      apply (simp add: h field_simps del: fact_Suc)
unfolding gbinomial_mult_fact'
apply (subst fact_Suc)
-      unfolding of_nat_mult
+      unfolding of_nat_mult
apply (subst mult_commute)
unfolding mult_assoc
unfolding gbinomial_mult_fact
+      done
also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
using eq0
-      unfolding h  setprod_nat_ivl_1_Suc
-      by simp
+      by (simp add: h setprod_nat_ivl_1_Suc)
also have "\<dots> = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
unfolding gbinomial_mult_fact ..
-    finally have ?thesis by (simp del: fact_Suc) }
-  ultimately show ?thesis by (cases k, auto)
+    finally have ?thesis by (simp del: fact_Suc)
+  }
+  ultimately show ?thesis by (cases k) auto
qed

-lemma binomial_symmetric: assumes kn: "k \<le> n"
+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
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
+  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
```