--- a/src/HOL/Library/Stirling.thy Wed Apr 10 13:34:55 2019 +0100
+++ b/src/HOL/Library/Stirling.thy Wed Apr 10 21:29:32 2019 +0100
@@ -156,7 +156,7 @@
next
case (Suc n)
have "(\<Sum>k\<le>Suc n. stirling (Suc n) k) = stirling (Suc n) 0 + (\<Sum>k\<le>n. stirling (Suc n) (Suc k))"
- by (simp only: sum_atMost_Suc_shift)
+ by (simp only: sum.atMost_Suc_shift)
also have "\<dots> = (\<Sum>k\<le>n. stirling (Suc n) (Suc k))"
by simp
also have "\<dots> = (\<Sum>k\<le>n. n * stirling n (Suc k) + stirling n k)"
@@ -166,7 +166,7 @@
also have "\<dots> = n * fact n + fact n"
proof -
have "n * (\<Sum>k\<le>n. stirling n (Suc k)) = n * ((\<Sum>k\<le>Suc n. stirling n k) - stirling n 0)"
- by (metis add_diff_cancel_left' sum_atMost_Suc_shift)
+ by (metis add_diff_cancel_left' sum.atMost_Suc_shift)
also have "\<dots> = n * (\<Sum>k\<le>n. stirling n k)"
by (cases n) simp_all
also have "\<dots> = n * fact n"
@@ -192,9 +192,9 @@
(of_nat (n * stirling n 0) * x ^ 0 +
(\<Sum>i\<le>n. of_nat (n * stirling n (Suc i)) * (x ^ Suc i))) +
(\<Sum>i\<le>n. of_nat (stirling n i) * (x ^ Suc i))"
- by (subst sum_atMost_Suc_shift) (simp add: sum.distrib ring_distribs)
+ by (subst sum.atMost_Suc_shift) (simp add: sum.distrib ring_distribs)
also have "\<dots> = pochhammer x (Suc n)"
- by (subst sum_atMost_Suc_shift [symmetric])
+ by (subst sum.atMost_Suc_shift [symmetric])
(simp add: algebra_simps sum.distrib sum_distrib_left pochhammer_Suc flip: Suc)
finally show ?case .
qed