--- a/src/HOL/Lattices_Big.thy Wed Aug 22 12:31:57 2018 +0200
+++ b/src/HOL/Lattices_Big.thy Wed Aug 22 16:41:10 2018 +0200
@@ -798,41 +798,26 @@
context linordered_ab_semigroup_add
begin
-lemma add_Min_commute:
+lemma Min_add_commute:
fixes k
assumes "finite N" and "N \<noteq> {}"
- shows "k + Min N = Min {k + m | m. m \<in> N}"
+ shows "Min ((+) k ` N) = k + Min N"
proof -
- have "\<And>x y. k + min x y = min (k + x) (k + y)"
- by (simp add: min_def not_le)
- (blast intro: antisym less_imp_le add_left_mono)
+ have "\<And>x y. k + min x y = min (k+x) (k+y)"
+ by(simp add: min_def antisym add_left_mono)
with assms show ?thesis
- using hom_Min_commute [of "plus k" N]
- by simp (blast intro: arg_cong [where f = Min])
+ using hom_Min_commute [of "(+) k" N, symmetric] by simp
qed
lemma Max_add_commute:
fixes k
assumes "finite N" and "N \<noteq> {}"
- shows "Max((\<lambda>x. x+k) ` N) = Max N + k"
+ shows "Max ((+) k ` N) = k + Max N"
proof -
- have "\<And>x y. max x y + k = max (x+k) (y+k)"
- by(simp add: max_def antisym add_right_mono)
+ have "\<And>x y. k + max x y = max (k+x) (k+y)"
+ by(simp add: max_def antisym add_left_mono)
with assms show ?thesis
- using hom_Max_commute [of "\<lambda>x. x+k" N, symmetric] by simp
-qed
-
-lemma add_Max_commute:
- fixes k
- assumes "finite N" and "N \<noteq> {}"
- shows "k + Max N = Max {k + m | m. m \<in> N}"
-proof -
- have "\<And>x y. k + max x y = max (k + x) (k + y)"
- by (simp add: max_def not_le)
- (blast intro: antisym less_imp_le add_left_mono)
- with assms show ?thesis
- using hom_Max_commute [of "plus k" N]
- by simp (blast intro: arg_cong [where f = Max])
+ using hom_Max_commute [of "(+) k" N, symmetric] by simp
qed
end