src/HOL/Library/Product_Order.thy
 changeset 69313 b021008c5397 parent 69260 0a9688695a1b child 69861 62e47f06d22c
```     1.1 --- a/src/HOL/Library/Product_Order.thy	Sun Nov 18 09:51:41 2018 +0100
1.2 +++ b/src/HOL/Library/Product_Order.thy	Sun Nov 18 18:07:51 2018 +0000
1.3 @@ -220,11 +220,11 @@
1.4  of two complete lattices:\<close>
1.5
1.6  lemma INF_prod_alt_def:
1.7 -  "INFIMUM A f = (INFIMUM A (fst \<circ> f), INFIMUM A (snd \<circ> f))"
1.8 +  "Inf (f ` A) = (Inf ((fst \<circ> f) ` A), Inf ((snd \<circ> f) ` A))"
1.9    unfolding Inf_prod_def by simp
1.10
1.11  lemma SUP_prod_alt_def:
1.12 -  "SUPREMUM A f = (SUPREMUM A (fst \<circ> f), SUPREMUM A (snd \<circ> f))"
1.13 +  "Sup (f ` A) = (Sup ((fst \<circ> f) ` A), Sup((snd \<circ> f) ` A))"
1.14    unfolding Sup_prod_def by simp
1.15
1.16
1.17 @@ -235,7 +235,7 @@
1.18  instance prod :: (complete_distrib_lattice, complete_distrib_lattice) complete_distrib_lattice
1.19  proof
1.20    fix A::"('a\<times>'b) set set"
1.21 -  show "INFIMUM A Sup \<le> SUPREMUM {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y} Inf"
1.22 +  show "Inf (Sup ` A) \<le> Sup (Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})"
1.23      by (simp add: Sup_prod_def Inf_prod_def INF_SUP_set)
1.24  qed
1.25
```