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