equal
deleted
inserted
replaced
43 |
43 |
44 definition ubs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set" |
44 definition ubs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set" |
45 where "ubs R S = Collect (isUb R S)" |
45 where "ubs R S = Collect (isUb R S)" |
46 |
46 |
47 |
47 |
48 subsection \<open>Rules about the Operators @{term leastP}, @{term ub} and @{term lub}\<close> |
48 subsection \<open>Rules about the Operators \<^term>\<open>leastP\<close>, \<^term>\<open>ub\<close> and \<^term>\<open>lub\<close>\<close> |
49 |
49 |
50 lemma leastPD1: "leastP P x \<Longrightarrow> P x" |
50 lemma leastPD1: "leastP P x \<Longrightarrow> P x" |
51 by (simp add: leastP_def) |
51 by (simp add: leastP_def) |
52 |
52 |
53 lemma leastPD2: "leastP P x \<Longrightarrow> x <=* Collect P" |
53 lemma leastPD2: "leastP P x \<Longrightarrow> x <=* Collect P" |
116 |
116 |
117 definition lbs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set" |
117 definition lbs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set" |
118 where "lbs R S = Collect (isLb R S)" |
118 where "lbs R S = Collect (isLb R S)" |
119 |
119 |
120 |
120 |
121 subsection \<open>Rules about the Operators @{term greatestP}, @{term isLb} and @{term isGlb}\<close> |
121 subsection \<open>Rules about the Operators \<^term>\<open>greatestP\<close>, \<^term>\<open>isLb\<close> and \<^term>\<open>isGlb\<close>\<close> |
122 |
122 |
123 lemma greatestPD1: "greatestP P x \<Longrightarrow> P x" |
123 lemma greatestPD1: "greatestP P x \<Longrightarrow> P x" |
124 by (simp add: greatestP_def) |
124 by (simp add: greatestP_def) |
125 |
125 |
126 lemma greatestPD2: "greatestP P x \<Longrightarrow> Collect P *<= x" |
126 lemma greatestPD2: "greatestP P x \<Longrightarrow> Collect P *<= x" |