equal
  deleted
  inserted
  replaced
  
    
    
|     18 unfolding convex_le_def by (fast intro: upper_le_refl lower_le_refl) |     18 unfolding convex_le_def by (fast intro: upper_le_refl lower_le_refl) | 
|     19  |     19  | 
|     20 lemma convex_le_trans: "\<lbrakk>t \<le>\<natural> u; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> t \<le>\<natural> v" |     20 lemma convex_le_trans: "\<lbrakk>t \<le>\<natural> u; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> t \<le>\<natural> v" | 
|     21 unfolding convex_le_def by (fast intro: upper_le_trans lower_le_trans) |     21 unfolding convex_le_def by (fast intro: upper_le_trans lower_le_trans) | 
|     22  |     22  | 
|     23 interpretation convex_le!: preorder convex_le |     23 interpretation convex_le: preorder convex_le | 
|     24 by (rule preorder.intro, rule convex_le_refl, rule convex_le_trans) |     24 by (rule preorder.intro, rule convex_le_refl, rule convex_le_trans) | 
|     25  |     25  | 
|     26 lemma upper_le_minimal [simp]: "PDUnit compact_bot \<le>\<natural> t" |     26 lemma upper_le_minimal [simp]: "PDUnit compact_bot \<le>\<natural> t" | 
|     27 unfolding convex_le_def Rep_PDUnit by simp |     27 unfolding convex_le_def Rep_PDUnit by simp | 
|     28  |     28  | 
|    176 lemma Rep_convex_principal: |    176 lemma Rep_convex_principal: | 
|    177   "Rep_convex_pd (convex_principal t) = {u. u \<le>\<natural> t}" |    177   "Rep_convex_pd (convex_principal t) = {u. u \<le>\<natural> t}" | 
|    178 unfolding convex_principal_def |    178 unfolding convex_principal_def | 
|    179 by (simp add: Abs_convex_pd_inverse convex_le.ideal_principal) |    179 by (simp add: Abs_convex_pd_inverse convex_le.ideal_principal) | 
|    180  |    180  | 
|    181 interpretation convex_pd!: |    181 interpretation convex_pd: | 
|    182   ideal_completion convex_le pd_take convex_principal Rep_convex_pd |    182   ideal_completion convex_le pd_take convex_principal Rep_convex_pd | 
|    183 apply unfold_locales |    183 apply unfold_locales | 
|    184 apply (rule pd_take_convex_le) |    184 apply (rule pd_take_convex_le) | 
|    185 apply (rule pd_take_idem) |    185 apply (rule pd_take_idem) | 
|    186 apply (erule pd_take_convex_mono) |    186 apply (erule pd_take_convex_mono) |