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open LatPreInsts;
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(** complete lattices **)
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goal thy "is_inf x y (Inf {x, y})";
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br (bin_is_Inf_eq RS subst) 1;
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br Inf_is_Inf 1;
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qed "Inf_is_inf";
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goal thy "is_sup x y (Sup {x, y})";
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br (bin_is_Sup_eq RS subst) 1;
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br Sup_is_Sup 1;
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qed "Sup_is_sup";
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(** product lattices **)
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(* pairs *)
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goalw thy [is_inf_def, le_prod_def] "is_inf p q (fst p && fst q, snd p && snd q)";
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1573
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by (Simp_tac 1);
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1440
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by (safe_tac HOL_cs);
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by (REPEAT_FIRST (fn i => resolve_tac [inf_lb1, inf_lb2, inf_ub_lbs] i ORELSE atac i));
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qed "prod_is_inf";
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goalw thy [is_sup_def, le_prod_def] "is_sup p q (fst p || fst q, snd p || snd q)";
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1573
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by (Simp_tac 1);
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1440
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by (safe_tac HOL_cs);
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by (REPEAT_FIRST (fn i => resolve_tac [sup_ub1, sup_ub2, sup_lb_ubs] i ORELSE atac i));
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qed "prod_is_sup";
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(* functions *)
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goalw thy [is_inf_def, le_fun_def] "is_inf f g (%x. f x && g x)";
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by (safe_tac HOL_cs);
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br inf_lb1 1;
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br inf_lb2 1;
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br inf_ub_lbs 1;
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by (REPEAT_FIRST (fast_tac HOL_cs));
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qed "fun_is_inf";
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goalw thy [is_sup_def, le_fun_def] "is_sup f g (%x. f x || g x)";
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by (safe_tac HOL_cs);
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br sup_ub1 1;
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br sup_ub2 1;
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br sup_lb_ubs 1;
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by (REPEAT_FIRST (fast_tac HOL_cs));
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qed "fun_is_sup";
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(** dual lattices **)
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goalw thy [is_inf_def, le_dual_def] "is_inf x y (Abs_dual (Rep_dual x || Rep_dual y))";
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by (stac Abs_dual_inverse' 1);
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by (safe_tac HOL_cs);
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br sup_ub1 1;
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br sup_ub2 1;
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br sup_lb_ubs 1;
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ba 1;
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ba 1;
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qed "dual_is_inf";
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goalw thy [is_sup_def, le_dual_def] "is_sup x y (Abs_dual (Rep_dual x && Rep_dual y))";
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by (stac Abs_dual_inverse' 1);
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by (safe_tac HOL_cs);
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br inf_lb1 1;
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br inf_lb2 1;
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br inf_ub_lbs 1;
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ba 1;
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ba 1;
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qed "dual_is_sup";
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