src/HOL/Library/Quotient_Option.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Quotient_Option.thy	Fri Feb 19 13:54:19 2010 +0100
@@ -0,0 +1,80 @@
+(*  Title:      Quotient_Option.thy
+    Author:     Cezary Kaliszyk and Christian Urban
+*)
+theory Quotient_Option
+imports Main Quotient_Syntax
+begin
+
+section {* Quotient infrastructure for the option type. *}
+
+fun
+  option_rel
+where
+  "option_rel R None None = True"
+| "option_rel R (Some x) None = False"
+| "option_rel R None (Some x) = False"
+| "option_rel R (Some x) (Some y) = R x y"
+
+declare [[map option = (Option.map, option_rel)]]
+
+text {* should probably be in Option.thy *}
+lemma split_option_all:
+  shows "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>a. P (Some a))"
+  apply(auto)
+  apply(case_tac x)
+  apply(simp_all)
+  done
+
+lemma option_quotient[quot_thm]:
+  assumes q: "Quotient R Abs Rep"
+  shows "Quotient (option_rel R) (Option.map Abs) (Option.map Rep)"
+  unfolding Quotient_def
+  apply(simp add: split_option_all)
+  apply(simp add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
+  using q
+  unfolding Quotient_def
+  apply(blast)
+  done
+
+lemma option_equivp[quot_equiv]:
+  assumes a: "equivp R"
+  shows "equivp (option_rel R)"
+  apply(rule equivpI)
+  unfolding reflp_def symp_def transp_def
+  apply(simp_all add: split_option_all)
+  apply(blast intro: equivp_reflp[OF a])
+  apply(blast intro: equivp_symp[OF a])
+  apply(blast intro: equivp_transp[OF a])
+  done
+
+lemma option_None_rsp[quot_respect]:
+  assumes q: "Quotient R Abs Rep"
+  shows "option_rel R None None"
+  by simp
+
+lemma option_Some_rsp[quot_respect]:
+  assumes q: "Quotient R Abs Rep"
+  shows "(R ===> option_rel R) Some Some"
+  by simp
+
+lemma option_None_prs[quot_preserve]:
+  assumes q: "Quotient R Abs Rep"
+  shows "Option.map Abs None = None"
+  by simp
+
+lemma option_Some_prs[quot_preserve]:
+  assumes q: "Quotient R Abs Rep"
+  shows "(Rep ---> Option.map Abs) Some = Some"
+  apply(simp add: expand_fun_eq)
+  apply(simp add: Quotient_abs_rep[OF q])
+  done
+
+lemma option_map_id[id_simps]:
+  shows "Option.map id = id"
+  by (simp add: expand_fun_eq split_option_all)
+
+lemma option_rel_eq[id_simps]:
+  shows "option_rel (op =) = (op =)"
+  by (simp add: expand_fun_eq split_option_all)
+
+end