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