Theory Quotient_Option

Up to index of Isabelle/HOL/Library

theory Quotient_Option
imports Quotient_Syntax

(*  Title:      HOL/Library/Quotient_Option.thy
Author: Cezary Kaliszyk and Christian Urban
*)


header {* Quotient infrastructure for the option type *}

theory Quotient_Option
imports Main Quotient_Syntax
begin


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 "(∀x. P x) <-> P None ∧ (∀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