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theory QuotOption
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imports QuotMain
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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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fun
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option_map
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where
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"option_map f None = None"
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| "option_map f (Some x) = Some (f x)"
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3b21b24a5fb6
corrected map declarations for Sum and Prod; moved absrep_fun examples in separate file
Christian Urban <urbanc@in.tum.de>
diff
changeset
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declare [[map option = (option_map, option_rel)]]
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3b21b24a5fb6
corrected map declarations for Sum and Prod; moved absrep_fun examples in separate file
Christian Urban <urbanc@in.tum.de>
diff
changeset
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698
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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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923
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apply(unfold Quotient_def)
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apply(auto)
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apply(case_tac a, simp_all add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
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apply(case_tac a, simp_all add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
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apply(case_tac [!] r)
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apply(case_tac [!] s)
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apply(simp_all add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q] )
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698
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using q
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unfolding Quotient_def
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923
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apply(blast)+
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done
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698
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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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923
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apply(rule equivpI)
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unfolding reflp_def symp_def transp_def
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apply(auto)
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apply(case_tac [!] x)
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apply(simp_all add: equivp_reflp[OF a])
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apply(case_tac [!] y)
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apply(simp_all add: equivp_symp[OF a])
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apply(case_tac [!] z)
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698
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apply(simp_all)
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923
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apply(clarify)
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698
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apply(rule equivp_transp[OF a])
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apply(assumption)+
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done
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923
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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_respect]:
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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 \<equiv> id"
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829
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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apply (rule eq_reflection)
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923
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apply (auto simp add: expand_fun_eq)
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829
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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apply (case_tac x)
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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apply (auto)
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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done
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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923
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lemma option_rel_eq[id_simps]:
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925
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shows "option_rel (op =) \<equiv> (op =)"
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923
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apply(rule eq_reflection)
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apply(auto simp add: expand_fun_eq)
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apply(case_tac x)
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apply(case_tac [!] xa)
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apply(auto)
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829
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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done
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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92 |
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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end
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