(* Title: HOL/Basic_BNF_Least_Fixpoints.thy
Author: Jasmin Blanchette, TU Muenchen
Copyright 2014
Registration of basic types as BNF least fixpoints (datatypes).
*)
theory Basic_BNF_Least_Fixpoints
imports BNF_Least_Fixpoint
begin
subsection {* Size setup (TODO: Merge with rest of file) *}
lemma size_bool[code]: "size (b\<Colon>bool) = 0"
by (cases b) auto
lemma size_nat[simp, code]: "size (n\<Colon>nat) = n"
by (induct n) simp_all
declare prod.size[no_atp]
lemma size_sum_o_map: "size_sum g1 g2 \<circ> map_sum f1 f2 = size_sum (g1 \<circ> f1) (g2 \<circ> f2)"
by (rule ext) (case_tac x, auto)
lemma size_prod_o_map: "size_prod g1 g2 \<circ> map_prod f1 f2 = size_prod (g1 \<circ> f1) (g2 \<circ> f2)"
by (rule ext) auto
setup {*
BNF_LFP_Size.register_size_global @{type_name sum} @{const_name size_sum} @{thms sum.size}
@{thms size_sum_o_map}
#> BNF_LFP_Size.register_size_global @{type_name prod} @{const_name size_prod} @{thms prod.size}
@{thms size_prod_o_map}
*}
subsection {* FP sugar setup *}
definition xtor :: "'a \<Rightarrow> 'a" where
"xtor x = x"
lemma xtor_map: "f (xtor x) = xtor (f x)"
unfolding xtor_def by (rule refl)
lemma xtor_set: "f (xtor x) = f x"
unfolding xtor_def by (rule refl)
lemma xtor_rel: "R (xtor x) (xtor y) = R x y"
unfolding xtor_def by (rule refl)
lemma xtor_induct: "(\<And>x. P (xtor x)) \<Longrightarrow> P z"
unfolding xtor_def by assumption
lemma xtor_xtor: "xtor (xtor x) = x"
unfolding xtor_def by (rule refl)
lemmas xtor_inject = xtor_rel[of "op ="]
definition ctor_rec :: "'a \<Rightarrow> 'a" where
"ctor_rec x = x"
lemma ctor_rec: "g = id \<Longrightarrow> ctor_rec f (xtor x) = f ((id_bnf \<circ> g \<circ> id_bnf) x)"
unfolding ctor_rec_def id_bnf_def xtor_def comp_def id_def by hypsubst (rule refl)
lemma ctor_rec_o_map: "ctor_rec f \<circ> g = ctor_rec (f \<circ> (id_bnf \<circ> g \<circ> id_bnf))"
unfolding ctor_rec_def id_bnf_def comp_def by (rule refl)
lemma xtor_rel_induct: "(\<And>x y. vimage2p id_bnf id_bnf R x y \<Longrightarrow> IR (xtor x) (xtor y)) \<Longrightarrow> R \<le> IR"
unfolding xtor_def vimage2p_def id_bnf_def by default
lemma Inl_def_alt: "Inl \<equiv> (\<lambda>a. xtor (id_bnf (Inl a)))"
unfolding xtor_def id_bnf_def by (rule reflexive)
lemma Inr_def_alt: "Inr \<equiv> (\<lambda>a. xtor (id_bnf (Inr a)))"
unfolding xtor_def id_bnf_def by (rule reflexive)
lemma Pair_def_alt: "Pair \<equiv> (\<lambda>a b. xtor (id_bnf (a, b)))"
unfolding xtor_def id_bnf_def by (rule reflexive)
ML_file "Tools/BNF/bnf_lfp_basic_sugar.ML"
hide_const (open) xtor ctor_rec
hide_fact (open)
xtor_def xtor_map xtor_set xtor_rel xtor_induct xtor_xtor xtor_inject ctor_rec_def ctor_rec
ctor_rec_o_map xtor_rel_induct Inl_def_alt Inr_def_alt Pair_def_alt
end