(* Title: HOL/FunDef.thy
ID: $Id$
Author: Alexander Krauss, TU Muenchen
A package for general recursive function definitions.
*)
theory FunDef
imports Accessible_Part Datatype Recdef
uses
("Tools/function_package/sum_tools.ML")
("Tools/function_package/fundef_common.ML")
("Tools/function_package/fundef_lib.ML")
("Tools/function_package/inductive_wrap.ML")
("Tools/function_package/context_tree.ML")
("Tools/function_package/fundef_prep.ML")
("Tools/function_package/fundef_proof.ML")
("Tools/function_package/termination.ML")
("Tools/function_package/mutual.ML")
("Tools/function_package/pattern_split.ML")
("Tools/function_package/fundef_package.ML")
("Tools/function_package/fundef_datatype.ML")
("Tools/function_package/auto_term.ML")
begin
definition
THE_default :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a"
"THE_default d P = (if (\<exists>!x. P x) then (THE x. P x) else d)"
lemma THE_defaultI': "\<exists>!x. P x \<Longrightarrow> P (THE_default d P)"
by (simp add:theI' THE_default_def)
lemma THE_default1_equality:
"\<lbrakk>\<exists>!x. P x; P a\<rbrakk> \<Longrightarrow> THE_default d P = a"
by (simp add:the1_equality THE_default_def)
lemma THE_default_none:
"\<not>(\<exists>!x. P x) \<Longrightarrow> THE_default d P = d"
by (simp add:THE_default_def)
lemma fundef_ex1_existence:
assumes f_def: "f \<equiv> \<lambda>x. THE_default (d x) (\<lambda>y. (x,y)\<in>G)"
assumes ex1: "\<exists>!y. (x,y)\<in>G"
shows "(x, f x)\<in>G"
by (simp only:f_def, rule THE_defaultI', rule ex1)
lemma fundef_ex1_uniqueness:
assumes f_def: "f \<equiv> \<lambda>x. THE_default (d x) (\<lambda>y. (x,y)\<in>G)"
assumes ex1: "\<exists>!y. (x,y)\<in>G"
assumes elm: "(x, h x)\<in>G"
shows "h x = f x"
by (simp only:f_def, rule THE_default1_equality[symmetric], rule ex1, rule elm)
lemma fundef_ex1_iff:
assumes f_def: "f \<equiv> \<lambda>x. THE_default (d x) (\<lambda>y. (x,y)\<in>G)"
assumes ex1: "\<exists>!y. (x,y)\<in>G"
shows "((x, y)\<in>G) = (f x = y)"
apply (auto simp:ex1 f_def THE_default1_equality)
by (rule THE_defaultI', rule ex1)
lemma fundef_default_value:
assumes f_def: "f \<equiv> \<lambda>x. THE_default (d x) (\<lambda>y. (x,y)\<in>G)"
assumes graph: "\<And>x y. (x,y) \<in> G \<Longrightarrow> x \<in> D"
assumes "x \<notin> D"
shows "f x = d x"
proof -
have "\<not>(\<exists>y. (x, y) \<in> G)"
proof
assume "(\<exists>y. (x, y) \<in> G)"
with graph and `x\<notin>D` show False by blast
qed
hence "\<not>(\<exists>!y. (x, y) \<in> G)" by blast
thus ?thesis
unfolding f_def
by (rule THE_default_none)
qed
subsection {* Projections *}
consts
lpg::"(('a + 'b) * 'a) set"
rpg::"(('a + 'b) * 'b) set"
inductive lpg
intros
"(Inl x, x) : lpg"
inductive rpg
intros
"(Inr y, y) : rpg"
definition
"lproj x = (THE y. (x,y) : lpg)"
"rproj x = (THE y. (x,y) : rpg)"
lemma lproj_inl:
"lproj (Inl x) = x"
by (auto simp:lproj_def intro: the_equality lpg.intros elim: lpg.cases)
lemma rproj_inr:
"rproj (Inr x) = x"
by (auto simp:rproj_def intro: the_equality rpg.intros elim: rpg.cases)
use "Tools/function_package/sum_tools.ML"
use "Tools/function_package/fundef_common.ML"
use "Tools/function_package/fundef_lib.ML"
use "Tools/function_package/inductive_wrap.ML"
use "Tools/function_package/context_tree.ML"
use "Tools/function_package/fundef_prep.ML"
use "Tools/function_package/fundef_proof.ML"
use "Tools/function_package/termination.ML"
use "Tools/function_package/mutual.ML"
use "Tools/function_package/pattern_split.ML"
use "Tools/function_package/fundef_package.ML"
setup FundefPackage.setup
use "Tools/function_package/fundef_datatype.ML"
setup FundefDatatype.setup
use "Tools/function_package/auto_term.ML"
setup FundefAutoTerm.setup
lemmas [fundef_cong] =
let_cong if_cong image_cong INT_cong UN_cong bex_cong ball_cong imp_cong
lemma split_cong[fundef_cong]:
"\<lbrakk> \<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y; p = q \<rbrakk>
\<Longrightarrow> split f p = split g q"
by (auto simp:split_def)
end