src/HOL/FunDef.thy

added congruence rule for function composition

(*  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_core.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/auto_term.ML")
begin

section {* Definitions with default value *}

definition
  THE_default :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" where
  "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 == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
assumes ex1: "\<exists>!y. G x y"
shows "G x (f x)"
  by (simp only:f_def, rule THE_defaultI', rule ex1)





lemma fundef_ex1_uniqueness:
assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
assumes ex1: "\<exists>!y. G x y"
assumes elm: "G x (h x)"
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 == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
assumes ex1: "\<exists>!y. G x y"
shows "(G x y) = (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 == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
assumes graph: "\<And>x y. G x y \<Longrightarrow> D x"
assumes "\<not> D x"
shows "f x = d x"
proof -
  have "\<not>(\<exists>y. G x y)"
  proof
    assume "\<exists>y. G x y"
    hence "D x" using graph ..
    with `\<not> D x` show False ..
  qed
  hence "\<not>(\<exists>!y. G x y)" by blast
  
  thus ?thesis
    unfolding f_def
    by (rule THE_default_none)
qed



section {* Projections *}

inductive2 lpg :: "('a + 'b) \<Rightarrow> 'a \<Rightarrow> bool"
where
  "lpg (Inl x) x"
inductive2 rpg :: "('a + 'b) \<Rightarrow> 'b \<Rightarrow> bool"
where
  "rpg (Inr y) y"

definition "lproj x = (THE y. lpg x y)"
definition "rproj x = (THE y. rpg x y)"

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_core.ML"
use "Tools/function_package/termination.ML"
use "Tools/function_package/mutual.ML"
use "Tools/function_package/pattern_split.ML"
use "Tools/function_package/auto_term.ML"
use "Tools/function_package/fundef_package.ML"

setup FundefPackage.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)

lemma comp_cong[fundef_cong]:
  "f (g x) = f' (g' x')
  ==>  (f o g) x = (f' o g') x'"
unfolding o_apply .

end