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/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
lemma fundef_ex1_existence:
assumes f_def: "\<And>x. f x \<equiv> THE 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 theI', rule ex1)
lemma fundef_ex1_uniqueness:
assumes f_def: "\<And>x. f x \<equiv> THE 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 the1_equality[symmetric], rule ex1, rule elm)
lemma fundef_ex1_iff:
assumes f_def: "\<And>x. f x \<equiv> THE 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 the1_equality)
by (rule theI', rule ex1)
lemma True_implies: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
by simp
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/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