(* Title: HOL/Recdef.thy
ID: $Id$
Author: Konrad Slind and Markus Wenzel, TU Muenchen
*)
header {* TFL: recursive function definitions *}
theory Recdef
imports Wellfounded_Relations Datatype
uses
("../TFL/casesplit.ML")
("../TFL/utils.ML")
("../TFL/usyntax.ML")
("../TFL/dcterm.ML")
("../TFL/thms.ML")
("../TFL/rules.ML")
("../TFL/thry.ML")
("../TFL/tfl.ML")
("../TFL/post.ML")
("Tools/recdef_package.ML")
begin
lemma tfl_eq_True: "(x = True) --> x"
by blast
lemma tfl_rev_eq_mp: "(x = y) --> y --> x";
by blast
lemma tfl_simp_thm: "(x --> y) --> (x = x') --> (x' --> y)"
by blast
lemma tfl_P_imp_P_iff_True: "P ==> P = True"
by blast
lemma tfl_imp_trans: "(A --> B) ==> (B --> C) ==> (A --> C)"
by blast
lemma tfl_disj_assoc: "(a \<or> b) \<or> c == a \<or> (b \<or> c)"
by simp
lemma tfl_disjE: "P \<or> Q ==> P --> R ==> Q --> R ==> R"
by blast
lemma tfl_exE: "\<exists>x. P x ==> \<forall>x. P x --> Q ==> Q"
by blast
use "../TFL/casesplit.ML"
use "../TFL/utils.ML"
use "../TFL/usyntax.ML"
use "../TFL/dcterm.ML"
use "../TFL/thms.ML"
use "../TFL/rules.ML"
use "../TFL/thry.ML"
use "../TFL/tfl.ML"
use "../TFL/post.ML"
use "Tools/recdef_package.ML"
setup RecdefPackage.setup
lemmas [recdef_simp] =
inv_image_def
measure_def
lex_prod_def
same_fst_def
less_Suc_eq [THEN iffD2]
lemmas [recdef_cong] =
if_cong image_cong INT_cong UN_cong bex_cong ball_cong imp_cong
lemma let_cong [recdef_cong]:
"M = N ==> (!!x. x = N ==> f x = g x) ==> Let M f = Let N g"
by (unfold Let_def) blast
lemmas [recdef_wf] =
wf_trancl
wf_less_than
wf_lex_prod
wf_inv_image
wf_measure
wf_pred_nat
wf_same_fst
wf_empty
(* The following should really go into Datatype or Finite_Set, but
each one lacks the other theory as a parent . . . *)
lemma insert_None_conv_UNIV: "insert None (range Some) = UNIV"
by (rule set_ext, case_tac x, auto)
instance option :: (finite) finite
proof
have "finite (UNIV :: 'a set)" by (rule finite)
hence "finite (insert None (Some ` (UNIV :: 'a set)))" by simp
also have "insert None (Some ` (UNIV :: 'a set)) = UNIV"
by (rule insert_None_conv_UNIV)
finally show "finite (UNIV :: 'a option set)" .
qed
end