# Theory Old_Recdef

```(*  Title:      HOL/Library/Old_Recdef.thy
Author:     Konrad Slind and Markus Wenzel, TU Muenchen
*)

section ‹TFL: recursive function definitions›

theory Old_Recdef
imports Main
keywords
"recdef" :: thy_defn and
"permissive" "congs" "hints"
begin

subsection ‹Lemmas for TFL›

lemma tfl_wf_induct: "∀R. wf R ⟶
(∀P. (∀x. (∀y. (y,x)∈R ⟶ P y) ⟶ P x) ⟶ (∀x. P x))"
apply clarify
apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)
done

lemma tfl_cut_def: "cut f r x ≡ (λy. if (y,x) ∈ r then f y else undefined)"
unfolding cut_def .

lemma tfl_cut_apply: "∀f R. (x,a)∈R ⟶ (cut f R a)(x) = f(x)"
apply clarify
apply (rule cut_apply, assumption)
done

lemma tfl_wfrec:
"∀M R f. (f=wfrec R M) ⟶ wf R ⟶ (∀x. f x = M (cut f R x) x)"
apply clarify
apply (erule wfrec)
done

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 ∨ b) ∨ c ≡ a ∨ (b ∨ c)"
by simp

lemma tfl_disjE: "P ∨ Q ⟹ P ⟶ R ⟹ Q ⟶ R ⟹ R"
by blast

lemma tfl_exE: "∃x. P x ⟹ ∀x. P x ⟶ Q ⟹ Q"
by blast

ML_file ‹old_recdef.ML›

subsection ‹Rule 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 let_cong image_cong INF_cong SUP_cong bex_cong ball_cong imp_cong
map_cong filter_cong takeWhile_cong dropWhile_cong foldl_cong foldr_cong

lemmas [recdef_wf] =
wf_trancl
wf_less_than
wf_lex_prod
wf_inv_image
wf_measure
wf_measures
wf_pred_nat
wf_same_fst
wf_empty

end
```