(* Title: HOL/BNF_Util.thy
Author: Dmitriy Traytel, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Library for bounded natural functors.
*)
header {* Library for Bounded Natural Functors *}
theory BNF_Util
imports BNF_Cardinal_Arithmetic
begin
definition
fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"
where
"fun_rel A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
lemma fun_relI [intro]:
assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)"
shows "fun_rel A B f g"
using assms by (simp add: fun_rel_def)
lemma fun_relD:
assumes "fun_rel A B f g" and "A x y"
shows "B (f x) (g y)"
using assms by (simp add: fun_rel_def)
definition collect where
"collect F x = (\<Union>f \<in> F. f x)"
lemma fstI: "x = (y, z) \<Longrightarrow> fst x = y"
by simp
lemma sndI: "x = (y, z) \<Longrightarrow> snd x = z"
by simp
lemma bijI': "\<lbrakk>\<And>x y. (f x = f y) = (x = y); \<And>y. \<exists>x. y = f x\<rbrakk> \<Longrightarrow> bij f"
unfolding bij_def inj_on_def by auto blast
(* Operator: *)
definition "Gr A f = {(a, f a) | a. a \<in> A}"
definition "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
ML_file "Tools/BNF/bnf_util.ML"
ML_file "Tools/BNF/bnf_tactics.ML"
end