# Theory Simple_Nesting

theory Simple_Nesting
imports BNF_Corec
```(*  Title:      HOL/Corec_Examples/Tests/Simple_Nesting.thy
Author:     Aymeric Bouzy, Ecole polytechnique
Author:     Jasmin Blanchette, Inria, LORIA, MPII

Tests "corec"'s parsing of map functions.
*)

section ‹Tests "corec"'s Parsing of Map Functions›

theory Simple_Nesting
imports "HOL-Library.BNF_Corec"
begin

subsection ‹Corecursion via Map Functions›

consts
g :: 'a
g' :: 'a
g'' :: 'a
h :: 'a
h' :: 'a
q :: 'a
q' :: 'a

codatatype tree =
Node nat "tree list"

(* a direct, intuitive way to define a function *)
corec k0 where
"k0 x = Node (g x) (map (λy. if q y then g' y else k0 (h' y :: 'a)) (h (x :: 'a) :: nat list))"

(* another way -- this one is perhaps less intuitive but more systematic *)
corec k0' where
"k0' x = Node (g x) (map (λz. case z of Inl t ⇒ t | Inr (x' :: 'a) ⇒ k0' x')
(map (λy. if q y then Inl (g' y) else Inr (h' y)) (h (x :: 'a) :: nat list)))"

(* more examples, same story *)

corec k1 :: "nat ⇒ tree" where
"k1 x = Node (g x) (map (λy. k1 (h' y)) (h x :: nat list))"

corec k1' :: "nat ⇒ tree" where
"k1' x = Node (g x) (map (λz. case z of Inl t ⇒ t | Inr x' ⇒ k1' x')
(map (λy. Inr (h' y)) (h x :: nat list)))"

corec k2 :: "nat ⇒ tree" where
"k2 x = Node (g x) (map g' (h x :: nat list))"

corec k2' :: "nat ⇒ tree" where
"k2' x = Node (g x) (map (λz. case z of Inl t ⇒ t | Inr (x' :: nat) ⇒ k2' x')
(map (λy. Inl (g' y)) (h x :: nat list)))"

corec k3 :: "nat ⇒ tree" where
"k3 x = Node (g x) (h x)"

corec k3' :: "nat ⇒ tree" where
"k3' x = Node (g x) (map (λz. case z of Inl t ⇒ t | Inr (x' :: nat) ⇒ k3' x')
(map Inl (h x)))"

codatatype 'a y = Y 'a "'a y list"

corec hh where
"hh a = Y a (map hh [a])"

corec k where
"k a = Y a [k a, k undefined, k (a + a), undefined, k a]"

codatatype 'a x = X 'a "'a x option"

corec f where
"f a = X a (map_option f (Some a))"

corec gg where
"gg a = X a (Some (gg a))"

subsection ‹Some Friends›

codatatype u =
U (lab: nat) (sub: "u list")

definition u_id :: "u ⇒ u" where "u_id u = u"

friend_of_corec u_id where
"u_id u = U (lab u) (sub u)"

corec u_id_f :: "u ⇒ u" where
"u_id_f u = u_id (U (lab u) (map u_id_f (sub u)))"

corec (friend) u_id_g :: "u ⇒ u" where
"u_id_g u = U (lab u) (map (λu'. u_id (u_id_g u')) (sub u))"

corec (friend) u_id_g' :: "u ⇒ u" where
"u_id_g' u = U (lab u) (map (u_id o u_id_g') (sub u))"

corec (friend) u_id_g'' :: "u ⇒ u" where
"u_id_g'' u = U (lab u) (map ((λu'. u_id u') o u_id_g'') (sub u))"

corec u_id_h :: "u ⇒ u" where
"u_id_h u = u_id (u_id (U (lab u) (map (λu'. (u_id o u_id) (u_id (u_id (u_id_h u')))) (sub u))))"

corec u_id_h' :: "u ⇒ u" where
"u_id_h' u = u_id (u_id (U (lab u) (map (u_id o u_id o u_id_h') (sub u))))"

corec u_id_h'' :: "u ⇒ u" where
"u_id_h'' u = u_id (u_id (U (lab u) (map (u_id o (u_id o (λu'. u_id u')) o u_id_h'') (sub u))))"

definition u_K :: "u ⇒ u ⇒ u" where "u_K u v = u"

friend_of_corec u_K where
"u_K u v = U (lab u) (sub u)"

corec u_K_f :: "u ⇒ u" where
"u_K_f u = u_K (U (lab u) (map u_K_f (sub u))) undefined"

corec (friend) u_K_g :: "u ⇒ u" where
"u_K_g u = U (lab u) (map (λu'. u_K (u_K_g u') undefined) (sub u))"

corec (friend) u_K_g' :: "u ⇒ u" where
"u_K_g' u = U (lab u) (map ((λu'. u_K u' undefined) o u_K_g') (sub u))"

corec (friend) u_K_g'' :: "u ⇒ u" where
"u_K_g'' u = U (lab u) (map (u_K undefined o u_K_g'') (sub u))"

end
```