# Theory Stream_Friends

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

Friendly functions on streams.
*)

section ‹Friendly Functions on Streams›

theory Stream_Friends
imports "HOL-Library.BNF_Corec"
begin

codatatype 'a stream = SCons (shd: 'a) (stl: "'a stream")

corec pls :: "nat stream ⇒ nat stream ⇒ nat stream" where
"pls s s' = SCons (shd s + shd s') (pls (stl s) (stl s'))"

friend_of_corec pls :: "nat stream ⇒ nat stream ⇒ nat stream" where
"pls s s' = SCons (shd s + shd s') (pls (stl s) (stl s'))"
sorry

corec onetwo :: "nat stream" where
"onetwo = SCons 1 (SCons 2 onetwo)"

corec prd :: "nat stream ⇒ nat stream ⇒ nat stream" where
"prd xs ys = SCons (shd xs * shd ys) (pls (prd xs (stl ys)) (prd (stl xs) ys))"

friend_of_corec prd  where
"prd xs ys = SCons (shd xs * shd ys) (pls (prd xs (stl ys)) (prd (stl xs) ys))"
sorry

corec Exp :: "nat stream ⇒ nat stream" where
"Exp xs = SCons (2 ^^ shd xs) (prd (stl xs) (Exp xs))"

friend_of_corec Exp where
"Exp xs = SCons (2 ^^ shd xs) (prd (stl xs) (Exp xs))"
sorry

corec prd2 :: "nat stream ⇒ nat stream ⇒ nat stream" where
"prd2 xs ys = SCons (shd xs * shd ys) (pls (prd xs (stl ys)) (prd2 (stl xs) ys))"

fun sthe_default :: "'a stream ⇒ nat ⇒ 'a stream option ⇒ 'a stream" where
"sthe_default s _ None = s"
| "sthe_default _ _ (Some t) = t"

(* FIXME:
friend_of_corec sthe_default where
"sthe_default s n opt = SCons (shd (case opt of None ⇒ s | Some t ⇒ t)) (stl (case opt of None ⇒ s | Some t ⇒ t))"
sorry

corec funky0 :: "nat ⇒ nat stream" where
"funky0 n = SCons 0 (sthe_default undefined n (map_option funky0 undefined))"

consts funky0' :: "nat stream ⇒ nat stream"

friend_of_corec funky0' :: "nat stream ⇒ nat stream" where
"funky0' ns = SCons 0 (sthe_default undefined (shd ns) (map_option funky0' undefined))"
sorry

corec funky :: "nat ⇒ nat stream" where
"funky n = SCons 0 (sthe_default (funky (n + 1)) n (map_option funky (Some (n + 2))))"

corec funky' :: "nat ⇒ nat ⇒ nat stream" where
"funky' m n = SCons 0 (sthe_default (funky' (m - 1) (n + 1)) m (map_option (%(x, y). funky' (Suc x) (Suc y)) (Some (m - 2, n + 2))))"

corec funky'' :: "nat ⇒ nat stream" where
"funky'' n = SCons 0 (sthe_default (funky'' (n + 1)) n (Some (funky'' (n + 2))))"

corec phunky0 :: "nat ⇒ nat stream" where
"phunky0 n = sthe_default undefined n undefined"

fun lthe_default :: "'a stream ⇒ 'a stream option ⇒ 'a stream" where
"lthe_default s None = s"
| "lthe_default _ (Some t) = t"

friend_of_corec lthe_default where
"lthe_default s opt = SCons (shd (case opt of None ⇒ s | Some t ⇒ t)) (stl (case opt of None ⇒ s | Some t ⇒ t))"
sorry

corec phunky_debug :: "'a ⇒ 'a stream" where
"phunky_debug n = lthe_default (SCons n (lthe_default undefined (map_option phunky_debug undefined))) undefined"

corec phunky1 :: "nat ⇒ nat stream" where
"phunky1 n = sthe_default (SCons 0 (sthe_default (phunky1 (n + 1)) n (map_option phunky1 (Some (n + 2))))) n undefined"

corec phunky2 :: "nat ⇒ nat stream" where
"phunky2 n = sthe_default (SCons 0 (sthe_default (phunky2 (n + 1)) n (map_option phunky2 (Some (n + 2))))) n
(map_option (%m. SCons m (sthe_default (phunky2 (n + 1)) n (map_option phunky2 (Some (n + 2))))) (Some n))"

corec phunky3 :: "nat ⇒ nat stream" where
"phunky3 n = sthe_default (SCons 0 (sthe_default (phunky3 (n + 1)) n (map_option phunky3 (Some (n + 3))))) n
(Some (SCons n (sthe_default (phunky3 (n + 1)) n (map_option phunky3 (Some (n + 3))))))"
*)

end
```