src/HOL/Corec_Examples/Tests/Misc_Poly.thy

+(*  Title:      HOL/Corec_Examples/Tests/Misc_Poly.thy
+    Author:     Aymeric Bouzy, Ecole polytechnique
+    Author:     Jasmin Blanchette, Inria, LORIA, MPII
+    Copyright   2015, 2016
+
+Miscellaneous polymorphic examples.
+*)
+
+section {* Miscellaneous Polymorphic Examples *}
+
+theory Misc_Poly
+imports "~~/src/HOL/Library/BNF_Corec"
+begin
+
+codatatype ('a, 'b) T0 =
+  C1 (lab: "'a * 'b") (tl11: "('a, 'b) T0") (tl12: "('a, 'b) T0")
+| C2 (lab: "'a * 'b") (tl2: "('a, 'b) T0")
+| C3 (tl3: "('a, 'b) T0 list")
+
+codatatype stream = S (hd: nat) (tl: stream)
+
+corec test0 where
+  "test0 x y = (case (x, y) of
+  (S a1 s1, S a2 s2) \<Rightarrow> S (a1 + a2) (test0 s1 s2))"
+
+friend_of_corec test0 where
+  "test0 x y = (case (x, y) of
+  (S a1 s1, S a2 s2) \<Rightarrow> S (a1 + a2) (test0 s1 s2))"
+  sorry
+
+corec test01 where
+  "test01 x y = C2 (fst (lab x), snd (lab y)) (test01 (tl2 x) (tl2 y))"
+
+friend_of_corec test01 :: "('a, 'b) T0 \<Rightarrow> ('a, 'b) T0 \<Rightarrow> ('a, 'b) T0" where
+  "test01 x y = C2 (fst (lab x), snd (lab y)) (test01 (tl2 x) (tl2 y))"
+  sorry
+
+corec test02 where
+  "test02 x y = C2 (fst (lab x), snd (lab y)) (test01 (test02 x (tl2 y)) (test02 (tl2 x) y))"
+
+friend_of_corec test02 :: "('a, 'b) T0 \<Rightarrow> ('a, 'b) T0 \<Rightarrow> ('a, 'b) T0"  where
+  "test02 x y = C2 (fst (lab x), snd (lab y)) (test01 (test02 x (tl2 y)) (test02 (tl2 x) y))"
+  sorry
+
+corec test03 where
+  "test03 x = C2 (lab x) (C2 (lab x) (test02 x x))"
+
+friend_of_corec test03 where
+  "test03 x = C2 (lab x) (C2 (lab x) (test02 (test03 (tl2 x)) (test03 (tl2 x))))"
+  sorry
+
+corec test04 where
+  "test04 x = (case x of C1 a t1 t2 \<Rightarrow> C1 a (test04 t1) (test04 t2) | C2 a t \<Rightarrow> C2 a (test04 t) | C3 l \<Rightarrow> C3 l)"
+
+friend_of_corec test04 where
+  "test04 x = (case x of C1 a t1 t2 \<Rightarrow> C1 a (test04 t1) (test04 t2) | C2 a t \<Rightarrow> C2 a (test04 t) | C3 l \<Rightarrow> C3 l)"
+  sorry
+
+corec test05 where
+  "test05 x y = (case (x, y) of
+  (C1 a t11 t12, C1 b t21 t22) \<Rightarrow> C1 (fst a, snd b) (test05 t11 t21) (test05 t12 t22)
+| (C1 a t11 _, C2 b t2) \<Rightarrow> C2 (fst a, snd b) (test05 t11 t2)
+| (C2 a t1, C1 b _ t22) \<Rightarrow> C2 (fst a, snd b) (test05 t1 t22)
+| (C2 a t1, C2 b t2) \<Rightarrow> C2 (fst a, snd b) (test05 t1 t2)
+| (_, _) \<Rightarrow> C3 [])"
+
+friend_of_corec test05 :: "('a, 'b) T0 \<Rightarrow> ('a, 'b) T0 \<Rightarrow> ('a, 'b) T0" where
+  "test05 x y = (case (x, y) of
+  (C1 a t11 t12, C1 b t21 t22) \<Rightarrow> C1 (fst a, snd b) (test05 t11 t21) (test05 t12 t22)
+| (C1 a t11 _, C2 b t2) \<Rightarrow> C2 (fst a, snd b) (test05 t11 t2)
+| (C2 a t1, C1 b _ t22) \<Rightarrow> C2 (fst a, snd b) (test05 t1 t22)
+| (C2 a t1, C2 b t2) \<Rightarrow> C2 (fst a, snd b) (test05 t1 t2)
+| (_, _) \<Rightarrow> C3 [])"
+  sorry
+
+corec test06 where
+  "test06 x =
+  (if \<not> is_C1 x then
+    let tail = tl2 x in
+    C1 (lab x) (test06 tail) tail
+  else
+    C2 (lab x) (test06 (tl12 x)))"
+
+friend_of_corec test06 where
+  "test06 x =
+  (if \<not> is_C1 x then
+    let tail = tl2 x in
+    C1 (lab x) (test06 tail) tail
+  else
+    C2 (lab x) (test06 (tl12 x)))"
+    sorry
+
+corec test07 where
+  "test07 xs = C3 (map (\<lambda>x. test07 (tl3 x)) xs)"
+
+friend_of_corec test07 :: "('a, 'b) T0 list \<Rightarrow> ('a, 'b) T0" where
+  "test07 xs = C3 (map (\<lambda>x. test07 (tl3 x)) xs)"
+  sorry
+
+corec test08 where
+  "test08 xs = (case xs of
+    [] \<Rightarrow> C3 []
+  | T # r \<Rightarrow> C1 (lab T) (test08 r) T)"
+
+friend_of_corec test08 where
+  "test08 xs = (case xs of
+    [] \<Rightarrow> C3 []
+  | T # r \<Rightarrow> C1 (lab T) (test08 r) T)"
+  sorry
+
+corec test09 where
+  "test09 xs = (case xs of
+    [] \<Rightarrow> C3 []
+  | (C1 n T1 T2) # r \<Rightarrow> C1 n (test09 (T1 # r)) (test09 (T2 # r))
+  | _ # r \<Rightarrow> C3 [test09 r])"
+
+friend_of_corec test09 where
+  "test09 xs = (case xs of
+    [] \<Rightarrow> C3 []
+  | (C1 n T1 T2) # r \<Rightarrow> C1 n (test09 (T1 # r)) (test09 (T2 # r))
+  | _ # r \<Rightarrow> C3 [test09 r])"
+  sorry
+
+codatatype 'h tree =
+  Node (node: 'h) (branches: "'h tree list")
+
+consts integerize_list :: "'a list \<Rightarrow> int"
+
+corec f10 where
+  "f10 x y = Node
+  (integerize_list (branches x) + integerize_list (branches y))
+  (map (\<lambda>(x, y). f10 x y) (zip (branches x) (branches y)))"
+
+friend_of_corec f10 :: "int tree \<Rightarrow> int tree \<Rightarrow> int tree" where
+  "f10 x y = Node
+  (integerize_list (branches x) + integerize_list (branches y))
+  (map (\<lambda>(x, y). f10 x y) (zip (branches x) (branches y)))"
+  sorry
+
+codatatype llist =
+  Nil | LCons llist
+
+consts friend :: "llist \<Rightarrow> llist"
+
+friend_of_corec friend where
+  "friend x = Nil"
+  sorry
+
+corec f11 where
+  "f11 x = (case friend x of Nil \<Rightarrow> Nil | LCons y \<Rightarrow> LCons (f11 y))"
+
+corec f12 where
+  "f12 t = Node (node t) (map f12 (branches t))"
+
+friend_of_corec f12 where
+  "f12 t = Node (node t) (map f12 (branches t))"
+  sorry
+
+corec f13 where
+  "f13 n ts = Node n (map (%t. f13 (node t) (branches t)) ts)"
+
+friend_of_corec f13 where
+  "f13 n ts = Node n (map (%t. f13 (node t) (branches t)) ts)"
+  sorry
+
+corec f14 where
+  "f14 t_opt = Node undefined
+    (case map_option branches t_opt of
+      None \<Rightarrow> []
+    | Some ts \<Rightarrow> map (f14 o Some) ts)"
+
+friend_of_corec f14 :: "'a tree option \<Rightarrow> 'a tree" where
+  "f14 t_opt = Node undefined
+    (case map_option branches t_opt of
+      None \<Rightarrow> []
+    | Some ts \<Rightarrow> map (f14 o Some) ts)"
+  sorry
+
+corec f15 where
+  "f15 ts_opt = Node undefined
+    (case map_option (map branches) ts_opt of
+      None \<Rightarrow> []
+    | Some tss \<Rightarrow> map (f15 o Some) tss)"
+
+friend_of_corec f15 :: "'a tree list option \<Rightarrow> 'a tree" where
+  "f15 ts_opt = Node undefined
+    (case map_option (map branches) ts_opt of
+      None \<Rightarrow> []
+    | Some tss \<Rightarrow> map (f15 o Some) tss)"
+    sorry
+
+corec f16 where
+  "f16 ts_opt = Node undefined
+    (case ts_opt of
+      None \<Rightarrow> []
+    | Some ts \<Rightarrow> map (f16 o Some o branches) ts)"
+
+friend_of_corec f16 :: "'a tree list option \<Rightarrow> 'a tree" where
+  "f16 ts_opt = Node undefined
+    (case ts_opt of
+      None \<Rightarrow> []
+    | Some ts \<Rightarrow> map (f16 o Some o branches) ts)"
+    sorry
+
+corec f18 :: "'a tree \<Rightarrow> 'a tree" where
+  "f18 t = Node (node t) (map (f18 o f12) (branches t))"
+
+friend_of_corec f18 :: "'z tree \<Rightarrow> 'z tree" where
+  "f18 t = Node (node t) (map (f18 o f12) (branches t))"
+  sorry
+
+corec f19 where
+  "f19 t = Node (node t) (map (%f. f [t]) (map f13 [undefined]))"
+
+friend_of_corec f19 where
+  "f19 t = Node (node t) (map (%f. f [t]) (map f13 [undefined]))"
+  sorry
+
+datatype ('a, 'b, 'c) h = H1 (h_a: 'a) (h_tail: "('a, 'b, 'c) h") | H2 (h_b: 'b) (h_c: 'c) (h_tail: "('a, 'b, 'c) h") | H3
+
+term "map_h (map_option f12) (%n. n) f12"
+
+corec f20 where
+  "f20 x y = Node (node y) (case (map_h (map_option f12) (%n. n) f12 x) of
+    H1 None r \<Rightarrow> (f20 r y) # (branches y)
+  | H1 (Some t) r \<Rightarrow> (f20 r t) # (branches y)
+  | H2 n t r \<Rightarrow> (f20 r (Node n [])) # (branches y)
+  | H3 \<Rightarrow> branches y)"
+
+friend_of_corec f20 :: "('a tree option, 'a, 'a tree) h \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+  "f20 x y = Node (node y) (case (map_h (map_option f12) (%n. n) f12 x) of
+    H1 None r \<Rightarrow> (f20 r y) # (branches y)
+  | H1 (Some t) r \<Rightarrow> (f20 r t) # (branches y)
+  | H2 n t r \<Rightarrow> (f20 r (Node n [])) # (branches y)
+  | H3 \<Rightarrow> branches y)"
+  sorry
+
+corec f21 where
+  "f21 x xh =
+  Node (node x) (case xh of
+    H1 (Some a) yh \<Rightarrow> (f21 x (map_h (map_option (f20 yh)) id id yh)) # (branches a)
+  | H1 None yh \<Rightarrow> [f21 x yh]
+  | H2 b c yh \<Rightarrow> (f21 c (map_h id (%n. n + b) id yh)) # (branches x)
+  | H3 \<Rightarrow> branches x)"
+
+friend_of_corec f21 where
+  "f21 x xh =
+  Node (node x) (case xh of
+    H1 (Some a) yh \<Rightarrow> (f21 x (map_h (map_option (f20 yh)) (%t. t) (%t. t) yh)) # (branches a)
+  | H1 None yh \<Rightarrow> [f21 x yh]
+  | H2 b c yh \<Rightarrow> (f21 c (map_h (%t. t) (%n. n + b) (%t. t) yh)) # (branches x)
+  | H3 \<Rightarrow> branches x)"
+  sorry
+
+corec f22 :: "('a \<Rightarrow> 'h tree) \<Rightarrow> 'a list \<Rightarrow> 'h tree" where
+  "f22 f x = Node undefined (map f x)"
+
+friend_of_corec f22 :: "('h \<Rightarrow> 'h tree) \<Rightarrow> 'h list \<Rightarrow> 'h tree" where
+  "f22 f x = Node undefined (map f x)"
+  sorry
+
+corec f23 where
+  "f23 xh = Node undefined 
+    (if is_H1 xh then
+      (f23 (h_tail xh)) # (branches (h_a xh))
+    else if is_H1 xh then
+      (f23 (h_tail xh)) # (h_c xh) # (branches (h_b xh))
+    else
+      [])"
+
+friend_of_corec f23 where
+  "f23 xh = Node undefined 
+    (if is_H1 xh then
+      (f23 (h_tail xh)) # (branches (h_a xh))
+    else if is_H1 xh then
+      (f23 (h_tail xh)) # (h_c xh) # (branches (h_b xh))
+    else
+      [])"
+  sorry
+
+corec f24 where
+  "f24 xh = 
+    (if is_H1 xh then
+      Node 0 ((f24 (h_tail xh)) # (h_a xh 0))
+    else if is_H2 xh then
+      Node (h_b xh) ((f24 (h_tail xh)) # (h_c xh 0))
+    else
+      Node 0 [])"
+
+friend_of_corec f24 :: "(('b :: {zero}) \<Rightarrow> 'b tree list, 'b, 'b \<Rightarrow> 'b tree list) h \<Rightarrow> 'b tree" where
+  "f24 xh = 
+    (if is_H1 xh then
+      Node 0 ((f24 (h_tail xh)) # (h_a xh 0))
+    else if is_H2 xh then
+      Node (h_b xh) ((f24 (h_tail xh)) # (h_c xh 0))
+    else
+      Node 0 [])"
+  sorry
+
+codatatype ('a, 'b, 'c) s =
+  S (s1: 'a) (s2: 'b) (s3: 'c) (s4: "('a, 'b, 'c) s")
+
+corec (friend) ga :: "('a, 'b, nat) s \<Rightarrow> _" where
+  "ga s = S (s1 s) (s2 s) (s3 s) (s4 s)"
+
+corec (friend) gb :: "('a, nat, 'b) s \<Rightarrow> _" where
+  "gb s = S (s1 s) (s2 s) (s3 s) (s4 s)"
+
+consts foo :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a"
+
+corecursive (friend) gc :: "(nat, 'a, 'b) s \<Rightarrow> _" where
+  "gc s = S (s1 s) (s2 s) (s3 s) (foo (undefined :: 'a \<Rightarrow> 'a) s)"
+  sorry
+
+end