1(*  Title:      HOL/Corec_Examples/Tests/Misc_Poly.thy
2    Author:     Aymeric Bouzy, Ecole polytechnique
3    Author:     Jasmin Blanchette, Inria, LORIA, MPII
4    Copyright   2015, 2016
5
6Miscellaneous polymorphic examples.
7*)
8
9section \<open>Miscellaneous Polymorphic Examples\<close>
10
11theory Misc_Poly
12imports "HOL-Library.BNF_Corec"
13begin
14
15codatatype ('a, 'b) T0 =
16  C1 (lab: "'a * 'b") (tl11: "('a, 'b) T0") (tl12: "('a, 'b) T0")
17| C2 (lab: "'a * 'b") (tl2: "('a, 'b) T0")
18| C3 (tl3: "('a, 'b) T0 list")
19
20codatatype stream = S (hd: nat) (tl: stream)
21
22corec test0 where
23  "test0 x y = (case (x, y) of
24  (S a1 s1, S a2 s2) \<Rightarrow> S (a1 + a2) (test0 s1 s2))"
25
26friend_of_corec test0 where
27  "test0 x y = (case (x, y) of
28  (S a1 s1, S a2 s2) \<Rightarrow> S (a1 + a2) (test0 s1 s2))"
29  sorry
30
31corec test01 where
32  "test01 x y = C2 (fst (lab x), snd (lab y)) (test01 (tl2 x) (tl2 y))"
33
34friend_of_corec test01 :: "('a, 'b) T0 \<Rightarrow> ('a, 'b) T0 \<Rightarrow> ('a, 'b) T0" where
35  "test01 x y = C2 (fst (lab x), snd (lab y)) (test01 (tl2 x) (tl2 y))"
36  sorry
37
38corec test02 where
39  "test02 x y = C2 (fst (lab x), snd (lab y)) (test01 (test02 x (tl2 y)) (test02 (tl2 x) y))"
40
41friend_of_corec test02 :: "('a, 'b) T0 \<Rightarrow> ('a, 'b) T0 \<Rightarrow> ('a, 'b) T0"  where
42  "test02 x y = C2 (fst (lab x), snd (lab y)) (test01 (test02 x (tl2 y)) (test02 (tl2 x) y))"
43  sorry
44
45corec test03 where
46  "test03 x = C2 (lab x) (C2 (lab x) (test02 x x))"
47
48friend_of_corec test03 where
49  "test03 x = C2 (lab x) (C2 (lab x) (test02 (test03 (tl2 x)) (test03 (tl2 x))))"
50  sorry
51
52corec test04 where
53  "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)"
54
55friend_of_corec test04 where
56  "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)"
57  sorry
58
59corec test05 where
60  "test05 x y = (case (x, y) of
61  (C1 a t11 t12, C1 b t21 t22) \<Rightarrow> C1 (fst a, snd b) (test05 t11 t21) (test05 t12 t22)
62| (C1 a t11 _, C2 b t2) \<Rightarrow> C2 (fst a, snd b) (test05 t11 t2)
63| (C2 a t1, C1 b _ t22) \<Rightarrow> C2 (fst a, snd b) (test05 t1 t22)
64| (C2 a t1, C2 b t2) \<Rightarrow> C2 (fst a, snd b) (test05 t1 t2)
65| (_, _) \<Rightarrow> C3 [])"
66
67friend_of_corec test05 :: "('a, 'b) T0 \<Rightarrow> ('a, 'b) T0 \<Rightarrow> ('a, 'b) T0" where
68  "test05 x y = (case (x, y) of
69  (C1 a t11 t12, C1 b t21 t22) \<Rightarrow> C1 (fst a, snd b) (test05 t11 t21) (test05 t12 t22)
70| (C1 a t11 _, C2 b t2) \<Rightarrow> C2 (fst a, snd b) (test05 t11 t2)
71| (C2 a t1, C1 b _ t22) \<Rightarrow> C2 (fst a, snd b) (test05 t1 t22)
72| (C2 a t1, C2 b t2) \<Rightarrow> C2 (fst a, snd b) (test05 t1 t2)
73| (_, _) \<Rightarrow> C3 [])"
74  sorry
75
76corec test06 where
77  "test06 x =
78  (if \<not> is_C1 x then
79    let tail = tl2 x in
80    C1 (lab x) (test06 tail) tail
81  else
82    C2 (lab x) (test06 (tl12 x)))"
83
84friend_of_corec test06 where
85  "test06 x =
86  (if \<not> is_C1 x then
87    let tail = tl2 x in
88    C1 (lab x) (test06 tail) tail
89  else
90    C2 (lab x) (test06 (tl12 x)))"
91    sorry
92
93corec test07 where
94  "test07 xs = C3 (map (\<lambda>x. test07 (tl3 x)) xs)"
95
96friend_of_corec test07 :: "('a, 'b) T0 list \<Rightarrow> ('a, 'b) T0" where
97  "test07 xs = C3 (map (\<lambda>x. test07 (tl3 x)) xs)"
98  sorry
99
100corec test08 where
101  "test08 xs = (case xs of
102    [] \<Rightarrow> C3 []
103  | T # r \<Rightarrow> C1 (lab T) (test08 r) T)"
104
105friend_of_corec test08 where
106  "test08 xs = (case xs of
107    [] \<Rightarrow> C3 []
108  | T # r \<Rightarrow> C1 (lab T) (test08 r) T)"
109  sorry
110
111corec test09 where
112  "test09 xs = (case xs of
113    [] \<Rightarrow> C3 []
114  | (C1 n T1 T2) # r \<Rightarrow> C1 n (test09 (T1 # r)) (test09 (T2 # r))
115  | _ # r \<Rightarrow> C3 [test09 r])"
116
117friend_of_corec test09 where
118  "test09 xs = (case xs of
119    [] \<Rightarrow> C3 []
120  | (C1 n T1 T2) # r \<Rightarrow> C1 n (test09 (T1 # r)) (test09 (T2 # r))
121  | _ # r \<Rightarrow> C3 [test09 r])"
122  sorry
123
124codatatype 'h tree =
125  Node (node: 'h) (branches: "'h tree list")
126
127consts integerize_list :: "'a list \<Rightarrow> int"
128
129corec f10 where
130  "f10 x y = Node
131  (integerize_list (branches x) + integerize_list (branches y))
132  (map (\<lambda>(x, y). f10 x y) (zip (branches x) (branches y)))"
133
134friend_of_corec f10 :: "int tree \<Rightarrow> int tree \<Rightarrow> int tree" where
135  "f10 x y = Node
136  (integerize_list (branches x) + integerize_list (branches y))
137  (map (\<lambda>(x, y). f10 x y) (zip (branches x) (branches y)))"
138  sorry
139
140codatatype llist =
141  Nil | LCons llist
142
143consts friend :: "llist \<Rightarrow> llist"
144
145friend_of_corec friend where
146  "friend x = Nil"
147  sorry
148
149corec f11 where
150  "f11 x = (case friend x of Nil \<Rightarrow> Nil | LCons y \<Rightarrow> LCons (f11 y))"
151
152corec f12 where
153  "f12 t = Node (node t) (map f12 (branches t))"
154
155friend_of_corec f12 where
156  "f12 t = Node (node t) (map f12 (branches t))"
157  sorry
158
159corec f13 where
160  "f13 n ts = Node n (map (%t. f13 (node t) (branches t)) ts)"
161
162friend_of_corec f13 where
163  "f13 n ts = Node n (map (%t. f13 (node t) (branches t)) ts)"
164  sorry
165
166corec f14 where
167  "f14 t_opt = Node undefined
168    (case map_option branches t_opt of
169      None \<Rightarrow> []
170    | Some ts \<Rightarrow> map (f14 o Some) ts)"
171
172friend_of_corec f14 :: "'a tree option \<Rightarrow> 'a tree" where
173  "f14 t_opt = Node undefined
174    (case map_option branches t_opt of
175      None \<Rightarrow> []
176    | Some ts \<Rightarrow> map (f14 o Some) ts)"
177  sorry
178
179corec f15 where
180  "f15 ts_opt = Node undefined
181    (case map_option (map branches) ts_opt of
182      None \<Rightarrow> []
183    | Some tss \<Rightarrow> map (f15 o Some) tss)"
184
185friend_of_corec f15 :: "'a tree list option \<Rightarrow> 'a tree" where
186  "f15 ts_opt = Node undefined
187    (case map_option (map branches) ts_opt of
188      None \<Rightarrow> []
189    | Some tss \<Rightarrow> map (f15 o Some) tss)"
190    sorry
191
192corec f16 where
193  "f16 ts_opt = Node undefined
194    (case ts_opt of
195      None \<Rightarrow> []
196    | Some ts \<Rightarrow> map (f16 o Some o branches) ts)"
197
198friend_of_corec f16 :: "'a tree list option \<Rightarrow> 'a tree" where
199  "f16 ts_opt = Node undefined
200    (case ts_opt of
201      None \<Rightarrow> []
202    | Some ts \<Rightarrow> map (f16 o Some o branches) ts)"
203    sorry
204
205corec f18 :: "'a tree \<Rightarrow> 'a tree" where
206  "f18 t = Node (node t) (map (f18 o f12) (branches t))"
207
208friend_of_corec f18 :: "'z tree \<Rightarrow> 'z tree" where
209  "f18 t = Node (node t) (map (f18 o f12) (branches t))"
210  sorry
211
212corec f19 where
213  "f19 t = Node (node t) (map (%f. f [t]) (map f13 [undefined]))"
214
215friend_of_corec f19 where
216  "f19 t = Node (node t) (map (%f. f [t]) (map f13 [undefined]))"
217  sorry
218
219datatype ('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
220
221term "map_h (map_option f12) (%n. n) f12"
222
223corec f20 where
224  "f20 x y = Node (node y) (case (map_h (map_option f12) (%n. n) f12 x) of
225    H1 None r \<Rightarrow> (f20 r y) # (branches y)
226  | H1 (Some t) r \<Rightarrow> (f20 r t) # (branches y)
227  | H2 n t r \<Rightarrow> (f20 r (Node n [])) # (branches y)
228  | H3 \<Rightarrow> branches y)"
229
230friend_of_corec f20 :: "('a tree option, 'a, 'a tree) h \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
231  "f20 x y = Node (node y) (case (map_h (map_option f12) (%n. n) f12 x) of
232    H1 None r \<Rightarrow> (f20 r y) # (branches y)
233  | H1 (Some t) r \<Rightarrow> (f20 r t) # (branches y)
234  | H2 n t r \<Rightarrow> (f20 r (Node n [])) # (branches y)
235  | H3 \<Rightarrow> branches y)"
236  sorry
237
238corec f21 where
239  "f21 x xh =
240  Node (node x) (case xh of
241    H1 (Some a) yh \<Rightarrow> (f21 x (map_h (map_option (f20 yh)) id id yh)) # (branches a)
242  | H1 None yh \<Rightarrow> [f21 x yh]
243  | H2 b c yh \<Rightarrow> (f21 c (map_h id (%n. n + b) id yh)) # (branches x)
244  | H3 \<Rightarrow> branches x)"
245
246friend_of_corec f21 where
247  "f21 x xh =
248  Node (node x) (case xh of
249    H1 (Some a) yh \<Rightarrow> (f21 x (map_h (map_option (f20 yh)) (%t. t) (%t. t) yh)) # (branches a)
250  | H1 None yh \<Rightarrow> [f21 x yh]
251  | H2 b c yh \<Rightarrow> (f21 c (map_h (%t. t) (%n. n + b) (%t. t) yh)) # (branches x)
252  | H3 \<Rightarrow> branches x)"
253  sorry
254
255corec f22 :: "('a \<Rightarrow> 'h tree) \<Rightarrow> 'a list \<Rightarrow> 'h tree" where
256  "f22 f x = Node undefined (map f x)"
257
258friend_of_corec f22 :: "('h \<Rightarrow> 'h tree) \<Rightarrow> 'h list \<Rightarrow> 'h tree" where
259  "f22 f x = Node undefined (map f x)"
260  sorry
261
262corec f23 where
263  "f23 xh = Node undefined
264    (if is_H1 xh then
265      (f23 (h_tail xh)) # (branches (h_a xh))
266    else if is_H1 xh then
267      (f23 (h_tail xh)) # (h_c xh) # (branches (h_b xh))
268    else
269      [])"
270
271friend_of_corec f23 where
272  "f23 xh = Node undefined
273    (if is_H1 xh then
274      (f23 (h_tail xh)) # (branches (h_a xh))
275    else if is_H1 xh then
276      (f23 (h_tail xh)) # (h_c xh) # (branches (h_b xh))
277    else
278      [])"
279  sorry
280
281corec f24 where
282  "f24 xh =
283    (if is_H1 xh then
284      Node 0 ((f24 (h_tail xh)) # (h_a xh 0))
285    else if is_H2 xh then
286      Node (h_b xh) ((f24 (h_tail xh)) # (h_c xh 0))
287    else
288      Node 0 [])"
289
290friend_of_corec f24 :: "(('b :: {zero}) \<Rightarrow> 'b tree list, 'b, 'b \<Rightarrow> 'b tree list) h \<Rightarrow> 'b tree" where
291  "f24 xh =
292    (if is_H1 xh then
293      Node 0 ((f24 (h_tail xh)) # (h_a xh 0))
294    else if is_H2 xh then
295      Node (h_b xh) ((f24 (h_tail xh)) # (h_c xh 0))
296    else
297      Node 0 [])"
298  sorry
299
300codatatype ('a, 'b, 'c) s =
301  S (s1: 'a) (s2: 'b) (s3: 'c) (s4: "('a, 'b, 'c) s")
302
303corec (friend) ga :: "('a, 'b, nat) s \<Rightarrow> _" where
304  "ga s = S (s1 s) (s2 s) (s3 s) (s4 s)"
305
306corec (friend) gb :: "('a, nat, 'b) s \<Rightarrow> _" where
307  "gb s = S (s1 s) (s2 s) (s3 s) (s4 s)"
308
309consts foo :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a"
310
311corecursive (friend) gc :: "(nat, 'a, 'b) s \<Rightarrow> _" where
312  "gc s = S (s1 s) (s2 s) (s3 s) (foo (undefined :: 'a \<Rightarrow> 'a) s)"
313  sorry
314
315end
316