(* Title: HOL/Corec_Examples/Tests/Misc_Mono.thy
Author: Aymeric Bouzy, Ecole polytechnique
Author: Jasmin Blanchette, Inria, LORIA, MPII
Copyright 2015, 2016
Miscellaneous monomorphic examples.
*)
section \<open>Miscellaneous Monomorphic Examples\<close>
theory Misc_Mono
imports "HOL-Library.BNF_Corec"
begin
codatatype T0 =
C1 (lab: nat) (tl11: T0) (tl12: T0)
| C2 (lab: nat) (tl2: T0)
| C3 (tl3: "T0 list")
codatatype stream =
S (hd: nat) (tl: stream)
corec (friend) ff where
"ff x = S 0 (ff (ff x))"
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))"
apply (rule test0.code)
apply transfer_prover
done
corec test01 where
"test01 x y = C2 (lab x + lab y) (test01 (tl2 x) (tl2 y))"
friend_of_corec test01 where
"test01 x y = C2 (lab x + lab y) (test01 (tl2 x) (tl2 y))"
apply (rule test01.code)
sorry (* not parametric *)
corec test02 where
"test02 x y = C2 (lab x * lab y) (test01 (test02 x (tl2 y)) (test02 (tl2 x) y))"
friend_of_corec test02 where
"test02 x y = C2 (lab x * lab y) (test01 (test02 x (tl2 y)) (test02 (tl2 x) y))"
apply (rule test02.code)
sorry (* not parametric *)
corec test03 where
"test03 x = C2 (lab x) (C2 (lab x) (test02 (test03 (tl2 x)) (test03 (tl2 x))))"
friend_of_corec test03 where
"test03 x = C2 (lab x) (C2 (lab x) (test02 (test03 (tl2 x)) (test03 (tl2 x))))"
apply (rule test03.code)
sorry (* not parametric *)
corec (friend) test04a where
"test04a x = (case x of C1 a t1 t2 \<Rightarrow> C1 (a * a) (test04a t1) (test04a t2) | C2 a t \<Rightarrow> C2 (a * a) (test04a t) | C3 l \<Rightarrow> C3 l)"
corec test04 where
"test04 x = (case x of C1 a t1 t2 \<Rightarrow> C1 (a * a) (test04 t1) (test04 t2) | C2 a t \<Rightarrow> C2 (a * 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 * a) (test04 t1) (test04 t2) | C2 a t \<Rightarrow> C2 (a * a) (test04 t) | C3 l \<Rightarrow> C3 l)"
apply (rule test04.code)
apply transfer_prover
done
corec test05 where
"test05 x y = (case (x, y) of
(C1 a t11 t12, C1 b t21 t22) \<Rightarrow> C1 (a + b) (test05 t11 t21) (test05 t12 t22)
| (C1 a t11 _, C2 b t2) \<Rightarrow> C2 (a + b) (test05 t11 t2)
| (C2 a t1, C1 b _ t22) \<Rightarrow> C2 (a + b) (test05 t1 t22)
| (C2 a t1, C2 b t2) \<Rightarrow> C2 (a + b) (test05 t1 t2)
| (_, _) \<Rightarrow> C3 [])"
friend_of_corec test05 where
"test05 x y = (case (x, y) of
(C1 a t11 t12, C1 b t21 t22) \<Rightarrow> C1 (a + b) (test05 t11 t21) (test05 t12 t22)
| (C1 a t11 _, C2 b t2) \<Rightarrow> C2 (a + b) (test05 t11 t2)
| (C2 a t1, C1 b _ t22) \<Rightarrow> C2 (a + b) (test05 t1 t22)
| (C2 a t1, C2 b t2) \<Rightarrow> C2 (a + b) (test05 t1 t2)
| (_, _) \<Rightarrow> C3 [])"
apply (rule test05.code)
apply transfer_prover
done
corec test06 :: "T0 \<Rightarrow> T0" 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 :: "T0 \<Rightarrow> T0" 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)))"
apply (rule test06.code)
sorry (* not parametric *)
corec test07 where
"test07 xs = C3 (map (\<lambda>x. test07 (tl3 x)) xs)"
friend_of_corec test07 where
"test07 xs = C3 (map (\<lambda>x. test07 (tl3 x)) xs)"
apply (rule test07.code)
sorry (* not parametric *)
corec test08 where
"test08 xs = (case xs of
[] \<Rightarrow> C2 0 (test08 [])
| T # r \<Rightarrow> C1 1 (test08 r) T)"
friend_of_corec test08 where
"test08 xs = (case xs of
[] \<Rightarrow> C2 0 (test08 [])
| T # r \<Rightarrow> C1 1 (test08 r) T)"
apply (rule test08.code)
apply transfer_prover
done
corec test09 where
"test09 xs = test08 [case xs of
[] \<Rightarrow> C2 0 (test09 [])
| (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> C2 0 (test09 [])
| (C1 n T1 T2) # r \<Rightarrow> C1 n (test09 (T1 # r)) (test09 (T2 # r))
| _ # r \<Rightarrow> C3 [test09 r])"
defer
apply transfer_prover
sorry (* not sure the specifications are equal *)
codatatype tree =
Node (node: int) (branches: "tree list")
consts integerize_tree_list :: "'a list \<Rightarrow> int"
lemma integerize_tree_list_transfer[transfer_rule]:
"rel_fun (list_all2 R) (=) integerize_tree_list integerize_tree_list"
sorry
corec (friend) f10a where
"f10a x y = Node
(integerize_tree_list (branches x) + integerize_tree_list (branches y))
(map (\<lambda>(x, y). f10a x y) (zip (branches x) (branches y)))"
corec f10 where
"f10 x y = Node
(integerize_tree_list (branches x) + integerize_tree_list (branches y))
(map (\<lambda>(x, y). f10 x y) (zip (branches x) (branches y)))"
friend_of_corec f10 where
"f10 x y = Node
(integerize_tree_list (branches x) + integerize_tree_list (branches y))
(map (\<lambda>(x, y). f10 x y) (zip (branches x) (branches y)))"
apply (rule f10.code)
by transfer_prover+
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 :: "tree option \<Rightarrow> tree" where
"f14 t_opt = Node 0
(case map_option branches t_opt of
None \<Rightarrow> []
| Some ts \<Rightarrow> map (f14 o Some) ts)"
friend_of_corec f14 where
"f14 t_opt = Node 0
(case map_option branches t_opt of
None \<Rightarrow> []
| Some ts \<Rightarrow> map (f14 o Some) ts)"
sorry
corec f15 :: "tree list option \<Rightarrow> tree" where
"f15 ts_opt = Node 0
(case map_option (map branches) ts_opt of
None \<Rightarrow> []
| Some tss \<Rightarrow> map (f15 o Some) tss)"
friend_of_corec f15 where
"f15 ts_opt = Node 0
(case map_option (map branches) ts_opt of
None \<Rightarrow> []
| Some tss \<Rightarrow> map (f15 o Some) tss)"
sorry
corec f16 :: "tree list option \<Rightarrow> tree" where
"f16 ts_opt = Node 0
(case ts_opt of
None \<Rightarrow> []
| Some ts \<Rightarrow> map (f16 o Some o branches) ts)"
friend_of_corec f16 where
"f16 ts_opt = Node 0
(case ts_opt of
None \<Rightarrow> []
| Some ts \<Rightarrow> map (f16 o Some o branches) ts)"
sorry
corec f17 :: "tree list option \<Rightarrow> tree" where
"f17 ts_opt = Node 0 (case ts_opt of
None \<Rightarrow> []
| Some ts \<Rightarrow> [f17 (Some (map (List.hd o branches) ts))])"
(* not parametric
friend_of_corec f17 where
"f17 ts_opt = Node 0 (case ts_opt of
None \<Rightarrow> []
| Some ts \<Rightarrow> [f17 (Some (map (List.hd o branches) ts))])"
sorry
*)
corec f18 :: "tree \<Rightarrow> tree" where
"f18 t = Node (node t) (map (f18 o f12) (branches t))"
friend_of_corec f18 :: "tree \<Rightarrow> tree" where
"f18 t = Node (node t) (map (f18 o f12) (branches t))"
sorry
corec f19 :: "tree \<Rightarrow> tree" where
"f19 t = Node (node t) (map (%f. f [t]) (map f13 [1, 2, 3]))"
friend_of_corec f19 :: "tree \<Rightarrow> tree" where
"f19 t = Node (node t) (map (%f. f [t]) (map f13 [1, 2, 3]))"
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 :: "(tree option, int, tree) h \<Rightarrow> tree \<Rightarrow> 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)"
friend_of_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)"
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> tree) \<Rightarrow> 'a list \<Rightarrow> tree" where
"f22 f x = Node 0 (map f x)"
friend_of_corec f22:: "(nat \<Rightarrow> tree) \<Rightarrow> nat list \<Rightarrow> tree" where
"f22 f x = Node 0 (map f x)"
sorry
corec f23 where
"f23 xh = Node 0
(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 0
(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 :: "(nat \<Rightarrow> tree list, int, int \<Rightarrow> tree list) h \<Rightarrow> 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
corec f25 where
"f25 x = Node (node x) (map f25 ((id branches) x))"
codatatype ('a, 'b) y_type =
Y (lab: "'a \<Rightarrow> 'b") (y_tail: "('a, 'b) y_type")
corec f26 :: "(int, tree) y_type \<Rightarrow> tree \<Rightarrow> tree" where
"f26 y x = (case map_y_type f12 y of
Y f y' \<Rightarrow> Node (node x) ((f (node x)) # (map (f26 y') (branches x))))"
friend_of_corec f26 where
"f26 y x = (case map_y_type f12 y of
Y f y' \<Rightarrow> Node (node x) ((f (node x)) # (map (f26 y') (branches x))))"
sorry
consts int_of_list :: "'a list \<Rightarrow> int"
corec f27 :: "(int, tree) y_type \<Rightarrow> tree \<Rightarrow> tree" where
"f27 y x = Node (int_of_list (map (f26 (y_tail y)) (branches x))) [lab y (node x)]"
friend_of_corec f27 :: "(int, tree) y_type \<Rightarrow> tree \<Rightarrow> tree" where
"f27 y x = Node (int_of_list (map (f26 (y_tail y)) (branches x))) [lab y (node x)]"
sorry
corec f28 :: "(tree option list, (int \<Rightarrow> int) \<Rightarrow> int list \<Rightarrow> tree, tree) h \<Rightarrow> tree" where
"f28 xh = (case xh of
H3 \<Rightarrow> Node 0 []
| H1 l r \<Rightarrow> Node 0 ((f28 r) # map the (filter (%opt. case opt of None \<Rightarrow> False | Some _ \<Rightarrow> True) l))
| H2 f t r \<Rightarrow> Node (node t) (map (%t. f id [node t]) (branches t)))"
codatatype llist =
LNil | LCons (head: nat) (tail: llist)
inductive llist_in where
"llist_in (LCons x xs) x"
| "llist_in xs y \<Longrightarrow> llist_in (LCons x xs) y"
abbreviation "lset xs \<equiv> {x. llist_in xs x}"
corecursive lfilter where
"lfilter P xs = (if \<forall> x \<in> lset xs. \<not> P x then
LNil
else if P (head xs) then
LCons (head xs) (lfilter P (tail xs))
else
lfilter P (tail xs))"
proof (relation "measure (\<lambda>(P, xs). LEAST n. P (head ((tail ^^ n) xs)))", rule wf_measure, clarsimp)
fix P xs x
assume "llist_in xs x" "P x" "\<not> P (head xs)"
from this(1,2) obtain a where "P (head ((tail ^^ a) xs))"
by (atomize_elim, induct xs x rule: llist_in.induct) (auto simp: funpow_Suc_right
simp del: funpow.simps(2) intro: exI[of _ 0] exI[of _ "Suc i" for i])
with \<open>\<not> P (head xs)\<close>
have "(LEAST n. P (head ((tail ^^ n) xs))) = Suc (LEAST n. P (head ((tail ^^ Suc n) xs)))"
by (intro Least_Suc) auto
then show "(LEAST n. P (head ((tail ^^ n) (tail xs)))) < (LEAST n. P (head ((tail ^^ n) xs)))"
by (simp add: funpow_swap1[of tail])
qed
codatatype Stream =
SCons (head: nat) (tail: Stream)
corec map_Stream where
"map_Stream f s = SCons (f (head s)) (map_Stream f (tail s))"
friend_of_corec map_Stream where
"map_Stream f s = SCons (f (head s)) (map_Stream f (tail s))"
sorry
corec f29 where
"f29 f ll = SCons (head ll) (f29 f (map_Stream f (tail ll)))"
friend_of_corec f29 where
"f29 f ll = SCons (head ll) (f29 f (map_Stream f (tail ll)))"
sorry
corec f30 where
"f30 n m = (if n = 0 then SCons m (f30 m m) else f30 (n - 1) (n * m))"
corec f31 :: "llist \<Rightarrow> llist" where
"f31 x = (if x = LNil then LCons undefined (f31 undefined) else LCons undefined undefined)"
friend_of_corec f31 where
"f31 x = (if x = LNil then LCons undefined (f31 undefined) else LCons undefined undefined)"
sorry
corec f32 :: "tree \<Rightarrow> tree" where
"f32 t = Node (node t) (map ((\<lambda>t'. f18 t') o f32) (branches t))"
corec f33 :: "tree \<Rightarrow> tree" where
"f33 t = f18 (f18 (Node (node t) (map (\<lambda>t'. (f18 o f18) (f18 (f18 (f33 t')))) (branches t))))"
corec f34 :: "tree \<Rightarrow> tree" where
"f34 t = f18 (f18 (Node (node t) (map (f18 o f18 o f34) (branches t))))"
corec f35 :: "tree \<Rightarrow> tree" where
"f35 t = f18 (f18 (Node (node t) (map (f18 o (f18 o (\<lambda>t'. f18 t')) o f35) (branches t))))"
corec f37 :: "int \<Rightarrow> tree list \<Rightarrow> tree option \<Rightarrow> nat \<Rightarrow> tree" where
"f37 a x1 = undefined a x1"
end