author | wenzelm |
Fri, 27 Nov 2020 11:41:43 +0100 | |
changeset 72740 | 082200ee003d |
parent 67399 | eab6ce8368fa |
permissions | -rw-r--r-- |
62696 | 1 |
(* Title: HOL/Corec_Examples/Tests/Misc_Mono.thy |
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Author: Aymeric Bouzy, Ecole polytechnique |
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Author: Jasmin Blanchette, Inria, LORIA, MPII |
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Copyright 2015, 2016 |
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Miscellaneous monomorphic examples. |
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*) |
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section \<open>Miscellaneous Monomorphic Examples\<close> |
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theory Misc_Mono |
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66453
cc19f7ca2ed6
session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
wenzelm
parents:
63540
diff
changeset
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imports "HOL-Library.BNF_Corec" |
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begin |
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codatatype T0 = |
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C1 (lab: nat) (tl11: T0) (tl12: T0) |
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| C2 (lab: nat) (tl2: T0) |
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| C3 (tl3: "T0 list") |
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codatatype stream = |
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S (hd: nat) (tl: stream) |
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corec (friend) ff where |
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"ff x = S 0 (ff (ff x))" |
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corec test0 where |
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"test0 x y = (case (x, y) of |
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(S a1 s1, S a2 s2) \<Rightarrow> S (a1 + a2) (test0 s1 s2))" |
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friend_of_corec test0 where |
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"test0 x y = (case (x, y) of |
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(S a1 s1, S a2 s2) \<Rightarrow> S (a1 + a2) (test0 s1 s2))" |
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apply (rule test0.code) |
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apply transfer_prover |
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done |
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corec test01 where |
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"test01 x y = C2 (lab x + lab y) (test01 (tl2 x) (tl2 y))" |
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friend_of_corec test01 where |
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"test01 x y = C2 (lab x + lab y) (test01 (tl2 x) (tl2 y))" |
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apply (rule test01.code) |
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sorry (* not parametric *) |
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corec test02 where |
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"test02 x y = C2 (lab x * lab y) (test01 (test02 x (tl2 y)) (test02 (tl2 x) y))" |
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friend_of_corec test02 where |
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"test02 x y = C2 (lab x * lab y) (test01 (test02 x (tl2 y)) (test02 (tl2 x) y))" |
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apply (rule test02.code) |
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sorry (* not parametric *) |
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corec test03 where |
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"test03 x = C2 (lab x) (C2 (lab x) (test02 (test03 (tl2 x)) (test03 (tl2 x))))" |
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friend_of_corec test03 where |
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"test03 x = C2 (lab x) (C2 (lab x) (test02 (test03 (tl2 x)) (test03 (tl2 x))))" |
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apply (rule test03.code) |
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sorry (* not parametric *) |
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corec (friend) test04a where |
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"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)" |
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corec test04 where |
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"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)" |
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friend_of_corec test04 where |
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"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)" |
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apply (rule test04.code) |
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apply transfer_prover |
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done |
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corec test05 where |
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"test05 x y = (case (x, y) of |
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(C1 a t11 t12, C1 b t21 t22) \<Rightarrow> C1 (a + b) (test05 t11 t21) (test05 t12 t22) |
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| (C1 a t11 _, C2 b t2) \<Rightarrow> C2 (a + b) (test05 t11 t2) |
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| (C2 a t1, C1 b _ t22) \<Rightarrow> C2 (a + b) (test05 t1 t22) |
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| (C2 a t1, C2 b t2) \<Rightarrow> C2 (a + b) (test05 t1 t2) |
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| (_, _) \<Rightarrow> C3 [])" |
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friend_of_corec test05 where |
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"test05 x y = (case (x, y) of |
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(C1 a t11 t12, C1 b t21 t22) \<Rightarrow> C1 (a + b) (test05 t11 t21) (test05 t12 t22) |
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| (C1 a t11 _, C2 b t2) \<Rightarrow> C2 (a + b) (test05 t11 t2) |
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| (C2 a t1, C1 b _ t22) \<Rightarrow> C2 (a + b) (test05 t1 t22) |
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| (C2 a t1, C2 b t2) \<Rightarrow> C2 (a + b) (test05 t1 t2) |
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| (_, _) \<Rightarrow> C3 [])" |
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apply (rule test05.code) |
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apply transfer_prover |
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done |
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corec test06 :: "T0 \<Rightarrow> T0" where |
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"test06 x = |
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(if \<not> is_C1 x then |
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let tail = tl2 x in |
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C1 (lab x) (test06 tail) tail |
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else |
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C2 (lab x) (test06 (tl12 x)))" |
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friend_of_corec test06 :: "T0 \<Rightarrow> T0" where |
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"test06 x = |
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(if \<not> is_C1 x then |
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let tail = tl2 x in |
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C1 (lab x) (test06 tail) tail |
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else |
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C2 (lab x) (test06 (tl12 x)))" |
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apply (rule test06.code) |
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sorry (* not parametric *) |
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corec test07 where |
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"test07 xs = C3 (map (\<lambda>x. test07 (tl3 x)) xs)" |
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friend_of_corec test07 where |
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"test07 xs = C3 (map (\<lambda>x. test07 (tl3 x)) xs)" |
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apply (rule test07.code) |
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sorry (* not parametric *) |
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corec test08 where |
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"test08 xs = (case xs of |
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[] \<Rightarrow> C2 0 (test08 []) |
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| T # r \<Rightarrow> C1 1 (test08 r) T)" |
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friend_of_corec test08 where |
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"test08 xs = (case xs of |
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[] \<Rightarrow> C2 0 (test08 []) |
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| T # r \<Rightarrow> C1 1 (test08 r) T)" |
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apply (rule test08.code) |
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apply transfer_prover |
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done |
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corec test09 where |
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"test09 xs = test08 [case xs of |
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[] \<Rightarrow> C2 0 (test09 []) |
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| (C1 n T1 T2) # r \<Rightarrow> C1 n (test09 (T1 # r)) (test09 (T2 # r)) |
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| _ # r \<Rightarrow> C3 [test09 r]]" |
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friend_of_corec test09 where |
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"test09 xs = (case xs of |
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[] \<Rightarrow> C2 0 (test09 []) |
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| (C1 n T1 T2) # r \<Rightarrow> C1 n (test09 (T1 # r)) (test09 (T2 # r)) |
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| _ # r \<Rightarrow> C3 [test09 r])" |
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defer |
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apply transfer_prover |
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sorry (* not sure the specifications are equal *) |
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codatatype tree = |
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Node (node: int) (branches: "tree list") |
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consts integerize_tree_list :: "'a list \<Rightarrow> int" |
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lemma integerize_tree_list_transfer[transfer_rule]: |
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"rel_fun (list_all2 R) (=) integerize_tree_list integerize_tree_list" |
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sorry |
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corec (friend) f10a where |
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"f10a x y = Node |
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(integerize_tree_list (branches x) + integerize_tree_list (branches y)) |
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(map (\<lambda>(x, y). f10a x y) (zip (branches x) (branches y)))" |
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corec f10 where |
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"f10 x y = Node |
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(integerize_tree_list (branches x) + integerize_tree_list (branches y)) |
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(map (\<lambda>(x, y). f10 x y) (zip (branches x) (branches y)))" |
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friend_of_corec f10 where |
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"f10 x y = Node |
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(integerize_tree_list (branches x) + integerize_tree_list (branches y)) |
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(map (\<lambda>(x, y). f10 x y) (zip (branches x) (branches y)))" |
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apply (rule f10.code) |
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by transfer_prover+ |
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corec f12 where |
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"f12 t = Node (node t) (map f12 (branches t))" |
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friend_of_corec f12 where |
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"f12 t = Node (node t) (map f12 (branches t))" |
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sorry |
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corec f13 where |
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"f13 n ts = Node n (map (%t. f13 (node t) (branches t)) ts)" |
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friend_of_corec f13 where |
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"f13 n ts = Node n (map (%t. f13 (node t) (branches t)) ts)" |
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sorry |
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corec f14 :: "tree option \<Rightarrow> tree" where |
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"f14 t_opt = Node 0 |
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(case map_option branches t_opt of |
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None \<Rightarrow> [] |
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| Some ts \<Rightarrow> map (f14 o Some) ts)" |
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friend_of_corec f14 where |
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"f14 t_opt = Node 0 |
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(case map_option branches t_opt of |
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None \<Rightarrow> [] |
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| Some ts \<Rightarrow> map (f14 o Some) ts)" |
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sorry |
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corec f15 :: "tree list option \<Rightarrow> tree" where |
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"f15 ts_opt = Node 0 |
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(case map_option (map branches) ts_opt of |
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None \<Rightarrow> [] |
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| Some tss \<Rightarrow> map (f15 o Some) tss)" |
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friend_of_corec f15 where |
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"f15 ts_opt = Node 0 |
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(case map_option (map branches) ts_opt of |
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None \<Rightarrow> [] |
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| Some tss \<Rightarrow> map (f15 o Some) tss)" |
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sorry |
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corec f16 :: "tree list option \<Rightarrow> tree" where |
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"f16 ts_opt = Node 0 |
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(case ts_opt of |
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None \<Rightarrow> [] |
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| Some ts \<Rightarrow> map (f16 o Some o branches) ts)" |
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friend_of_corec f16 where |
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"f16 ts_opt = Node 0 |
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(case ts_opt of |
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None \<Rightarrow> [] |
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| Some ts \<Rightarrow> map (f16 o Some o branches) ts)" |
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sorry |
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corec f17 :: "tree list option \<Rightarrow> tree" where |
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"f17 ts_opt = Node 0 (case ts_opt of |
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None \<Rightarrow> [] |
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| Some ts \<Rightarrow> [f17 (Some (map (List.hd o branches) ts))])" |
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(* not parametric |
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friend_of_corec f17 where |
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"f17 ts_opt = Node 0 (case ts_opt of |
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None \<Rightarrow> [] |
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| Some ts \<Rightarrow> [f17 (Some (map (List.hd o branches) ts))])" |
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sorry |
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*) |
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corec f18 :: "tree \<Rightarrow> tree" where |
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"f18 t = Node (node t) (map (f18 o f12) (branches t))" |
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friend_of_corec f18 :: "tree \<Rightarrow> tree" where |
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"f18 t = Node (node t) (map (f18 o f12) (branches t))" |
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sorry |
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corec f19 :: "tree \<Rightarrow> tree" where |
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"f19 t = Node (node t) (map (%f. f [t]) (map f13 [1, 2, 3]))" |
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friend_of_corec f19 :: "tree \<Rightarrow> tree" where |
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"f19 t = Node (node t) (map (%f. f [t]) (map f13 [1, 2, 3]))" |
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sorry |
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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 |
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253 |
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term "map_h (map_option f12) (%n. n) f12" |
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corec f20 :: "(tree option, int, tree) h \<Rightarrow> tree \<Rightarrow> tree" where |
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"f20 x y = Node (node y) (case (map_h (map_option f12) (%n. n) f12 x) of |
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H1 None r \<Rightarrow> (f20 r y) # (branches y) |
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| H1 (Some t) r \<Rightarrow> (f20 r t) # (branches y) |
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| H2 n t r \<Rightarrow> (f20 r (Node n [])) # (branches y) |
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| H3 \<Rightarrow> branches y)" |
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friend_of_corec f20 where |
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"f20 x y = Node (node y) (case (map_h (map_option f12) (%n. n) f12 x) of |
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H1 None r \<Rightarrow> (f20 r y) # (branches y) |
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| H1 (Some t) r \<Rightarrow> (f20 r t) # (branches y) |
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| H2 n t r \<Rightarrow> (f20 r (Node n [])) # (branches y) |
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| H3 \<Rightarrow> branches y)" |
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sorry |
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||
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corec f21 where |
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"f21 x xh = |
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Node (node x) (case xh of |
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H1 (Some a) yh \<Rightarrow> (f21 x (map_h (map_option (f20 yh)) id id yh)) # (branches a) |
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| H1 None yh \<Rightarrow> [f21 x yh] |
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| H2 b c yh \<Rightarrow> (f21 c (map_h id (%n. n + b) id yh)) # (branches x) |
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| H3 \<Rightarrow> branches x)" |
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278 |
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friend_of_corec f21 where |
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280 |
"f21 x xh = |
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281 |
Node (node x) (case xh of |
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282 |
H1 (Some a) yh \<Rightarrow> (f21 x (map_h (map_option (f20 yh)) (%t. t) (%t. t) yh)) # (branches a) |
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| H1 None yh \<Rightarrow> [f21 x yh] |
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| H2 b c yh \<Rightarrow> (f21 c (map_h (%t. t) (%n. n + b) (%t. t) yh)) # (branches x) |
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| H3 \<Rightarrow> branches x)" |
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sorry |
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287 |
||
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corec f22 :: "('a \<Rightarrow> tree) \<Rightarrow> 'a list \<Rightarrow> tree" where |
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"f22 f x = Node 0 (map f x)" |
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||
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friend_of_corec f22:: "(nat \<Rightarrow> tree) \<Rightarrow> nat list \<Rightarrow> tree" where |
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"f22 f x = Node 0 (map f x)" |
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293 |
sorry |
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294 |
||
295 |
corec f23 where |
|
296 |
"f23 xh = Node 0 |
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297 |
(if is_H1 xh then |
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298 |
(f23 (h_tail xh)) # (branches (h_a xh)) |
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299 |
else if is_H1 xh then |
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300 |
(f23 (h_tail xh)) # (h_c xh) # (branches (h_b xh)) |
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301 |
else |
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302 |
[])" |
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303 |
||
304 |
friend_of_corec f23 where |
|
305 |
"f23 xh = Node 0 |
|
306 |
(if is_H1 xh then |
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307 |
(f23 (h_tail xh)) # (branches (h_a xh)) |
|
308 |
else if is_H1 xh then |
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309 |
(f23 (h_tail xh)) # (h_c xh) # (branches (h_b xh)) |
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310 |
else |
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311 |
[])" |
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312 |
sorry |
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313 |
||
314 |
corec f24 where |
|
315 |
"f24 xh = |
|
316 |
(if is_H1 xh then |
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317 |
Node 0 ((f24 (h_tail xh)) # (h_a xh 0)) |
|
318 |
else if is_H2 xh then |
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319 |
Node (h_b xh) ((f24 (h_tail xh)) # (h_c xh 0)) |
|
320 |
else |
|
321 |
Node 0 [])" |
|
322 |
||
323 |
friend_of_corec f24 :: "(nat \<Rightarrow> tree list, int, int \<Rightarrow> tree list) h \<Rightarrow> tree" where |
|
324 |
"f24 xh = |
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325 |
(if is_H1 xh then |
|
326 |
Node 0 ((f24 (h_tail xh)) # (h_a xh 0)) |
|
327 |
else if is_H2 xh then |
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328 |
Node (h_b xh) ((f24 (h_tail xh)) # (h_c xh 0)) |
|
329 |
else |
|
330 |
Node 0 [])" |
|
331 |
sorry |
|
332 |
||
333 |
corec f25 where |
|
334 |
"f25 x = Node (node x) (map f25 ((id branches) x))" |
|
335 |
||
336 |
codatatype ('a, 'b) y_type = |
|
337 |
Y (lab: "'a \<Rightarrow> 'b") (y_tail: "('a, 'b) y_type") |
|
338 |
||
339 |
corec f26 :: "(int, tree) y_type \<Rightarrow> tree \<Rightarrow> tree" where |
|
340 |
"f26 y x = (case map_y_type f12 y of |
|
341 |
Y f y' \<Rightarrow> Node (node x) ((f (node x)) # (map (f26 y') (branches x))))" |
|
342 |
||
343 |
friend_of_corec f26 where |
|
344 |
"f26 y x = (case map_y_type f12 y of |
|
345 |
Y f y' \<Rightarrow> Node (node x) ((f (node x)) # (map (f26 y') (branches x))))" |
|
346 |
sorry |
|
347 |
||
348 |
consts int_of_list :: "'a list \<Rightarrow> int" |
|
349 |
||
350 |
corec f27 :: "(int, tree) y_type \<Rightarrow> tree \<Rightarrow> tree" where |
|
351 |
"f27 y x = Node (int_of_list (map (f26 (y_tail y)) (branches x))) [lab y (node x)]" |
|
352 |
||
353 |
friend_of_corec f27 :: "(int, tree) y_type \<Rightarrow> tree \<Rightarrow> tree" where |
|
354 |
"f27 y x = Node (int_of_list (map (f26 (y_tail y)) (branches x))) [lab y (node x)]" |
|
355 |
sorry |
|
356 |
||
357 |
corec f28 :: "(tree option list, (int \<Rightarrow> int) \<Rightarrow> int list \<Rightarrow> tree, tree) h \<Rightarrow> tree" where |
|
358 |
"f28 xh = (case xh of |
|
359 |
H3 \<Rightarrow> Node 0 [] |
|
360 |
| H1 l r \<Rightarrow> Node 0 ((f28 r) # map the (filter (%opt. case opt of None \<Rightarrow> False | Some _ \<Rightarrow> True) l)) |
|
361 |
| H2 f t r \<Rightarrow> Node (node t) (map (%t. f id [node t]) (branches t)))" |
|
362 |
||
363 |
codatatype llist = |
|
364 |
LNil | LCons (head: nat) (tail: llist) |
|
365 |
||
366 |
inductive llist_in where |
|
367 |
"llist_in (LCons x xs) x" |
|
368 |
| "llist_in xs y \<Longrightarrow> llist_in (LCons x xs) y" |
|
369 |
||
370 |
abbreviation "lset xs \<equiv> {x. llist_in xs x}" |
|
371 |
||
372 |
corecursive lfilter where |
|
373 |
"lfilter P xs = (if \<forall> x \<in> lset xs. \<not> P x then |
|
374 |
LNil |
|
375 |
else if P (head xs) then |
|
376 |
LCons (head xs) (lfilter P (tail xs)) |
|
377 |
else |
|
378 |
lfilter P (tail xs))" |
|
379 |
proof (relation "measure (\<lambda>(P, xs). LEAST n. P (head ((tail ^^ n) xs)))", rule wf_measure, clarsimp) |
|
380 |
fix P xs x |
|
381 |
assume "llist_in xs x" "P x" "\<not> P (head xs)" |
|
382 |
from this(1,2) obtain a where "P (head ((tail ^^ a) xs))" |
|
383 |
by (atomize_elim, induct xs x rule: llist_in.induct) (auto simp: funpow_Suc_right |
|
384 |
simp del: funpow.simps(2) intro: exI[of _ 0] exI[of _ "Suc i" for i]) |
|
385 |
with \<open>\<not> P (head xs)\<close> |
|
386 |
have "(LEAST n. P (head ((tail ^^ n) xs))) = Suc (LEAST n. P (head ((tail ^^ Suc n) xs)))" |
|
387 |
by (intro Least_Suc) auto |
|
388 |
then show "(LEAST n. P (head ((tail ^^ n) (tail xs)))) < (LEAST n. P (head ((tail ^^ n) xs)))" |
|
389 |
by (simp add: funpow_swap1[of tail]) |
|
390 |
qed |
|
391 |
||
392 |
codatatype Stream = |
|
393 |
SCons (head: nat) (tail: Stream) |
|
394 |
||
395 |
corec map_Stream where |
|
396 |
"map_Stream f s = SCons (f (head s)) (map_Stream f (tail s))" |
|
397 |
||
398 |
friend_of_corec map_Stream where |
|
399 |
"map_Stream f s = SCons (f (head s)) (map_Stream f (tail s))" |
|
400 |
sorry |
|
401 |
||
402 |
corec f29 where |
|
403 |
"f29 f ll = SCons (head ll) (f29 f (map_Stream f (tail ll)))" |
|
404 |
||
405 |
friend_of_corec f29 where |
|
406 |
"f29 f ll = SCons (head ll) (f29 f (map_Stream f (tail ll)))" |
|
407 |
sorry |
|
408 |
||
409 |
corec f30 where |
|
410 |
"f30 n m = (if n = 0 then SCons m (f30 m m) else f30 (n - 1) (n * m))" |
|
411 |
||
412 |
corec f31 :: "llist \<Rightarrow> llist" where |
|
413 |
"f31 x = (if x = LNil then LCons undefined (f31 undefined) else LCons undefined undefined)" |
|
414 |
||
415 |
friend_of_corec f31 where |
|
416 |
"f31 x = (if x = LNil then LCons undefined (f31 undefined) else LCons undefined undefined)" |
|
417 |
sorry |
|
418 |
||
419 |
corec f32 :: "tree \<Rightarrow> tree" where |
|
420 |
"f32 t = Node (node t) (map ((\<lambda>t'. f18 t') o f32) (branches t))" |
|
421 |
||
422 |
corec f33 :: "tree \<Rightarrow> tree" where |
|
423 |
"f33 t = f18 (f18 (Node (node t) (map (\<lambda>t'. (f18 o f18) (f18 (f18 (f33 t')))) (branches t))))" |
|
424 |
||
425 |
corec f34 :: "tree \<Rightarrow> tree" where |
|
426 |
"f34 t = f18 (f18 (Node (node t) (map (f18 o f18 o f34) (branches t))))" |
|
427 |
||
428 |
corec f35 :: "tree \<Rightarrow> tree" where |
|
429 |
"f35 t = f18 (f18 (Node (node t) (map (f18 o (f18 o (\<lambda>t'. f18 t')) o f35) (branches t))))" |
|
430 |
||
431 |
corec f37 :: "int \<Rightarrow> tree list \<Rightarrow> tree option \<Rightarrow> nat \<Rightarrow> tree" where |
|
432 |
"f37 a x1 = undefined a x1" |
|
433 |
||
434 |
end |