# Theory Misc_Primrec

theory Misc_Primrec
imports Misc_Datatype
```(*  Title:      HOL/Datatype_Examples/Misc_Primrec.thy
Author:     Jasmin Blanchette, TU Muenchen

Miscellaneous primitive recursive function definitions.
*)

section ‹Miscellaneous Primitive Recursive Function Definitions›

theory Misc_Primrec
imports Misc_Datatype
begin

primrec nat_of_simple :: "simple ⇒ nat" where
"nat_of_simple X1 = 1" |
"nat_of_simple X2 = 2" |
"nat_of_simple X3 = 3" |
"nat_of_simple X4 = 4"

primrec simple_of_simple' :: "simple' ⇒ simple" where
"simple_of_simple' (X1' _) = X1" |
"simple_of_simple' (X2' _) = X2" |
"simple_of_simple' (X3' _) = X3" |
"simple_of_simple' (X4' _) = X4"

primrec inc_simple'' :: "nat ⇒ simple'' ⇒ simple''" where
"inc_simple'' k (X1'' n i) = X1'' (n + k) (i + int k)" |
"inc_simple'' _ X2'' = X2''"

primrec myapp :: "'a mylist ⇒ 'a mylist ⇒ 'a mylist" where
"myapp MyNil ys = ys" |
"myapp (MyCons x xs) ys = MyCons x (myapp xs ys)"

primrec myrev :: "'a mylist ⇒ 'a mylist" where
"myrev MyNil = MyNil" |
"myrev (MyCons x xs) = myapp (myrev xs) (MyCons x MyNil)"

primrec shuffle_sp :: "('a :: ord, 'b :: ord, 'c, 'd) some_passive ⇒ ('d, 'a, 'b, 'c) some_passive" where
"shuffle_sp (SP1 sp) = SP1 (shuffle_sp sp)" |
"shuffle_sp (SP2 a) = SP3 a" |
"shuffle_sp (SP3 b) = SP4 b" |
"shuffle_sp (SP4 c) = SP5 c" |
"shuffle_sp (SP5 d) = SP2 d"

primrec
hf_size :: "hfset ⇒ nat"
where
"hf_size (HFset X) = 1 + sum id (fset (fimage hf_size X))"

primrec rename_lam :: "(string ⇒ string) ⇒ lambda ⇒ lambda" where
"rename_lam f (Var s) = Var (f s)" |
"rename_lam f (App l l') = App (rename_lam f l) (rename_lam f l')" |
"rename_lam f (Abs s l) = Abs (f s) (rename_lam f l)" |
"rename_lam f (Let SL l) = Let (fimage (map_prod f (rename_lam f)) SL) (rename_lam f l)"

primrec (in loc)
sum_i1 :: "('b::{zero,plus}) I1 ⇒ 'b" and
sum_i2 :: "'b I2 ⇒ 'b"
where
"sum_i1 (I11 n i) = n + sum_i1 i" |
"sum_i1 (I12 n i) = n + sum_i2 i" |
"sum_i2 I21 = 0" |
"sum_i2 (I22 i j) = sum_i1 i + sum_i2 j"

context loc
begin

primrec forest_of_mylist :: "'a tree mylist ⇒ 'a forest" where
"forest_of_mylist MyNil = FNil" |
"forest_of_mylist (MyCons t ts) = FCons t (forest_of_mylist ts)"

primrec mylist_of_forest :: "'a forest ⇒ 'a tree mylist" where
"mylist_of_forest FNil = MyNil" |
"mylist_of_forest (FCons t ts) = MyCons t (mylist_of_forest ts)"

definition frev :: "'a forest ⇒ 'a forest" where
"frev = forest_of_mylist ∘ myrev ∘ mylist_of_forest"

primrec
mirror_tree :: "'a tree ⇒ 'a tree" and
mirror_forest :: "'a forest ⇒ 'a forest"
where
"mirror_tree TEmpty = TEmpty" |
"mirror_tree (TNode x ts) = TNode x (mirror_forest ts)" |
"mirror_forest FNil = FNil" |
"mirror_forest (FCons t ts) = frev (FCons (mirror_tree t) (mirror_forest ts))"

primrec
mylist_of_tree' :: "'a tree' ⇒ 'a mylist" and
mylist_of_branch :: "'a branch ⇒ 'a mylist"
where
"mylist_of_tree' TEmpty' = MyNil" |
"mylist_of_tree' (TNode' b b') = myapp (mylist_of_branch b) (mylist_of_branch b')" |
"mylist_of_branch (Branch x t) = MyCons x (mylist_of_tree' t)"

end

primrec
id_tree :: "'a bin_rose_tree ⇒ 'a bin_rose_tree" and
id_trees1 :: "'a bin_rose_tree mylist ⇒ 'a bin_rose_tree mylist" and
id_trees2 :: "'a bin_rose_tree mylist ⇒ 'a bin_rose_tree mylist"
where
"id_tree (BRTree a ts ts') = BRTree a (id_trees1 ts) (id_trees2 ts')" |
"id_trees1 MyNil = MyNil" |
"id_trees1 (MyCons t ts) = MyCons (id_tree t) (id_trees1 ts)" |
"id_trees2 MyNil = MyNil" |
"id_trees2 (MyCons t ts) = MyCons (id_tree t) (id_trees2 ts)"

primrec
trunc_tree :: "'a bin_rose_tree ⇒ 'a bin_rose_tree" and
trunc_trees1 :: "'a bin_rose_tree mylist ⇒ 'a bin_rose_tree mylist" and
trunc_trees2 :: "'a bin_rose_tree mylist ⇒ 'a bin_rose_tree mylist"
where
"trunc_tree (BRTree a ts ts') = BRTree a (trunc_trees1 ts) (trunc_trees2 ts')" |
"trunc_trees1 MyNil = MyNil" |
"trunc_trees1 (MyCons t ts) = MyCons (id_tree t) MyNil" |
"trunc_trees2 MyNil = MyNil" |
"trunc_trees2 (MyCons t ts) = MyCons (id_tree t) MyNil"

primrec
is_ground_exp :: "('a, 'b) exp ⇒ bool" and
is_ground_trm :: "('a, 'b) trm ⇒ bool" and
is_ground_factor :: "('a, 'b) factor ⇒ bool"
where
"is_ground_exp (Term t) ⟷ is_ground_trm t" |
"is_ground_exp (Sum t e) ⟷ is_ground_trm t ∧ is_ground_exp e" |
"is_ground_trm (Factor f) ⟷ is_ground_factor f" |
"is_ground_trm (Prod f t) ⟷ is_ground_factor f ∧ is_ground_trm t" |
"is_ground_factor (C _) ⟷ True" |
"is_ground_factor (V _) ⟷ False" |
"is_ground_factor (Paren e) ⟷ is_ground_exp e"

primrec map_ftreeA :: "('a ⇒ 'a) ⇒ 'a ftree ⇒ 'a ftree" where
"map_ftreeA f (FTLeaf x) = FTLeaf (f x)" |
"map_ftreeA f (FTNode g) = FTNode (map_ftreeA f ∘ g)"

primrec map_ftreeB :: "('a ⇒ 'b) ⇒ 'a ftree ⇒ 'b ftree" where
"map_ftreeB f (FTLeaf x) = FTLeaf (f x)" |
"map_ftreeB f (FTNode g) = FTNode (map_ftreeB f ∘ g ∘ the_inv f)"

end
```