src/HOL/BNF/Examples/Misc_Primrec.thy
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(*  Title:      HOL/BNF/Examples/Misc_Primrec.thy
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2013
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Miscellaneous primitive recursive function definitions.
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*)
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header {* Miscellaneous Primitive Recursive Function Definitions *}
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theory Misc_Primrec
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imports Misc_Datatype
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begin
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primrec_new nat_of_simple :: "simple \<Rightarrow> nat" where
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  "nat_of_simple X1 = 1" |
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  "nat_of_simple X2 = 2" |
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  "nat_of_simple X3 = 3" |
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  "nat_of_simple X4 = 4"
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primrec_new simple_of_simple' :: "simple' \<Rightarrow> simple" where
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  "simple_of_simple' (X1' _) = X1" |
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  "simple_of_simple' (X2' _) = X2" |
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  "simple_of_simple' (X3' _) = X3" |
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  "simple_of_simple' (X4' _) = X4"
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primrec_new inc_simple'' :: "nat \<Rightarrow> simple'' \<Rightarrow> simple''" where
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  "inc_simple'' k (X1'' n i) = X1'' (n + k) (i + int k)" |
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  "inc_simple'' _ X2'' = X2''"
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primrec_new myapp :: "'a mylist \<Rightarrow> 'a mylist \<Rightarrow> 'a mylist" where
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  "myapp MyNil ys = ys" |
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  "myapp (MyCons x xs) ys = MyCons x (myapp xs ys)"
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primrec_new myrev :: "'a mylist \<Rightarrow> 'a mylist" where
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  "myrev MyNil = MyNil" |
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  "myrev (MyCons x xs) = myapp (myrev xs) (MyCons x MyNil)"
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primrec_new shuffle_sp :: "('a, 'b, 'c, 'd) some_passive \<Rightarrow> ('d, 'a, 'b, 'c) some_passive" where
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  "shuffle_sp (SP1 sp) = SP1 (shuffle_sp sp)" |
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  "shuffle_sp (SP2 a) = SP3 a" |
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  "shuffle_sp (SP3 b) = SP4 b" |
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  "shuffle_sp (SP4 c) = SP5 c" |
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  "shuffle_sp (SP5 d) = SP2 d"
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primrec_new
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  hf_size :: "hfset \<Rightarrow> nat"
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where
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  "hf_size (HFset X) = 1 + setsum id (fset (fimage hf_size X))"
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primrec_new rename_lam :: "(string \<Rightarrow> string) \<Rightarrow> lambda \<Rightarrow> lambda" where
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  "rename_lam f (Var s) = Var (f s)" |
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  "rename_lam f (App l l') = App (rename_lam f l) (rename_lam f l')" |
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  "rename_lam f (Abs s l) = Abs (f s) (rename_lam f l)" |
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  "rename_lam f (Let SL l) = Let (fimage (map_pair f (rename_lam f)) SL) (rename_lam f l)"
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primrec_new
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  sum_i1 :: "('a\<Colon>{zero,plus}) I1 \<Rightarrow> 'a" and
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  sum_i2 :: "'a I2 \<Rightarrow> 'a"
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where
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  "sum_i1 (I11 n i) = n + sum_i1 i" |
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  "sum_i1 (I12 n i) = n + sum_i2 i" |
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  "sum_i2 I21 = 0" |
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  "sum_i2 (I22 i j) = sum_i1 i + sum_i2 j"
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primrec_new forest_of_mylist :: "'a tree mylist \<Rightarrow> 'a forest" where
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  "forest_of_mylist MyNil = FNil" |
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  "forest_of_mylist (MyCons t ts) = FCons t (forest_of_mylist ts)"
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primrec_new mylist_of_forest :: "'a forest \<Rightarrow> 'a tree mylist" where
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  "mylist_of_forest FNil = MyNil" |
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  "mylist_of_forest (FCons t ts) = MyCons t (mylist_of_forest ts)"
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definition frev :: "'a forest \<Rightarrow> 'a forest" where
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  "frev = forest_of_mylist \<circ> myrev \<circ> mylist_of_forest"
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primrec_new
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  mirror_tree :: "'a tree \<Rightarrow> 'a tree" and
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  mirror_forest :: "'a forest \<Rightarrow> 'a forest"
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where
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  "mirror_tree TEmpty = TEmpty" |
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  "mirror_tree (TNode x ts) = TNode x (mirror_forest ts)" |
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  "mirror_forest FNil = FNil" |
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  "mirror_forest (FCons t ts) = frev (FCons (mirror_tree t) (mirror_forest ts))"
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primrec_new
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  mylist_of_tree' :: "'a tree' \<Rightarrow> 'a mylist" and
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  mylist_of_branch :: "'a branch \<Rightarrow> 'a mylist"
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where
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  "mylist_of_tree' TEmpty' = MyNil" |
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  "mylist_of_tree' (TNode' b b') = myapp (mylist_of_branch b) (mylist_of_branch b')" |
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  "mylist_of_branch (Branch x t) = MyCons x (mylist_of_tree' t)"
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primrec_new
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  is_ground_exp :: "('a, 'b) exp \<Rightarrow> bool" and
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  is_ground_trm :: "('a, 'b) trm \<Rightarrow> bool" and
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  is_ground_factor :: "('a, 'b) factor \<Rightarrow> bool"
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where
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  "is_ground_exp (Term t) \<longleftrightarrow> is_ground_trm t" |
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  "is_ground_exp (Sum t e) \<longleftrightarrow> is_ground_trm t \<and> is_ground_exp e" |
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  "is_ground_trm (Factor f) \<longleftrightarrow> is_ground_factor f" |
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  "is_ground_trm (Prod f t) \<longleftrightarrow> is_ground_factor f \<and> is_ground_trm t" |
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  "is_ground_factor (C _) \<longleftrightarrow> True" |
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  "is_ground_factor (V _) \<longleftrightarrow> False" |
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  "is_ground_factor (Paren e) \<longleftrightarrow> is_ground_exp e"
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primrec_new map_ftreeA :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where
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  "map_ftreeA f (FTLeaf x) = FTLeaf (f x)" |
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  "map_ftreeA f (FTNode g) = FTNode (map_ftreeA f \<circ> g)"
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primrec_new map_ftreeB :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a ftree \<Rightarrow> 'b ftree" where
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  "map_ftreeB f (FTLeaf x) = FTLeaf (f x)" |
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  "map_ftreeB f (FTNode g) = FTNode (map_ftreeB f \<circ> g \<circ> the_inv f)"
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end