# Theory Lifting_Code_Dt_Test

(*  Title:      HOL/Quotient_Examples/Lifting_Code_Dt_Test.thy
Author:     Ondrej Kuncar, TU Muenchen

Miscellaneous lift_definition(code_dt) definitions (for testing purposes).
*)

theory Lifting_Code_Dt_Test
imports Main
begin

(* basic examples *)

typedef bool2 = "{x. x}" by auto

setup_lifting type_definition_bool2

lift_definition(code_dt) f1 ::  is "Some True" by simp

lift_definition(code_dt) f2 ::  is "[True]" by simp

lift_definition(code_dt) f3 ::  is "(True, 42)" by simp

lift_definition(code_dt) f4 ::  is "Inr True" by simp

lift_definition(code_dt) f5 :: "'a  (bool2 × 'a) option" is "λx. Some (True, x)" by simp

(* ugly (i.e., sensitive to rewriting done in my tactics) definition of T *)

typedef 'a T = "{ x::'a. (y::'a) z::'a. (w::'a). (z = z)  eq_onp top y y
rel_prod (eq_onp top) (eq_onp top) (x, y) (x, y)  pred_prod top top (w, w) }"
by auto

setup_lifting type_definition_T

lift_definition(code_dt) f6 ::  is "Some True" by simp

lift_definition(code_dt) f7 :: "(bool T × int) option" is "Some (True, 42)" by simp

lift_definition(code_dt) f8 :: "bool T  int  (bool T × int) option"
is "λx y. if x then Some (x, y) else None" by simp

lift_definition(code_dt) f9 :: "nat  ((bool T × int) option) list × nat"
is "λx. ([Some (True, 42)], x)" by simp

(* complicated nested datatypes *)

(* stolen from Datatype_Examples *)
datatype 'a tree = Empty | Node 'a "'a tree list"

datatype 'a ttree = TEmpty | TNode 'a "'a ttree list tree"

datatype 'a tttree = TEmpty | TNode 'a

lift_definition(code_dt) f10 :: "int  int T tree" is "λi. Node i [Node i Nil, Empty]" by simp

lift_definition(code_dt) f11 :: "int  int T ttree"
is "λi. ttree.TNode i (Node [ttree.TNode i Empty] [])" by simp

lift_definition(code_dt) f12 ::  is "λi. tttree.TNode i Empty" by simp

(* Phantom type variables *)

datatype 'a phantom = PH1 | PH2

datatype ('a, 'b) phantom2 = PH21 'a | PH22 "'a option"

lift_definition(code_dt) f13 ::  is "λi. PH1" by auto

lift_definition(code_dt) f14 :: "int  (int T, nat T) phantom2" is "λi. PH22 (Some i)" by auto

(* Mutual datatypes *)

datatype 'a M1 = Empty 'a | CM "'a M2"
and 'a M2 = CM2 "'a M1"

lift_definition(code_dt) f15 :: "int  int T M1" is "λi. Empty i" by auto

(* Codatatypes *)

codatatype 'a stream = S 'a "'a stream"

primcorec
sconst :: "'a  'a stream" where
"sconst a = S a (sconst a)"

lift_definition(code_dt) f16 ::  is "λi. sconst i"  unfolding pred_stream_def
by auto

(* Sort constraints *)

datatype ('a::finite, 'b::finite) F = F 'a | F2 'b

instance T :: (finite) finite by standard (transfer, auto)

lift_definition(code_dt) f17 :: "bool  (bool T, 'b::finite) F" is "λb. F b" by auto

export_code f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17