Nominal/nominal_dt_rawperm.ML
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(*  Title:      nominal_dt_rawperm.ML
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    Author:     Cezary Kaliszyk
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    Author:     Christian Urban
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  Definitions of the raw permutations and
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  proof that the raw datatypes are in the
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  pt-class.
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*)
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signature NOMINAL_DT_RAWPERM =
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sig
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  val define_raw_perms: string list -> typ list -> term list -> thm -> theory -> 
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    (term list * thm list * thm list) * theory
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end
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structure Nominal_Dt_RawPerm: NOMINAL_DT_RAWPERM =
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struct
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(** proves the two pt-type class properties **)
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fun prove_permute_zero lthy induct perm_defs perm_fns =
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let
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  val perm_types = map (body_type o fastype_of) perm_fns
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  val perm_indnames = Datatype_Prop.make_tnames perm_types
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  fun single_goal ((perm_fn, T), x) =
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    HOLogic.mk_eq (perm_fn $ @{term "0::perm"} $ Free (x, T), Free (x, T))
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  val goals =
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    HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
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      (map single_goal (perm_fns ~~ perm_types ~~ perm_indnames)))
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  val simps = HOL_basic_ss addsimps (@{thm permute_zero} :: perm_defs)
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  val tac = (Datatype_Aux.indtac induct perm_indnames 
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             THEN_ALL_NEW asm_simp_tac simps) 1
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in
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  Goal.prove lthy perm_indnames [] goals (K tac)
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  |> Datatype_Aux.split_conj_thm
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end
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fun prove_permute_plus lthy induct perm_defs perm_fns =
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let
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  val p = Free ("p", @{typ perm})
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  val q = Free ("q", @{typ perm})
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  val perm_types = map (body_type o fastype_of) perm_fns
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  val perm_indnames = Datatype_Prop.make_tnames perm_types
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  fun single_goal ((perm_fn, T), x) = HOLogic.mk_eq 
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      (perm_fn $ (mk_plus p q) $ Free (x, T), perm_fn $ p $ (perm_fn $ q $ Free (x, T)))
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  val goals =
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    HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
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      (map single_goal (perm_fns ~~ perm_types ~~ perm_indnames)))
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  val simps = HOL_basic_ss addsimps (@{thm permute_plus} :: perm_defs)
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  val tac = (Datatype_Aux.indtac induct perm_indnames
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             THEN_ALL_NEW asm_simp_tac simps) 1
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in
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  Goal.prove lthy ("p" :: "q" :: perm_indnames) [] goals (K tac)
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  |> Datatype_Aux.split_conj_thm 
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end
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fun mk_perm_eq ty_perm_assoc cnstr = 
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let
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  fun lookup_perm p (ty, arg) = 
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    case (AList.lookup (op=) ty_perm_assoc ty) of
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      SOME perm => perm $ p $ arg
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    | NONE => Const (@{const_name permute}, perm_ty ty) $ p $ arg
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  val p = Free ("p", @{typ perm})
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  val (arg_tys, ty) =
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    fastype_of cnstr
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    |> strip_type
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  val arg_names = Name.variant_list ["p"] (Datatype_Prop.make_tnames arg_tys)
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  val args = map Free (arg_names ~~ arg_tys)
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  val lhs = lookup_perm p (ty, list_comb (cnstr, args))
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  val rhs = list_comb (cnstr, map (lookup_perm p) (arg_tys ~~ args))
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  val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))  
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in
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  (Attrib.empty_binding, eq)
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end
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fun define_raw_perms full_ty_names tys constrs induct_thm thy =
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let
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  val perm_fn_names = full_ty_names
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    |> map Long_Name.base_name
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    |> map (prefix "permute_")
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  val perm_fn_types = map perm_ty tys
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  val perm_fn_frees = map Free (perm_fn_names ~~ perm_fn_types)
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  val perm_fn_binds = map (fn s => (Binding.name s, NONE, NoSyn)) perm_fn_names
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  val perm_eqs = map (mk_perm_eq (tys ~~ perm_fn_frees)) constrs
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  val lthy =
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    Class.instantiation (full_ty_names, [], @{sort pt}) thy
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  val ((perm_funs, perm_eq_thms), lthy') =
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    Primrec.add_primrec perm_fn_binds perm_eqs lthy;
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  val perm_zero_thms = prove_permute_zero lthy' induct_thm perm_eq_thms perm_funs
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  val perm_plus_thms = prove_permute_plus lthy' induct_thm perm_eq_thms perm_funs
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  fun tac _ (_, _, simps) =
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    Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac simps)
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  fun morphism phi (fvs, dfs, simps) =
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    (map (Morphism.term phi) fvs, 
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     map (Morphism.thm phi) dfs, 
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     map (Morphism.thm phi) simps);
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in
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  lthy'
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  |> Class.prove_instantiation_exit_result morphism tac 
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       (perm_funs, perm_eq_thms, perm_zero_thms @ perm_plus_thms)
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end
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end (* structure *)
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