theory NewParser+
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imports "../Nominal-General/Nominal2_Base" +
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        "../Nominal-General/Nominal2_Eqvt" +
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        "../Nominal-General/Nominal2_Supp" +
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        "Perm" "NewFv" "NewAlpha" "Tacs" "Equivp" "Lift"+
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begin+
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+
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section{* Interface for nominal_datatype *}+
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+
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+
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ML {* +
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(* nominal datatype parser *)+
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local+
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  structure P = OuterParse;+
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  structure S = Scan+
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+
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  fun triple1 ((x, y), z) = (x, y, z)+
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  fun triple2 (x, (y, z)) = (x, y, z)+
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  fun tuple ((x, y, z), u) = (x, y, z, u)+
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  fun tswap (((x, y), z), u) = (x, y, u, z)+
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in+
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+
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val _ = OuterKeyword.keyword "bind"+
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val _ = OuterKeyword.keyword "bind_set"+
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val _ = OuterKeyword.keyword "bind_res"+
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+
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val anno_typ = S.option (P.name --| P.$$$ "::") -- P.typ+
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+
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val bind_mode = P.$$$ "bind" || P.$$$ "bind_set" || P.$$$ "bind_res"+
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+
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val bind_clauses = +
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  P.enum "," (bind_mode -- S.repeat1 P.term -- (P.$$$ "in" |-- S.repeat1 P.name) >> triple1)+
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+
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val cnstr_parser =+
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  P.binding -- S.repeat anno_typ -- bind_clauses -- P.opt_mixfix >> tswap+
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+
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(* datatype parser *)+
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val dt_parser =+
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  (P.type_args -- P.binding -- P.opt_mixfix >> triple1) -- +
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    (P.$$$ "=" |-- P.enum1 "|" cnstr_parser) >> tuple+
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+
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(* binding function parser *)+
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val bnfun_parser = +
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  S.optional (P.$$$ "binder" |-- P.fixes -- SpecParse.where_alt_specs) ([], [])+
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+
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(* main parser *)+
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val main_parser =+
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  P.and_list1 dt_parser -- bnfun_parser >> triple2+
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+
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end+
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*}+
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+
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ML {*+
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fun get_cnstrs dts =+
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  map (fn (_, _, _, constrs) => constrs) dts+
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+
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fun get_typed_cnstrs dts =+
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  flat (map (fn (_, bn, _, constrs) => +
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   (map (fn (bn', _, _) => (Binding.name_of bn, Binding.name_of bn')) constrs)) dts)+
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+
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fun get_cnstr_strs dts =+
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  map (fn (bn, _, _) => Binding.name_of bn) (flat (get_cnstrs dts))+
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+
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fun get_bn_fun_strs bn_funs =+
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  map (fn (bn_fun, _, _) => Binding.name_of bn_fun) bn_funs+
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*}+
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+
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+
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ML {* +
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fun add_primrec_wrapper funs eqs lthy = +
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  if null funs then (([], []), lthy)+
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  else +
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   let +
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     val eqs' = map (fn (_, eq) => (Attrib.empty_binding, eq)) eqs+
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     val funs' = map (fn (bn, ty, mx) => (bn, SOME ty, mx)) funs+
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     val ((funs'', eqs''), lthy') = Primrec.add_primrec funs' eqs' lthy+
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     val phi = ProofContext.export_morphism lthy' lthy+
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   in +
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     ((map (Morphism.term phi) funs'', map (Morphism.thm phi) eqs''), lthy')+
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   end+
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*}+
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+
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ML {*+
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fun add_datatype_wrapper dt_names dts =+
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let+
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  val conf = Datatype.default_config+
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in+
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  Local_Theory.theory_result (Datatype.add_datatype conf dt_names dts)+
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end+
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*}+
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+
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+
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text {* Infrastructure for adding "_raw" to types and terms *}+
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+
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ML {*+
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fun add_raw s = s ^ "_raw"+
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fun add_raws ss = map add_raw ss+
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fun raw_bind bn = Binding.suffix_name "_raw" bn+
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+
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fun replace_str ss s = +
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  case (AList.lookup (op=) ss s) of +
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     SOME s' => s'+
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   | NONE => s+
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+
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fun replace_typ ty_ss (Type (a, Ts)) = Type (replace_str ty_ss a, map (replace_typ ty_ss) Ts)+
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  | replace_typ ty_ss T = T  +
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+
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fun raw_dts ty_ss dts =+
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let+
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  fun raw_dts_aux1 (bind, tys, mx) =+
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    (raw_bind bind, map (replace_typ ty_ss) tys, mx)+
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+
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  fun raw_dts_aux2 (ty_args, bind, mx, constrs) =+
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    (ty_args, raw_bind bind, mx, map raw_dts_aux1 constrs)+
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in+
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  map raw_dts_aux2 dts+
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end+
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+
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fun replace_aterm trm_ss (Const (a, T)) = Const (replace_str trm_ss a, T)+
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  | replace_aterm trm_ss (Free (a, T)) = Free (replace_str trm_ss a, T)+
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  | replace_aterm trm_ss trm = trm+
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+
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fun replace_term trm_ss ty_ss trm =+
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  trm |> Term.map_aterms (replace_aterm trm_ss) |> map_types (replace_typ ty_ss) +
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*}+
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+
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ML {*+
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fun rawify_dts dt_names dts dts_env =+
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let+
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  val raw_dts = raw_dts dts_env dts+
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  val raw_dt_names = add_raws dt_names+
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in+
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  (raw_dt_names, raw_dts)+
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end +
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*}+
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+
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ML {*+
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fun rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs =+
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let+
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  val bn_funs' = map (fn (bn, ty, mx) => +
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    (raw_bind bn, replace_typ dts_env ty, mx)) bn_funs+
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  +
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  val bn_eqs' = map (fn (attr, trm) => +
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    (attr, replace_term (cnstrs_env @ bn_fun_env) dts_env trm)) bn_eqs+
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in+
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  (bn_funs', bn_eqs') +
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end +
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*}+
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+
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ML {* +
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fun rawify_bclauses dts_env cnstrs_env bn_fun_env bclauses =+
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let+
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  fun rawify_bnds bnds = +
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    map (apfst (Option.map (replace_term (cnstrs_env @ bn_fun_env) dts_env))) bnds+
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+
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  fun rawify_bclause (BEmy n) = BEmy n+
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    | rawify_bclause (BLst (bnds, bdys)) = BLst (rawify_bnds bnds, bdys)+
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    | rawify_bclause (BSet (bnds, bdys)) = BSet (rawify_bnds bnds, bdys)+
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    | rawify_bclause (BRes (bnds, bdys)) = BRes (rawify_bnds bnds, bdys)+
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in+
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  map (map (map rawify_bclause)) bclauses+
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end+
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*}+
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+
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(* strip_bn_fun takes a bn function defined by the user as a union or+
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   append of elements and returns those elements together with bn functions+
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   applied *)+
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ML {*+
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fun strip_bn_fun t =+
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  case t of+
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    Const (@{const_name sup}, _) $ l $ r => strip_bn_fun l @ strip_bn_fun r+
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  | Const (@{const_name append}, _) $ l $ r => strip_bn_fun l @ strip_bn_fun r+
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  | Const (@{const_name insert}, _) $ (Const (@{const_name atom}, _) $ Bound i) $ y =>+
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      (i, NONE) :: strip_bn_fun y+
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  | Const (@{const_name Cons}, _) $ (Const (@{const_name atom}, _) $ Bound i) $ y =>+
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      (i, NONE) :: strip_bn_fun y+
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  | Const (@{const_name bot}, _) => []+
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  | Const (@{const_name Nil}, _) => []+
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  | (f as Free _) $ Bound i => [(i, SOME f)]+
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  | _ => error ("Unsupported binding function: " ^ (PolyML.makestring t))+
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*}+
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+
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ML {*+
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fun find [] _ = error ("cannot find element")+
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  | find ((x, z)::xs) y = if (Long_Name.base_name x) = y then z else find xs y+
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*}+
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+
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ML {*+
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fun prep_bn lthy dt_names dts eqs = +
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let+
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  fun aux eq = +
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  let+
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    val (lhs, rhs) = eq+
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      |> strip_qnt_body "all" +
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      |> HOLogic.dest_Trueprop+
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      |> HOLogic.dest_eq+
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    val (bn_fun, [cnstr]) = strip_comb lhs+
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    val (_, ty) = dest_Free bn_fun+
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    val (ty_name, _) = dest_Type (domain_type ty)+
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    val dt_index = find_index (fn x => x = ty_name) dt_names+
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    val (cnstr_head, cnstr_args) = strip_comb cnstr+
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    val rhs_elements = strip_bn_fun rhs+
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    val included = map (apfst (fn i => length (cnstr_args) - i - 1)) rhs_elements+
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  in+
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    (dt_index, (bn_fun, (cnstr_head, included)))+
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  end+
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  fun order dts i ts = +
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  let+
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    val dt = nth dts i+
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    val cts = map (fn (x, _, _) => Binding.name_of x) ((fn (_, _, _, x) => x) dt)+
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    val ts' = map (fn (x, y) => (fst (dest_Const x), y)) ts+
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  in+
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    map (find ts') cts+
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  end+
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+
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  val unordered = AList.group (op=) (map aux eqs)+
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  val unordered' = map (fn (x, y) =>  (x, AList.group (op=) y)) unordered+
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  val ordered = map (fn (x, y) => (x, map (fn (v, z) => (v, order dts x z)) y)) unordered' +
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  val ordered' = flat (map (fn (ith, l) => map (fn (bn, data) => (bn, ith, data)) l) ordered)+
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+
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  val _ = tracing ("eqs\n" ^ cat_lines (map (Syntax.string_of_term lthy) eqs))+
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  (*val _ = tracing ("map eqs\n" ^ @{make_string} (map aux2 eqs))*)+
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  val _ = tracing ("ordered'\n" ^ @{make_string} ordered')+
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in+
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  ordered'+
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end+
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*}+
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+
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ML {* add_primrec_wrapper *}+
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ML {*+
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fun raw_nominal_decls dts bn_funs bn_eqs binds lthy =+
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let+
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  val thy = ProofContext.theory_of lthy+
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  val thy_name = Context.theory_name thy+
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+
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  val dt_names = map (fn (_, s, _, _) => Binding.name_of s) dts+
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  val dt_full_names = map (Long_Name.qualify thy_name) dt_names +
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  val dt_full_names' = add_raws dt_full_names+
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  val dts_env = dt_full_names ~~ dt_full_names'+
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+
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  val cnstrs = get_cnstr_strs dts+
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  val cnstrs_ty = get_typed_cnstrs dts+
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  val cnstrs_full_names = map (Long_Name.qualify thy_name) cnstrs+
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  val cnstrs_full_names' = map (fn (x, y) => Long_Name.qualify thy_name +
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    (Long_Name.qualify (add_raw x) (add_raw y))) cnstrs_ty+
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  val cnstrs_env = cnstrs_full_names ~~ cnstrs_full_names'+
−
+
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  val bn_fun_strs = get_bn_fun_strs bn_funs+
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  val bn_fun_strs' = add_raws bn_fun_strs+
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  val bn_fun_env = bn_fun_strs ~~ bn_fun_strs'+
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  val bn_fun_full_env = map (pairself (Long_Name.qualify thy_name)) +
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    (bn_fun_strs ~~ bn_fun_strs')+
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  +
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  val (raw_dt_names, raw_dts) = rawify_dts dt_names dts dts_env+
−
+
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  val (raw_bn_funs, raw_bn_eqs) = rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs +
−
   +
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  val raw_bclauses = rawify_bclauses dts_env cnstrs_env bn_fun_full_env binds +
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  val raw_bns = prep_bn lthy dt_full_names' raw_dts (map snd raw_bn_eqs)+
−
+
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  val (raw_dt_names, lthy1) = add_datatype_wrapper raw_dt_names raw_dts lthy+
−
  val ((raw_bn_funs, raw_bn_eqs), lthy2) = add_primrec_wrapper raw_bn_funs raw_bn_eqs lthy1+
−
+
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  val morphism_2_0 = ProofContext.export_morphism lthy2 lthy+
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  fun export_fun f (t, n , l) = (f t, n, map (map (apsnd (Option.map f))) l);+
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  val bn_funs_decls = map (export_fun (Morphism.term morphism_2_0)) raw_bns;+
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in+
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  (raw_dt_names, raw_bn_eqs, raw_bclauses, bn_funs_decls, lthy2)+
−
end+
−
*}+
−
+
−
lemma equivp_hack: "equivp x"+
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sorry+
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ML {*+
−
fun equivp_hack ctxt rel =+
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let+
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  val thy = ProofContext.theory_of ctxt+
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  val ty = domain_type (fastype_of rel)+
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  val cty = ctyp_of thy ty+
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  val ct = cterm_of thy rel+
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in+
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  Drule.instantiate' [SOME cty] [SOME ct] @{thm equivp_hack}+
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end+
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*}+
−
+
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ML {* val cheat_equivp = Unsynchronized.ref false *}+
−
ML {* val cheat_fv_rsp = Unsynchronized.ref false *}+
−
ML {* val cheat_alpha_bn_rsp = Unsynchronized.ref false *}+
−
ML {* val cheat_supp_eq = Unsynchronized.ref false *}+
−
+
−
ML {*+
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fun remove_loop t =+
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  let val _ = HOLogic.dest_eq (HOLogic.dest_Trueprop (prop_of t)) in t end+
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  handle TERM _ => @{thm eqTrueI} OF [t]+
−
*}+
−
+
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text {* +
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  nominal_datatype2 does the following things in order:+
−
+
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Parser.thy/raw_nominal_decls+
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  1) define the raw datatype+
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  2) define the raw binding functions +
−
+
−
Perm.thy/define_raw_perms+
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  3) define permutations of the raw datatype and show that the raw type is +
−
     in the pt typeclass+
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      +
−
Lift.thy/define_fv_alpha_export, Fv.thy/define_fv & define_alpha+
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  4) define fv and fv_bn+
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  5) define alpha and alpha_bn+
−
+
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Perm.thy/distinct_rel+
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  6) prove alpha_distincts (C1 x \<notsimeq> C2 y ...)             (Proof by cases; simp)+
−
+
−
Tacs.thy/build_rel_inj+
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  6) prove alpha_eq_iff    (C1 x = C2 y \<leftrightarrow> P x y ...)+
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     (left-to-right by intro rule, right-to-left by cases; simp)+
−
Equivp.thy/prove_eqvt+
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  7) prove bn_eqvt (common induction on the raw datatype)+
−
  8) prove fv_eqvt (common induction on the raw datatype with help of above)+
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Rsp.thy/build_alpha_eqvts+
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  9) prove alpha_eqvt and alpha_bn_eqvt+
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     (common alpha-induction, unfolding alpha_gen, permute of #* and =)+
−
Equivp.thy/build_alpha_refl & Equivp.thy/build_equivps+
−
 10) prove that alpha and alpha_bn are equivalence relations+
−
     (common induction and application of 'compose' lemmas)+
−
Lift.thy/define_quotient_types+
−
 11) define quotient types+
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Rsp.thy/build_fvbv_rsps+
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 12) prove bn respects     (common induction and simp with alpha_gen)+
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Rsp.thy/prove_const_rsp+
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 13) prove fv respects     (common induction and simp with alpha_gen)+
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 14) prove permute respects    (unfolds to alpha_eqvt)+
−
Rsp.thy/prove_alpha_bn_rsp+
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 15) prove alpha_bn respects+
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     (alpha_induct then cases then sym and trans of the relations)+
−
Rsp.thy/prove_alpha_alphabn+
−
 16) show that alpha implies alpha_bn (by unduction, needed in following step)+
−
Rsp.thy/prove_const_rsp+
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 17) prove respects for all datatype constructors+
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     (unfold eq_iff and alpha_gen; introduce zero permutations; simp)+
−
Perm.thy/quotient_lift_consts_export+
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 18) define lifted constructors, fv, bn, alpha_bn, permutations+
−
Perm.thy/define_lifted_perms+
−
 19) lift permutation zero and add properties to show that quotient type is in the pt typeclass+
−
Lift.thy/lift_thm+
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 20) lift permutation simplifications+
−
 21) lift induction+
−
 22) lift fv+
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 23) lift bn+
−
 24) lift eq_iff+
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 25) lift alpha_distincts+
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 26) lift fv and bn eqvts+
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Equivp.thy/prove_supports+
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 27) prove that union of arguments supports constructors+
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Equivp.thy/prove_fs+
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 28) show that the lifted type is in fs typeclass     (* by q_induct, supports *)+
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Equivp.thy/supp_eq+
−
 29) prove supp = fv+
−
*}+
−
+
−
ML {*+
−
(* for testing porposes - to exit the procedure early *)+
−
exception TEST of Proof.context+
−
+
−
val (STEPS, STEPS_setup) = Attrib.config_int "STEPS" (K 10);+
−
+
−
fun get_STEPS ctxt = Config.get ctxt STEPS+
−
*}+
−
+
−
setup STEPS_setup+
−
+
−
+
−
ML {*+
−
fun nominal_datatype2 dts bn_funs bn_eqs bclauses lthy =+
−
let+
−
  (* definition of the raw datatype and raw bn-functions *)+
−
  val (raw_dt_names, raw_bn_eqs, raw_bclauses, raw_bns, lthy1) =+
−
    if get_STEPS lthy > 1 then raw_nominal_decls dts bn_funs bn_eqs bclauses lthy+
−
    else raise TEST lthy+
−
+
−
  val dtinfo = Datatype.the_info (ProofContext.theory_of lthy1) (hd raw_dt_names);+
−
  val {descr, sorts, ...} = dtinfo;+
−
  val raw_tys = map (fn (i, _) => nth_dtyp descr sorts i) descr;+
−
  val all_typs = map (fn i => Datatype_Aux.typ_of_dtyp descr sorts (Datatype_Aux.DtRec i)) (map fst descr)+
−
  val all_full_tnames = map (fn (_, (n, _, _)) => n) descr;+
−
+
−
  val dtinfos = map (Datatype.the_info (ProofContext.theory_of lthy1)) all_full_tnames;+
−
  +
−
  val inject_thms = flat (map #inject dtinfos);+
−
  val distinct_thms = flat (map #distinct dtinfos);+
−
  val rel_dtinfos = List.take (dtinfos, (length dts)); +
−
  val rel_distinct = map #distinct rel_dtinfos;  (* thm list list *)+
−
  val induct_thm = #induct dtinfo;+
−
  val exhaust_thms = map #exhaust dtinfos;+
−
+
−
  (* definitions of raw permutations *)+
−
  val ((raw_perm_def, raw_perm_simps, perms), lthy2) =+
−
    if get_STEPS lthy > 2 +
−
    then Local_Theory.theory_result (define_raw_perms descr sorts induct_thm (length dts)) lthy1+
−
    else raise TEST lthy1+
−
+
−
  val (_, lthy2a) = Local_Theory.note ((Binding.empty,+
−
    [Attrib.internal (fn _ => Nominal_ThmDecls.eqvt_add)]), raw_perm_def) lthy2;+
−
  val thy = Local_Theory.exit_global lthy2a;+
−
  val thy_name = Context.theory_name thy+
−
+
−
  (* definition of raw fv_functions *)+
−
  val lthy3 = Theory_Target.init NONE thy;+
−
  val raw_bn_funs = map (fn (f, _, _) => f) raw_bns;+
−
+
−
  val ((fv, fvbn), fv_def, lthy3a) = +
−
    if get_STEPS lthy > 3 +
−
    then define_raw_fv descr sorts raw_bns raw_bclauses lthy3+
−
    else raise TEST lthy3+
−
  +
−
+
−
  (* definition of raw alphas *)+
−
  val (((alpha_ts, alpha_intros), (alpha_cases, alpha_induct)), lthy4) =+
−
    if get_STEPS lthy > 4 +
−
    then define_raw_alpha dtinfo raw_bns raw_bclauses fv lthy3a+
−
    else raise TEST lthy3a+
−
  +
−
  val (alpha_ts_nobn, alpha_ts_bn) = chop (length fv) alpha_ts+
−
  +
−
  val dts_names = map (fn (i, (s, _, _)) => (s, i)) (#descr dtinfo);+
−
  val bn_tys = map (domain_type o fastype_of) raw_bn_funs;+
−
  val bn_nos = map (dtyp_no_of_typ dts_names) bn_tys;+
−
  val bns = raw_bn_funs ~~ bn_nos;+
−
  val rel_dists = flat (map (distinct_rel lthy4 alpha_cases)+
−
    (rel_distinct ~~ alpha_ts_nobn));+
−
  val rel_dists_bn = flat (map (distinct_rel lthy4 alpha_cases)+
−
    ((map (fn i => nth rel_distinct i) bn_nos) ~~ alpha_ts_bn))+
−
  +
−
  (* definition of raw_alpha_eq_iff  lemmas *)+
−
  val alpha_eq_iff = build_rel_inj alpha_intros (inject_thms @ distinct_thms) alpha_cases lthy4+
−
  val alpha_eq_iff_simp = map remove_loop alpha_eq_iff;+
−
  val _ = map tracing (map PolyML.makestring alpha_eq_iff_simp);+
−
+
−
  (* proving equivariance lemmas *)+
−
  val _ = warning "Proving equivariance";+
−
  val (bv_eqvt, lthy5) = prove_eqvt raw_tys induct_thm (raw_bn_eqs @ raw_perm_def) (map fst bns) lthy4+
−
  val (fv_eqvt, lthy6) = prove_eqvt raw_tys induct_thm (fv_def @ raw_perm_def) (fv @ fvbn) lthy5+
−
  val (alpha_eqvt, lthy6a) = Nominal_Eqvt.equivariance alpha_ts alpha_induct alpha_intros lthy6;+
−
+
−
  (* proving alpha equivalence *)+
−
  val _ = warning "Proving equivalence";+
−
  val fv_alpha_all = combine_fv_alpha_bns (fv, fvbn) (alpha_ts_nobn, alpha_ts_bn) bn_nos;+
−
  val reflps = build_alpha_refl fv_alpha_all alpha_ts induct_thm alpha_eq_iff_simp lthy6a;+
−
  val alpha_equivp =+
−
    if !cheat_equivp then map (equivp_hack lthy6a) alpha_ts+
−
    else build_equivps alpha_ts reflps alpha_induct+
−
      inject_thms alpha_eq_iff_simp distinct_thms alpha_cases alpha_eqvt lthy6a;+
−
  val qty_binds = map (fn (_, b, _, _) => b) dts;+
−
  val qty_names = map Name.of_binding qty_binds;+
−
  val qty_full_names = map (Long_Name.qualify thy_name) qty_names+
−
  val (qtys, lthy7) = define_quotient_types qty_binds all_typs alpha_ts_nobn alpha_equivp lthy6a;+
−
  val const_names = map Name.of_binding (flat (map (fn (_, _, _, t) => map (fn (b, _, _) => b) t) dts));+
−
  val raw_consts =+
−
    flat (map (fn (i, (_, _, l)) =>+
−
      map (fn (cname, dts) =>+
−
        Const (cname, map (Datatype_Aux.typ_of_dtyp descr sorts) dts --->+
−
          Datatype_Aux.typ_of_dtyp descr sorts (Datatype_Aux.DtRec i))) l) descr);+
−
  val (consts, const_defs, lthy8) = quotient_lift_consts_export qtys (const_names ~~ raw_consts) lthy7;+
−
  val _ = warning "Proving respects";+
−
  val bns_rsp_pre' = build_fvbv_rsps alpha_ts alpha_induct raw_bn_eqs (map fst bns) lthy8;+
−
  val (bns_rsp_pre, lthy9) = fold_map (+
−
    fn (bn_t, _) => prove_const_rsp qtys Binding.empty [bn_t] (fn _ =>+
−
       resolve_tac bns_rsp_pre' 1)) bns lthy8;+
−
  val bns_rsp = flat (map snd bns_rsp_pre);+
−
+
−
  fun fv_rsp_tac _ = if !cheat_fv_rsp then Skip_Proof.cheat_tac thy+
−
    else fvbv_rsp_tac alpha_induct fv_def lthy8 1;+
−
  val fv_rsps = prove_fv_rsp fv_alpha_all alpha_ts fv_rsp_tac lthy9;+
−
  val (fv_rsp_pre, lthy10) = fold_map+
−
    (fn fv => fn ctxt => prove_const_rsp qtys Binding.empty [fv]+
−
    (fn _ => asm_simp_tac (HOL_ss addsimps fv_rsps) 1) ctxt) (fv @ fvbn) lthy9;+
−
  val fv_rsp = flat (map snd fv_rsp_pre);+
−
  val (perms_rsp, lthy11) = prove_const_rsp qtys Binding.empty perms+
−
    (fn _ => asm_simp_tac (HOL_ss addsimps alpha_eqvt) 1) lthy10;+
−
  fun alpha_bn_rsp_tac _ = if !cheat_alpha_bn_rsp then Skip_Proof.cheat_tac thy+
−
    else+
−
      let val alpha_bn_rsp_pre = prove_alpha_bn_rsp alpha_ts alpha_induct (alpha_eq_iff_simp @ rel_dists @ rel_dists_bn) alpha_equivp exhaust_thms alpha_ts_bn lthy11 in asm_simp_tac (HOL_ss addsimps alpha_bn_rsp_pre) 1 end;+
−
  val (alpha_bn_rsps, lthy11a) = fold_map (fn cnst => prove_const_rsp qtys Binding.empty [cnst]+
−
    alpha_bn_rsp_tac) alpha_ts_bn lthy11+
−
  fun const_rsp_tac _ =+
−
    let val alpha_alphabn = prove_alpha_alphabn alpha_ts alpha_induct alpha_eq_iff_simp alpha_ts_bn lthy11a+
−
      in constr_rsp_tac alpha_eq_iff_simp (fv_rsp @ bns_rsp @ reflps @ alpha_alphabn) 1 end+
−
  val (const_rsps, lthy12) = fold_map (fn cnst => prove_const_rsp qtys Binding.empty [cnst]+
−
    const_rsp_tac) raw_consts lthy11a+
−
    val qfv_names = map (unsuffix "_raw" o Long_Name.base_name o fst o dest_Const) (fv @ fvbn)+
−
  val (qfv_ts, qfv_defs, lthy12a) = quotient_lift_consts_export qtys (qfv_names ~~ (fv @ fvbn)) lthy12;+
−
  val (qfv_ts_nobn, qfv_ts_bn) = chop (length perms) qfv_ts;+
−
  val qbn_names = map (fn (b, _ , _) => Name.of_binding b) bn_funs+
−
  val (qbn_ts, qbn_defs, lthy12b) = quotient_lift_consts_export qtys (qbn_names ~~ raw_bn_funs) lthy12a;+
−
  val qalpha_bn_names = map (unsuffix "_raw" o Long_Name.base_name o fst o dest_Const) alpha_ts_bn+
−
  val (qalpha_ts_bn, qalphabn_defs, lthy12c) = +
−
    quotient_lift_consts_export qtys (qalpha_bn_names ~~ alpha_ts_bn) lthy12b;+
−
  val _ = warning "Lifting permutations";+
−
  val thy = Local_Theory.exit_global lthy12c;+
−
  val perm_names = map (fn x => "permute_" ^ x) qty_names+
−
  val thy' = define_lifted_perms qtys qty_full_names (perm_names ~~ perms) raw_perm_simps thy;+
−
  val lthy13 = Theory_Target.init NONE thy';+
−
  val q_name = space_implode "_" qty_names;+
−
  fun suffix_bind s = Binding.qualify true q_name (Binding.name s);+
−
  val _ = warning "Lifting induction";+
−
  val constr_names = map (Long_Name.base_name o fst o dest_Const) consts;+
−
  val q_induct = Rule_Cases.name constr_names (lift_thm qtys lthy13 induct_thm);+
−
  fun note_suffix s th ctxt =+
−
    snd (Local_Theory.note ((suffix_bind s, []), th) ctxt);+
−
  fun note_simp_suffix s th ctxt =+
−
    snd (Local_Theory.note ((suffix_bind s, [Attrib.internal (K Simplifier.simp_add)]), th) ctxt);+
−
  val (_, lthy14) = Local_Theory.note ((suffix_bind "induct",+
−
    [Attrib.internal (K (Rule_Cases.case_names constr_names))]), +
−
    [Rule_Cases.name constr_names q_induct]) lthy13;+
−
  val q_inducts = Project_Rule.projects lthy13 (1 upto (length fv)) q_induct+
−
  val (_, lthy14a) = Local_Theory.note ((suffix_bind "inducts", []), q_inducts) lthy14;+
−
  val q_perm = map (lift_thm qtys lthy14) raw_perm_def;+
−
  val lthy15 = note_simp_suffix "perm" q_perm lthy14a;+
−
  val q_fv = map (lift_thm qtys lthy15) fv_def;+
−
  val lthy16 = note_simp_suffix "fv" q_fv lthy15;+
−
  val q_bn = map (lift_thm qtys lthy16) raw_bn_eqs;+
−
  val lthy17 = note_simp_suffix "bn" q_bn lthy16;+
−
  val _ = warning "Lifting eq-iff";+
−
  (*val _ = map tracing (map PolyML.makestring alpha_eq_iff);*)+
−
  val eq_iff_unfolded0 = map (Local_Defs.unfold lthy17 @{thms alphas}) alpha_eq_iff_simp+
−
  val eq_iff_unfolded1 = map (Local_Defs.unfold lthy17 @{thms Pair_eqvt}) eq_iff_unfolded0+
−
  val q_eq_iff_pre0 = map (lift_thm qtys lthy17) eq_iff_unfolded1;+
−
  val q_eq_iff_pre1 = map (Local_Defs.fold lthy17 @{thms Pair_eqvt}) q_eq_iff_pre0+
−
  val q_eq_iff_pre2 = map (Local_Defs.fold lthy17 @{thms alphas}) q_eq_iff_pre1+
−
  val q_eq_iff = map (Local_Defs.unfold lthy17 (Quotient_Info.id_simps_get lthy17)) q_eq_iff_pre2+
−
  val (_, lthy18) = Local_Theory.note ((suffix_bind "eq_iff", []), q_eq_iff) lthy17;+
−
  val q_dis = map (lift_thm qtys lthy18) rel_dists;+
−
  val lthy19 = note_simp_suffix "distinct" q_dis lthy18;+
−
  val q_eqvt = map (lift_thm qtys lthy19) (bv_eqvt @ fv_eqvt);+
−
  val (_, lthy20) = Local_Theory.note ((Binding.empty,+
−
    [Attrib.internal (fn _ => Nominal_ThmDecls.eqvt_add)]), q_eqvt) lthy19;+
−
  val _ = warning "Supports";+
−
  val supports = map (prove_supports lthy20 q_perm) consts;+
−
  val fin_supp = HOLogic.conj_elims (prove_fs lthy20 q_induct supports qtys);+
−
  val thy3 = Local_Theory.exit_global lthy20;+
−
  val _ = warning "Instantiating FS";+
−
  val lthy21 = Theory_Target.instantiation (qty_full_names, [], @{sort fs}) thy3;+
−
  fun tac _ = Class.intro_classes_tac [] THEN (ALLGOALS (resolve_tac fin_supp))+
−
  val lthy22 = Class.prove_instantiation_instance tac lthy21+
−
  val fv_alpha_all = combine_fv_alpha_bns (qfv_ts_nobn, qfv_ts_bn) (alpha_ts_nobn, qalpha_ts_bn) bn_nos;+
−
  val (names, supp_eq_t) = supp_eq fv_alpha_all;+
−
  val _ = warning "Support Equations";+
−
  fun supp_eq_tac' _ = if !cheat_supp_eq then Skip_Proof.cheat_tac thy else+
−
    supp_eq_tac q_induct q_fv q_perm q_eq_iff lthy22 1;+
−
  val q_supp = HOLogic.conj_elims (Goal.prove lthy22 names [] supp_eq_t supp_eq_tac') handle e =>+
−
    let val _ = warning ("Support eqs failed") in [] end;+
−
  val lthy23 = note_suffix "supp" q_supp lthy22;+
−
in+
−
  (0, lthy23)+
−
end handle TEST ctxt => (0, ctxt)+
−
*}+
−
+
−
section {* Preparing and parsing of the specification *}+
−
+
−
ML {* +
−
(* parsing the datatypes and declaring *)+
−
(* constructors in the local theory    *)+
−
fun prepare_dts dt_strs lthy = +
−
let+
−
  val thy = ProofContext.theory_of lthy+
−
  +
−
  fun mk_type full_tname tvrs =+
−
    Type (full_tname, map (fn a => TVar ((a, 0), [])) tvrs)+
−
+
−
  fun prep_cnstr full_tname tvs (cname, anno_tys, mx, _) =+
−
  let+
−
    val tys = map (Syntax.read_typ lthy o snd) anno_tys+
−
    val ty = mk_type full_tname tvs+
−
  in+
−
    ((cname, tys ---> ty, mx), (cname, tys, mx))+
−
  end+
−
  +
−
  fun prep_dt (tvs, tname, mx, cnstrs) = +
−
  let+
−
    val full_tname = Sign.full_name thy tname+
−
    val (cnstrs', cnstrs'') = +
−
      split_list (map (prep_cnstr full_tname tvs) cnstrs)+
−
  in+
−
    (cnstrs', (tvs, tname, mx, cnstrs''))+
−
  end +
−
+
−
  val (cnstrs, dts) = split_list (map prep_dt dt_strs)+
−
in+
−
  lthy+
−
  |> Local_Theory.theory (Sign.add_consts_i (flat cnstrs))+
−
  |> pair dts+
−
end+
−
*}+
−
+
−
ML {*+
−
(* parsing the binding function specification and *)+
−
(* declaring the functions in the local theory    *)+
−
fun prepare_bn_funs bn_fun_strs bn_eq_strs lthy =+
−
let+
−
  val ((bn_funs, bn_eqs), _) = +
−
    Specification.read_spec bn_fun_strs bn_eq_strs lthy+
−
+
−
  fun prep_bn_fun ((bn, T), mx) = (bn, T, mx) +
−
  +
−
  val bn_funs' = map prep_bn_fun bn_funs+
−
in+
−
  lthy+
−
  |> Local_Theory.theory (Sign.add_consts_i bn_funs')+
−
  |> pair (bn_funs', bn_eqs) +
−
end +
−
*}+
−
+
−
text {* associates every SOME with the index in the list; drops NONEs *}+
−
ML {*+
−
fun indexify xs =+
−
let+
−
  fun mapp _ [] = []+
−
    | mapp i (NONE :: xs) = mapp (i + 1) xs+
−
    | mapp i (SOME x :: xs) = (x, i) :: mapp (i + 1) xs+
−
in +
−
  mapp 0 xs +
−
end+
−
+
−
fun index_lookup xs x =+
−
  case AList.lookup (op=) xs x of+
−
    SOME x => x+
−
  | NONE => error ("Cannot find " ^ x ^ " as argument annotation.");+
−
*}+
−
+
−
ML {*+
−
fun prepare_bclauses dt_strs lthy = +
−
let+
−
  val annos_bclauses =+
−
    get_cnstrs dt_strs+
−
    |> map (map (fn (_, antys, _, bns) => (map fst antys, bns)))+
−
+
−
  fun prep_binder env bn_str =+
−
    case (Syntax.read_term lthy bn_str) of+
−
      Free (x, _) => (NONE, index_lookup env x)+
−
    | Const (a, T) $ Free (x, _) => (SOME (Const (a, T)), index_lookup env x)+
−
    | _ => error ("The term " ^ bn_str ^ " is not allowed as binding function.")+
−
 +
−
  fun prep_body env bn_str = index_lookup env bn_str+
−
+
−
  fun prep_mode "bind"     = BLst +
−
    | prep_mode "bind_set" = BSet +
−
    | prep_mode "bind_res" = BRes +
−
+
−
  fun prep_bclause env (mode, binders, bodies) = +
−
  let+
−
    val binders' = map (prep_binder env) binders+
−
    val bodies' = map (prep_body env) bodies+
−
  in  +
−
    prep_mode mode (binders', bodies')+
−
  end+
−
+
−
  fun prep_bclauses (annos, bclause_strs) = +
−
  let+
−
    val env = indexify annos (* for every label, associate the index *)+
−
  in+
−
    map (prep_bclause env) bclause_strs+
−
  end+
−
in+
−
  map (map prep_bclauses) annos_bclauses+
−
end+
−
*}+
−
+
−
text {* +
−
  adds an empty binding clause for every argument+
−
  that is not already part of a binding clause+
−
*}+
−
+
−
ML {*+
−
fun included i bcs = +
−
let+
−
  fun incl (BEmy j) = i = j+
−
    | incl (BLst (bns, bds)) = (member (op =) (map snd bns) i) orelse (member (op =) bds i)+
−
    | incl (BSet (bns, bds)) = (member (op =) (map snd bns) i) orelse (member (op =) bds i)+
−
    | incl (BRes (bns, bds)) = (member (op =) (map snd bns) i) orelse (member (op =) bds i)+
−
in+
−
  exists incl bcs +
−
end+
−
*}+
−
+
−
ML {* +
−
fun complete dt_strs bclauses = +
−
let+
−
  val args = +
−
    get_cnstrs dt_strs+
−
    |> map (map (fn (_, antys, _, _) => length antys))+
−
+
−
  fun complt n bcs = +
−
  let+
−
    fun add bcs i = (if included i bcs then [] else [BEmy i]) +
−
  in+
−
    bcs @ (flat (map_range (add bcs) n))+
−
  end+
−
in+
−
  map2 (map2 complt) args bclauses+
−
end+
−
*}+
−
+
−
ML {*+
−
fun nominal_datatype2_cmd (dt_strs, bn_fun_strs, bn_eq_strs) lthy = +
−
let+
−
  fun prep_typ (tvs, tname, mx, _) = (tname, length tvs, mx)+
−
  val lthy0 = +
−
    Local_Theory.theory (Sign.add_types (map prep_typ dt_strs)) lthy+
−
  val (dts, lthy1) = prepare_dts dt_strs lthy0+
−
  val ((bn_funs, bn_eqs), lthy2) = prepare_bn_funs bn_fun_strs bn_eq_strs lthy1+
−
  val bclauses = prepare_bclauses dt_strs lthy2+
−
  val bclauses' = complete dt_strs bclauses +
−
in+
−
  nominal_datatype2 dts bn_funs bn_eqs bclauses' lthy |> snd+
−
end+
−
+
−
+
−
(* Command Keyword *)+
−
+
−
val _ = OuterSyntax.local_theory "nominal_datatype" "test" OuterKeyword.thy_decl+
−
  (main_parser >> nominal_datatype2_cmd)+
−
*}+
−
+
−
(*+
−
atom_decl name+
−
+
−
nominal_datatype lam =+
−
  Var name+
−
| App lam lam+
−
| Lam x::name t::lam  bind_set x in t+
−
| Let p::pt t::lam bind_set "bn p" in t+
−
and pt =+
−
  P1 name+
−
| P2 pt pt+
−
binder+
−
 bn::"pt \<Rightarrow> atom set"+
−
where+
−
  "bn (P1 x) = {atom x}"+
−
| "bn (P2 p1 p2) = bn p1 \<union> bn p2"+
−
+
−
find_theorems Var_raw+
−
+
−
+
−
+
−
thm lam_pt.bn+
−
thm lam_pt.fv[simplified lam_pt.supp(1-2)]+
−
thm lam_pt.eq_iff+
−
thm lam_pt.induct+
−
thm lam_pt.perm+
−
+
−
nominal_datatype exp =+
−
  EVar name+
−
| EUnit+
−
| EPair q1::exp q2::exp+
−
| ELetRec l::lrbs e::exp     bind "b_lrbs l" in e l+
−
+
−
and fnclause =+
−
  K x::name p::pat f::exp    bind_res "b_pat p" in f+
−
+
−
and fnclauses =+
−
  S fnclause+
−
| ORs fnclause fnclauses+
−
+
−
and lrb =+
−
  Clause fnclauses+
−
+
−
and lrbs =+
−
  Single lrb+
−
| More lrb lrbs+
−
+
−
and pat =+
−
  PVar name+
−
| PUnit+
−
| PPair pat pat+
−
+
−
binder+
−
  b_lrbs      :: "lrbs \<Rightarrow> atom list" and+
−
  b_pat       :: "pat \<Rightarrow> atom set" and+
−
  b_fnclauses :: "fnclauses \<Rightarrow> atom list" and+
−
  b_fnclause  :: "fnclause \<Rightarrow> atom list" and+
−
  b_lrb       :: "lrb \<Rightarrow> atom list"+
−
+
−
where+
−
  "b_lrbs (Single l) = b_lrb l"+
−
| "b_lrbs (More l ls) = append (b_lrb l) (b_lrbs ls)"+
−
| "b_pat (PVar x) = {atom x}"+
−
| "b_pat (PUnit) = {}"+
−
| "b_pat (PPair p1 p2) = b_pat p1 \<union> b_pat p2"+
−
| "b_fnclauses (S fc) = (b_fnclause fc)"+
−
| "b_fnclauses (ORs fc fcs) = append (b_fnclause fc) (b_fnclauses fcs)"+
−
| "b_lrb (Clause fcs) = (b_fnclauses fcs)"+
−
| "b_fnclause (K x pat exp) = [atom x]"+
−
+
−
thm exp_fnclause_fnclauses_lrb_lrbs_pat.bn+
−
thm exp_fnclause_fnclauses_lrb_lrbs_pat.fv+
−
thm exp_fnclause_fnclauses_lrb_lrbs_pat.eq_iff+
−
thm exp_fnclause_fnclauses_lrb_lrbs_pat.induct+
−
thm exp_fnclause_fnclauses_lrb_lrbs_pat.perm+
−
+
−
nominal_datatype ty =+
−
  Vr "name"+
−
| Fn "ty" "ty"+
−
and tys =+
−
  Al xs::"name fset" t::"ty" bind_res xs in t+
−
+
−
thm ty_tys.fv[simplified ty_tys.supp]+
−
thm ty_tys.eq_iff+
−
+
−
*)+
−
+
−
(* some further tests *)+
−
+
−
(*+
−
nominal_datatype ty2 =+
−
  Vr2 "name"+
−
| Fn2 "ty2" "ty2"+
−
+
−
nominal_datatype tys2 =+
−
  All2 xs::"name fset" ty::"ty2" bind_res xs in ty+
−
+
−
nominal_datatype lam2 =+
−
  Var2 "name"+
−
| App2 "lam2" "lam2 list"+
−
| Lam2 x::"name" t::"lam2" bind x in t+
−
*)+
−
+
−
+
−
+
−
end+
−
+
−
+
−
+
−