(* Title: HOLCF/Tools/pcpodef_package.ML
Author: Brian Huffman
Primitive domain definitions for HOLCF, similar to Gordon/HOL-style
typedef (see also ~~/src/HOL/Tools/typedef_package.ML).
*)
signature PCPODEF_PACKAGE =
sig
val pcpodef_proof: (bool * binding) * (binding * string list * mixfix) * term
* (binding * binding) option -> theory -> Proof.state
val pcpodef_proof_cmd: (bool * binding) * (binding * string list * mixfix) * string
* (binding * binding) option -> theory -> Proof.state
val cpodef_proof: (bool * binding) * (binding * string list * mixfix) * term
* (binding * binding) option -> theory -> Proof.state
val cpodef_proof_cmd: (bool * binding) * (binding * string list * mixfix) * string
* (binding * binding) option -> theory -> Proof.state
end;
structure PcpodefPackage :> PCPODEF_PACKAGE =
struct
(** type definitions **)
(* prepare_cpodef *)
fun declare_type_name a = Variable.declare_constraints (Logic.mk_type (TFree (a, dummyS)));
fun adm_const T = Const (@{const_name adm}, (T --> HOLogic.boolT) --> HOLogic.boolT);
fun mk_adm (x, T, P) = adm_const T $ absfree (x, T, P);
fun prepare_pcpodef prep_term pcpo def name (t, vs, mx) raw_set opt_morphs thy =
let
val _ = Theory.requires thy "Pcpodef" "pcpodefs";
val ctxt = ProofContext.init thy;
val full = Sign.full_name thy;
val full_name = full name;
val bname = Binding.name_of name;
(*rhs*)
val set = prep_term (ctxt |> fold declare_type_name vs) raw_set;
val setT = Term.fastype_of set;
val rhs_tfrees = Term.add_tfrees set [];
val oldT = HOLogic.dest_setT setT handle TYPE _ =>
error ("Not a set type: " ^ quote (Syntax.string_of_typ ctxt setT));
(*goal*)
val goal_UU_mem = HOLogic.mk_Trueprop (HOLogic.mk_mem (Const (@{const_name UU}, oldT), set));
val goal_nonempty =
HOLogic.mk_Trueprop (HOLogic.mk_exists ("x", oldT, HOLogic.mk_mem (Free ("x", oldT), set)));
val goal_admissible =
HOLogic.mk_Trueprop (mk_adm ("x", oldT, HOLogic.mk_mem (Free ("x", oldT), set)));
(*lhs*)
val defS = Sign.defaultS thy;
val lhs_tfrees = map (fn v => (v, the_default defS (AList.lookup (op =) rhs_tfrees v))) vs;
val lhs_sorts = map snd lhs_tfrees;
val tname = Binding.map_name (Syntax.type_name mx) t;
val full_tname = full tname;
val newT = Type (full_tname, map TFree lhs_tfrees);
val (Rep_name, Abs_name) =
(case opt_morphs of
NONE => (Binding.prefix_name "Rep_" name, Binding.prefix_name "Abs_" name)
| SOME morphs => morphs);
val RepC = Const (full Rep_name, newT --> oldT);
fun lessC T = Const (@{const_name sq_le}, T --> T --> HOLogic.boolT);
val less_def = Logic.mk_equals (lessC newT,
Abs ("x", newT, Abs ("y", newT, lessC oldT $ (RepC $ Bound 1) $ (RepC $ Bound 0))));
fun make_po tac thy1 =
let
val ((_, {type_definition, set_def, ...}), thy2) = thy1
|> TypedefPackage.add_typedef def (SOME name) (t, vs, mx) set opt_morphs tac;
val lthy3 = thy2
|> TheoryTarget.instantiation ([full_tname], lhs_tfrees, @{sort po});
val less_def' = Syntax.check_term lthy3 less_def;
val ((_, (_, less_definition')), lthy4) = lthy3
|> Specification.definition (NONE,
((Binding.prefix_name "less_" (Binding.suffix_name "_def" name), []), less_def'));
val ctxt_thy = ProofContext.init (ProofContext.theory_of lthy4);
val less_definition = singleton (ProofContext.export lthy4 ctxt_thy) less_definition';
val thy5 = lthy4
|> Class.prove_instantiation_instance
(K (Tactic.rtac (@{thm typedef_po} OF [type_definition, less_definition]) 1))
|> LocalTheory.exit_global;
in ((type_definition, less_definition, set_def), thy5) end;
fun make_cpo admissible (type_def, less_def, set_def) theory =
let
val admissible' = fold_rule (the_list set_def) admissible;
val cpo_thms = map (Thm.transfer theory) [type_def, less_def, admissible'];
val theory' = theory
|> AxClass.prove_arity (full_tname, lhs_sorts, @{sort cpo})
(Tactic.rtac (@{thm typedef_cpo} OF cpo_thms) 1);
val cpo_thms' = map (Thm.transfer theory') cpo_thms;
in
theory'
|> Sign.add_path (Binding.name_of name)
|> PureThy.add_thms
([((Binding.prefix_name "adm_" name, admissible'), []),
((Binding.prefix_name "cont_" Rep_name, @{thm typedef_cont_Rep} OF cpo_thms'), []),
((Binding.prefix_name "cont_" Abs_name, @{thm typedef_cont_Abs} OF cpo_thms'), []),
((Binding.prefix_name "lub_" name, @{thm typedef_lub} OF cpo_thms'), []),
((Binding.prefix_name "thelub_" name, @{thm typedef_thelub} OF cpo_thms'), []),
((Binding.prefix_name "compact_" name, @{thm typedef_compact} OF cpo_thms'), [])])
|> snd
|> Sign.parent_path
end;
fun make_pcpo UU_mem (type_def, less_def, set_def) theory =
let
val UU_mem' = fold_rule (the_list set_def) UU_mem;
val pcpo_thms = map (Thm.transfer theory) [type_def, less_def, UU_mem'];
val theory' = theory
|> AxClass.prove_arity (full_tname, lhs_sorts, @{sort pcpo})
(Tactic.rtac (@{thm typedef_pcpo} OF pcpo_thms) 1);
val pcpo_thms' = map (Thm.transfer theory') pcpo_thms;
in
theory'
|> Sign.add_path (Binding.name_of name)
|> PureThy.add_thms
([((Binding.suffix_name "_strict" Rep_name, @{thm typedef_Rep_strict} OF pcpo_thms'), []),
((Binding.suffix_name "_strict" Abs_name, @{thm typedef_Abs_strict} OF pcpo_thms'), []),
((Binding.suffix_name "_strict_iff" Rep_name, @{thm typedef_Rep_strict_iff} OF pcpo_thms'), []),
((Binding.suffix_name "_strict_iff" Abs_name, @{thm typedef_Abs_strict_iff} OF pcpo_thms'), []),
((Binding.suffix_name "_defined" Rep_name, @{thm typedef_Rep_defined} OF pcpo_thms'), []),
((Binding.suffix_name "_defined" Abs_name, @{thm typedef_Abs_defined} OF pcpo_thms'), [])])
|> snd
|> Sign.parent_path
end;
fun pcpodef_result UU_mem admissible =
make_po (Tactic.rtac exI 1 THEN Tactic.rtac UU_mem 1)
#-> (fn defs => make_cpo admissible defs #> make_pcpo UU_mem defs);
fun cpodef_result nonempty admissible =
make_po (Tactic.rtac nonempty 1)
#-> make_cpo admissible;
in
if pcpo
then (goal_UU_mem, goal_admissible, pcpodef_result)
else (goal_nonempty, goal_admissible, cpodef_result)
end
handle ERROR msg =>
cat_error msg ("The error(s) above occurred in cpodef " ^ quote (Binding.str_of name));
(* proof interface *)
local
fun gen_pcpodef_proof prep_term pcpo ((def, name), typ, set, opt_morphs) thy =
let
val (goal1, goal2, make_result) =
prepare_pcpodef prep_term pcpo def name typ set opt_morphs thy;
fun after_qed [[th1, th2]] = ProofContext.theory (make_result th1 th2);
in Proof.theorem_i NONE after_qed [[(goal1, []), (goal2, [])]] (ProofContext.init thy) end;
in
fun pcpodef_proof x = gen_pcpodef_proof Syntax.check_term true x;
fun pcpodef_proof_cmd x = gen_pcpodef_proof Syntax.read_term true x;
fun cpodef_proof x = gen_pcpodef_proof Syntax.check_term false x;
fun cpodef_proof_cmd x = gen_pcpodef_proof Syntax.read_term false x;
end;
(** outer syntax **)
local structure P = OuterParse and K = OuterKeyword in
val typedef_proof_decl =
Scan.optional (P.$$$ "(" |--
((P.$$$ "open" >> K false) -- Scan.option P.binding || P.binding >> (fn s => (true, SOME s)))
--| P.$$$ ")") (true, NONE) --
(P.type_args -- P.binding) -- P.opt_infix -- (P.$$$ "=" |-- P.term) --
Scan.option (P.$$$ "morphisms" |-- P.!!! (P.binding -- P.binding));
fun mk_pcpodef_proof pcpo ((((((def, opt_name), (vs, t)), mx), A), morphs)) =
(if pcpo then pcpodef_proof_cmd else cpodef_proof_cmd)
((def, the_default (Binding.map_name (Syntax.type_name mx) t) opt_name), (t, vs, mx), A, morphs);
val _ =
OuterSyntax.command "pcpodef" "HOLCF type definition (requires admissibility proof)" K.thy_goal
(typedef_proof_decl >>
(Toplevel.print oo (Toplevel.theory_to_proof o mk_pcpodef_proof true)));
val _ =
OuterSyntax.command "cpodef" "HOLCF type definition (requires admissibility proof)" K.thy_goal
(typedef_proof_decl >>
(Toplevel.print oo (Toplevel.theory_to_proof o mk_pcpodef_proof false)));
end;
end;