(* Title: HOL/HOLCF/Tools/cpodef.ML
Author: Brian Huffman
Primitive domain definitions for HOLCF, similar to Gordon/HOL-style
typedef (see also ~~/src/HOL/Tools/typedef.ML).
*)
signature CPODEF =
sig
type cpo_info =
{ below_def: thm, adm: thm, cont_Rep: thm, cont_Abs: thm,
lub: thm, compact: thm }
type pcpo_info =
{ Rep_strict: thm, Abs_strict: thm,
Rep_bottom_iff: thm, Abs_bottom_iff: thm }
val add_podef: binding * (string * sort) list * mixfix ->
term -> (binding * binding) option -> tactic -> theory ->
(Typedef.info * thm) * theory
val add_cpodef: binding * (string * sort) list * mixfix ->
term -> (binding * binding) option -> tactic * tactic -> theory ->
(Typedef.info * cpo_info) * theory
val add_pcpodef: binding * (string * sort) list * mixfix ->
term -> (binding * binding) option -> tactic * tactic -> theory ->
(Typedef.info * cpo_info * pcpo_info) * theory
val cpodef_proof:
(binding * (string * sort) list * mixfix) * term
* (binding * binding) option -> theory -> Proof.state
val cpodef_proof_cmd:
(binding * (string * string option) list * mixfix) * string
* (binding * binding) option -> theory -> Proof.state
val pcpodef_proof:
(binding * (string * sort) list * mixfix) * term
* (binding * binding) option -> theory -> Proof.state
val pcpodef_proof_cmd:
(binding * (string * string option) list * mixfix) * string
* (binding * binding) option -> theory -> Proof.state
end
structure Cpodef : CPODEF =
struct
(** type definitions **)
type cpo_info =
{ below_def: thm, adm: thm, cont_Rep: thm, cont_Abs: thm,
lub: thm, compact: thm }
type pcpo_info =
{ Rep_strict: thm, Abs_strict: thm,
Rep_bottom_iff: thm, Abs_bottom_iff: thm }
(* building terms *)
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 below_const T = Const (@{const_name below}, T --> T --> HOLogic.boolT)
(* proving class instances *)
fun prove_cpo
(name: binding)
(newT: typ)
(Rep_name: binding, Abs_name: binding)
(type_definition: thm) (* type_definition Rep Abs A *)
(below_def: thm) (* op << == %x y. Rep x << Rep y *)
(admissible: thm) (* adm (%x. x : set) *)
(thy: theory)
=
let
val cpo_thms = map (Thm.transfer thy) [type_definition, below_def, admissible]
val (full_tname, Ts) = dest_Type newT
val lhs_sorts = map (snd o dest_TFree) Ts
val tac = rtac (@{thm typedef_cpo} OF cpo_thms) 1
val thy = Axclass.prove_arity (full_tname, lhs_sorts, @{sort cpo}) (K tac) thy
(* transfer thms so that they will know about the new cpo instance *)
val cpo_thms' = map (Thm.transfer thy) cpo_thms
fun make thm = Drule.zero_var_indexes (thm OF cpo_thms')
val cont_Rep = make @{thm typedef_cont_Rep}
val cont_Abs = make @{thm typedef_cont_Abs}
val lub = make @{thm typedef_lub}
val compact = make @{thm typedef_compact}
val (_, thy) =
thy
|> Sign.add_path (Binding.name_of name)
|> Global_Theory.add_thms
([((Binding.prefix_name "adm_" name, admissible), []),
((Binding.prefix_name "cont_" Rep_name, cont_Rep ), []),
((Binding.prefix_name "cont_" Abs_name, cont_Abs ), []),
((Binding.prefix_name "lub_" name, lub ), []),
((Binding.prefix_name "compact_" name, compact ), [])])
||> Sign.parent_path
val cpo_info : cpo_info =
{ below_def = below_def, adm = admissible, cont_Rep = cont_Rep,
cont_Abs = cont_Abs, lub = lub, compact = compact }
in
(cpo_info, thy)
end
fun prove_pcpo
(name: binding)
(newT: typ)
(Rep_name: binding, Abs_name: binding)
(type_definition: thm) (* type_definition Rep Abs A *)
(below_def: thm) (* op << == %x y. Rep x << Rep y *)
(bottom_mem: thm) (* bottom : set *)
(thy: theory)
=
let
val pcpo_thms = map (Thm.transfer thy) [type_definition, below_def, bottom_mem]
val (full_tname, Ts) = dest_Type newT
val lhs_sorts = map (snd o dest_TFree) Ts
val tac = rtac (@{thm typedef_pcpo} OF pcpo_thms) 1
val thy = Axclass.prove_arity (full_tname, lhs_sorts, @{sort pcpo}) (K tac) thy
val pcpo_thms' = map (Thm.transfer thy) pcpo_thms
fun make thm = Drule.zero_var_indexes (thm OF pcpo_thms')
val Rep_strict = make @{thm typedef_Rep_strict}
val Abs_strict = make @{thm typedef_Abs_strict}
val Rep_bottom_iff = make @{thm typedef_Rep_bottom_iff}
val Abs_bottom_iff = make @{thm typedef_Abs_bottom_iff}
val (_, thy) =
thy
|> Sign.add_path (Binding.name_of name)
|> Global_Theory.add_thms
([((Binding.suffix_name "_strict" Rep_name, Rep_strict), []),
((Binding.suffix_name "_strict" Abs_name, Abs_strict), []),
((Binding.suffix_name "_bottom_iff" Rep_name, Rep_bottom_iff), []),
((Binding.suffix_name "_bottom_iff" Abs_name, Abs_bottom_iff), [])])
||> Sign.parent_path
val pcpo_info =
{ Rep_strict = Rep_strict, Abs_strict = Abs_strict,
Rep_bottom_iff = Rep_bottom_iff, Abs_bottom_iff = Abs_bottom_iff }
in
(pcpo_info, thy)
end
(* prepare_cpodef *)
fun prepare prep_term name (tname, raw_args, _) raw_set opt_morphs thy =
let
val _ = Theory.requires thy (Context.theory_name @{theory}) "cpodefs"
(*rhs*)
val tmp_ctxt =
Proof_Context.init_global thy
|> fold (Variable.declare_typ o TFree) raw_args
val set = prep_term tmp_ctxt raw_set
val tmp_ctxt' = tmp_ctxt |> Variable.declare_term set
val setT = Term.fastype_of set
val oldT = HOLogic.dest_setT setT handle TYPE _ =>
error ("Not a set type: " ^ quote (Syntax.string_of_typ tmp_ctxt setT))
(*lhs*)
val lhs_tfrees = map (Proof_Context.check_tfree tmp_ctxt') raw_args
val full_tname = Sign.full_name thy tname
val newT = Type (full_tname, map TFree lhs_tfrees)
val morphs = opt_morphs
|> the_default (Binding.prefix_name "Rep_" name, Binding.prefix_name "Abs_" name)
in
(newT, oldT, set, morphs)
end
fun add_podef typ set opt_morphs tac thy =
let
val name = #1 typ
val ((full_tname, info as ({Rep_name, ...}, {type_definition, ...})), thy) = thy
|> Typedef.add_typedef_global typ set opt_morphs tac
val oldT = #rep_type (#1 info)
val newT = #abs_type (#1 info)
val lhs_tfrees = map dest_TFree (snd (dest_Type newT))
val RepC = Const (Rep_name, newT --> oldT)
val below_eqn = Logic.mk_equals (below_const newT,
Abs ("x", newT, Abs ("y", newT, below_const oldT $ (RepC $ Bound 1) $ (RepC $ Bound 0))))
val ((_, (_, below_ldef)), lthy) = thy
|> Class.instantiation ([full_tname], lhs_tfrees, @{sort po})
|> Specification.definition (NONE,
((Binding.prefix_name "below_" (Thm.def_binding name), []), below_eqn))
val ctxt_thy = Proof_Context.init_global (Proof_Context.theory_of lthy)
val below_def = singleton (Proof_Context.export lthy ctxt_thy) below_ldef
val thy = lthy
|> Class.prove_instantiation_exit
(K (rtac (@{thm typedef_po} OF [type_definition, below_def]) 1))
in ((info, below_def), thy) end
fun prepare_cpodef
(prep_term: Proof.context -> 'a -> term)
(typ: binding * (string * sort) list * mixfix)
(raw_set: 'a)
(opt_morphs: (binding * binding) option)
(thy: theory)
: term * term * (thm -> thm -> theory -> (Typedef.info * cpo_info) * theory) =
let
val name = #1 typ
val (newT, oldT, set, morphs) =
prepare prep_term name typ raw_set opt_morphs thy
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)))
fun cpodef_result nonempty admissible thy =
let
val ((info as (_, {type_definition, ...}), below_def), thy) = thy
|> add_podef typ set opt_morphs (rtac nonempty 1)
val (cpo_info, thy) = thy
|> prove_cpo name newT morphs type_definition below_def admissible
in
((info, cpo_info), thy)
end
in
(goal_nonempty, goal_admissible, cpodef_result)
end
handle ERROR msg =>
cat_error msg ("The error(s) above occurred in cpodef " ^ Binding.print (#1 typ))
fun prepare_pcpodef
(prep_term: Proof.context -> 'a -> term)
(typ: binding * (string * sort) list * mixfix)
(raw_set: 'a)
(opt_morphs: (binding * binding) option)
(thy: theory)
: term * term * (thm -> thm -> theory -> (Typedef.info * cpo_info * pcpo_info) * theory) =
let
val name = #1 typ
val (newT, oldT, set, morphs) =
prepare prep_term name typ raw_set opt_morphs thy
val goal_bottom_mem =
HOLogic.mk_Trueprop (HOLogic.mk_mem (Const (@{const_name bottom}, oldT), set))
val goal_admissible =
HOLogic.mk_Trueprop (mk_adm ("x", oldT, HOLogic.mk_mem (Free ("x", oldT), set)))
fun pcpodef_result bottom_mem admissible thy =
let
val tac = rtac exI 1 THEN rtac bottom_mem 1
val ((info as (_, {type_definition, ...}), below_def), thy) = thy
|> add_podef typ set opt_morphs tac
val (cpo_info, thy) = thy
|> prove_cpo name newT morphs type_definition below_def admissible
val (pcpo_info, thy) = thy
|> prove_pcpo name newT morphs type_definition below_def bottom_mem
in
((info, cpo_info, pcpo_info), thy)
end
in
(goal_bottom_mem, goal_admissible, pcpodef_result)
end
handle ERROR msg =>
cat_error msg ("The error(s) above occurred in pcpodef " ^ Binding.print (#1 typ))
(* tactic interface *)
fun add_cpodef typ set opt_morphs (tac1, tac2) thy =
let
val (goal1, goal2, cpodef_result) =
prepare_cpodef Syntax.check_term typ set opt_morphs thy
val thm1 = Goal.prove_global thy [] [] goal1 (K tac1)
handle ERROR msg => cat_error msg
("Failed to prove non-emptiness of " ^ quote (Syntax.string_of_term_global thy set))
val thm2 = Goal.prove_global thy [] [] goal2 (K tac2)
handle ERROR msg => cat_error msg
("Failed to prove admissibility of " ^ quote (Syntax.string_of_term_global thy set))
in cpodef_result thm1 thm2 thy end
fun add_pcpodef typ set opt_morphs (tac1, tac2) thy =
let
val (goal1, goal2, pcpodef_result) =
prepare_pcpodef Syntax.check_term typ set opt_morphs thy
val thm1 = Goal.prove_global thy [] [] goal1 (K tac1)
handle ERROR msg => cat_error msg
("Failed to prove non-emptiness of " ^ quote (Syntax.string_of_term_global thy set))
val thm2 = Goal.prove_global thy [] [] goal2 (K tac2)
handle ERROR msg => cat_error msg
("Failed to prove admissibility of " ^ quote (Syntax.string_of_term_global thy set))
in pcpodef_result thm1 thm2 thy end
(* proof interface *)
local
fun gen_cpodef_proof prep_term prep_constraint
((b, raw_args, mx), set, opt_morphs) thy =
let
val ctxt = Proof_Context.init_global thy
val args = map (apsnd (prep_constraint ctxt)) raw_args
val (goal1, goal2, make_result) =
prepare_cpodef prep_term (b, args, mx) set opt_morphs thy
fun after_qed [[th1, th2]] = Proof_Context.background_theory (snd o make_result th1 th2)
| after_qed _ = raise Fail "cpodef_proof"
in Proof.theorem NONE after_qed [[(goal1, []), (goal2, [])]] ctxt end
fun gen_pcpodef_proof prep_term prep_constraint
((b, raw_args, mx), set, opt_morphs) thy =
let
val ctxt = Proof_Context.init_global thy
val args = map (apsnd (prep_constraint ctxt)) raw_args
val (goal1, goal2, make_result) =
prepare_pcpodef prep_term (b, args, mx) set opt_morphs thy
fun after_qed [[th1, th2]] = Proof_Context.background_theory (snd o make_result th1 th2)
| after_qed _ = raise Fail "pcpodef_proof"
in Proof.theorem NONE after_qed [[(goal1, []), (goal2, [])]] ctxt end
in
fun cpodef_proof x = gen_cpodef_proof Syntax.check_term (K I) x
fun cpodef_proof_cmd x = gen_cpodef_proof Syntax.read_term Typedecl.read_constraint x
fun pcpodef_proof x = gen_pcpodef_proof Syntax.check_term (K I) x
fun pcpodef_proof_cmd x = gen_pcpodef_proof Syntax.read_term Typedecl.read_constraint x
end
(** outer syntax **)
val typedef_proof_decl =
(Parse.type_args_constrained -- Parse.binding) -- Parse.opt_mixfix --
(@{keyword "="} |-- Parse.term) --
Scan.option
(@{keyword "morphisms"} |-- Parse.!!! (Parse.binding -- Parse.binding))
fun mk_pcpodef_proof pcpo ((((args, t), mx), A), morphs) =
(if pcpo then pcpodef_proof_cmd else cpodef_proof_cmd)
((t, args, mx), A, morphs)
val _ =
Outer_Syntax.command @{command_spec "pcpodef"}
"HOLCF type definition (requires admissibility proof)"
(typedef_proof_decl >>
(Toplevel.print oo (Toplevel.theory_to_proof o mk_pcpodef_proof true)))
val _ =
Outer_Syntax.command @{command_spec "cpodef"}
"HOLCF type definition (requires admissibility proof)"
(typedef_proof_decl >>
(Toplevel.print oo (Toplevel.theory_to_proof o mk_pcpodef_proof false)))
end