respectfulness theorem has to be proved if a new constant is lifted by quotient_definition
(* Title: HOL/Tools/Quotient/quotient_type.ML
Author: Cezary Kaliszyk and Christian Urban
Definition of a quotient type.
*)
signature QUOTIENT_TYPE =
sig
val add_quotient_type: ((string list * binding * mixfix) * (typ * term * bool) *
((binding * binding) option)) * thm -> local_theory -> Quotient_Info.quotients * local_theory
val quotient_type: ((string list * binding * mixfix) * (typ * term * bool) *
((binding * binding) option)) list -> Proof.context -> Proof.state
val quotient_type_cmd: (((((string list * binding) * mixfix) * string) * (bool * string)) *
(binding * binding) option) list -> Proof.context -> Proof.state
end;
structure Quotient_Type: QUOTIENT_TYPE =
struct
(*** definition of quotient types ***)
val mem_def1 = @{lemma "y : Collect S ==> S y" by simp}
val mem_def2 = @{lemma "S y ==> y : Collect S" by simp}
(* constructs the term lambda (c::rty => bool). EX (x::rty). c = rel x *)
fun typedef_term rel rty lthy =
let
val [x, c] =
[("x", rty), ("c", HOLogic.mk_setT rty)]
|> Variable.variant_frees lthy [rel]
|> map Free
in
HOLogic.Collect_const (HOLogic.mk_setT rty) $ (lambda c (HOLogic.exists_const rty $
lambda x (HOLogic.mk_conj (rel $ x $ x,
HOLogic.mk_eq (c, HOLogic.Collect_const rty $ (rel $ x))))))
end
(* makes the new type definitions and proves non-emptyness *)
fun typedef_make (vs, qty_name, mx, rel, rty) equiv_thm lthy =
let
val typedef_tac =
EVERY1 (map rtac [@{thm part_equivp_typedef}, equiv_thm])
in
Typedef.add_typedef false NONE (qty_name, map (rpair dummyS) vs, mx)
(typedef_term rel rty lthy) NONE typedef_tac lthy
end
(* tactic to prove the quot_type theorem for the new type *)
fun typedef_quot_type_tac equiv_thm ((_, typedef_info): Typedef.info) =
let
val rep_thm = #Rep typedef_info RS mem_def1
val rep_inv = #Rep_inverse typedef_info
val abs_inv = #Abs_inverse typedef_info
val rep_inj = #Rep_inject typedef_info
in
(rtac @{thm quot_type.intro} THEN' RANGE [
rtac equiv_thm,
rtac rep_thm,
rtac rep_inv,
rtac abs_inv THEN' rtac mem_def2 THEN' atac,
rtac rep_inj]) 1
end
(* proves the quot_type theorem for the new type *)
fun typedef_quot_type_thm (rel, abs, rep, equiv_thm, typedef_info) lthy =
let
val quot_type_const = Const (@{const_name "quot_type"},
fastype_of rel --> fastype_of abs --> fastype_of rep --> @{typ bool})
val goal = HOLogic.mk_Trueprop (quot_type_const $ rel $ abs $ rep)
in
Goal.prove lthy [] [] goal
(K (typedef_quot_type_tac equiv_thm typedef_info))
end
(* main function for constructing a quotient type *)
fun add_quotient_type (((vs, qty_name, mx), (rty, rel, partial), opt_morphs), equiv_thm) lthy =
let
val part_equiv =
if partial
then equiv_thm
else equiv_thm RS @{thm equivp_implies_part_equivp}
(* generates the typedef *)
val ((qty_full_name, typedef_info), lthy1) =
typedef_make (vs, qty_name, mx, rel, rty) part_equiv lthy
(* abs and rep functions from the typedef *)
val Abs_ty = #abs_type (#1 typedef_info)
val Rep_ty = #rep_type (#1 typedef_info)
val Abs_name = #Abs_name (#1 typedef_info)
val Rep_name = #Rep_name (#1 typedef_info)
val Abs_const = Const (Abs_name, Rep_ty --> Abs_ty)
val Rep_const = Const (Rep_name, Abs_ty --> Rep_ty)
(* more useful abs and rep definitions *)
val abs_const = Const (@{const_name quot_type.abs},
(rty --> rty --> @{typ bool}) --> (Rep_ty --> Abs_ty) --> rty --> Abs_ty)
val rep_const = Const (@{const_name quot_type.rep}, (Abs_ty --> Rep_ty) --> Abs_ty --> rty)
val abs_trm = abs_const $ rel $ Abs_const
val rep_trm = rep_const $ Rep_const
val (rep_name, abs_name) =
(case opt_morphs of
NONE => (Binding.prefix_name "rep_" qty_name, Binding.prefix_name "abs_" qty_name)
| SOME morphs => morphs)
val ((abs_t, (_, abs_def)), lthy2) = lthy1
|> Local_Theory.define ((abs_name, NoSyn), ((Thm.def_binding abs_name, []), abs_trm))
val ((rep_t, (_, rep_def)), lthy3) = lthy2
|> Local_Theory.define ((rep_name, NoSyn), ((Thm.def_binding rep_name, []), rep_trm))
(* quot_type theorem *)
val quot_thm = typedef_quot_type_thm (rel, Abs_const, Rep_const, part_equiv, typedef_info) lthy3
(* quotient theorem *)
val quotient_thm_name = Binding.prefix_name "Quotient_" qty_name
val quotient_thm =
(quot_thm RS @{thm quot_type.Quotient})
|> fold_rule [abs_def, rep_def]
(* name equivalence theorem *)
val equiv_thm_name = Binding.suffix_name "_equivp" qty_name
(* storing the quotients *)
val quotients = {qtyp = Abs_ty, rtyp = rty, equiv_rel = rel, equiv_thm = equiv_thm}
fun qinfo phi = Quotient_Info.transform_quotients phi quotients
fun abs_rep phi = Quotient_Info.transform_abs_rep phi {abs = abs_t, rep = rep_t}
val lthy4 = lthy3
|> Local_Theory.declaration {syntax = false, pervasive = true}
(fn phi => Quotient_Info.update_quotients qty_full_name (qinfo phi)
#> Quotient_Info.update_abs_rep qty_full_name (abs_rep phi))
|> (snd oo Local_Theory.note)
((equiv_thm_name,
if partial then [] else [Attrib.internal (K Quotient_Info.equiv_rules_add)]),
[equiv_thm])
|> (snd oo Local_Theory.note)
((quotient_thm_name, [Attrib.internal (K Quotient_Info.quotient_rules_add)]),
[quotient_thm])
in
(quotients, lthy4)
end
(* sanity checks for the quotient type specifications *)
fun sanity_check ((vs, qty_name, _), (rty, rel, _), _) =
let
val rty_tfreesT = map fst (Term.add_tfreesT rty [])
val rel_tfrees = map fst (Term.add_tfrees rel [])
val rel_frees = map fst (Term.add_frees rel [])
val rel_vars = Term.add_vars rel []
val rel_tvars = Term.add_tvars rel []
val qty_str = Binding.print qty_name ^ ": "
val illegal_rel_vars =
if null rel_vars andalso null rel_tvars then []
else [qty_str ^ "illegal schematic variable(s) in the relation."]
val dup_vs =
(case duplicates (op =) vs of
[] => []
| dups => [qty_str ^ "duplicate type variable(s) on the lhs: " ^ commas_quote dups])
val extra_rty_tfrees =
(case subtract (op =) vs rty_tfreesT of
[] => []
| extras => [qty_str ^ "extra type variable(s) on the lhs: " ^ commas_quote extras])
val extra_rel_tfrees =
(case subtract (op =) vs rel_tfrees of
[] => []
| extras => [qty_str ^ "extra type variable(s) in the relation: " ^ commas_quote extras])
val illegal_rel_frees =
(case rel_frees of
[] => []
| xs => [qty_str ^ "illegal variable(s) in the relation: " ^ commas_quote xs])
val errs = illegal_rel_vars @ dup_vs @ extra_rty_tfrees @ extra_rel_tfrees @ illegal_rel_frees
in
if null errs then () else error (cat_lines errs)
end
(* check for existence of map functions *)
fun map_check ctxt (_, (rty, _, _), _) =
let
fun map_check_aux rty warns =
(case rty of
Type (_, []) => warns
| Type (s, _) =>
if Symtab.defined (Enriched_Type.entries ctxt) s then warns else s :: warns
| _ => warns)
val warns = map_check_aux rty []
in
if null warns then ()
else warning ("No map function defined for " ^ commas warns ^
". This will cause problems later on.")
end
(*** interface and syntax setup ***)
(* the ML-interface takes a list of tuples consisting of:
- the name of the quotient type
- its free type variables (first argument)
- its mixfix annotation
- the type to be quotient
- the partial flag (a boolean)
- the relation according to which the type is quotient
- optional names of morphisms (rep/abs)
it opens a proof-state in which one has to show that the
relations are equivalence relations
*)
fun quotient_type quot_list lthy =
let
(* sanity check *)
val _ = List.app sanity_check quot_list
val _ = List.app (map_check lthy) quot_list
fun mk_goal (rty, rel, partial) =
let
val equivp_ty = ([rty, rty] ---> @{typ bool}) --> @{typ bool}
val const =
if partial then @{const_name part_equivp} else @{const_name equivp}
in
HOLogic.mk_Trueprop (Const (const, equivp_ty) $ rel)
end
val goals = map (mk_goal o #2) quot_list
fun after_qed [thms] = fold (snd oo add_quotient_type) (quot_list ~~ thms)
in
Proof.theorem NONE after_qed [map (rpair []) goals] lthy
end
fun quotient_type_cmd specs lthy =
let
fun parse_spec (((((vs, qty_name), mx), rty_str), (partial, rel_str)), opt_morphs) lthy =
let
val rty = Syntax.read_typ lthy rty_str
val tmp_lthy1 = Variable.declare_typ rty lthy
val rel =
Syntax.parse_term tmp_lthy1 rel_str
|> Type.constraint (rty --> rty --> @{typ bool})
|> Syntax.check_term tmp_lthy1
val tmp_lthy2 = Variable.declare_term rel tmp_lthy1
in
(((vs, qty_name, mx), (rty, rel, partial), opt_morphs), tmp_lthy2)
end
val (spec', _) = fold_map parse_spec specs lthy
in
quotient_type spec' lthy
end
val partial = Scan.optional (Parse.reserved "partial" -- @{keyword ":"} >> K true) false
val quotspec_parser =
Parse.and_list1
((Parse.type_args -- Parse.binding) --
(* FIXME Parse.type_args_constrained and standard treatment of sort constraints *)
Parse.opt_mixfix -- (@{keyword "="} |-- Parse.typ) --
(@{keyword "/"} |-- (partial -- Parse.term)) --
Scan.option (@{keyword "morphisms"} |-- Parse.!!! (Parse.binding -- Parse.binding)))
val _ =
Outer_Syntax.local_theory_to_proof @{command_spec "quotient_type"}
"quotient type definitions (require equivalence proofs)"
(quotspec_parser >> quotient_type_cmd)
end;