--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/quotient_typ.ML Sat Dec 12 01:44:56 2009 +0100
@@ -0,0 +1,226 @@
+signature QUOTIENT =
+sig
+ exception LIFT_MATCH of string
+
+ val quotient_type: ((binding * mixfix) * (typ * term)) list -> Proof.context -> Proof.state
+ val quotient_type_cmd: (((bstring * mixfix) * string) * string) list -> Proof.context -> Proof.state
+
+end;
+
+structure Quotient: QUOTIENT =
+struct
+
+exception LIFT_MATCH of string
+
+(* wrappers for define, note and theorem_i *)
+fun define (name, mx, rhs) lthy =
+let
+ val ((rhs, (_ , thm)), lthy') =
+ Local_Theory.define ((name, mx), (Attrib.empty_binding, rhs)) lthy
+in
+ ((rhs, thm), lthy')
+end
+
+fun note (name, thm, attrs) lthy =
+let
+ val ((_,[thm']), lthy') = Local_Theory.note ((name, attrs), [thm]) lthy
+in
+ (thm', lthy')
+end
+
+fun internal_attr at = Attrib.internal (K at)
+
+fun theorem after_qed goals ctxt =
+let
+ val goals' = map (rpair []) goals
+ fun after_qed' thms = after_qed (the_single thms)
+in
+ Proof.theorem_i NONE after_qed' [goals'] ctxt
+end
+
+
+(* definition of quotient types *)
+(********************************)
+
+(* 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
+ lambda c
+ (HOLogic.exists_const rty $
+ lambda x (HOLogic.mk_eq (c, (rel $ x))))
+end
+
+(* makes the new type definitions and proves non-emptyness*)
+fun typedef_make (qty_name, mx, rel, rty) lthy =
+let
+ val typedef_tac =
+ EVERY1 [rewrite_goal_tac @{thms mem_def},
+ rtac @{thm exI},
+ rtac @{thm exI},
+ rtac @{thm refl}]
+ val tfrees = map fst (Term.add_tfreesT rty [])
+in
+ Local_Theory.theory_result
+ (Typedef.add_typedef false NONE
+ (qty_name, tfrees, 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 unfold_mem = MetaSimplifier.rewrite_rule [@{thm mem_def}]
+ val rep_thm = #Rep typedef_info |> unfold_mem
+ val rep_inv = #Rep_inverse typedef_info
+ val abs_inv = #Abs_inverse typedef_info |> unfold_mem
+ val rep_inj = #Rep_inject typedef_info
+in
+ EVERY1 [rtac @{thm QUOT_TYPE.intro},
+ rtac equiv_thm,
+ rtac rep_thm,
+ rtac rep_inv,
+ rtac abs_inv,
+ rtac @{thm exI},
+ rtac @{thm refl},
+ rtac rep_inj]
+end
+
+(* proves the QUOT_TYPE theorem *)
+fun typedef_quot_type_thm (rel, abs, rep, equiv_thm, typedef_info) lthy =
+let
+ val quot_type_const = Const (@{const_name "QUOT_TYPE"}, dummyT)
+ val goal = HOLogic.mk_Trueprop (quot_type_const $ rel $ abs $ rep)
+ |> Syntax.check_term lthy
+in
+ Goal.prove lthy [] [] goal
+ (K (typedef_quot_type_tac equiv_thm typedef_info))
+end
+
+(* proves the quotient theorem *)
+fun typedef_quotient_thm (rel, abs, rep, abs_def, rep_def, quot_type_thm) lthy =
+let
+ val quotient_const = Const (@{const_name "Quotient"}, dummyT)
+ val goal = HOLogic.mk_Trueprop (quotient_const $ rel $ abs $ rep)
+ |> Syntax.check_term lthy
+
+ val typedef_quotient_thm_tac =
+ EVERY1 [K (rewrite_goals_tac [abs_def, rep_def]),
+ rtac @{thm QUOT_TYPE.Quotient},
+ rtac quot_type_thm]
+in
+ Goal.prove lthy [] [] goal
+ (K typedef_quotient_thm_tac)
+end
+
+(* main function for constructing the quotient type *)
+fun mk_typedef_main (((qty_name, mx), (rty, rel)), equiv_thm) lthy =
+let
+ (* generates typedef *)
+ val ((_, typedef_info), lthy1) = typedef_make (qty_name, mx, rel, rty) lthy
+
+ (* abs and rep functions *)
+ val abs_ty = #abs_type typedef_info
+ val rep_ty = #rep_type typedef_info
+ val abs_name = #Abs_name typedef_info
+ val rep_name = #Rep_name typedef_info
+ val abs = Const (abs_name, rep_ty --> abs_ty)
+ val rep = Const (rep_name, abs_ty --> rep_ty)
+
+ (* ABS and REP definitions *)
+ val ABS_const = Const (@{const_name "QUOT_TYPE.abs"}, dummyT )
+ val REP_const = Const (@{const_name "QUOT_TYPE.rep"}, dummyT )
+ val ABS_trm = Syntax.check_term lthy1 (ABS_const $ rel $ abs)
+ val REP_trm = Syntax.check_term lthy1 (REP_const $ rep)
+ val ABS_name = Binding.prefix_name "abs_" qty_name
+ val REP_name = Binding.prefix_name "rep_" qty_name
+ val (((ABS, ABS_def), (REP, REP_def)), lthy2) =
+ lthy1 |> define (ABS_name, NoSyn, ABS_trm)
+ ||>> define (REP_name, NoSyn, REP_trm)
+
+ (* quot_type theorem *)
+ val quot_thm = typedef_quot_type_thm (rel, abs, rep, equiv_thm, typedef_info) lthy2
+ val quot_thm_name = Binding.prefix_name "QUOT_TYPE_" qty_name
+
+ (* quotient theorem *)
+ val quotient_thm = typedef_quotient_thm (rel, ABS, REP, ABS_def, REP_def, quot_thm) lthy2
+ val quotient_thm_name = Binding.prefix_name "Quotient_" qty_name
+
+ (* storing the quot-info *)
+ val qty_str = fst (Term.dest_Type abs_ty)
+ val lthy3 = quotdata_update qty_str
+ (Logic.varifyT abs_ty, Logic.varifyT rty, rel, equiv_thm) lthy2
+ (* FIXME: varifyT should not be used *)
+ (* FIXME: the relation needs to be a string, since its type needs *)
+ (* FIXME: to recalculated in for example REGULARIZE *)
+
+in
+ lthy3
+ |> note (quot_thm_name, quot_thm, [])
+ ||>> note (quotient_thm_name, quotient_thm, [internal_attr quotient_rules_add])
+ ||>> note (Binding.suffix_name "_equivp" qty_name, equiv_thm, [internal_attr equiv_rules_add])
+end
+
+
+
+
+(* interface and syntax setup *)
+
+(* the ML-interface takes a list of 4-tuples consisting of *)
+(* *)
+(* - the name of the quotient type *)
+(* - its mixfix annotation *)
+(* - the type to be quotient *)
+(* - the relation according to which the type is quotient *)
+
+fun quotient_type quot_list lthy =
+let
+ fun mk_goal (rty, rel) =
+ let
+ val equivp_ty = ([rty, rty] ---> @{typ bool}) --> @{typ bool}
+ in
+ HOLogic.mk_Trueprop (Const (@{const_name equivp}, equivp_ty) $ rel)
+ end
+
+ val goals = map (mk_goal o snd) quot_list
+
+ fun after_qed thms lthy =
+ fold_map mk_typedef_main (quot_list ~~ thms) lthy |> snd
+in
+ theorem after_qed goals lthy
+end
+
+fun quotient_type_cmd spec lthy =
+let
+ fun parse_spec (((qty_str, mx), rty_str), rel_str) =
+ let
+ val qty_name = Binding.name qty_str
+ val rty = Syntax.read_typ lthy rty_str
+ val rel = Syntax.read_term lthy rel_str
+ in
+ ((qty_name, mx), (rty, rel))
+ end
+in
+ quotient_type (map parse_spec spec) lthy
+end
+
+val quotspec_parser =
+ OuterParse.and_list1
+ (OuterParse.short_ident -- OuterParse.opt_infix --
+ (OuterParse.$$$ "=" |-- OuterParse.typ) --
+ (OuterParse.$$$ "/" |-- OuterParse.term))
+
+val _ = OuterKeyword.keyword "/"
+
+val _ =
+ OuterSyntax.local_theory_to_proof "quotient"
+ "quotient type definitions (requires equivalence proofs)"
+ OuterKeyword.thy_goal (quotspec_parser >> quotient_type_cmd)
+
+end; (* structure *)
+
+open Quotient