haftmann@37743
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(* Title: HOL/Tools/Quotient/quotient_typ.ML
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kaliszyk@35222
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Author: Cezary Kaliszyk and Christian Urban
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urbanc@35806
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Definition of a quotient type.
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*)
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signature QUOTIENT_TYPE =
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sig
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val add_quotient_type: ((string list * binding * mixfix) * (typ * term * bool)) * thm
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-> local_theory -> Quotient_Info.quotients * local_theory
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val quotient_type: ((string list * binding * mixfix) * (typ * term * bool)) list
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-> Proof.context -> Proof.state
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val quotient_type_cmd: ((((string list * binding) * mixfix) * string) * (bool * string)) list
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-> Proof.context -> Proof.state
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end;
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structure Quotient_Type: QUOTIENT_TYPE =
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struct
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(*** definition of quotient types ***)
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val mem_def1 = @{lemma "y : Collect S ==> S y" by simp}
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val mem_def2 = @{lemma "S y ==> y : Collect S" by simp}
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(* constructs the term lambda (c::rty => bool). EX (x::rty). c = rel x *)
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fun typedef_term rel rty lthy =
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let
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val [x, c] =
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[("x", rty), ("c", HOLogic.mk_setT rty)]
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|> Variable.variant_frees lthy [rel]
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|> map Free
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in
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HOLogic.Collect_const (HOLogic.mk_setT rty) $ (lambda c (HOLogic.exists_const rty $
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lambda x (HOLogic.mk_conj (rel $ x $ x,
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HOLogic.mk_eq (c, HOLogic.Collect_const rty $ (rel $ x))))))
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end
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(* makes the new type definitions and proves non-emptyness *)
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fun typedef_make (vs, qty_name, mx, rel, rty) equiv_thm lthy =
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wenzelm@41700
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let
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val typedef_tac =
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EVERY1 (map rtac [@{thm part_equivp_typedef}, equiv_thm])
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in
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(* FIXME: purely local typedef causes at the moment
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problems with type variables
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Typedef.add_typedef false NONE (qty_name, vs, mx)
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(typedef_term rel rty lthy) NONE typedef_tac lthy
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*)
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(* FIXME should really use local typedef here *)
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Local_Theory.background_theory_result
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(Typedef.add_typedef_global false NONE
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(qty_name, map (rpair dummyS) vs, mx)
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urbanc@35806
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(typedef_term rel rty lthy)
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urbanc@35806
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NONE typedef_tac) lthy
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end
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(* tactic to prove the quot_type theorem for the new type *)
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fun typedef_quot_type_tac equiv_thm ((_, typedef_info): Typedef.info) =
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let
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val rep_thm = #Rep typedef_info RS mem_def1
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val rep_inv = #Rep_inverse typedef_info
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val abs_inv = #Abs_inverse typedef_info
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val rep_inj = #Rep_inject typedef_info
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in
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(rtac @{thm quot_type.intro} THEN' RANGE [
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rtac equiv_thm,
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rtac rep_thm,
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rtac rep_inv,
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rtac abs_inv THEN' rtac mem_def2 THEN' atac,
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rtac rep_inj]) 1
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end
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(* proves the quot_type theorem for the new type *)
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fun typedef_quot_type_thm (rel, abs, rep, equiv_thm, typedef_info) lthy =
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let
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val quot_type_const = Const (@{const_name "quot_type"},
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fastype_of rel --> fastype_of abs --> fastype_of rep --> @{typ bool})
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val goal = HOLogic.mk_Trueprop (quot_type_const $ rel $ abs $ rep)
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in
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Goal.prove lthy [] [] goal
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(K (typedef_quot_type_tac equiv_thm typedef_info))
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end
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(* main function for constructing a quotient type *)
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fun add_quotient_type (((vs, qty_name, mx), (rty, rel, partial)), equiv_thm) lthy =
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let
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val part_equiv =
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if partial
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then equiv_thm
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else equiv_thm RS @{thm equivp_implies_part_equivp}
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(* generates the typedef *)
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val ((qty_full_name, typedef_info), lthy1) =
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typedef_make (vs, qty_name, mx, rel, rty) part_equiv lthy
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(* abs and rep functions from the typedef *)
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val Abs_ty = #abs_type (#1 typedef_info)
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val Rep_ty = #rep_type (#1 typedef_info)
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val Abs_name = #Abs_name (#1 typedef_info)
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val Rep_name = #Rep_name (#1 typedef_info)
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val Abs_const = Const (Abs_name, Rep_ty --> Abs_ty)
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val Rep_const = Const (Rep_name, Abs_ty --> Rep_ty)
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(* more useful abs and rep definitions *)
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val abs_const = Const (@{const_name quot_type.abs},
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(rty --> rty --> @{typ bool}) --> (Rep_ty --> Abs_ty) --> rty --> Abs_ty)
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val rep_const = Const (@{const_name quot_type.rep}, (Abs_ty --> Rep_ty) --> Abs_ty --> rty)
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val abs_trm = abs_const $ rel $ Abs_const
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val rep_trm = rep_const $ Rep_const
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val abs_name = Binding.prefix_name "abs_" qty_name
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val rep_name = Binding.prefix_name "rep_" qty_name
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val ((_, (_, abs_def)), lthy2) = lthy1
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|> Local_Theory.define ((abs_name, NoSyn), (Attrib.empty_binding, abs_trm))
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val ((_, (_, rep_def)), lthy3) = lthy2
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|> Local_Theory.define ((rep_name, NoSyn), (Attrib.empty_binding, rep_trm))
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(* quot_type theorem *)
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val quot_thm = typedef_quot_type_thm (rel, Abs_const, Rep_const, part_equiv, typedef_info) lthy3
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(* quotient theorem *)
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val quotient_thm_name = Binding.prefix_name "Quotient_" qty_name
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val quotient_thm =
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(quot_thm RS @{thm quot_type.Quotient})
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|> fold_rule [abs_def, rep_def]
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(* name equivalence theorem *)
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val equiv_thm_name = Binding.suffix_name "_equivp" qty_name
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(* storing the quotients *)
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val quotients = {qtyp = Abs_ty, rtyp = rty, equiv_rel = rel, equiv_thm = equiv_thm}
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fun qinfo phi = Quotient_Info.transform_quotients phi quotients
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val lthy4 = lthy3
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|> Local_Theory.declaration {syntax = false, pervasive = true}
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(fn phi => Quotient_Info.update_quotients qty_full_name (qinfo phi))
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|> (snd oo Local_Theory.note)
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((equiv_thm_name,
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if partial then [] else [Attrib.internal (K Quotient_Info.equiv_rules_add)]),
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[equiv_thm])
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|> (snd oo Local_Theory.note)
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((quotient_thm_name, [Attrib.internal (K Quotient_Info.quotient_rules_add)]),
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[quotient_thm])
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in
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(quotients, lthy4)
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end
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(* sanity checks for the quotient type specifications *)
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fun sanity_check ((vs, qty_name, _), (rty, rel, _)) =
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let
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val rty_tfreesT = map fst (Term.add_tfreesT rty [])
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val rel_tfrees = map fst (Term.add_tfrees rel [])
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val rel_frees = map fst (Term.add_frees rel [])
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val rel_vars = Term.add_vars rel []
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val rel_tvars = Term.add_tvars rel []
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val qty_str = Binding.print qty_name ^ ": "
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val illegal_rel_vars =
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if null rel_vars andalso null rel_tvars then []
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else [qty_str ^ "illegal schematic variable(s) in the relation."]
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val dup_vs =
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(case duplicates (op =) vs of
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[] => []
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| dups => [qty_str ^ "duplicate type variable(s) on the lhs: " ^ commas_quote dups])
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val extra_rty_tfrees =
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(case subtract (op =) vs rty_tfreesT of
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[] => []
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| extras => [qty_str ^ "extra type variable(s) on the lhs: " ^ commas_quote extras])
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val extra_rel_tfrees =
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(case subtract (op =) vs rel_tfrees of
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[] => []
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wenzelm@41700
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| extras => [qty_str ^ "extra type variable(s) in the relation: " ^ commas_quote extras])
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val illegal_rel_frees =
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(case rel_frees of
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[] => []
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| xs => [qty_str ^ "illegal variable(s) in the relation: " ^ commas_quote xs])
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wenzelm@41700
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val errs = illegal_rel_vars @ dup_vs @ extra_rty_tfrees @ extra_rel_tfrees @ illegal_rel_frees
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in
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if null errs then () else error (cat_lines errs)
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end
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(* check for existence of map functions *)
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wenzelm@46211
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fun map_check thy (_, (rty, _, _)) =
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let
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fun map_check_aux rty warns =
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(case rty of
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Type (_, []) => warns
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| Type (s, _) =>
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wenzelm@46211
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if is_some (Quotient_Info.lookup_quotmaps_global thy s) then warns else s :: warns
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| _ => warns)
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val warns = map_check_aux rty []
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in
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if null warns then ()
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else warning ("No map function defined for " ^ commas warns ^
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". This will cause problems later on.")
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end
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(*** interface and syntax setup ***)
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(* the ML-interface takes a list of 5-tuples consisting of:
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- the name of the quotient type
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- its free type variables (first argument)
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- its mixfix annotation
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- the type to be quotient
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- the partial flag (a boolean)
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- the relation according to which the type is quotient
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it opens a proof-state in which one has to show that the
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relations are equivalence relations
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*)
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fun quotient_type quot_list lthy =
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let
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(* sanity check *)
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val _ = List.app sanity_check quot_list
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wenzelm@46211
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val _ = List.app (map_check (Proof_Context.theory_of lthy)) quot_list
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wenzelm@41700
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fun mk_goal (rty, rel, partial) =
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let
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val equivp_ty = ([rty, rty] ---> @{typ bool}) --> @{typ bool}
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val const =
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wenzelm@41700
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if partial then @{const_name part_equivp} else @{const_name equivp}
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wenzelm@41700
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in
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wenzelm@41700
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HOLogic.mk_Trueprop (Const (const, equivp_ty) $ rel)
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end
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wenzelm@41700
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wenzelm@41700
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val goals = map (mk_goal o snd) quot_list
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fun after_qed [thms] = fold (snd oo add_quotient_type) (quot_list ~~ thms)
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in
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wenzelm@46153
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Proof.theorem NONE after_qed [map (rpair []) goals] lthy
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end
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kaliszyk@35222
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wenzelm@41700
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fun quotient_type_cmd specs lthy =
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wenzelm@41700
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let
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wenzelm@41700
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fun parse_spec ((((vs, qty_name), mx), rty_str), (partial, rel_str)) lthy =
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wenzelm@41700
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let
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wenzelm@41700
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val rty = Syntax.read_typ lthy rty_str
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wenzelm@41700
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val lthy1 = Variable.declare_typ rty lthy
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wenzelm@41700
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val rel =
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wenzelm@41700
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Syntax.parse_term lthy1 rel_str
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wenzelm@41700
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|> Type.constraint (rty --> rty --> @{typ bool})
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wenzelm@41700
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|> Syntax.check_term lthy1
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wenzelm@41700
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val lthy2 = Variable.declare_term rel lthy1
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wenzelm@41700
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in
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wenzelm@41700
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(((vs, qty_name, mx), (rty, rel, partial)), lthy2)
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wenzelm@41700
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end
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kaliszyk@35222
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wenzelm@41700
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val (spec', lthy') = fold_map parse_spec specs lthy
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kaliszyk@35222
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in
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wenzelm@41700
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quotient_type spec' lthy'
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kaliszyk@35222
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end
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kaliszyk@35222
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kaliszyk@37493
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val partial = Scan.optional (Parse.reserved "partial" -- Parse.$$$ ":" >> K true) false
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kaliszyk@35222
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val quotspec_parser =
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kaliszyk@37493
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Parse.and_list1
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kaliszyk@37493
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((Parse.type_args -- Parse.binding) --
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277 |
Parse.opt_mixfix -- (Parse.$$$ "=" |-- Parse.typ) --
|
kaliszyk@37493
|
278 |
(Parse.$$$ "/" |-- (partial -- Parse.term)))
|
kaliszyk@35222
|
279 |
|
wenzelm@36970
|
280 |
val _ = Keyword.keyword "/"
|
kaliszyk@35222
|
281 |
|
kaliszyk@35222
|
282 |
val _ =
|
wenzelm@41700
|
283 |
Outer_Syntax.local_theory_to_proof "quotient_type"
|
wenzelm@41700
|
284 |
"quotient type definitions (require equivalence proofs)"
|
wenzelm@41700
|
285 |
Keyword.thy_goal (quotspec_parser >> quotient_type_cmd)
|
kaliszyk@35222
|
286 |
|
wenzelm@46151
|
287 |
end;
|