(* Title: HOL/Tools/Lifting/lifting_term.ML
Author: Ondrej Kuncar
Proves Quotient theorem.
*)
signature LIFTING_TERM =
sig
exception QUOT_THM of typ * typ * Pretty.T
exception PARAM_QUOT_THM of typ * Pretty.T
exception MERGE_TRANSFER_REL of Pretty.T
exception CHECK_RTY of typ * typ
val prove_quot_thm: Proof.context -> typ * typ -> thm
val abs_fun: Proof.context -> typ * typ -> term
val equiv_relation: Proof.context -> typ * typ -> term
val prove_param_quot_thm: Proof.context -> typ -> thm * (typ * thm) list * Proof.context
val generate_parametrized_relator: Proof.context -> typ -> term * term list
val merge_transfer_relations: Proof.context -> cterm -> thm
val parametrize_transfer_rule: Proof.context -> thm -> thm
end
structure Lifting_Term: LIFTING_TERM =
struct
open Lifting_Util
infix 0 MRSL
exception QUOT_THM_INTERNAL of Pretty.T
exception QUOT_THM of typ * typ * Pretty.T
exception PARAM_QUOT_THM of typ * Pretty.T
exception MERGE_TRANSFER_REL of Pretty.T
exception CHECK_RTY of typ * typ
fun match ctxt err ty_pat ty =
let
val thy = Proof_Context.theory_of ctxt
in
Sign.typ_match thy (ty_pat, ty) Vartab.empty
handle Type.TYPE_MATCH => err ctxt ty_pat ty
end
fun equiv_match_err ctxt ty_pat ty =
let
val ty_pat_str = Syntax.string_of_typ ctxt ty_pat
val ty_str = Syntax.string_of_typ ctxt ty
in
raise QUOT_THM_INTERNAL (Pretty.block
[Pretty.str ("The quotient type " ^ quote ty_str),
Pretty.brk 1,
Pretty.str ("and the quotient type pattern " ^ quote ty_pat_str),
Pretty.brk 1,
Pretty.str "don't match."])
end
fun get_quot_data ctxt s =
case Lifting_Info.lookup_quotients ctxt s of
SOME qdata => qdata
| NONE => raise QUOT_THM_INTERNAL (Pretty.block
[Pretty.str ("No quotient type " ^ quote s),
Pretty.brk 1,
Pretty.str "found."])
fun get_quot_thm ctxt s =
let
val thy = Proof_Context.theory_of ctxt
in
Thm.transfer thy (#quot_thm (get_quot_data ctxt s))
end
fun get_pcrel_info ctxt s =
case #pcr_info (get_quot_data ctxt s) of
SOME pcr_info => pcr_info
| NONE => raise QUOT_THM_INTERNAL (Pretty.block
[Pretty.str ("No parametrized correspondce relation for " ^ quote s),
Pretty.brk 1,
Pretty.str "found."])
fun get_pcrel_def ctxt s =
let
val thy = Proof_Context.theory_of ctxt
in
Thm.transfer thy (#pcrel_def (get_pcrel_info ctxt s))
end
fun get_pcr_cr_eq ctxt s =
let
val thy = Proof_Context.theory_of ctxt
in
Thm.transfer thy (#pcr_cr_eq (get_pcrel_info ctxt s))
end
fun get_rel_quot_thm ctxt s =
let
val thy = Proof_Context.theory_of ctxt
in
(case Lifting_Info.lookup_quot_maps ctxt s of
SOME map_data => Thm.transfer thy (#rel_quot_thm map_data)
| NONE => raise QUOT_THM_INTERNAL (Pretty.block
[Pretty.str ("No relator for the type " ^ quote s),
Pretty.brk 1,
Pretty.str "found."]))
end
fun get_rel_distr_rules ctxt s tm =
let
val thy = Proof_Context.theory_of ctxt
in
(case Lifting_Info.lookup_relator_distr_data ctxt s of
SOME rel_distr_thm => (
case tm of
Const (@{const_name POS}, _) => map (Thm.transfer thy) (#pos_distr_rules rel_distr_thm)
| Const (@{const_name NEG}, _) => map (Thm.transfer thy) (#neg_distr_rules rel_distr_thm)
)
| NONE => raise QUOT_THM_INTERNAL (Pretty.block
[Pretty.str ("No relator distr. data for the type " ^ quote s),
Pretty.brk 1,
Pretty.str "found."]))
end
fun is_id_quot thm = (prop_of thm = prop_of @{thm identity_quotient})
fun check_raw_types (provided_rty_name, rty_of_qty_name) qty_name =
if provided_rty_name <> rty_of_qty_name then
raise QUOT_THM_INTERNAL (Pretty.block
[Pretty.str ("The type " ^ quote provided_rty_name),
Pretty.brk 1,
Pretty.str ("is not a raw type for the quotient type " ^ quote qty_name ^ ";"),
Pretty.brk 1,
Pretty.str ("the correct raw type is " ^ quote rty_of_qty_name ^ ".")])
else
()
fun zip_Tvars ctxt type_name rty_Tvars qty_Tvars =
case try (get_rel_quot_thm ctxt) type_name of
NONE => rty_Tvars ~~ qty_Tvars
| SOME rel_quot_thm =>
let
fun quot_term_absT quot_term =
let
val (_, abs, _, _) = (dest_Quotient o HOLogic.dest_Trueprop) quot_term
in
fastype_of abs
end
fun equiv_univ_err ctxt ty_pat ty =
let
val ty_pat_str = Syntax.string_of_typ ctxt ty_pat
val ty_str = Syntax.string_of_typ ctxt ty
in
raise QUOT_THM_INTERNAL (Pretty.block
[Pretty.str ("The type " ^ quote ty_str),
Pretty.brk 1,
Pretty.str ("and the relator type pattern " ^ quote ty_pat_str),
Pretty.brk 1,
Pretty.str "don't unify."])
end
fun raw_match (TVar (v, S), T) subs =
(case Vartab.defined subs v of
false => Vartab.update_new (v, (S, T)) subs
| true => subs)
| raw_match (Type (_, Ts), Type (_, Us)) subs =
raw_matches (Ts, Us) subs
| raw_match _ subs = subs
and raw_matches (T :: Ts, U :: Us) subs = raw_matches (Ts, Us) (raw_match (T, U) subs)
| raw_matches _ subs = subs
val rty = Type (type_name, rty_Tvars)
val qty = Type (type_name, qty_Tvars)
val rel_quot_thm_concl = (Logic.strip_imp_concl o prop_of) rel_quot_thm
val schematic_rel_absT = quot_term_absT rel_quot_thm_concl;
val ctxt' = Variable.declare_typ schematic_rel_absT ctxt
val thy = Proof_Context.theory_of ctxt'
val absT = rty --> qty
val schematic_absT = Logic.type_map (singleton (Variable.polymorphic ctxt')) absT
val maxidx = Term.maxidx_of_typs [schematic_rel_absT, schematic_absT]
val _ = Sign.typ_unify thy (schematic_rel_absT, schematic_absT) (Vartab.empty,maxidx)
handle Type.TUNIFY => equiv_univ_err ctxt schematic_rel_absT schematic_absT
val subs = raw_match (schematic_rel_absT, absT) Vartab.empty
val rel_quot_thm_prems = (Logic.strip_imp_prems o prop_of) rel_quot_thm
in
map (dest_funT o
Envir.subst_type subs o
quot_term_absT)
rel_quot_thm_prems
end
fun instantiate_rtys ctxt (Type (rty_name, _)) (qty as Type (qty_name, _)) =
let
val quot_thm = get_quot_thm ctxt qty_name
val (Type (rs, rtys), qty_pat) = quot_thm_rty_qty quot_thm
val _ = check_raw_types (rty_name, rs) qty_name
val qtyenv = match ctxt equiv_match_err qty_pat qty
in
map (Envir.subst_type qtyenv) rtys
end
| instantiate_rtys _ _ _ = error "instantiate_rtys: not Type"
fun prove_schematic_quot_thm ctxt (rty, qty) =
(case (rty, qty) of
(Type (s, tys), Type (s', tys')) =>
if s = s'
then
let
val args = map (prove_schematic_quot_thm ctxt) (zip_Tvars ctxt s tys tys')
in
if forall is_id_quot args
then
@{thm identity_quotient}
else
args MRSL (get_rel_quot_thm ctxt s)
end
else
let
val rtys' = instantiate_rtys ctxt rty qty
val args = map (prove_schematic_quot_thm ctxt) (tys ~~ rtys')
in
if forall is_id_quot args
then
get_quot_thm ctxt s'
else
let
val quot_thm = get_quot_thm ctxt s'
val rel_quot_thm = args MRSL (get_rel_quot_thm ctxt s)
in
[rel_quot_thm, quot_thm] MRSL @{thm Quotient_compose}
end
end
| (_, Type (s', tys')) =>
(case try (get_quot_thm ctxt) s' of
SOME quot_thm =>
let
val rty_pat = (fst o quot_thm_rty_qty) quot_thm
in
prove_schematic_quot_thm ctxt (rty_pat, qty)
end
| NONE =>
let
val rty_pat = Type (s', map (fn _ => TFree ("a",[])) tys')
in
prove_schematic_quot_thm ctxt (rty_pat, qty)
end)
| _ => @{thm identity_quotient})
handle QUOT_THM_INTERNAL pretty_msg => raise QUOT_THM (rty, qty, pretty_msg)
fun force_qty_type thy qty quot_thm =
let
val (_, qty_schematic) = quot_thm_rty_qty quot_thm
val match_env = Sign.typ_match thy (qty_schematic, qty) Vartab.empty
fun prep_ty thy (x, (S, ty)) =
(ctyp_of thy (TVar (x, S)), ctyp_of thy ty)
val ty_inst = Vartab.fold (cons o (prep_ty thy)) match_env []
in
Thm.instantiate (ty_inst, []) quot_thm
end
fun check_rty_type ctxt rty quot_thm =
let
val thy = Proof_Context.theory_of ctxt
val (rty_forced, _) = quot_thm_rty_qty quot_thm
val rty_schematic = Logic.type_map (singleton (Variable.polymorphic ctxt)) rty
val _ = Sign.typ_match thy (rty_schematic, rty_forced) Vartab.empty
handle Type.TYPE_MATCH => raise CHECK_RTY (rty_schematic, rty_forced)
in
()
end
(*
The function tries to prove that rty and qty form a quotient.
Returns: Quotient theorem; an abstract type of the theorem is exactly
qty, a representation type of the theorem is an instance of rty in general.
*)
fun prove_quot_thm ctxt (rty, qty) =
let
val thy = Proof_Context.theory_of ctxt
val schematic_quot_thm = prove_schematic_quot_thm ctxt (rty, qty)
val quot_thm = force_qty_type thy qty schematic_quot_thm
val _ = check_rty_type ctxt rty quot_thm
in
quot_thm
end
fun abs_fun ctxt (rty, qty) =
quot_thm_abs (prove_quot_thm ctxt (rty, qty))
fun equiv_relation ctxt (rty, qty) =
quot_thm_rel (prove_quot_thm ctxt (rty, qty))
val get_fresh_Q_t =
let
val Q_t = @{term "Trueprop (Quotient R Abs Rep T)"}
val frees_Q_t = Term.add_free_names Q_t []
val tfrees_Q_t = rev (Term.add_tfree_names Q_t [])
in
fn ctxt =>
let
fun rename_free_var tab (Free (name, typ)) = Free (the_default name (AList.lookup op= tab name),typ)
| rename_free_var _ t = t
fun rename_free_vars tab = map_aterms (rename_free_var tab)
fun rename_free_tvars tab =
map_types (map_type_tfree (fn (name, sort) => TFree (the_default name (AList.lookup op= tab name), sort)))
val (new_frees_Q_t, ctxt) = Variable.variant_fixes frees_Q_t ctxt
val tab_frees = frees_Q_t ~~ new_frees_Q_t
val (new_tfrees_Q_t, ctxt) = Variable.invent_types (replicate (length tfrees_Q_t) []) ctxt
val tab_tfrees = tfrees_Q_t ~~ (fst o split_list) new_tfrees_Q_t
val renamed_Q_t = rename_free_vars tab_frees Q_t
val renamed_Q_t = rename_free_tvars tab_tfrees renamed_Q_t
in
(renamed_Q_t, ctxt)
end
end
fun prove_param_quot_thm ctxt ty =
let
fun generate (ty as Type (s, tys)) (table_ctxt as (table, ctxt)) =
if null tys
then
let
val thy = Proof_Context.theory_of ctxt
val instantiated_id_quot_thm = instantiate' [SOME (ctyp_of thy ty)] [] @{thm identity_quotient}
in
(instantiated_id_quot_thm, (table, ctxt))
end
else
let
val (args, table_ctxt) = fold_map generate tys table_ctxt
in
(args MRSL (get_rel_quot_thm ctxt s), table_ctxt)
end
| generate ty (table, ctxt) =
if AList.defined (op=) table ty
then (the (AList.lookup (op=) table ty), (table, ctxt))
else
let
val thy = Proof_Context.theory_of ctxt
val (Q_t, ctxt') = get_fresh_Q_t ctxt
val Q_thm = Thm.assume (cterm_of thy Q_t)
val table' = (ty, Q_thm)::table
in
(Q_thm, (table', ctxt'))
end
val (param_quot_thm, (table, ctxt)) = generate ty ([], ctxt)
in
(param_quot_thm, rev table, ctxt)
end
handle QUOT_THM_INTERNAL pretty_msg => raise PARAM_QUOT_THM (ty, pretty_msg)
fun generate_parametrized_relator ctxt ty =
let
val orig_ctxt = ctxt
val (quot_thm, table, ctxt) = prove_param_quot_thm ctxt ty
val parametrized_relator = quot_thm_crel quot_thm
val args = map (fn (_, q_thm) => quot_thm_crel q_thm) table
val exported_terms = Variable.exportT_terms ctxt orig_ctxt (parametrized_relator :: args)
in
(hd exported_terms, tl exported_terms)
end
(* Parametrization *)
local
fun get_lhs rule = (Thm.dest_fun o Thm.dest_arg o strip_imp_concl o cprop_of) rule;
fun no_imp _ = raise CTERM ("no implication", []);
infix 0 else_imp
fun (cv1 else_imp cv2) ct =
(cv1 ct
handle THM _ => cv2 ct
| CTERM _ => cv2 ct
| TERM _ => cv2 ct
| TYPE _ => cv2 ct);
fun first_imp cvs = fold_rev (curry op else_imp) cvs no_imp
fun rewr_imp rule ct =
let
val rule1 = Thm.incr_indexes (#maxidx (Thm.rep_cterm ct) + 1) rule;
val lhs_rule = get_lhs rule1;
val rule2 = Thm.rename_boundvars (Thm.term_of lhs_rule) (Thm.term_of ct) rule1;
val lhs_ct = Thm.dest_fun ct
in
Thm.instantiate (Thm.match (lhs_rule, lhs_ct)) rule2
handle Pattern.MATCH => raise CTERM ("rewr_imp", [lhs_rule, lhs_ct])
end
fun rewrs_imp rules = first_imp (map rewr_imp rules)
fun map_interrupt f l =
let
fun map_interrupt' _ [] l = SOME (rev l)
| map_interrupt' f (x::xs) l = (case f x of
NONE => NONE
| SOME v => map_interrupt' f xs (v::l))
in
map_interrupt' f l []
end
in
fun merge_transfer_relations ctxt ctm =
let
val ctm = Thm.dest_arg ctm
val tm = term_of ctm
val rel = (hd o get_args 2) tm
fun same_constants (Const (n1,_)) (Const (n2,_)) = n1 = n2
| same_constants _ _ = false
fun prove_extra_assms ctxt ctm distr_rule =
let
fun prove_assm assm = try (Goal.prove ctxt [] [] (term_of assm))
(fn _ => SOLVED' (REPEAT_ALL_NEW (resolve_tac (Transfer.get_transfer_raw ctxt))) 1)
fun is_POS_or_NEG ctm =
case (head_of o term_of o Thm.dest_arg) ctm of
Const (@{const_name POS}, _) => true
| Const (@{const_name NEG}, _) => true
| _ => false
val inst_distr_rule = rewr_imp distr_rule ctm
val extra_assms = filter_out is_POS_or_NEG (cprems_of inst_distr_rule)
val proved_assms = map_interrupt prove_assm extra_assms
in
Option.map (curry op OF inst_distr_rule) proved_assms
end
handle CTERM _ => NONE
fun cannot_merge_error_msg () = Pretty.block
[Pretty.str "Rewriting (merging) of this term has failed:",
Pretty.brk 1,
Syntax.pretty_term ctxt rel]
in
case get_args 2 rel of
[Const (@{const_name "HOL.eq"}, _), _] => rewrs_imp @{thms neg_eq_OO pos_eq_OO} ctm
| [_, Const (@{const_name "HOL.eq"}, _)] => rewrs_imp @{thms neg_OO_eq pos_OO_eq} ctm
| [_, trans_rel] =>
let
val (rty', qty) = (relation_types o fastype_of) trans_rel
val r = (fst o dest_Type) rty'
val q = (fst o dest_Type) qty
in
if r = q then
let
val distr_rules = get_rel_distr_rules ctxt r (head_of tm)
val distr_rule = get_first (prove_extra_assms ctxt ctm) distr_rules
in
case distr_rule of
NONE => raise MERGE_TRANSFER_REL (cannot_merge_error_msg ())
| SOME distr_rule => (map (merge_transfer_relations ctxt) (cprems_of distr_rule))
MRSL distr_rule
end
else
let
val pcrel_def = get_pcrel_def ctxt q
val pcrel_const = (head_of o fst o Logic.dest_equals o prop_of) pcrel_def
in
if same_constants pcrel_const (head_of trans_rel) then
let
val unfolded_ctm = Thm.rhs_of (Conv.arg1_conv (Conv.arg_conv (Conv.rewr_conv pcrel_def)) ctm)
val distr_rule = rewrs_imp @{thms POS_pcr_rule NEG_pcr_rule} unfolded_ctm
val result = (map (merge_transfer_relations ctxt) (cprems_of distr_rule)) MRSL distr_rule
val fold_pcr_rel = Conv.rewr_conv (Thm.symmetric pcrel_def)
in
Conv.fconv_rule (HOLogic.Trueprop_conv (Conv.combination_conv
(Conv.arg_conv (Conv.arg_conv fold_pcr_rel)) fold_pcr_rel)) result
end
else
raise MERGE_TRANSFER_REL (Pretty.str "Non-parametric correspondence relation used.")
end
end
end
handle QUOT_THM_INTERNAL pretty_msg => raise MERGE_TRANSFER_REL pretty_msg
end
fun parametrize_transfer_rule ctxt thm =
let
fun parametrize_relation_conv ctm =
let
val (rty, qty) = (relation_types o fastype_of) (term_of ctm)
in
case (rty, qty) of
(Type (r, rargs), Type (q, qargs)) =>
if r = q then
if forall op= (rargs ~~ qargs) then
Conv.all_conv ctm
else
all_args_conv parametrize_relation_conv ctm
else
if forall op= (rargs ~~ (instantiate_rtys ctxt rty qty)) then
let
val pcr_cr_eq = (Thm.symmetric o mk_meta_eq) (get_pcr_cr_eq ctxt q)
in
Conv.rewr_conv pcr_cr_eq ctm
end
handle QUOT_THM_INTERNAL _ => Conv.all_conv ctm
else
(let
val pcrel_def = Thm.symmetric (get_pcrel_def ctxt q)
in
(Conv.rewr_conv pcrel_def then_conv all_args_conv parametrize_relation_conv) ctm
end
handle QUOT_THM_INTERNAL _ =>
(Conv.arg1_conv (all_args_conv parametrize_relation_conv)) ctm)
| _ => Conv.all_conv ctm
end
in
Conv.fconv_rule (HOLogic.Trueprop_conv (Conv.fun2_conv parametrize_relation_conv)) thm
end
end;