(* Title: HOL/Tools/Lifting/lifting_def.ML
Author: Ondrej Kuncar
Definitions for constants on quotient types.
*)
signature LIFTING_DEF =
sig
val add_lift_def:
(binding * mixfix) -> typ -> term -> thm -> local_theory -> local_theory
val lift_def_cmd:
(binding * string option * mixfix) * string -> local_theory -> Proof.state
val can_generate_code_cert: thm -> bool
end;
structure Lifting_Def: LIFTING_DEF =
struct
open Lifting_Util
infix 0 MRSL
(* Generation of the code certificate from the rsp theorem *)
fun get_body_types (Type ("fun", [_, U]), Type ("fun", [_, V])) = get_body_types (U, V)
| get_body_types (U, V) = (U, V)
fun get_binder_types (Type ("fun", [T, U]), Type ("fun", [V, W])) = (T, V) :: get_binder_types (U, W)
| get_binder_types _ = []
fun get_binder_types_by_rel (Const (@{const_name "fun_rel"}, _) $ _ $ S) (Type ("fun", [T, U]), Type ("fun", [V, W])) =
(T, V) :: get_binder_types_by_rel S (U, W)
| get_binder_types_by_rel _ _ = []
fun get_body_type_by_rel (Const (@{const_name "fun_rel"}, _) $ _ $ S) (Type ("fun", [_, U]), Type ("fun", [_, V])) =
get_body_type_by_rel S (U, V)
| get_body_type_by_rel _ (U, V) = (U, V)
fun force_rty_type ctxt rty rhs =
let
val thy = Proof_Context.theory_of ctxt
val rhs_schematic = singleton (Variable.polymorphic ctxt) rhs
val rty_schematic = fastype_of rhs_schematic
val match = Sign.typ_match thy (rty_schematic, rty) Vartab.empty
in
Envir.subst_term_types match rhs_schematic
end
fun unabs_def ctxt def =
let
val (_, rhs) = Thm.dest_equals (cprop_of def)
fun dest_abs (Abs (var_name, T, _)) = (var_name, T)
| dest_abs tm = raise TERM("get_abs_var",[tm])
val (var_name, T) = dest_abs (term_of rhs)
val (new_var_names, ctxt') = Variable.variant_fixes [var_name] ctxt
val thy = Proof_Context.theory_of ctxt'
val refl_thm = Thm.reflexive (cterm_of thy (Free (hd new_var_names, T)))
in
Thm.combination def refl_thm |>
singleton (Proof_Context.export ctxt' ctxt)
end
fun unabs_all_def ctxt def =
let
val (_, rhs) = Thm.dest_equals (cprop_of def)
val xs = strip_abs_vars (term_of rhs)
in
fold (K (unabs_def ctxt)) xs def
end
val map_fun_unfolded =
@{thm map_fun_def[abs_def]} |>
unabs_def @{context} |>
unabs_def @{context} |>
Local_Defs.unfold @{context} [@{thm comp_def}]
fun unfold_fun_maps ctm =
let
fun unfold_conv ctm =
case (Thm.term_of ctm) of
Const (@{const_name "map_fun"}, _) $ _ $ _ =>
(Conv.arg_conv unfold_conv then_conv Conv.rewr_conv map_fun_unfolded) ctm
| _ => Conv.all_conv ctm
in
(Conv.fun_conv unfold_conv) ctm
end
fun unfold_fun_maps_beta ctm =
let val try_beta_conv = Conv.try_conv (Thm.beta_conversion false)
in
(unfold_fun_maps then_conv try_beta_conv) ctm
end
fun prove_rel ctxt rsp_thm (rty, qty) =
let
val ty_args = get_binder_types (rty, qty)
fun disch_arg args_ty thm =
let
val quot_thm = Lifting_Term.prove_quot_thm ctxt args_ty
in
[quot_thm, thm] MRSL @{thm apply_rsp''}
end
in
fold disch_arg ty_args rsp_thm
end
exception CODE_CERT_GEN of string
fun simplify_code_eq ctxt def_thm =
Local_Defs.unfold ctxt [@{thm o_def}, @{thm map_fun_def}, @{thm id_def}] def_thm
(*
quot_thm - quotient theorem (Quotient R Abs Rep T).
returns: whether the Lifting package is capable to generate code for the abstract type
represented by quot_thm
*)
fun can_generate_code_cert quot_thm =
case quot_thm_rel quot_thm of
Const (@{const_name HOL.eq}, _) => true
| Const (@{const_name invariant}, _) $ _ => true
| _ => false
fun generate_code_cert ctxt def_thm rsp_thm (rty, qty) =
let
val thy = Proof_Context.theory_of ctxt
val quot_thm = Lifting_Term.prove_quot_thm ctxt (get_body_types (rty, qty))
val fun_rel = prove_rel ctxt rsp_thm (rty, qty)
val abs_rep_thm = [quot_thm, fun_rel] MRSL @{thm Quotient_rep_abs}
val abs_rep_eq =
case (HOLogic.dest_Trueprop o prop_of) fun_rel of
Const (@{const_name HOL.eq}, _) $ _ $ _ => abs_rep_thm
| Const (@{const_name invariant}, _) $ _ $ _ $ _ => abs_rep_thm RS @{thm invariant_to_eq}
| _ => raise CODE_CERT_GEN "relation is neither equality nor invariant"
val unfolded_def = Conv.fconv_rule (Conv.arg_conv unfold_fun_maps_beta) def_thm
val unabs_def = unabs_all_def ctxt unfolded_def
val rep = (cterm_of thy o quot_thm_rep) quot_thm
val rep_refl = Thm.reflexive rep RS @{thm meta_eq_to_obj_eq}
val repped_eq = [rep_refl, unabs_def RS @{thm meta_eq_to_obj_eq}] MRSL @{thm cong}
val code_cert = [repped_eq, abs_rep_eq] MRSL @{thm trans}
in
simplify_code_eq ctxt code_cert
end
fun generate_trivial_rep_eq ctxt def_thm =
let
val unfolded_def = Conv.fconv_rule (Conv.arg_conv unfold_fun_maps_beta) def_thm
val code_eq = unabs_all_def ctxt unfolded_def
val simp_code_eq = simplify_code_eq ctxt code_eq
in
simp_code_eq
end
fun generate_rep_eq ctxt def_thm rsp_thm (rty, qty) =
if body_type rty = body_type qty then
SOME (generate_trivial_rep_eq ctxt def_thm)
else
let
val (rty_body, qty_body) = get_body_types (rty, qty)
val quot_thm = Lifting_Term.prove_quot_thm ctxt (rty_body, qty_body)
in
if can_generate_code_cert quot_thm then
SOME (generate_code_cert ctxt def_thm rsp_thm (rty, qty))
else
NONE
end
fun generate_abs_eq ctxt def_thm rsp_thm quot_thm =
let
fun refl_tac ctxt =
let
fun intro_reflp_tac (t, i) =
let
val concl_pat = Drule.strip_imp_concl (cprop_of @{thm reflpD})
val insts = Thm.first_order_match (concl_pat, t)
in
rtac (Drule.instantiate_normalize insts @{thm reflpD}) i
end
handle Pattern.MATCH => no_tac
val fun_rel_meta_eq = mk_meta_eq @{thm fun_rel_eq}
val conv = Conv.bottom_conv (K (Conv.try_conv (Conv.rewr_conv fun_rel_meta_eq))) ctxt
val rules = Lifting_Info.get_reflp_preserve_rules ctxt
in
EVERY' [CSUBGOAL intro_reflp_tac,
CONVERSION conv,
REPEAT_ALL_NEW (resolve_tac rules)]
end
fun try_prove_prem ctxt prop =
SOME (Goal.prove ctxt [] [] prop (fn {context, ...} => refl_tac context 1))
handle ERROR _ => NONE
val abs_eq_with_assms =
let
val (rty, qty) = quot_thm_rty_qty quot_thm
val rel = quot_thm_rel quot_thm
val ty_args = get_binder_types_by_rel rel (rty, qty)
val body_type = get_body_type_by_rel rel (rty, qty)
val quot_ret_thm = Lifting_Term.prove_quot_thm ctxt body_type
val rep_abs_folded_unmapped_thm =
let
val rep_id = [quot_thm, def_thm] MRSL @{thm Quotient_Rep_eq}
val ctm = Thm.dest_equals_lhs (cprop_of rep_id)
val unfolded_maps_eq = unfold_fun_maps ctm
val t1 = [quot_thm, def_thm, rsp_thm] MRSL @{thm Quotient_rep_abs_fold_unmap}
val prems_pat = (hd o Drule.cprems_of) t1
val insts = Thm.first_order_match (prems_pat, cprop_of unfolded_maps_eq)
in
unfolded_maps_eq RS (Drule.instantiate_normalize insts t1)
end
in
rep_abs_folded_unmapped_thm
|> fold (fn _ => fn thm => thm RS @{thm fun_relD2}) ty_args
|> (fn x => x RS (@{thm Quotient_rel_abs2} OF [quot_ret_thm]))
end
val prems = prems_of abs_eq_with_assms
val indexed_prems = map_index (apfst (fn x => x + 1)) prems
val indexed_assms = map (apsnd (try_prove_prem ctxt)) indexed_prems
val proved_assms = map (apsnd the) (filter (is_some o snd) indexed_assms)
val abs_eq = fold_rev (fn (i, assms) => fn thm => assms RSN (i, thm)) proved_assms abs_eq_with_assms
in
simplify_code_eq ctxt abs_eq
end
fun define_code_using_abs_eq abs_eq_thm lthy =
if null (Logic.strip_imp_prems(prop_of abs_eq_thm)) then
(snd oo Local_Theory.note) ((Binding.empty, [Code.add_default_eqn_attrib]), [abs_eq_thm]) lthy
else
lthy
fun define_code_using_rep_eq maybe_rep_eq_thm lthy =
case maybe_rep_eq_thm of
SOME rep_eq_thm =>
let
val add_abs_eqn_attribute =
Thm.declaration_attribute (fn thm => Context.mapping (Code.add_abs_eqn thm) I)
val add_abs_eqn_attrib = Attrib.internal (K add_abs_eqn_attribute);
in
(snd oo Local_Theory.note) ((Binding.empty, [add_abs_eqn_attrib]), [rep_eq_thm]) lthy
end
| NONE => lthy
fun has_constr ctxt quot_thm =
let
val thy = Proof_Context.theory_of ctxt
val abs_fun = quot_thm_abs quot_thm
in
if is_Const abs_fun then
Code.is_constr thy ((fst o dest_Const) abs_fun)
else
false
end
fun has_abstr ctxt quot_thm =
let
val thy = Proof_Context.theory_of ctxt
val abs_fun = quot_thm_abs quot_thm
in
if is_Const abs_fun then
Code.is_abstr thy ((fst o dest_Const) abs_fun)
else
false
end
fun define_code abs_eq_thm maybe_rep_eq_thm (rty, qty) lthy =
let
val (rty_body, qty_body) = get_body_types (rty, qty)
in
if rty_body = qty_body then
if null (Logic.strip_imp_prems(prop_of abs_eq_thm)) then
(snd oo Local_Theory.note) ((Binding.empty, [Code.add_default_eqn_attrib]), [abs_eq_thm]) lthy
else
(snd oo Local_Theory.note) ((Binding.empty, [Code.add_default_eqn_attrib]), [the maybe_rep_eq_thm]) lthy
else
let
val body_quot_thm = Lifting_Term.prove_quot_thm lthy (rty_body, qty_body)
in
if has_constr lthy body_quot_thm then
define_code_using_abs_eq abs_eq_thm lthy
else if has_abstr lthy body_quot_thm then
define_code_using_rep_eq maybe_rep_eq_thm lthy
else
lthy
end
end
(*
Defines an operation on an abstract type in terms of a corresponding operation
on a representation type.
var - a binding and a mixfix of the new constant being defined
qty - an abstract type of the new constant
rhs - a term representing the new constant on the raw level
rsp_thm - a respectfulness theorem in the internal tagged form (like '(R ===> R ===> R) f f'),
i.e. "(Lifting_Term.equiv_relation (fastype_of rhs, qty)) $ rhs $ rhs"
*)
fun add_lift_def var qty rhs rsp_thm lthy =
let
val rty = fastype_of rhs
val quot_thm = Lifting_Term.prove_quot_thm lthy (rty, qty)
val absrep_trm = quot_thm_abs quot_thm
val rty_forced = (domain_type o fastype_of) absrep_trm
val forced_rhs = force_rty_type lthy rty_forced rhs
val lhs = Free (Binding.print (#1 var), qty)
val prop = Logic.mk_equals (lhs, absrep_trm $ forced_rhs)
val (_, prop') = Local_Defs.cert_def lthy prop
val (_, newrhs) = Local_Defs.abs_def prop'
val ((_, (_ , def_thm)), lthy') =
Local_Theory.define (var, ((Thm.def_binding (#1 var), []), newrhs)) lthy
val transfer_thm = ([quot_thm, rsp_thm, def_thm] MRSL @{thm Quotient_to_transfer})
|> Raw_Simplifier.rewrite_rule (Transfer.get_relator_eq lthy')
val abs_eq_thm = generate_abs_eq lthy' def_thm rsp_thm quot_thm
val maybe_rep_eq_thm = generate_rep_eq lthy' def_thm rsp_thm (rty_forced, qty)
fun qualify defname suffix = Binding.qualified true suffix defname
val lhs_name = (#1 var)
val rsp_thm_name = qualify lhs_name "rsp"
val abs_eq_thm_name = qualify lhs_name "abs_eq"
val rep_eq_thm_name = qualify lhs_name "rep_eq"
val transfer_thm_name = qualify lhs_name "transfer"
val transfer_attr = Attrib.internal (K Transfer.transfer_add)
in
lthy'
|> (snd oo Local_Theory.note) ((rsp_thm_name, []), [rsp_thm])
|> (snd oo Local_Theory.note) ((transfer_thm_name, [transfer_attr]), [transfer_thm])
|> (snd oo Local_Theory.note) ((abs_eq_thm_name, []), [abs_eq_thm])
|> (case maybe_rep_eq_thm of
SOME rep_eq_thm => (snd oo Local_Theory.note) ((rep_eq_thm_name, []), [rep_eq_thm])
| NONE => I)
|> define_code abs_eq_thm maybe_rep_eq_thm (rty_forced, qty)
end
fun mk_readable_rsp_thm_eq tm lthy =
let
val ctm = cterm_of (Proof_Context.theory_of lthy) tm
fun norm_fun_eq ctm =
let
fun abs_conv2 cv = Conv.abs_conv (K (Conv.abs_conv (K cv) lthy)) lthy
fun erase_quants ctm' =
case (Thm.term_of ctm') of
Const ("HOL.eq", _) $ _ $ _ => Conv.all_conv ctm'
| _ => (Conv.binder_conv (K erase_quants) lthy then_conv
Conv.rewr_conv @{thm fun_eq_iff[symmetric, THEN eq_reflection]}) ctm'
in
(abs_conv2 erase_quants then_conv Thm.eta_conversion) ctm
end
fun simp_arrows_conv ctm =
let
val unfold_conv = Conv.rewrs_conv
[@{thm fun_rel_eq_invariant[THEN eq_reflection]}, @{thm fun_rel_eq_rel[THEN eq_reflection]},
@{thm fun_rel_def[THEN eq_reflection]}]
val left_conv = simp_arrows_conv then_conv Conv.try_conv norm_fun_eq
fun binop_conv2 cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
val invariant_commute_conv = Conv.bottom_conv
(K (Conv.try_conv (Conv.rewrs_conv (Lifting_Info.get_invariant_commute_rules lthy)))) lthy
in
case (Thm.term_of ctm) of
Const (@{const_name "fun_rel"}, _) $ _ $ _ =>
(binop_conv2 left_conv simp_arrows_conv then_conv unfold_conv) ctm
| _ => invariant_commute_conv ctm
end
val unfold_ret_val_invs = Conv.bottom_conv
(K (Conv.try_conv (Conv.rewr_conv @{thm invariant_same_args}))) lthy
val simp_conv = Trueprop_conv (Conv.fun2_conv simp_arrows_conv)
val univq_conv = Conv.rewr_conv @{thm HOL.all_simps(6)[symmetric, THEN eq_reflection]}
val univq_prenex_conv = Conv.top_conv (K (Conv.try_conv univq_conv)) lthy
val beta_conv = Thm.beta_conversion true
val eq_thm =
(simp_conv then_conv univq_prenex_conv then_conv beta_conv then_conv unfold_ret_val_invs) ctm
in
Object_Logic.rulify(eq_thm RS Drule.equal_elim_rule2)
end
fun rename_to_tnames ctxt term =
let
fun all_typs (Const ("all", _) $ Abs (_, T, t)) = T :: all_typs t
| all_typs _ = []
fun rename (Const ("all", T1) $ Abs (_, T2, t)) (new_name :: names) =
(Const ("all", T1) $ Abs (new_name, T2, rename t names))
| rename t _ = t
val (fixed_def_t, _) = yield_singleton (Variable.importT_terms) term ctxt
val new_names = Datatype_Prop.make_tnames (all_typs fixed_def_t)
in
rename term new_names
end
(*
lifting_definition command. It opens a proof of a corresponding respectfulness
theorem in a user-friendly, readable form. Then add_lift_def is called internally.
*)
fun lift_def_cmd (raw_var, rhs_raw) lthy =
let
val ((binding, SOME qty, mx), lthy') = yield_singleton Proof_Context.read_vars raw_var lthy
val rhs = (Syntax.check_term lthy' o Syntax.parse_term lthy') rhs_raw
fun try_to_prove_refl thm =
let
val lhs_eq =
thm
|> prop_of
|> Logic.dest_implies
|> fst
|> strip_all_body
|> try HOLogic.dest_Trueprop
in
case lhs_eq of
SOME (Const ("HOL.eq", _) $ _ $ _) => SOME (@{thm refl} RS thm)
| _ => NONE
end
val rsp_rel = Lifting_Term.equiv_relation lthy' (fastype_of rhs, qty)
val rty_forced = (domain_type o fastype_of) rsp_rel;
val forced_rhs = force_rty_type lthy' rty_forced rhs;
val internal_rsp_tm = HOLogic.mk_Trueprop (rsp_rel $ forced_rhs $ forced_rhs)
val readable_rsp_thm_eq = mk_readable_rsp_thm_eq internal_rsp_tm lthy'
val maybe_proven_rsp_thm = try_to_prove_refl readable_rsp_thm_eq
val (readable_rsp_tm, _) = Logic.dest_implies (prop_of readable_rsp_thm_eq)
val readable_rsp_tm_tnames = rename_to_tnames lthy' readable_rsp_tm
fun after_qed thm_list lthy =
let
val internal_rsp_thm =
case thm_list of
[] => the maybe_proven_rsp_thm
| [[thm]] => Goal.prove lthy [] [] internal_rsp_tm
(fn _ => rtac readable_rsp_thm_eq 1 THEN Proof_Context.fact_tac [thm] 1)
in
add_lift_def (binding, mx) qty rhs internal_rsp_thm lthy
end
in
case maybe_proven_rsp_thm of
SOME _ => Proof.theorem NONE after_qed [] lthy'
| NONE => Proof.theorem NONE after_qed [[(readable_rsp_tm_tnames,[])]] lthy'
end
fun quot_thm_err ctxt (rty, qty) pretty_msg =
let
val error_msg = cat_lines
["Lifting failed for the following types:",
Pretty.string_of (Pretty.block
[Pretty.str "Raw type:", Pretty.brk 2, Syntax.pretty_typ ctxt rty]),
Pretty.string_of (Pretty.block
[Pretty.str "Abstract type:", Pretty.brk 2, Syntax.pretty_typ ctxt qty]),
"",
(Pretty.string_of (Pretty.block
[Pretty.str "Reason:", Pretty.brk 2, pretty_msg]))]
in
error error_msg
end
fun check_rty_err ctxt (rty_schematic, rty_forced) (raw_var, rhs_raw) =
let
val (_, ctxt') = yield_singleton Proof_Context.read_vars raw_var ctxt
val rhs = (Syntax.check_term ctxt' o Syntax.parse_term ctxt') rhs_raw
val error_msg = cat_lines
["Lifting failed for the following term:",
Pretty.string_of (Pretty.block
[Pretty.str "Term:", Pretty.brk 2, Syntax.pretty_term ctxt rhs]),
Pretty.string_of (Pretty.block
[Pretty.str "Type:", Pretty.brk 2, Syntax.pretty_typ ctxt rty_schematic]),
"",
(Pretty.string_of (Pretty.block
[Pretty.str "Reason:",
Pretty.brk 2,
Pretty.str "The type of the term cannot be instancied to",
Pretty.brk 1,
Pretty.quote (Syntax.pretty_typ ctxt rty_forced),
Pretty.str "."]))]
in
error error_msg
end
fun lift_def_cmd_with_err_handling (raw_var, rhs_raw) lthy =
(lift_def_cmd (raw_var, rhs_raw) lthy
handle Lifting_Term.QUOT_THM (rty, qty, msg) => quot_thm_err lthy (rty, qty) msg)
handle Lifting_Term.CHECK_RTY (rty_schematic, rty_forced) =>
check_rty_err lthy (rty_schematic, rty_forced) (raw_var, rhs_raw)
(* parser and command *)
val liftdef_parser =
((Parse.binding -- (@{keyword "::"} |-- (Parse.typ >> SOME) -- Parse.opt_mixfix')) >> Parse.triple2)
--| @{keyword "is"} -- Parse.term
val _ =
Outer_Syntax.local_theory_to_proof @{command_spec "lift_definition"}
"definition for constants over the quotient type"
(liftdef_parser >> lift_def_cmd_with_err_handling)
end; (* structure *)